ultimate strength of beams vqth by judith gloede …

ULTIMATE STRENGTH OF BEAMS VQTH

REINFORCED RECTANGULAR OPENInGS

by

Judith Gloede Congdon BA BSc

A thesis submitted to the Faculty of Graduate Studies and Research in partial fulfillment of the requirements for the degree of Master of Engineering

Department of Civil Engineering and Applied Mechanics McGill University Montreal PQ July 1969

Judith G10ede Congdon 1970

ii

SUMMARY

Results are presented for tests to destruction performed

as part of this study on one IOWF21 and ten 14WF38 beams

each containing a single rectangular opening in the web The

openings in all but one of the beams were reinforced with

straight horizontal reinforcing plates welded above and below

the opening on each side of the web The variables investishy

gated include moment to shear ratiO opening depth to beam

depth ratiO opening length to depth ratio and reinforcing

size

An ultimate strength analysis is offered based on failure

by development of a four hinge mechanism the hinges occurring

at cross-sections at the edges of the opening A simple to

use approximate method of solution is also offered and a proshy

cedure for design is suggested

The experimental results show the theory to be reasonably

accurate at high moment to shear ratios but conservative at

high values of shear The approximate method is less conshy

servative in this region

iii

ACKNOVlLEDGEMENTS

The writer wishes to express her appreciation to those

who provided assistance during the course of this study In

particular thanks are due to

Dr RGRedwood who acted as research director and

provided constant guidance and encouragement during the

course of this investigation

Messrs B Cockayne and G Matsell and the remainder of

the technical staff who helped in the fabrication and testing

of the test beams and in the preparation of tensile coupons

The staff of the Department of Metallurgical Engineershy

ing who generously offered access to the 20k Instron Testing

Machine for the testing of tensile coupons

The writer t s husband Vayne who provided considerable

help and encouragement and who also typed this manuscript

This investigation was made possible by the financial

assistro1ce of the National Research Council of Canada and

by the Canadian Steel Industries Construction Council

iv

NOTATION

a half length of opening

Af area of flange

Ar area of reinforcing

Aw area of web

b width of flange

c width of one pair of reinforcing bars (including web)

d depth of beam

E modulus of elasticity

Est strain hardening modulus

f subscript denoting stress or strain after load increment

FI stress resultant at high moment edge of opening

F2 stress resultant at low moment edge of opening

G shear modulus

h half depth of opening

i subscript denoting stress or strain before load increment

k I location of stress reversal at high moment edgeof opening

location of stress reversal at low moment edgeof opening

L length of moment arm to center of opening

M applied bending moment

Mp plastic bending moment

M p reduced plastic moment

x

v

p applied load

q thickness of reinforcing

R corner radius of opening

s half remaining clear web at opening

t thickness of flange

u length of web stub

shear force

plastic shear force

w thickness of web

extension of reinforcing past edge of opening

distance from the boundary of the opening to the stress resultant at the high moment edge of opening

distance from the boundary of the opening to the stress resultant at the low moment edge of opening

z plastic section modulus

proportionality factor 1= i(~2(2~ - 1)21 ~ proportionality factor f= kl + k2 - 1 - ~J pound45 strain at 45 0 to the longitudinal direction

pound-45 strain at -45 0 to the longitudinal direction

pound x strain in the longitudinal direction

6 xp plastic component of ~x

pound y strain corresponding to yielding

~yy strain in the direction of the beam depth

euro yp plastic component of euroyy

vi

~z strain in the transverse direction

plastic component of poundz

strain corresponding to hardening

the initiation of strain

~p equivalent strain

~pr reference value of ~p

~ xy shearing strain

~ normal stress

yield stress

yield stress for flange

yield stress for reinforcing

yield stress for web

shear stress

AISC American Institute of Steel Construction

ASTM American SOCiety of Testing Materials

CISC Canadian Institute of Steel Construction

ASCE American Society of Civil Engineers

vii

LIST OF FIGURES

Page

1 Types of Reinforcing 93

2 Stress Distribution at Opening 94

3 Idealized Stress-Strain Curve for Structural Steel 94

4 Interaction Curve - Unperforated Beam 95

5 Load - Deflection Curves 96

6 Interaction Curve - Perforated Beam 97

7 Hinge Locations and Cross-Section of Member 98

8 Stress Distributions and Resultant Forces 99

9 Interaction Curve Showing Low and High Shear Regions 100

10 Interaction Curve - 14WF38 Nominal 7 x lOin Opening Itn X iReinforcing 101

11 Interaction Curve - 14WF3~ Nominal 7f1 X lot Opening 213 x t Reinforcing 102

12 Interaction Curve - 14WF38 Nominal 7 xlot Opening No Reinforcing 103

13 Interaction Curve - 14WF38 Nominal 7 x 14 Opening It x lIt Reinforcing 104

14 Interaction Curve - 14WF38 Nominal 9 x 13t Opening li x -n Reinforcing 105

15 Interaction Curve - 10WF21 Nominal 5~ n x Bi Opening li x i lt Reinforcing 106J

16 Test Beam Dimensions and Dial Gauge Locations 107



19 Lateral Bracing System 110

viii

Page

20 Photographs of Lateral Bracing System III

21 Strain Gauge Locations 112

22 Tensile Coupons 113

23 Stress-Strain Curve from Instron Testing Machine 113

24 Photographs of Test Beams after Collapse 114

25 Comparison Photographs of Test Beams 115


27a Stress Distribution at High Moment Edge of OpeningBeam 2C 117

27b Stress Distribution at Low Moment Edge of OpeningBeam 2C 117

28a Stress Distribution at Centerline of OpeningBeam 2C 118

28b Bending Stress Distribution Across Flange at High Moment Edge of Opening - Beam 2C 118

29a Bending Stress Distribution Across Length of Opening at Top of Flange Beam 2C 119

29b Bending Stress Distribution Across Length of Opening at Top of Reinforcing - Beam 2C 119

30a Stress Distribution at High MOment Edge of Opening - Beam 3B 120

30b Stress Distribution at Low Moment Edge of Opening - Beam 3B 120

31a Stress Distribution at High Moment Edge of Opening - Beam 2D 121

31b stress Distribution at High Moment Edge of Opening - Beam 4B 121

32 Load-Relative Deflection Curve - Beam lA 122

33 Load-Relative Deflection Curve - Beam 2A 122

ix

Page

34 Load-Relative Deflection Curve - Beam 2B 123

35 Load-Relative Deflection Curve - Beam 2C 123

36 Load-Relative Deflection Curve - Beam 2D 124







43 Interaction Curves - Test Beam lA 128

44 Interaction Curves - Test Beams 2A and 2B 129

45 Interaction Curves - Test Beams 2C and 2D 130

46 Interaction Curves - Test Beam 3A 131

47 Interaction Curves - Test Beam 3B 132




51 Lateral Deflection at High Moment Edge of OpeningBeam 2C 136

52 Flange Buckling at High Moment Edge of OpeningBeam 2C 137

53 Shear Stresses at End of Reinforcing 138

54 Variable Reinforcing Size 139

55 Variable Moment to Shear Ratio 140


x

Page


58 Variable Aspect Ratio 143

59 Variable Opening Depth to Beam Depth M4

xi

LIST OF TABLES

Page

1 Beam Properties 145

2 Tensile Test Results 146

3a Shear Values Corresponding to Yielding 147

3b Shear Values Corresponding to Strain Hardening 147

4 Shear Values Corresponding to Whitewash Flaking 148

5 Correlation Between Experiment and Theory 149

6 Correlation Between Experiment and Approximate Theory 149

7 Correlation Between Experimental Failure and Complete Yielding 150

8 Correlation Between Theoretical Failure and Complete Yielding 150

SUJn1[ARY

ACKNOWLEDGEMENTS

NOTATION

LIST OF FIGURES

LIST OF TABLES

TABLE OF CONTENTS

Page

ii

iii

iv

vii

xi

CHAPTER I INTRODUCTION

11 General Background 1


13 Elastic and Ultimate Strength Analysis 4

CHAPTER 11 ULTINfATE STHEUGTH BEHAVIOR

21 Behavior of Unperforated Beams 8

22 Behavior of Beams with Web Openings 12

23 Previous Ultimate Strength Investigations 17

24 Scope of the Investigation 25

CHAPTER III ULTIMATE STRENGTH ANALYSIS

31 Assumptions 27

32 Low Shear Solution - Case I 29

33 High Shear Solution - Case 11 35

34 Limits of Interaction Curves 36

CHAPTER IV APPROXIMATE METHOD

41 General Remarks 41

42 Development of the Method 42

43 Limitations on the Approximate Method 46

44 Summary and Discussion 50

Page CHAPTER V EXPERIMENTAL INVESTIGATION

51 Selection of Test Beams 56

52 Experimental Setup 59

53 Testing Procedure 62

54 Determination of Yield Stresses 63

CHAPTER VI EXPERIMENTAL RESULTS

61 Behavior of Beams during Testing 66

62 Stress Distributions and Failure Loads 71

63 other Factors 72

CHAPTER VII ANALYSIS OF RESULTS

71 Order of Onset of Yielding 77

72 Stress Distributions 78

73 Failure Loads 80

74 Influence of Reinforcing and Other Variables 82

CHAPTER VIII CONCLUSIONS AND RECO~Th1ENDATIONS

81 Conclusions 85

82 Recommendations for Design 88

83 Recommendations for Future Work 91

FIGURES 93

TABLES 145

APPENDIX A Computer Program for Interaction Curve 151

APPENDIX B Plasticity Relationships 158

BIBLIOGRAPHY 161

1

CHAPTER I

INTRODUCTION

11 General Background

It has become common practice to cut openings in the

webs of beams to permit the passage of utility ducts By

passing these utilities through rather than under the beams

the height of each floor in a building can be reduced thereshy

by effecting a considerable saving in cost particularly in

the case of multistorey structures However cutting an

opening in the web of a beam may considerably reduce the

strength of the beam in the vicinity of the opening If

the opening is located at some position in the beam where

stresses are low this may cause no special problems but

if it is located in a high stress region the designer is

faced with the problem of finding an economioal way to inshy

crease the strength of the beam so that failure will not

result at the opening under working loads

Two alternate approaches are possible in this case

The size of the entire member in question may be increased

on the basis of providing sufficient strength at the openshy

ing so that failure will not ocour The other approach is

to reinforce the original member in the vicinity of the

opening so that the loads can be carried without inducing

2

high stresses The size of the member itself then need not

be increased

Just as the decision whether to pass utilities through

rather than under floor beams must be based primarily on

economic considerations the decision whether to reinforce

an opening or increase the size of the entire member when

the opening is located in a high stress region must also be

based on economics Since many methods of reinforcing an

opening are possible the relative costs and effects of each

type of reinforcing deserve consideration

12 Types of Reinforcing

Reinforcing for an opening in the web of a beam may be

of three basic types

1 reinforcing to resist high bending stresses and low shear

stresses

2 reinforcing to resist low bending stresses and high shear

stresses

3 reinforcing to resist both high bending and shear stresses

Two examples of the first type are illustrated in Figures

la and lb By increasing the areas of the sections above

and below the centroid they provide additional moment

resisting capacity to the section However since shear

stresses are considered to be carried only by the web of

the section this type of reinforcing cannot increase the

3

shear capacity above that given by the shear capacity of

the uncut section times the ratio of the net web area at

the opening to the initial web area of the beam Reinforcshy

ing types to resist high shear stresses are those that

increase the web area so that more web is available to

resist these stresses Web doubler plates as shovm in

Figure lc are typical of this type of reinforcing although

their use is usually very limited due to welding difficulties

This type of reinforcing also increases moment capacity by

increasing the area above and below the opening Another

type of shear reinforcing uses vertical or inclined web

stiffeners to carry the shear forces past the opening A

typical arrangement of inclined web stiffeners is shown in

Figure Id The third type of reinforcing is essentially a

combination of the first two examples of which can be seen

in Figures le and If The area above and below the opening

is increased to give added moment capaCity and the effective

web area is increased by the addition of vertical or inclined

shear plates to provide added shear capacity to the section

A combination of the reinforcing arrangements in Figures la

and Id would be another example of this type Many other

arrangements of web opening reinforcing are pOSSible but

those shown in Figure 1 and mentioned above are typical of

the three main types

Since the problem of cutting and reinforcing an opening

4

in a section is basically one of economics the cost of the

process of reinforcing an opening must be considered This

cost may be divided into three parts the cost of supplying

and cutting the material to be used in reinforcing the openshy

ing that of bending and fitting this material and welding

this material into place Since the cost of welding is

paramount among the several expenses it is advantageous to

keep the welding to a minimum in the selection of a reinforcshy

ing type This tends to make the high shear type of reinforcshy

ing uneconomical and essentially limits the more practical

opening reinforcings to those of the type considered in

Figures la and lb The present investigation is devoted to

this type of reinforcing and specifically to that arrangement

given in Figure lb

13 Elastic and Ultimate Strength Analysis

BaSically two types of analysis were available for this

study - elastic and ultimate strength The elastic analysiS

of the stresses in a beam containing an opening in its web

consists of the superposition of the stresses occurring in

an unperforated beam and the stresses resulting from forces

applied to the boundary of the opening in such a way as to

satisfy certain boundary conditions at the opening This

approach finds its basis in an important work presented by

NIMuskhelishvili(l) in the 1930s and was used by SR

5

Heller Jr et al(2) in 1962 and by JEBower(3) in 1966 to

investigate the elastic stresses around openings in wideshy

flange beams Bowers work also outlined the widely used

approximate Vierendeel analysis based on the assumption that

the beam behaves like a Vierendeel truss in the vicinity of

the opening The shear force carried by the section causes

secondary bending moments to occur within the tee sections

formed by the flange and the remaining part of the web above

and below the opening A point of contraflexure for the

secondary bending stresses is assumed to occur at the midshy

length of the opening The forces acting at the opening and

the resultant stress distributions are shown in Figure 2 It~

is assumed that the shear is carried equally by the tee secshy

tions above and below the opening and that the secondary and

primary bending stresses are additive

Bower also performed an experimental study(4) in 1966

to verify the results of his analysis A series of 16WF36

beams each containing one opening either circular or

rectangular with corner radii of 14 inch were tested at

varying moment to shear ratios The beams were implemented

with electrical resistance strain gauges in the vicinity of

the openings so that strains could be measured and compared

to those predicted by theory The results of Bowers tests

showed that very high stress concentrations occur at the

corners of rectangular openings while smaller stress concenshy

6

trations occur above and below circular openings where the

tee section is the smallest The findings for circular

openings confirm those given by So(5) in 1963 while those

for rectangular openings were confirmed by Chen(6) in 1967

Bower showed that for both types of openings investigated

the elastic analysis predicts all stresses and stress concenshy

trations with reasonable accuracy except for the octahedral

shear stresses away from the boundary of the opening The

elastic analYSiS however is complicated and requirea the

use of a computer and is incapable of dealing with the presshy

ence of notches or other irregularities at the opening The

approximate Vierendeel method was found to predict bending

stresses and octahedral shear stresses with acceptable

accuracy while failing to predict the stress concentrations

in both cases However the approximate method is relatively

simple and lends itself easily to hand calculations Since

the elastic analysis is not practical particularly for

design purposes because of the length of the calculations

involved it has been suggested by Bower and others(7) that

the Vierendeel analysis be used in its place with stress

concentrations neglected when an elastic analysis is desired

A design based on this method would result in a beam whose

response is not purely elastic under working loads

An analysis based on complete yielding of the sections

where high stress concentrations exist under elastic condishy

7

--~~

~ tions would eliminate this problem Such a plastic or

ultimate strength analysis would take advantage of the

additional strength of the perforated beam between the onset

of yielding and the complete plastification of the section

or sections that would cause failure or uncontrolled deformshy

ations of the beam Besides eliminating the problem of stress

concentrations in the elastic range a plastic analysis also

has the advantage of being simpler to use than the complete

elastic analysis and of not being rendered unusuable when

notches or irregularities are present at the opening since

the method depends only on the reasonably accurate prediction

of the sections where complete yielding or plastic hinges

will occur These regions can generally be determined withshy

out difficulty particularly in the case of rectangular

openings Consideration of these factors suggest an ultishy

mate strength analysis as the most rational approach to

the problem

8

CHAPTER 11

ULTIMATE STRENGTH BEHAVIOR

21 Behavior of Unperforated Beams

Ultimate strength analysis truces advantage of the ducshy

tility of structural steel This ductility mruces it possible

for the steel to undergo large deformations beyond the elastic

limit before failure occurs As can be seen from the idealized

stress-strain curve in Figure 3 the material is elastic up

to the yield level but after this level is reached the strain

increases extensively without any further increase in stress

Beoause of this attainment of the yield stress at the outershy

most fiber of a beam in bending does not cause the failure of

the member Rather the member has reserve strength that

permits increase of load up to the point where the entire

cross-section has reaohed the yield stress ie when a plastio

hinge has fonned at the yielded section Since rotation is

free to occur at plastic hinge locations when sufficient

hinges have formed so that the structure forms a mechanism

and is unstable under the applied loads collapse will occur

For a simply supported beam of uniform cross-section

formation of only one plastio hinge is suffioient to cause

collapse If no shear or axial forces are acting the plastic

moment or maximum moment capacity of the section is easily

9

obtained and is equal to the first moment of the area of the

section above or below the centroid about the centroid times

the yield stress of the material This is writte~ as

where Z is the plastic section modulus of the section and ~y

is the yield stress

The presence of a moment-gradient (shear) has the effect

of lowering the plastic moment capacity of a member Since

the shear force is carried by the web of the member the

moment carrying capacity of the web and therefore that of

the member must be reduced since the presence of shear

causes a reduction in the normal stress carrying capacity of

the web The yielding criterion of von Mises is generally

accepted as being the most applicable to structural steel

and can be explained physically in either of two ways

Henckys interpretation states that when the energy of disshy

tortion reaches a maximum value yielding results A preferred

explanation offered by Prandtl states that when the shear

stress on the octahedral plane reaches some maximum value

yielding occurs(8) Von Mises t yield criteria can be

expressed mathematically as

10

where ltSyiS the yield stress ltS the normal stress and the

shear stress The effect of shear on moment capacity can

best be illustrated by an interaction curve as shown in

Figure 4 where the axes have been non-dimensionalized by

diVision by Mp and Vp Vp is the maximum shear carrying

capacity of the section under pure shear and is obtained

from equation (2) as

where = Vw( d-2t) V is the shear force and w( d-2t) the

clear web area Equation (3) presupposes that the flanges

of the section carry no shear The broken line in Figure 4

illustrates the interaction between moment and shear for an

unperforated wide-flange section as predicted by equation (2)

Such an interaction curve shows for any given moment value

the corresponding shear value that a member can safely sustain

Any value less than those on the interaction curve (those

bounded by the VV and MM axes and the interaction curve)p p represents a combination of moment and shear which the member

can safely carry Values on or outside the interaction curve

represent those combinations which would cause collapse of the

member The reduction in strength due to shear is however (9)fairly small for unperforated sections and OISC and

AISc(lO) design codes for buildings allow it to be neglectshy

11

ed for design purposes since the ooouranoe of strain hardenshy

ing makes possible the attainment of moments greater than Mp

in the presence of shear The allowable moment and shear

values thus permitted are those bounded by the solid line in

Figure 4

The inorease in strength due to strain hardening oan best

be illustrated by considering the load-defleotion curves in

Figure 5 The load-defleotion ourve for a beam in pure bendshy

ing is given in Figure 5a When the plastio moment of the

seotion is reaohed the defleotions beoome exoessive and

increase with no further inorease in load This ooours

beoause the member forms a meohanism with the formation of

a plastio hinge at the attainment of Mp However when shear

is present the load-defleotion ourve does not beoome horishy

zontal when the reduoed Mp Mp as predioted by equation (2)

and given by the interaction curve in Figure 4 is reached

but rather oontinues to climb at a lesser slope reaching

values equal to or greater than the full Mp given by equation

(1) This oan be seen in Figure 5b The increase in strength

above Mp under moment gradient is due to strain hardening

and is usually neglected in simple plastiC theory although

it can be predicted but the procedure is complicated for

all but very simple cases Strain hardening occurs in the

presence of shear because yielding occurs in localized areas

or slip bands so that the material within these bands strain

12

hardens before adjacent areas reach the point of yield Alshy

though simple plastic theory neglects strain hardening the

significance of this phenomenon should not be underestimated

since all rolled sections are proportioned such that the limit

of shear carrying capacity of the web lies within the strain

hardening range(ll)

22 Behavior of Beams with Web Openings

The introduction of an opening into the web of a wideshy

flange section has two effects on the interaction curve The

first and more immediately apparent of these effects is the

reduction of shear and moment capacity by the reduction of the

area available to resist these forces The moment capacity

is reduced to

(4)

where w is the web thickness and h is the half depth of the

opening The shear capacity is reduced to

v = w( d-2t-2h) $-3

This change in the interaction curve is shown by the broken

line in Figure 6 The second effect caused by the presence

of an opening is the strong interaction between moment and

shear due to the member behaving like a Vierendeel truss in

13

the vicinity of the opening This is the same type of

behavior described in Section 13 except that the member

is analyzed in the plastic rather than the elastic range

The secondary bending moments caused by the shear force are

added to the primary bending moments thus causing the critical

sections to yield completely at lower loads than would be exshy

pected were this interaction effect ignored This is shown

by the line in Figure 6 where the effects of shear as given

by equation (2) have also been included

The addition of horizontal bar reinforcing above and

below an opening in a wide-flange beam increases the maxishy

mum moment capacity of the section (no shear acting) above

that of an unreinforced opening by increasing the area of

the section capable of resisting moment The effects of

interaction between moment and shear are also reduced by the

addition of reinforcing since the reinforcing is of such a

type as to be particularly well suited to resist secondary

bending moments due to Vierendeel action The maximum shear

carrying capacity of a reinforced opening may reach

(Vv) = d-2t-2h (6)P max d-2t

which is the maximum shear carrying capacity of the section

assuming no interaction and is derived from equation (5) by

division by equation (3) This can represent a considerable

14

increase in strength over the shear capacity of the unreinshy

forced opening which is normally much below (VVp)max because

secondary bending moments have to be resisted to a large

extent by the web since there is no reinforcing to perform

this task

The presence of an unreinforced rectangular opening in

the web of a wide-flange section oauses ohanges in the failure

modes of the beam as well as in the interaction ourve Sevshy

eral investigations have confirmed these ohanges in failure

modes(461213) Vllien subjected to pure bending the member

is found to fail by oomplete yielding of the tee sections

above and below the opening Load defleotion curves for this

case are similar to those for an unperforated beam under pure

bending in that with the attainment of Mp the curve becomes

horizontal and collapse eventually occurs with no further

increase in load

Vhen the perforated beam is subjected to bending with

shear large relative displacements occur between the ends

of the opening and at the same time localized plastiC binges

form at each of the four corners of the opening by complete

yielding of the tee section at these locations The load-

deflection curve in this situation is similar to the load-

deflection curve of an unperforated member under moment-

gradient except that the second portion of the curve after

15

the bend is not always as straight as that for the unpershy

forated beam and in fact may possess considerable

curvature in some cases(13) The value of the moment at the

opening at which the load-relative deflection curve bends

is not that of the unperforated section but instead is a

reduced value obtained by considering both reduced area and

moment-shear interaction as described previously_ Again

the increase in load above the value corresponding to the

predicted moment is due to strain hardening for the same

reasons as cited previously_ This strain hardening effect

makes it exceedingly difficult to ascertain at what value of

shear or moment an experimental test beam should be considershy

ed to have failed so that this failure load can be compared

to one predicted by theory It can be said with certainty

that this value should lie somewhere between the load at

which the load-relative deflection curve starts to bend from

its initial slope and the ultimate load at which the beam

suffers total collapse This total collapse will be either

by web buokling at the opening or by oraoking of one or more

of the corners of the opening after which deflections become

uncontrolled and the load carrying capacity of the beam

decreases

The use of the collapse load as the failure load would

however not be justified unless the effects of strain hardenshy

ing and the possibility ot tearing and buokling are incorporatshy

16

ed into the analysis Also the deflections become so exshy

cessive as to make the beam unserviceable before this load

is reached Indeed a rational definition would have to be

such as to have the failure load fall within the region

where the load-relative deflection curve bends from its

initial slope to some new slope One possible means of

defining failure in this type of situation is suggested in

ASCE tlOommentary on Plastic Design in Steel(14) and

is perhaps the most rational approach to the problem The

initial slope of the load-relative deflection curve is

multiplied by some constant factor to obtain a new slope

A line having this new elope is then drawn tangent to the

upper portion of the load-relative deflection curve and the

load at which this new slope intersects the initial slope is

taken to be the failure load This is analogous to considershy

ing strain hardening in some simple case where the slope of

the theoretical load-deflection curve in the strain hardening

range is equal to a constant times the initial slope of the

curve when the beam is still elastic If this method is to

be used a method of determining a suitable constant must

be chosen possibly from experiment

The modes of failure of a beam containing a reinforced

opening are expected to be the same as those with an unreinshy

forced opening under both pure bending and bending with shear()

Similarly the load-deflection curves or the load-relative

17

deflection curves would be the same in both cases Thus for

cases of moment gradient the same situation exists for both

reinforced and unreinforced openings where the load continues

to increase past the attainment of the predicted moment

capacity of the section due to strain hardening and the same

difficulty exists for determining failure loads for beams

with reinforced openings as for the unreinforced case

The method suggested in ASCE and described previousshy

ly for determining failure load was adopted for the purposes

of correlating experimental and theoretioal values of failure

load in this study Since the load-relative defleotion curves

for some of the test beams approached a fairly constant slope

in the strain hardening range it was decided to determine

the slopes and constant multiplication factors for these

cases and apply the same factor to all of the test beams

since all were similar It was thus decided to use a value

of thirty times the initial slope for the final slope in

determining experimental failure loads

23 Previous Ultimate Strength Investigations

While considerable theoretical and experimental work

has been done in the determination of elastic stresses around

openings in plates and beams relatively little has been done

ooncerning ultimate strength behavior In 1958 wJworley(12)

18

carried out an investigation of the ultimate strength of

aluminum alloy I-beams containing web openings of various

shapes Rectangular elliptical and triangular openings

were studied with the main objects of the research being

to determine which shape of opening resulted in the smallest

loss of strength and to verify the validity of an upper

bound theorem in predicting the fully plastiC load carrying

capaCity of the test beams The upper bound plastiC theorem

states that of all the possible mechanisms or combinations

of plastic hinges sufficient to cause collapse the one that

ensures collapse at the lowest load is the correct mechanism

under which the member will actually fail Worleys experishy

mental results were in fairly good agreement with the ultimate

loads predicted by the upper bound theorem and were the most

accurate for the case of rectangular openings where hinge

locations were more easily predicted than for the other openshy

ing types Of the various opening shapes tested it was found

that the elliptical opening was the strongest on a maximum

area removed baSiS while the triangular was stronger on a

maximum depth basiS The practical need for these two shapes

especially the triangular is however questionable Worley

excluded the effects of shear and axial force in his analysis

and for rectangular openings assumed a four hinge mechanism

with hinges considered at the corners of the opening

19

In 1963 W-CSo(5) presented a study of large circular

openings in wide-flange beams including an ultimate strength

analysis based on the assumption of a plastic hinge over the

center of the opening Sots analysis however neglected

secondary bending effects due to Vierendeel action in the

vioinity of the opening and also ignored shear yielding

effects in the web The experimental investigation performshy

ed in the ultimate strength range consisted of the testing

of two l4WF30 beams to collapse The first beam was tested

in pure bending so that shear and Vierendeel action were not

present and the ultimate strength predioted by Sos analysis

would be expeoted to be oorreot since the same strength would

be predicted in this case whether or not these effects were

considered Deviation between experiment and theory was 3

The second beam was tested under moment gradient but due to

the relative location of opening and maximum moment (the

beam was simply supported and subjected to a single concenshy

trated load) the beam was expected to and did fail under

the load and not at the opening

S-y Cheng(15) in 1966 considered bending shear and

axial forces in his investigation of the ultimate strength

of extended circular openings in wide-flange beams At high

values of shear a four hinge mechanism was assumed while at

low shears hinges were assumed above and below the opening

20

where the tee section was smallest - similar to the failure

mode in pure bending Stress distributions were assumed at

hinge locations for each case Two l4WF30 beams were tested

to destruction one in pure bending and the other with a

moment to shear ratio of 24 inches which is a high value of

shear for the size beam used While reasonable agreement

between theory and experiment was obtained for both beams

tested an inherent fault in Chengs work lies in its inshy

ability to predict a continuous variation of failure loads

as the shear varies from a low to a high value It is

impossible for the change from a one to a four hinge mechanism

to occur suddenly with only a slight increase in shear

Experiments in the ultimate strength region were performshy

ed by I-C Chen(6) in 1967 on two 14WF30 beams each containing

two large openings in their webs One simply supported beam

containing two 8 inch diameter circular openings spaced 8

inches apart was found to fail in the manner of a be~u conshy

taining only one opening failure being associated with the

opening subjected to the largest moment both being under

the same shear force In the second beam containing two

8x12 inch reotangular openings spaced 4 inches apart the two

openings failed as a unit with four hinges forming at the

outer oorners opoundthe openings and the web between them evenshy

tually buckling No theoretioal ultimate strength analysis

was offered by Chen

21

In 1968 JEBower(16) proposed an ultimate strength

theory for beams containing a single rectangular opening in

their webs A point of contraflexure for Vierendeel stresses

was assumed to occur at the midlength of the opening and

three alternate stress distributions were proposed for

complete yielding of the low moment edge of opening The

first of these stress distributions assumed no Vierendeel

action and was quickly discounted as giving unrealistically

high predictions of strength The two remaining stress

distributions both took into account secondary bending moments

due to shear the first assuming localized web yielding in

shear and the second assuming uniform shear distribution

over part of the web with this area yielding in combined

bending and shear The first of these lower bound solutions

was based on an unrealistic stress distribution and proved

to give unrealistically low predictions of the beam strength

The second lower bound solution which was finally adopted

by Bower was based on a more realistic stress distribution

but was restricted in that the point of contraflexure was

assumed at the midlength of the opening which is not always

the case

Experiments were performed on four 16WF36 beams each

oontaining one rectangular opening as part of this work

Collapse loads are oited for each of the test beams and are

22

compared with predictions from the second lower bound analysis

However since all of the beams were tested at relatively high

ratios of shear to moment considerable strain hardening

would have occurred before collapse and experimental results

which take advantage of the reserved strength due to this

strain hardening (even though accompanied by large deformashy

tions) cannot justifiably be compared to a theory that ignores

this effect

A more general ultimate strength analysis for beams with

rectangular openings was offered by RGRedwood(17) also in

1968 A point of contraflexure was assumed to occur someshy

where within the tee section above and below the opening but

its location was not dictated as in Bowers analysis A

four hinge mechanism was assumed on the basis of previous

tests(13) with stress distributions assumed at both high

and low moment edges of the opening Shear stresses were

assumed carried by the entire web of the section with web

yielding occurring in combined bending and shear

Redwoods analysis was correlated with his previous

experimental investigations and found to be conservative

when shears were large But again the effects of strain

hardening were included in the experimental collapse loads

and not in the theory If the experimental failure loads

could be related to the occurrence of full yielding at the

23

critical sections a more valid comparison with the theory

would result Such a comparison was taken into account by

Redwood in a later paper(la) In this paper failure is defined

in a way similar to that suggested in the previous section

Another recent paper by Redwood(19) deals with beams

containing multiple unreinforced openings in their webs A

method is offered for determining the minimum spacing required

between openings to prevent their interaction and failure

as a unit An analysis is also given for determining the

strength of a member when failure is associated with more

than one opening and this is combined with Redwoods analysis

offered previously(17) for failure associated with only one

opening The theory is compared to the experimental work

performed by Chen(6) in 1967 (7)In 1964 EPSegner Jr conducted an investigation

of several types of reinforcing for rectangular openings

testing six wide-flange beams in four different sizes each

containing two or more openings The openings in five of

the six beams were reinforced each With one of five main

types of reinforcing The openings were positioned at

several different moment to shear ratios and each of the

beams was tested to destruction Unfortunately little can

be said about the performance of the different types of

reinforcing under varying moment-shear conditions because

24

of the large number of variables included in each of the tests

Perhaps Segners most significant contribution is to be

found in his cost comparison of the various types of reinshy

forcing for rectangular openings Estimates cited for six

different fabricators clearly indicate that shear-type reinshy

forCing or reinforcing designed to carry high shear forces

around the opening is economically non-feasible except in

unusual circumstances The most economical type of reinshy

forcing included in this study consisted of two flat plates

bent to fit the shape of the opening and welded inside the

opening with a small gap between them occurring at the midshy

depth of the opening This is the same type as shown in

Figure la Another type of reinforcing included in Segnerts

experimental work but unfortunately omitted from his eco~

nomic comparison consisted of straight horizontal reinforcshy

ing plates welded above and below the opening at the openingis

boundary The plates extended past the vertical edges of the

opening to allow for anchorage of the plate Considerable

welding problems were said to have been encountered by placing

the reinforcing flush with the upper and lower edges of the

opening since the fillet was likely to overrun into the radii

of the re-entrant corners but this could easily be overcome

by leaving a small web stub between the plate and the opening

to allow for welding This type including the modification

25

for welding purposes is that given in Figure lb

Although this reinforcing type was not included in the

economic comparison presented by Segner it would seem to be

of only slightly greater cost than that in Figure la

Approximately the same amount of reinforcing material is

needed for these two types and while the straight bar reinshy

forcing requires more welding it does not require bending

of the plates and eliminates the worry of fit between the

plate bends and the opening radii After giving careful

consideration to the economics and fabrication problems of

reinforcing a rectangular opening in the web of a beam it

was decided to devote the present investigation to reinforcshy

ing of the horizontal straight bar type

24 Scope of the Investigation

An ultimate strength analysis is presented for wideshy

flange beams containing large rectangular openings in their

webs and reinforced with straight bar reinforcing above and

below the openings The investigation includes the effect

on beam strength of variable opening depth to beam depth

opening length to depth reinforcing Size and moment to

shear ratio One lOWF21 and ten l4WF38 beams were tested to

collapse Each of the test beams contained one rectangular

opening centered at the middepth of the beam and ten of the

26

eleven openings were reinforced Deflections were measured

at several pOints for all of the test beams and each beam

was painted with a brittle coating of whitewash in the region

of the opening to obtain an indication of the order of onset

of yielding Four of the test beams were also equipped with

electrical resistance strain gauges at critical sections The

experimental investigation encompassed each of the variables

of the theoretical work

27

CHAPTER III

ULTI1iATE STRENGTH ANALYSIS

31 Assumptions

Based on the discussions in the previous chapters the

following assumptions are made in this analysis of reinforcshy

ed rectangular openings with horizontal reinforcing bars

1 Failure of the member takes place by formation of a four

hinge mechanism with hinge locations being at the corners

of the opening These locations are shown in Figure 7a

a cross-section through the opening being given in Figure

7b

2 A point of contraflexure occurs somewhere within the

length of the tee section Its location is the same

above and below the opening but this location is not

dictated

3 Stress distributions at high and low moment edges of the

opening are those shown in Figure 8b Since the stresses

are antisymmetric with respect to the vertical axis of

the beam only the stresses for the tee-section above

the opening are indicated The resultant forces acting

on the tee-section are shown in Figure 8a

4 Shear stresses are carried only by the web of the teeshy

section and are uniformly distributed across this web at

28

hinge locations when hinges are fully formed While

the shear stress at the boundary of the opening must be

zero this assumption introduces only very slight error

into the theory since these stresses can increase from

zero to their maximum value at a very small distance from

the opening 1 s boundary_ Due to the presence of the reinshy

forcing bar it is likely that most of the shear will be

carried by the region of web between the reinforcing bar

and the flange particularly when the secondary bending

stresses are very high This is further discussed in

Section 34 and introduces no significant error in the

development of this method

5 Yielding in the web occurs lliLder combined bending and

shear according to the von Mises criterion stated in

equation (2) Yielding in the flange and reinforcing is

in direct tension or compression

On the basis of the assumed stress distributions equishy

librium equations can be obtained for the tee-section at

values of the shear force which may vary in some cases

from zero up to the maximum as given by equation (5) At

the high moment edge of the opening section (1) in Figure

Bb stress reversal will occur within the web when the shear

force is relatively low (case I) but will occur within the

flange when the shear force is high (case II) These two

29

cases will be repounderred to as low and high shear conditions

respectively_ Three different locations of stress reversal

are possible under the low stress conditions

1 Reversal in the web stub below the reinforcing This

occurs at very low values of shear

2 Reversal in that portion of the web to which the reinshy

forcing is attached

3 Reversal in the clear web between reinforcing and flange

The range of shear values corresponding to reversal in

this region is very small

At section (2) the low moment edge of the opening for

practical situations reversal will occur only in the flange

32 Low Shear Solution - Case I

Consider equilibrium of the portion of the beam to the

right of Section (2) in Figure 7a Taking moments about

section (2) gives

where V denotes shear force as before F2 is the resultant

force at section (2) and Y2 is the distance from the edge of

the opening to the resultant normal F2bull All other symbols

used in this section and not specifically defined here are

defined in Figures 7 and 8 Taking moments about section (1)

30

for that portion of the beam to the right of this section

yields

(8)

where Fl and Yl are comparable to F2 and Y2 respectively

Equilibrium of the tee-section in Figure 8a requires that

(9)

Oonsideration of the assumed stress distribution at section (1)

for stress reversal occurring in the web stub below the reinshy

forcing ie for 0 =klS ~ u gives on integrating the

stresses

Fl = Af ~yf + SW(1-2~)G + Ar ~yr (10)

FIYl = Af ~ Yf(s~) + ~S2WG(1-2ki) + ArJ yr(u1) (11)

where Af = bt the area of one flange and Ar = q(c-w) the

area of one pair of reinforcing bars Separate yield stress

values are used for flange web and reinforcing because of

the large variation possible in these in a single beam A

discussion of the determination of these stresses is included

in Chapter 5 Yield stresses are designated as ~ yf for the

flange G yw for the web and ~ yr for the reinforcing

31

and ~ the normal stress carrying capacity of the web is

calculated from

t2 _ r- 2 3(V)2 ~ - yw-4sw (12)

and is another way of stating the von Mises yield criterion

given by equation (2) From consideration of the stress

distribution at section (2) for stress reversal in the flange

F2 = Af ltS yf(1-2k2) + sw ~ + Ar c yr (13)

F2Y2 = Al~Yf [S(1-2k2)~(1-4k2+2k~)J -1-s2Wlti +Ar~ yr(u~) (14)

Although explicit expressions for shear and moment capacity

cannot be determined from the above equations it is possible

to obtain from them that portion of the interaction curve

relating combinations of shear and moment at midlength of

opening to cause collapse of the beam for shear values in

the assumed range Consideration of other locations of stress

reversal will yield expressions from which the remainder of

the interaction curve can be determined

Specifically for 0 ~ kls ~ u kl can be determined as

a function of k2 from equations (9) (10) and (13) and is

given by

32

From equations (7) (8) (11) and (14) a quadratic expression

is obtained for k2 in terms of known quantities and V

Z [AfCSyf ] Vak2 (sw cs ) s + t + k2 ( -d+2h) + A G = 0 (16 )

f yf

For a given value of V equations (12) (15) and (16) can be

used to determine the corresponding values of kl and k2bull Fl

can then be calculated from equation (10) and FIYl from equashy

tion (11) The moment at mid1ength of opening is then detershy

mined from

(17)

Thus for a given value of V a corresponding value for M can

be obtained The values of kl and k2 must be checked when

calculated to determine whether initial assumptions of

o ~ k1s ~ u and 0 ~ k2 ~ 10 have been met Only one solution

to the quadratic equation in k2 will fulfill these requireshy

ments unless both solutions are equal The above equations

will yield a value for M for a given value of V starting at

V = 0 and continuing for increments in V until the location

of stress reversal at section (1) moves into the region where

the reinforcing is attached ie for kls gt u When this

occurs neither solution to equation (16) will yield values

of k1 and k2 that satisfy the initial assumptions and the

33

initial equations for stress resultants at section (1) must

be replaced by new equations reflecting this change Quite

simply equations (10) and (11) are replaced by

respectively for u ~ kls ~ (u+q) Equations (13) and (14)

remain unchanged for section (2) since reversal will occur

only in the flange at this section Thus for u ~ klS ~ (u+q)

ie for reversal within the reinforcing equations (9)

(13) and (18) yield

Similarly from equations (7) (8) (14) and (19)

- (d-2h)]

va u 2( c-w) Go s ( c-w) G

+ + [ _----oiOioV] [ yr IJ = 0 (21)AfGyf Af csyf sw cs- + s c-w) csyr shy

The same procedure is followed as in the previous case

first V being assumed ~ being obtained from equation (12)

34

k2 from equation (21) kl from (20) Fl from (18) FIJl from

(19) and finally from equation (17) the value of M correshy

sponding to the assumed value of V Again the values

obtained for kl and k2 must be checked to assure that the

initial conditions of u ~ kls ~ (u+q) and 0 ~ k2 ~ 10 are

met These equations will yield admissible values of kl

and k2 for values of V increasing from the last admissible

value obtained by solution of the previous set of equations

and continuing to where kls reaches the value (u+q) ie

for stress reversal in the clear web at section (1) ~nen

(u+q) ~ kls ~ s equations (18) and (19) are replaced by

(22)

and equations (9) (13) and (22) give

(24)

While from equations (7) (8) (14) and (23)

(25)

35

For a given value of V a corresponding value for M is

determined in the same manner as for the two previous cases

Acceptable solutions will be obtainable as long as the stress

reversal at section (1) falls within the clear web However

when this reversal occurs in the flange the so-called high

shear solution is indicated

33 High Shear Solution -Case 11

When the shear force is high stress reversal at section

(1) occurs in the flange and the equations for resultant

normal force and moment arm from the edge of the opening

become

Fl = Af ltS f(2k -1) - sw~ - A C (26)Y 1 r yr

The stress reversal at section (2) will still occur in the

flange and equations (13) and (14) are used with the above

equations and equations (7) (8) and (9) to obtain

(28)

(29)

36

For a given value of V a corresponding M is obtained from

the above equations and equations (12) and (17)

By starting with a value of V = 0 and increasing this

by a small increment with each successive iteration correshy

sponding values of M can be found for the full range of V

values by use of the equations in this and the previous

section An interaction curve can then be plotted relating

V and M over their full range of values A typical intershy

action curve is shown in Figure 9 for a beam containing a

reinforced opening where the curve has been nondimensionalized

by division of V and M by Vp and Mp respectively Also

indicated in this curve are the four regions corresponding

to the different locations of stress reversal at the high

moment edge of the opening

34 Limits of Interaction Curves

It is quite possible for the MM values for a certainp range of VV values on the interaction curve to be greaterp than 10 as can be seen in Figure 9 This simply means

that the reinforced opening can be stronger than the unperfoshy

rated beam for pure bending and for bending with very low

shear However values of MM greater than 10 have no realp practical significance because in such cases failure of the

beam would occur not at the opening but at some unperforated

37

portion of the beam near the opening Therefore 10 is

taken as the maximum value of MM and a horizontal line isp drawn at this value from the MM axis to intersect thep interaction curve This is shown in Figure 9

The interaction curve is also limited with respect to

the maximum VV value permissible Since shear forces canp only be carried by the remaining part of the web the maximum

plastic shear capacity at the opening is equal to V thep plastic shear capacity of the uncut section times the ratio

of the remaining clear web to the initial clear web This

is written as

d-2t-2h (6)= d-2t

In other words when the web of the section at the opening

has yielded in shear its normal stress carrying capacity

~ becomes zero The value of V at which this occurs can be

determined from equation (12) and the ratio of this shear

value to Vp is given by equation (6) Vmen this limiting

value of shear is reached it is no longer possible to obtain

additional points on the interaction curve because any

further increment in V would yield an imaginary solution for

~ Rather when(VVp)max is reached a line is drawn from

this last point on the interaction curve perpendicular to

38

the vVp axis This line is the final portion of the intershy

action curve and represents the actual limiting value of

This limiting value of VVp is however not always

reached for all practical sizes of beam opening and reinshy

forcing It is possible for the equations in the previous

two sections to yield values of kl and k2 that no longer

conform to initial assumptions (for any position of stress

reversal) at a shear value less than the maximum predicted

by equation (6) This occurred in some of the cases tested

to confirm the consistency of these equations and was found

to be caused by an imaginary solution to the quadratic in k2

occurring while stress reversal at section (1) was still in

the clear web of the section Stress reversal at section (2)

was as assumed in the flange Consideration of other

locations of stress reversal did not give real answers This

condition was then investigated and it was found that because

of an interdependence of opening length and reinforcing area

either the opening length must be less than a given value or

the reinforcing area greater than another value in order

for the interaction curve to be able to pass this region

The limiting half length of opening a is given by

(30)

39

and is obtained by combining equations (7) (8) (14) (23)

and (24) with ~ equal to zero By eliminating small terms

this may be simplified to

(31)

Alternatively the minimum area of reinforcing required is

given by

A =2 ( 32) r 13

If this requirement is met it will be assured that the

maximum shear capacity of the section will be reached

Physically the interdependence of reinforcing area and length

of opening can be interpreted to mean that as opening length

is increased the secondary bending stresses which are a

function of shear and opening length increase at sections

(1) and (2) to such an extent that unless the reinforcing

area is increased so that it can carry these high stresses

the lower portion of the web is forced to carry such high

bending stresses that little or no shear can be carried by

this portion of the web The shear stress can then no longshy

er be considered to be uniform across the entire web of the

tee section but instead is restricted to the clear web

between flange and reinforcing or to a portion of this area

40

Reaching the maximum value of vVas given by equationp

(6) is not a necessity either for the interaction curve or

for the performance of a given beam and opening A given

interaction curve is still correct if it terminates before

reaching this value of vVp This occurs when the MMp value

decreases rapidly for very small increments in vvp and

becomes zero In this case the last portion of the intershy

action curve is very nearly a vertical line This line then

represents the actual maximum shear capacity for the given

beam opening and reinforcing dimensions Only when a beam

is to perform under high shear conditions and it is desired

to develop its maximum shear capacity is it necessary to

ensure that Ar is large enough so that this value can be

reached

41

CHAPTER IV

APPROXIMATE METHOD

41 General Remarks

If a particular beam and opening size is being investishy

gated over a very limited range of shear values it is a

fairly straightforward matter to calculate the moment capacity

of the beam corresponding to the given shear values using

the equations presented in the previous chapter hven if the

location of stress reversal were consistently assumed in the

wrong region the calculations would have to be repeated at

most four times for a particular shear value However it is

quickly realized that for a large range of shear values or

for a complete interaction curve it would be extremely

laborious to perform these calculations by hand and recourse

to automatic computation would seem almost essential A

computer program has been developed to perform the necessary

calculations for a complete interaction curve for a specified

beam opening and reinforcing size and is given in Appendix A

Since the practical development of a complete interaction

curve is seriously limited by the length of calculations

involved a simple to use approximate method is suggested for

determining the strength of a perforated beam at various

moment to shear ratios In the development of this approxishy

42

mate method the high shear region only is considered for

reasons which will become evident

42 DeveloEment of the Method

Combination of equations (3) (12) and (28) result in

the following expression

(33)

where for simplici ty ~I~- - ~Y= ~~ t the specified minimum

yield stress of the material Similarly equations (1) (17)

(26) (28) and (29) can be combined to give

M d(kl -k2) + t(k~-ki) (34) ~ = a-i) + wtf2t)~

Xf

and from equations (12) (28) and (29) kl and k2 are related

by

(35 )

If it is now assumed that the flange thickness is significantshy

ly less than half of the remaining clear web tlaquo s the

above expressions can be readily simplified Since (u+~) is

43

of the same order of magnitude as t it is also assumed that

(u+~) laquo By eliminating small terms and substituting for

kl+k -l-x = ~ equation (35) can be simplified to2 f

(36)

This simple quadratic can readily be evaluated for~ In

general both solutions will be real although for a wide

range of beam and opening sizes investigated the solution

corresponding to a minus in the quadratic formula was found

to fall within the region of low shear on the interaction

curve (as defined in the previous chapter) while the solution

corresponding to a plus consistently fell in the medium to

high shear region thus satisfying the initial assumptions

This latter solution was therefore considered to be the

correct one and the corresponding value for ~is given by A 2as [ 2 ] 12_2s2( r) + w2(s2~) _ 4A2

IS = If f)Af 3 r ( 37 ) - (s2 + 4a2 )

T

which may be rewritten as

_2~(Ar + l(Aw) ( 2h)2 1 ( r 2 ~ C = 1+01 If) 2 If l-er ~ - 16 X)(l+o)~ (38)

where Aw = wd and 0lt = i(~)2(k - 1)2

Equation (34) can also be simplified by eliminating small

44

terms and by considering that for the transition from low to

high shear to occur kl must equal unity Equation (34) then

becomes

M (39 )M= p AW

1 + 4A f

Similarly equation (33) can be simplified to

(40)

For zero or very small values of ~ the above equation can

yield values of VV greater than the maximum VVp value as p

given by equation (6) This implies that some portion of the

shear is carried by the flanges of the beam Since the flanges

can in fact carry some shear and since the theory as given

in Chapter III is sufficiently conservative for large shear

values (as can be seen from the experimental results given

in Chapter VI) equation (40) is not considered to predict

excessive values of shear capacity and is adapted for use in

this method of analysis The shear capacity is then given

by equation (40) and the corresponding maximum M~Ap value at

this shear value is given by equation (39) where ~ is given

by equation (38)

The maximum shear capacity of the section is given by

45

equation (40) when ~ is equal to zero Solution of equation

(38) for Ar when ~ equals zero gives the required reinforcing

area necessary to reach the maximum shear capacity of a secshy

tion This is given by

- AW(l 2h) flA (41)r - T -a-~

Equation (41) reduoes to equation (32) in Chapter III when

substitution is made for ~ When the reinforcing area

conforms to the above requirement the maximum shear capacshy

ityas given by

V 2hr = (I-a-) (42) p

will be reached and the corresponding maximum moment capacshy

ity at this value of shear is given by

Ar 1 - A

M fr= A p 1 + w

4Af Thus for a given beam opening and reinforcing size if

equation (41) is satisfied equations (42) and (43) together

fix one point on the approximate interaction curve while if

equation (41) is not satisfied this point is fixed by equashy

tions (38) (39) and (40) In either case a line is dropped

from this point perpendicular to the vvp axis thus forming

46

part of the approximate interaction curve The remainder of

the curve is obtained by connecting the point given by equashy

tion (42) and (43) or by equations (38) (39) and (40) with

a point on the M~Ip axis corresponding to the maximum moment

capacity of the section This maximum value of blfvl isp simply the plastic moment of the reinforced perforated secshy

tion divided by the plastic moment of the unperforated secshy

tion and is identical to the same point obtained in the

previous chapter with no shear force acting The maximum

value of Mi1p is given by

Z - wh2 + Arlt 2h+2u+q) = ---------w----------shyZ

or using a consistent approximation for Z this can be

rewri tten as

M A (45 ) M=

p 1 + w4Af

An approximate interaotion curve is given in Figure 9 for

comparison with the interaction ourve obtained by the method

desoribed in Chapter Ill

43 Limitations of the AEproximate Method

The maximum praotical M~lp value for the approximate

47

interaction curve is limited to 10 for the same reasons that

the more exact curve is so limited and a horizontal line

drawn from the MA~p axis at 10 and intersecting the approxshy

imate curve forms its upper portion The maximum value of

VVp for the approximate curve is limited to VVp max given

by equation (42) This limitation has been discussed in the

previous section

Equation (41) specifies the minimum size of reinforcing

necessary to reach the maximum shear capacity of the section

as given by equation (42) yVhen the reinforcing area conforms

to this minimum requirement equation (43) is used to predict

the moment capacity of the section corresponding to the shear

value of equation (42) Examination of equation (43) reveals

that as the reinforcing area is increased past the minimum

given by equation (41) the moment capacity decreases rather

than increases as would be expected from physical considershy

ations or from examination of interaction curves obtained

by the methods of Chapter Ill For sizes of reinforcing

only slightly greater than that given by equation (41)

equation (43) will be sufficiently accurate and will not

significantly underestimate the moment capacity of the secshy

tion although it may be deSirable to replace Ar in equation

(43) with the minimum Ar given by equation (41) Equation

(43) then becomes

48

A 2h l1 - ~(l-a)J~

M ( 46)~= A 1 + W

4Af This may be written more simply as

aw 1 - 73Af

= ---1-shyA 1 + w

4Af The corresponding shear value is given by equation (42)

However as Ar increases more and more above the value

given by equation (41) the moment capacity of the section

will be increasingly underestimated and if more accurate

values are desired recourse must be made to more accurate

methods Examination of equation (38) reveals that as Ar

increases past the value given by equation (41) 0 becomes

negative up to the point where

when the solution for cgt becomes imaginary For values of

Ar lying between those given by equations (41) and (48) ~

is negative and equations (39) and (42) can be used to

accurately predict the point on the approximate interaction

curve from which a perpendicular line should be dropped to

the VVp axis and a line drawn to the point given by equation

(44) on the MM axisp

49

Al though for this case a value exists for (gt equation

(42) rather than (40) is used to determine the shear capacshy

ity of the section This is the only logical approach beshy

cause Ar is greater than that required to reach the maximum

shear capacity and increasing the reinforcing area although

it cannot increase the shear capacity of the section past

that given by equation (42) also should not serve to decrease

this capacity

When Ar is greater than the value given by equation (48)

equation (39) cannot be used to obtain a more accurate preshy

diction of moment capacity because 0 as given by equation (38)

will not be real Consideration of the equations used in

deriving the approximate method leads to the conclusion that

will be imaginary only when the maximum shear capacity of

the section is reached while klslaquou+q) Equation (48) shows

the relation between reinforcing area and size of opening for

this to occur It can be seen that small opening size or

large reinforcing size will eause the maximum shear capacity

to be reached while kls (u+q) When this occurs equation

(42) predicts the shear capacity of the section since Ar is

greater than that required by equation (41) to reach this

maximum shear value At this value of shear ~ equals zero

and equations (17) (20) and (21) in Chapter III reduce to

(49a)

(50)

50

A q (5la )

M = _l_l_qdA_rf=--[2_U_2_+__(2_U_+_q_)_+_2_h_(_2U_+_q_)_-_2_(_k_l_S_)2_-_4_k_l_S_h_J_-_~_~_(_~~)

Mp Aw 1 + 1iA

f

By considering t laquo d equations (49a) and (5la) can be further

simplified to

(49)

(51)

Equations (49) (50) and (51) can then be used with equation (42)

to determine the pOint of intersection of the two lines composing

the approximate interaction curve

44 Summary and Discussion

Determination of the approximate interaction curve can be

summarized as follows

Case I

The maximum shear capacity of the section will be reached

if

A 2 AW(l_ 2h) fT (41)r 4 d 4a where a = ~(h)2(~ _ 1)2 The maximum shear capacity is then4 a 2h bull

given by

V (1 _ ~h) (42)Vp =

51

and the moment which can be attained with this shear acting

is Ar

1- ri- = Af (43)

p 1 + w 41f

where Ar is that given by the right hand side of equation (41)

Case 11

If equation (41) is not satisfied the maximum shear

capacity will not be reached and the shear capacity is given

by

(40)

The moment capacity which can be attained with this shear

acting is

1 1 + w

4Af where ~ is given by

A A _ ~( r) + l( w) ( 2h)2 1 ( r)2 0 ( )~ - 1+ltgtlt Af 2 If l-T 1+CgtI - 16 -x (l+olt)l 38

Case Ill

If Ar is significantly greater than that given by equashy

tion (41) the maximum shear capacity will still be given by

52

but if desired a more accurate estimate of the moment which

can be attained with this shear acting can be obtained as

follows

Case IlIA

For (48)

MMp at maximum VVp is given by

Ar 1 - I - ~

~ = __f_~_ ill A

P 1+ w4If

where r will be negative and is given by

(38)

Case IIIB

For A2 gt (aw) 2

+ (~)2w2 (48)r 3 ~

at maximum VV is given byp A A

1+-f(2h)__1_ JL (ha) (2dh ) (1+ 2dh )Af d 2[j Af (51)

A1 + w

4Af

53

where kl and k2 are given by

+l (50)s

(49)

and only that solution satisfying the conditions 0 ~ k2 ~ 10

and u ~ kls ~ (u+q) is correct

The appropriate one of these three cases can be used to

determine a point on the approximate interaction curve A

perpendicular line is then dropped from this point to the

vVp axis and forms part of the curve The remainder of the

curve is obtained by drawing a straight line from the initial

point to a point on the MMp axis given by equation (44) or

(45) and by then cutting off the approximate interaction curve

at the maximum Mlyenip value of 10 as described in the first

paragraph of this section

Complete and approximate interaction curves for each

size of beam opening wld reinforcing included in the expershy

imental part of this investigation are shown in Figures 10

through 15 A cross-section of the member is shovm on each

curve for reference and the method used in obtaining each of

the approximate curves is stated On Figures 11 14 and 15

the reinforcing size was greater than that given by equation

(41) and both Case I and Case III were used to obtain approxshy

54

imate curves for comparison It can be seen that only when

the reinforcing area was significantly greater than that

given by equation (41) (Figure 11) was there a large difshy

ference in the Case I and Case III values

For the partioular case of unreinforced openings by

substituting for Ar equal to zero in equations (39) and (40)

these reduoe to

M AW 2h 1

1 - 2If(l-or~ r= A (52)

p 1 + w4If

v (2h)JTr = I-d~ p

where ~= i(~)2(~ - 1)2 as before These equations are

identioal to those given by R_GRedwood(17) for unreinforced

openings Again a perpendioular is dropped from the point

defined by these two equations to the vVp axis and another

line is dravln from this point to a point on the MMp axis

given by

111 Z - wh2 (54)r = z

p

thus oompleting the approximate interaction curve for the

case of an unreinforoed opening Using the same approxishy

mations as used previously equation (54) oan be written

55

MM = 1 (55) p

which agrees with Redwoods work and gives a slightly highshy

er value for the maximum MM value A complete and approxshyp imate interaction curve for a section containing an unreinshy

forced opening is given in Figure 12 where the equations

given by Redwood(17) were used for the complete curve

56

CHAPTER V

EXPERIMENTAL INVESTIGATION

51 Selection of Test Beams

As discussed in Chapter I it was decided to limit this

investigation to rectangular openings reinforced with straight

horizontal strips welded above and below the opening It

was also decided to use a standard wide-flange beam rather

than a welded section because of the usual use of these in

building structures where openings are most oommonly requirshy

ed ASTM A 36 structural steel was used throughout all

of the experimental work because this material was used in

most of the previous investigations related to this work

The size of the test beams was chosen on the basis of

economy ease of handling and ease with which reinforcing

bars could be welded in place It was decided to use a secshy

tion that was compact at the specified yield stress and

also at the higher yield stresses expected in the test

beams In the selection of test beams only sections with

low depth to thickness ratios were conSidered in order to

avoid the problem of web instability The flange width to

thickness ratio was also kept as low as possible in order to

avoid premature compression flange buckling On the basis

of these considerations and after a preliminary test pershy

57

formed on a 10VF21 section it was decided to carry out the

remainder of the experimental investigation using a 14WF38

section of A 36 steel The properties of the test beams

used are listed in Table 1 A nominal opening depth to beam

depth ratio of 05 was chosen because of the practical need

for openings of approximately this depth in larger beams and

also because investigations of unreinforced openings were in

this range One test beam included in this work had a nominal

opening depth to beam depth ratio of 0643 A basic opening

length to depth ratiO or aspect ratiO of 15 was chosen

again from practical considerations Two beams were includshy

ed with aspect ratios of 20

The corner radii of the opening had to be large enough

to avoid excessive stress concentrations that might cause

premature cracking of the corners and yet small enough to

not appreciably decrease the effective area of the opening

In previous investigations corner radii ranged from 14 to

one inch A 58 inch radius was used in this study for the

14WF38 beams and a 316 inch radius for the 10WF21 beam

The area of reinforcing for each opening was chosen so

that the moment capacity of the unperforated member would be

equalled or exceeded and the full shear capacity of the net

section could be utilized The moment capacity condition

requires that

58

wh2 gt- (56)2h + 2u + q

This follows from equation (44) The reinforcing area reshy

quired to meet the shear condition is given by equation (32)

in Ohapter 111 It was found that for the 14WF38 beam used

in this investigation with 2hd equal to 05 ~~d ah equal

to 15 the area of reinforcing required to meet these condishy

tions was approximately ~vice that given by the right hand

side of equation (56) One beam was tested in the high shear

range with reinforcing such that the interaction curve fell

short of VVp msx Again it should be mentioned that the

length u is that required for welding the reinforcing in

place and should be as small as possible The width of the

reinforcing bars q was chosen so that the width to thickshy

ness ratio did not exceed 85 as specified by the design

code(910) for ultimate strength design The anchorage

length of the reinforcing bars was also designed by use of

the AISC and OISO design codes on the basis of develshy

oping the full strength of the reinforcing bars at the edges

of the opening The ultimate strength loads were taken as

167 times those for elastic design as recommended by the

codes

The openings in all of the test beams were flame cut

the 10WF2l beam number lA having the corners drilled before

59

cutting and then having the rough edges ground to the proper

size Beams numbered 2A 2B and 3A were entirely poundlame cut

including corners and were not poundinished poundurther The remainshy

ing beams had drilled corners with the rest of the opening

being poundlame cut They were not finished poundUrther

All test beams were simply supported and were loaded with

a single concentrated load The openings were positioned so

as to achieve the desired moment to shear ratios and all

openings were centered at the middepth opound the beam All beams

were locally reinpoundorced with cover plates so as to ensure

poundailure at the openings and web stipoundfeners were added under

the load and at reaction points Local reinpoundorcing was deshy

signed on the basis opound resisting the loads predicted by the

interaction curves plus the expected increase in loads above

this due to strain hardening Beam Sizes opening locations

and local reinpoundorcing are shown in Figures 16 17 and 18 poundor

each opound the test beams

52 Experimental SetuE

All beams were tested in a Baldwin-Tate-Emery 400k

Universal Testing Machine which is a mechanically operated

hydraulic testing machine opound 400k capacity The ends opound the

beam were supported by a specially reinpoundorced 14 inch wideshy

poundlange section that also poundormed part of the lateral bracing

60

system This is shown in Figures 19a and 20a Lateral bracshy

ing consisted of support against horizontal displacement and

rotation at pOints of load and at end reactions Bracing

for the beams numbered lA 2A and 3A was achieved by the

attachment of horizontal 8 inch wide-flange beams to the

vertical portion of the end supports with the 8 inch secshy

tions running the entire length of the test beams on each

side and positioned so that pins could be inserted at bracshy

ing points between the 8 inch sections and the top flange of

the test beam to allow for only vertical displacements of the

test beam The pins were given a heavy coating of grease to

minimize friction This type of bracing proved satisfactory

but made it very difficult to see parts of the test beam durshy

ing the test To alleviate this visual problem beam 2B was

tested with the lateral bracing described above on only one

side of the beam while for the other side vertical bars

attached rigidly to the 8 inch sections and bent to pass

over the test beam were held tightly against the upper flange

of the test beam at bracing points This arrangement proved

unsatisfactory and was not used for subsequent tests

For testing the remaining beams short 8 inch wide-flange

sections were attached vertically to the vertical portion of

the end supports and greased pins were used to provide latershy

al bracing at these locations (Figures 19b and 20b) Lateral

61

bracing at the point of load was provided by roller bearings

- two on each side of the test beam spaced 6 inches apart

vertically and bearing against the web stiffener of the beam

The roller bearings were attached to triangular sections

which in turn were attached to a wide-flange section held

rigidly to the laboratory floor See Figures 19c 20c

and 20d This type of lateral bracing provided nearly

frictionless vertical displacement of the beam while preshy

venting rotation and horizontal displacements and proved

to be very effective Access to the beam during testing

was not at all limited by this bracing system

Deflection dial gauges with a least reading of 0001

inch were used to determine deflections at points of load

end supports edges of openings 2 inches outside each openshy

ing edge and at centerline of beam when possible All dial

gauges were rigidly fixed to non-yielding supports Their

positions are indicated in Figures 16 17 and 18 Four of

the test beams were equipped with electrical resistance strain

gauges in the vicinity of the opening The locations of

these gauges are shown in Figure 21 Each test beam was

painted with an opaque brittle coating of whitewash - a

mixture of lime and water of the consistency of light cream

applied in the vicinity of the opening at least one day before

testing During testing this coating of whitewash flaked

62

off in minute pieces in areas undergoing plastic strains thus

indicating visually which sections of the beam had yielded

The flaking occurred at loads higher than those correspondshy

ing to yielding of each section and did not give an accurate

measure of the onset of yielding but instead gave an estishy

mate of the order in which each of the sections yielded

53 Testing Procedure

Essentially the same procedure was followed in testing

each of the beams The beam was first placed in position on

its end supports a fixed roller under one end and a free

roller under the other The bema was made level and centershy

ed with respect to its loading point and end supports

Lateral bracing was then fixed in place and the head of the

machine brought into contact with the specially hardened

roller through vhich the load was applied to the beam Dial

gauges were then put in position and strain gauge leads

connected to the Budd Automatic Strain Indicator A small

initial load was then applied to the beam and zero readings

were taken for all strain and dial gauges

At least four load increments of either lOk or 20k were

applied within the elastic range with all gauge readings beshy

ing recorded for each load increment When the first inshy

elastic response of the beam was noted either from strain

63

gauge readings on the beams that were so equipped or by

deflection vdth time of any of the dial gauges the load

increment was reduced to 5k and after each load increment was

applied all plastic flow was allowed to take place before

readings were taken To determine whether plastic flow had

ceased at each load increment dial gauge readings were plotshy

ted against time When the S-shaped curve reached its upper

portion and levelled out plastic flow had ceased and readshy

ings were taken During this time required for plastic flow

as much as 30 minutes for higher loads it was important to

hold the load on the beam constant Loading continued in

this manner until the ultinlate collapse of the test beam

either by web buckling at the opening or by tearing of one

or more of the openings corners When this occurred the

beam was no longer able to maintain the applied load The

load was then removed and the beam was taken out of the test

frame

54 Determination of Yield Stresses

In order to accurately access the expected strength of

each of the test beams it was necessary to determine their

actual yield stresses This was accomplished by the testing

of tensile coupons cut from the flange ana web of the beams

and from the reinforcing strip used at the openings The

64

dimensions of a typical tensile coupon are given in Figure 22a

and the locations from which such coupons were taken are shown

in Figure 22b The beams from which coupons were to be taken

were ordered an extra 12 inches long so that this section could

be cut off and tensile coupons made and tested before the testshy

ing of the actual beam Reinforcing strip was ordered an extra

36 inches long for the same purpose Test beams 2A 2B and

3A were all cut from the same beam and were all reinforced

from the same reinforcing strip Test beams 2C 2D 3B 4A

4B 5A and 6A were all cut from the same beam Reinforcing

for beams 2C 2D 4A and 4B was all cut from the same strip

Each of the tensile coupons was tested in either a 20k Riehle

or a 20k Instron Testing Machine Coupons tested in the

Riehle machine were tested under nearly constant load rate

An extensometer was attached to the coupons to automatically

record the stress-strain curves Since the crosshead

position could not be fixed on this particular machine

a static yield stress could not be obtained and therepoundore

only the lower yield stress is given in Table 2 Tensile

coupons tested in the Instron machine were loaded at a

constant cross head speed of 002 inches per minute When

the plastiC plateau was reached the crosshead position was

fixed and the load was allowed to drop off After about five

minutes the static yield stress had been reached and no

65

further change in load could be seen on the automatically

recorded stress-strain curve Straining was then allowed to

continue into the strain hardening range A typical stressshy

strain curve for a coupon tested in this machine is given in

Figure 23 Lower yield and static yield stresses are given

for coupons tested in the Instron machine in Table 2

Either the static yield or the lower yield stresses can

be used in determinimg the expected strength of a test beam

depending on which more closely approaches the actual test

conditions of the beam In this investigation the beams

were tested under loading increments with the loads being

held constant until plastic flow had ceased Since it was

thought that the method of obtaining the lower yield stresses

more closely paralleled the actual testing procedure these

stresses were used in evaluating the expected strength of

the test beams

66

CHAPTER VI

EXPERIMENTAL RESULTS

61 Behavior of Beams during Testing

Since the behavior of all the beams during testing was

quite similar a complete description of only one such typshy

ical beam number 20 is given here Deviations from this

typical behavior are discussed at the end of this section

Beam 20 was first tested elastically to a load of 80k

in lOk increments and was then unloaded Strain gauge and

dial gauge readings were taken at each load increment and

were analyzed After it was determined from these readings

that strain gauges and automatic recording equipment were

in good working order the beam was reloaded this time to

be tested to destruction This check was run for only this

one test beam all others being tested continuously through

elastic and plastic ranges

An initial load of 5k was again applied to the beam and

strain and dial gauge readings taken The load was then inshy

creased to 80k in 40k increments (lOk or 20k increments were

used in all other tests because separate elastic tests were

not performed on these) and then to 120k in lOk increments

With the load held stationary a slight increase of deflecshy

tion with time was noted on the dial gauges at this load so

67

increments were decreased to 5k bull

At l30k slight lateral deflection of the beam at the

opening was noted although this deflection never became exshy

cessive throughout the remainder of the test Strain gauge

readings first indicated yielding at both the bottom edge of

the upper reinforcing bar at the low moment edge of the openshy

ing (Figure 21 gauge 25) and at the top edge of the top

flange at the high moment edge of the opening (gauge 17) at

a l35k load When the load was increased to 140k yielding

occurred at the other top edge of the top flange at the high

moment edge of the opening (gauge 19) At a load of 145k

yielding had occurred at the center of the top flange at the

high moment edge of the opening (gauge 18) and also in the

web at the centerline of the opening (rosette gauge 10 11

12) At 155k the web at the high moment edge of the openshy

ing had yielded (rosette gauge 13 14 15) as had the web at

the low moment edge of the upper reinforcing bar (rosette

gauge 4 5 6) Slight displacement between the ends of the

opening could be seen at a load of l60k and at l65k the

lower edge of the upper reinforcing bar at the high moment

edge of the opening (gauge 20) had yielded

At l70k the upper edge of the upper reinforcing bar at

the high moment edge of the opening had yielded (gauge 21)

thus making the yielding at the high moment edge of the openshy

68

ing complete (based on average strain values for the flange

and reinforcing) While at this load the first flaking of

the whitewash was noted at both the upper low moment and

the lower high moment corners of the opening The web at the

middepth of the beam between the low moment ends of the reinshy

forcing yielded at 175k (rosette gauge 1 2 3) and whiteshy

wash flaking was detected on the underside of the top flange

at the high moment edge of the opening At lSOk the web at

the low moment edge of the opening (rosette gauge 7 S 9)

yielded and whitewash flaking occurred along the web immedishy

ately beneath the upper reinforcing bar from the low moment

corner of the opening to beyond the low moment end of the

reinforcing bar and also in the web above and below the

opening At 185k whitewash flaking occurred at the lower

low moment corner of the opening and in the web immediately

above the lower reinforcing bar from the corner of the openshy

ing to the low moment end of the reinforcing bar Yielding

occurred at the center of the top flange at the low moment

edge of the opening (gauge 23) at 190k and also at the top

edge of the upper reinforcing bar at the low moment edge of

the opening (gauge 26) This made yielding at the low moment

edge of the opening complete (based on average strain values

for the flange and reinforcing) At this same load whiteshy

wash flaking occurred in the web above and below the low

69

moment end of the upper reinforcing bar

The first strain hardening was also indicated from

strain gauge readings at the load of 190k and occurred in

the web at the centerline of the opening (rosette gauge 10

11 12) At 195k whitewash flaking occurred in the web

between the low moment ends of the upper and lower reinforcshy

ing bars Yielding occurred at the lower edge of the top


200k and strain hardening occurred at the upper edge of the

top flange at the high moment edge of the opening (gauge 17)

and at the lower edge of the upper reinforcing bar at the

high moment edge of the opening (gauge 20)

At 211k a crack developed in the lower low moment corshy

ner of the opening and it was no longer possible to maintain

the applied load The beam was then unloaded and removed

from the test frame The crack which occurred at the corner

radius was one half inch long and was inclined toward the low

moment end of the lower reinforcing bar A photograph of this

beam after completion of the test can be seen in Figure 24

Of the remaining beams in the test series only 2D 3B

and 4B were implemented with strain gauges the others having

only the brittle coating of whitewash to indicate the order

of onset of yielding The first of these beams to be tested

lA was given too thick a coating of whitewash and the flaking

70

of this whitewash during testing was not as satisfactory as

on other tests Similar problems were encountered with 2A

which had to be given a second coating of whitewash because

most of the first coat was rubbed off due to improper handshy

ling of the beam before testing The flaking off of the whiteshy

wash on beam 2B during testing was also not as good as was

expected although the whitewash had not been applied too

thickly This beam was tested on an extremely humid day

and it is thought that the problem with the whitewash flaking

was due to this excessive humidity

The order of onset of yielding as indicated by strain

gauge readings is given for each of the strain gauged beams

(20 2D 3B 4B) in Table 3a while the order of onset of

strain hardening is given for these beams in Table 3b Loads

corresponding to complete yielding of the cross-sections are

also given The order of whitewash flaking for each of the

beams is given in Table 4 Ultimate loads are also given

along with the mode of collapse of the beam

Of the eleven beams tested all but one were tested to

collapse The test of beam lA was terminated when deflecshy

tions became excessive although no web buckling or tearing

of corners had occurred and the beam was still able to mainshy

tain the applied load Only one of the beams tested to

collapse ultimately failed by web buckling This was the

beam with the unreinforced opening 6A All of the other

71

beams suffered ultimate collapse by the tearing of one or two

of the corners of the opening Beams 2A 2B 2C 3A 4A 4B

and 5A developed cracks at the lower low moment corners of the

openings Beam 3B cracked at the upper high moment corner of

the opening and beam 2D developed cracks at both the lower low

moment and the upper high moment corners of the opening simulshy

taneously Photographs of each of the test beams after collapse

are shovm in Figure 24 Comparison photographs of the beams for

variable moment to shear ratio reinforcing size aspect ratio

and opening depth to beam depth are given in Figures 25 and 26

62 Stress Distributions and Failure Loads

Based on measured strains stresses were calculated for

beams 2C 2D 3B and 4B Elastic stresses were determined

from the usual elastic theory while plastiC stresses were

determined from plasticity theory presented by Hill(8) and

reduced to the form given by Bower(l6) Initiation of strain

hardening was also determined from this method which is

summarized in Appendix B

Stress distributions at the high and low moment edges

of the opening of beam 2C are shown in Figure 27 while those

at the centerline of the opening and across the flange at the

high moment edge of the opening are given in Figure 28 The

variations in bending stress across the length of the opening

72

at the center of the top flange and at the top of the upper

reinforcing bar are shown in Figure 29 Figure 30 shows the

stress distribution at the high and low moment edges of the

opening for beam 3B and the stress distributions at the high

moment edge of the opening of beams 2D a~d 4B are given in

Figure 31

Experimental failure loads for each of the test beams

were determined from the load-deflection curves for each

test as described in Chapter 11 These curves and their

corresponding failure loads are given in Figures 32 through

42 Interaction curves based on the analysis presented in

Chapter III were dravn for each of the test beams using the

actual measured dimensions and yield stresses for each beam

The experimentally deterrQined failure loads were plotted on

these curves which are given in Figures 43 through 50

Approximate interaction curves for nominal dimensions and

yield stresses are also included in these figures

63 Other Factors

Of the eleven beams tested only one underwent signifshy

icant lateral buckling during testing Beam 2B buckled

laterally at the high moment edge of the opening due to the

inadequate lateral support provided as described in Chapter

V However final collapse of this beam was due to the tearshy

73

ing of one of the corners of the opening and not to the

lateral buckling Also the lateral buckling was not

significant until after the very high loads associated with

strain hardening were reached The modified lateral support

system shown in Figures 19b and c and 20bc and d and discussshy

ed in the previous chapter effectively prevented lateral

buckling of the remaining test beams and no further diffishy

oulties were encountered due to this factor

For each of the test beams lateral deflections were

largest at the high moment edge of the opening although they

never became significant except in the case of beam 2B A

curve showing lateral deflection at the high moment edge of

the opening versus shear force at the opening is given for a

typical test beam 20 in Figure 51 The shear corresponding

to the experimental failure load is indicated on the curve

and the laterally unsupported length of the beam is given

The laterally unsupported length of the test beams did not in

any case exceed the lengths specified by the design codes(9 10)

Local buckling of the top flange occurred at the high

moment edge of the opening for several of the beams tested

although this buckling always occurred at very high loads

after considerable strain hardening had taken place Figure

52 shows the difference in strain readings between the upper

and lower edges of the flange at the high moment edge of the

74

opening versus the shear force at the opening for a typical

test beam 20 The distribution of stresses across the width

of the flange at the high moment edge of the opening (Figure

28b) show a relatively uniform distribution up to yielding

at this location However the effects of local flangebull

buckling can be seen by the considerably higher compression

stresses at the edges of the flange at very high values of

shear

The nine beams containing reinforced openings which were

tested to destruction all exhibited an effect which was preshy

viously unexpected The strain gauge readings for rosette

gauges 1 2 3 and 4 5 6 on beam 20 (Figure 21) as well as

the whitewash flruring on all nine of these beams indicated

widespread yielding of the web between the low moment ends of

the reinforcing bars This effect occurred at loads near the

experimental failure loads of the beams but did not interact

with the development of hinges at the corners of the openings

because of the extension of the reinforcing bars well past

the edges of the openings In addition to the shear yielding

in the web between the low moment ends of the reinforcing

bars the whitewash flaking on beams 2D and 4A also indicated

the same type of yielding occurring at slightly higher loads

in the web between the high moment ends of the reinforcing

bars The whitewash flaking in these areas was sufficient

so that it can be seen in some of the photos in Figure 24

75

The shear yielding that occurs in the web between the

ends of the reinforcing bars may be due at least in part to

the development of the strength of the reinforcing bar withshy

in a short part of its anchorage length Considering the

reinforcing bar in compression above the opening it can be

seen that since its strength must increase from zero at the

ends the unbalanced compression force in the beam causes a

shearing force to occur in the web which at the low moment

end of the reinforcing is additive to the existing shears

(due to the vertical shear force on the beam) below the reinshy

forcing and subtracted from those above This is illustrated

in Figure 53 In the same manner the shears resulting from

the tension in the lower reinforcing bar act in the same

direction as and are added to those in the web between the

reinforcing bars while being subtracted from those in the web

between reinforcing and flange At the high moment ends of

the reinforcing bars the same effect can occur since the top

reinforcing bar is usually in tension while the lower bar is

in compression Again the shears between the reinforcing bars

are increased while those between reinforcing bar and flange

are decreased This would account for the yielding in the web

between the low moment ends of the reinforcing bars for all

of the reinforced beams tested to collapse as well as explaining

76

the yielding in the web between the high moment ends of the

reinforcing bars for beams 2D and 4A

77

CHAPTER VII

ANALYSIS OF RESULTS

71 Order of Onset of Yielding

Comparison of the order of onset of yielding as detershy

mined by strain gauge readings and whitewash flaking in

Tables 3 and 4 indicates that the whitewash flaking is a

reliable method of estimating the order of onset of yieldshy

ing Actually since the strain gauges were only at scattershy

ed locations while the whitewash covered the entire area

around the opening the whitewash flaking is a considerably

more complete guide to the order of onset of yielding It

can also be said that yielding begins first at the corners of

the opening as indicated by the whitewash consistently

flaking first at these locations even though it was impossishy

ble to confirm this by strain gauge readings since gauges

could not be fitted in these locations

The yielding patterns clearly indicate that failure of

the beams occurred by complete yielding of the cross-sections

at the edges of the openings ie at the assumed hinge

locations The formation of hinges at these locations was

accompanied or followed by the shear yielding of all or part

of the web along the length of the opening and by the shear

yielding of the web between the low moment ends of the reinshy

78

forcing bars and in two cases also between the high moment

ends of the reinforcing bars The increase in load past the

formation of the first hinge is accompanied by localized

strain hardening Since the corners of the opening yield

considerably before other locations strain hardening probshy

ably begins at these corners well before the first hinge is

completely formed Yielding in the web above and below the

opening does not seem to become complete until considerable

rotation has occurred at the hinge locations The complete

yielding of the web in these areas would indicate that the

full shear capacity of the member is utilized

72 Stress Distributions

Experimental bending and shear stresses are shown for

each of the strain gauged beams (20 2D 3B 4B) in Figures

27 through 31 Straight lines have been drawn through pOints

representing measured stresses although this does not mean

that the variation is necessarily linear Examination of

each of these stress distributions shows that stresses inshy

creased in proportion to increasing loads while the stresses

were still elastic While in the elastic range the neutral

axis remained in approximately the same pOSition but startshy

ed to move after the first yielding had occurred at each

cross-section This is as would be expected

79

Since strain gauge readings were not available for the

edges of the opening it is impossible to follow the complete

progression of yielding for the various cross-sections from

strain gauge readings However since it is lcnown from the

whitewash flaking on the test beams that yielding occurs

first at the corners of the opening where stress concentrashy

tions are high it can be said that at the low moment edges

of the openings (beams 2C and 3B) yielding began first at the

corners of the opening and then progressed to the reinforcshy

ing bars and web before the flange yielded At the high

moment edges of the openings yielding in the flange occurrshy

ed at lower shear values This would be expected from conshy

siderations of the total stresses (primary bending stresses

plus secondary bending stresses due to Vierendeel action) at

the cross-section in question At the flange the secondary

bending stresses are added to the primary bending stresses

at the high moment edge of the opening but subtract from the

primary bending stresses at the low moment edge of the openshy

ing The experimentally determined stress distributions at

the high and low moment edges of the openings are completely

compatible with those assumed in the development of the

theory in Chapter III (Figure 8) Bending stress distribushy

tions along the length of the opening at the top of the upper

reinforcing bar and at the centerline of the top flange show

80

the expected high stresses at the edges of the opening due

to the secondary bending stresses (Figure 29)

Comparison of stress distributions with those presented

by Bower(16) for beams with unreinforced rectangular openings

shows that the addition of horizontal reinforcing bars above

and below the opening changes the position of the neutral

aXiS but does not alter the location where the highest

stresses occur and plastic hinges consequently form The

order of the onset of yielding is also similar to that

found by Bower as was expected

73 Failure Loads

In Table 5 experimentally determined failure loads are

compared to those predicted by the theory presented in

Chapter III for each of the test be~~s Examination of this

table shows that although the theory may be considered someshy

what conservative in high shear for low shear the correlashy

tion between theory and experimental is good and in no case

did the theory overestimate the actual strength of the beam

Table 6 shows the comparison between experimental failure

loads and those predicted by the approximate analysis for

nominal beam dimensions and yield stresses It can be seen

from this table that the approximate method is less conservshy

ative in the high shear region while still not overestimating

81

the strength of the beam An added margin of safety also

exists because the additional strength of the beam due to

strain hardening has not been taken into account This extra

strength is an added safeguard against total collapse even

though it is accompanied by excessive deformations and finalshy

ly involves tearing or pOSSibly buckling

Experimentally determined failure loads are compared

with loads corresponding to complete yielding of cross-secshy

tions for the strain gauged beams (20 2D 3B 4B) in Table

7 It can be seen from this comparison that experimentally

determined failure loads are close to those corresponding to

the complete yielding of the cross-section at the high moment

edge of the opening while loads corresponding to the complete

yielding of the cross-section at the low moment edge of the

opening are considerably higher This can be accounted for

by the considerable amount of strain hardening that is

suspected of occurring at the corners of the opening before

complete yielding of the low moment edge and possibly also

the high moment edge of the opening occurs Complete yieldshy

ing of cross-sections at the high and low moment edges of

the openings are also compared with the theoretical loads

predicted by the analysis of Chapter III in Table 8 Again

the correlation is reasonably good

The performance of each of the test beams was as expectshy

82

ed With the exception of the shear yielding which occurred in

the web at the ends of the reinforcing bars as discussed in

Section 63 The method used in determining experimental

failure loads has been shown to be justifiable on the basis

of the good correlation between these loads and those correshy

sponding to complete yielding of cross-sections at high and

low moment edges of the opening (considering that strain

hardening has not been taken into account) On the basis of

comparison with experimental failure loads the theory develshy

oped in Chapter III has been shown to adequately predict the

performance of a given beam under varying moment to shear

ratios (although conservatively at high shear) The approxshy

imate method as developed in Chapter IV has been shown to

predict beam strength with accuracy adequate for design

purposes (assuming a suitable factor of safety is used)

14 Influence of Reinforcing and Other Variables

As can be seen from comparison of the experimental

results from test beams 2A 3A and 6A and those for beams 2B

and 3B there is a definite increase in beam strength with

the addition of horizontal reinforcing bars As predicted by

the theoretical analysis of Chapter Ill the maximum shear

capacity of a perforated section as given by equation (6)

can be reached by the addition of a certain minimum amount

83

of reinforcing as given by equation (32) If no reinforcing

or an amount less than that required by equation (32) is used

the maximum shear capacity cannot be reached This is conshy

firmed by the result of the tests on beams 2A 3A and 6A

For higher moment to shear ratios the increase in strength

with increased reinforcing size is adequately predicted by

the analysis in Chapter 111 (and also by the approximate

method of Chapter IV) This is confirmed by the tests on

beams 2B and 3B The increase in strength with the addition

of reinforcing bars and with the further increase in the

size of this reinforcing cen be seen by superimposing the

interaction curves of Figures 10 11 and 12 as has been done

in Figure 54

The influence of the magnitude of moment to shear ratio

on the strength of a beam of given size opening and reinforcshy

ing dimenSions is adequately predicted by interaction curves

generated by the method of Chapter 111 and by the approximate

interaction curves as in Chapter IV This is confirmed by

test series 2A 2B 20 2D series 3A 3B and by series 4A

4B See Figures 55 56 and 57

An increase in aspect ratio everything else being held

constant results in a decrease in strength of the beam This

is predicted by the interaction curves in Figures 10 and 13

which are superimposed in Figure 58 for easy reference and

84

is confirmed by the tests on beams in series 2A 4A and

series 2B 4B An increase in hole depth also has the effect

of lowering the strength of the member all other variables

being held constant This is predicted by the interaction

curves in Figures 10 and 14 which are superimposed in Figure

59 and is confirmed by the tests on beams 20 and 5A

The test results have also been plotted in Figures 54

through 59 but because of the variation in the sizes and

yield stresses of the actual test beams these results have

been plotted on the curves (which are based on nominal

dimensions and yields) on the basis of percent difference

be~veen the experimental failure loads of the test beams

and the values predicted by the exact interaction curves

(from measured beam dimensions and yield stresses)

85

CHAPTER VIII

CONCLUSIONS AND RECOMMENDATIONS

81 Conclusions

On the basis of the results and discussion presented in

the previous chapters the following conclusions can be

drawn

1 The flaking off of the brittle coating of whitewash on

the beams during testing is an accurate indication of

the order of onset of yielding although it does not inshy

dicate at what loads yielding actually occurs

2 High stress concentrations exist at the corners of

rectangular openings even when horizontal reinforcing

bars have been added above and below the opening

3 Failure of a beam under moment gradient containing a

single rectangular opening reinforced with horizontal

reinforcing bars above and below the opening occurs by

the formation of a four hinge mechanism the hinges

occurring at cross-sections at the edges of the opening

Ultimate collapse of such a beam occurs by the tearing

of one or more of the corners of the opening provided

the beam has sufficient lateral support to prevent

lateral buckling

4 With the development of the four hinge mechanism large

86

relative displacements take place between the ends of the

opening accompanied by full or partial shear yielding in

the web above and below the opening This yielding when

it occurs over the entire web area is an indication that

the full shear capacity of the section is utilized

5 Because of the shear yielding that occurs in the web beshy

tween the ends of the reinforcing bars the extension of

these reinforcing bars past the ends of the opening (while

designed on the oasis of developing the weld strength)

should not be less than a certain length (as yet undetershy

mined but probably about three inches for the size beam

and openings tested) due to the possible interaction of

the web yielding with the yielding at the corners of the

opening

6 The stress distributions assumed in the analysis in Chapshy

ter III are completely compatible with those found at the

high and low moment edges of the strain gauged test beams

The large influence of the secondary bending moments (due

to Vierendeel action) is confirmed by these stress

distributions

7 The interaction curve resulting from the analysis in

Chapter III predicts with reasonable accuracy the strength

of a beam with given opening and reinforcing size at any

combination of moment and shear This method is however

87

conservative at high shear values

8 The approximate method developed in Chapter IV also preshy

dicts with reasonable accuracy the strength of a beam

with given opening and reinforcing size for the full

range of moment and shear values and is less conservshy

ative at high values of shear

9 Due to the fact that the theory neglects the effects of

strain hardening there is a built in margin of safety

against complete collapse of the member although an

increase in load past the predicted failure load is

accompanied by excessive deformations

10 The correlation obtained between experimentally detershy

mined failure loads and loads corresponding to complete

yielding of cross-sections at the high moment edge of

the opening for the strain gauged beams suggests that

the method used in determining the experimental failure

loads is justifiable Comparison of experimental failure

loads with loads corresponding to complete yielding of

the cross-section at the low moment edge of the opening

would not be expected to be as good because of the strain

hardening that is assumed to occur at the corners of the

opening before this section is completely yielded

11 The addition of horizontal reinforcing bars above and

below a rectangular opening in a beam definitely increases

88

the strength of the beam If greater than a predictshy

able minimum area this reinforcing will make it possible

to achieve the full shear strength of the section as

given by equation (6) Any further increase in reinforcshy

ing size results in a further increase in the moment

capacity of the member for a given value of the shear

force

12 For a given size beam an increase in aspect ratio (openshy

ing length to depth) results in a decrease in the strength

of the beam over the full range of moment to shear ratios

if all other factors are held constant An increase in

the opening depth to beam depth ratio also has the same

effect all other variables being constant

13 The test performed on beam 6A containing the unreinshy

forced rectangular opening further confirmed the analysis

presented by Redwood(17 18) for beams containing single

unreinforced rectangular openings in their webs

82 Recommendations for Design

Based on the good agreement between experimental results

and failure loads predicted by the approximate interaction

curve it is recommended that the approximate method as

described in Chapter IV be used for design purposes assuming

that a suitable factor of safety is used The ease with

89

which the necessary calculations can be carried out makes

this method extremely practical The use of this method can

be outlined as follows When it is decided to cut an openshy

ing in the web of a beam for the passage of utilities the

necessary size of opening should first be determined and an

approximate interaction curve constructed for the beam conshy

taining an unreinforced opening by using e~uations (52) (53)

and (55) Mp and Vp should be calculated from equations (1)

and (3) respectively A line should then be drawn from the

origin at the particular moment to shear ratio which repshy

resents the actual position of the proposed opening in the

beam and extended to intersect the approximate interaction

curve The intersection of this line with the interaction

curve then represents the capacity of the beam if the openshy

ing is not to be reinforced If this capacity is less than

that required it is necessary to reinforce the opening

and reinforcing should be chosen on the basis of satisfying

the inequality of equation (41) The reinforcing should be

chosen so that the right hand side of equation (41) is not

greatly exceeded An approximate interaction curve for a

beam containing an opening with this size reinforcing can

then be constructed by using equations (42) (43) and (45)

where Ar is given by the right hand side of equation (41)

The intersection of the MV line with this approximate intershy

go

action curve then gives the capacity of the beam with this

size reinforcing If this exceeds the required capacity of

the section this size reinforcing should be adopted

However if the required capacity of the beam is greater

than that given by the approximate interaction curve two

possibilities may exist If the required shear is greater

than the maximum shear capacity of the section as given by

the vertical line on the approximate interaction curve or

by equation (42) then either shear reinforcing must be

added or the depth of the section increased in order to

provide the required strength If however the required

shear capacity is not greater than that given by equation

(42) but the point representing the required capacity of the

beam on the MV line lies outside the apprOximate intershy

action curve the area of reinforcing should be increased

above the value given by equation (48) and a new approximate

interaction curve constructed using equations (42) (45)

(49) (50) and (51) If the required capacity is still

greater than that given by this new approximate interaction

curve the reinforcing size would have to be further increased

until the reinforcing is such that the required strength is

provided The reinforcing size that meets this requirement

should then be adopted for use

91

83 Reoommendations for Future Work

With the completion of this study on the ultimate

strength of beams containing single rectangular openings

reinforced with straight reinforcing bars above and below

the openings it is logical that attention would turn to

similar studies for other commonly used opening shapes in

particular circular and extended circular openings with

horizontal reinforcing bars Also of interest would be an

ultimate strength investigation of reinforced closely spaced

multiple openings of rectangular or circular shape

Since the decision to cut and reinforce an opening in

the web of a beam is based primarily on economic considershy

ations an investigation into reducing the cost of reinshy

forced openings would be very much justified As the cost

of welding is paramount among the other factors involved in

reinforcing an opening reducing the e~mount of welding is

desirable If the horizontal reinforcing bars used in the

present study were doubled in size and welded to only one

side of the web above and below the opening the cost of

reinforcing would be almost halved while the same area of

reinforcing would be available for resisting the shear and

moment forces on the section However other factors such

as twisting moments would have to be taken into conSideration

due to the asymmetry of the section about its vertical axis

92

that would result from this type of reinforcing arrangement

An investigation of the ultimate strength of wide poundlange

sections containing large rectangular openings reinforced

with one-sided straight horizontal bar reinforcing has

already been initiated at McGill University

More information is also needed on the anchorage of the

reinforcing bars It would be desirable to determine at

what length such reinforcing is capable opound assuming its full

load An investigation of the required extension of reinshy

forcing bars past the ends of the opening to prevent preshy

mature poundailure due to interaction between hinge formation

and web yielding at reinforcing ends is also necessary

93

1)

) )J + 1 +1 I I

I

( a) (b)

I + --+--~+ -f--- -shy

I (c) (d)

II I I

I I j+shy

) 1)I l J I

(e) ( f)

Types of Reinforcing

Figure 1

94

primary bending stresses

secondary bending stresses

Stress Distribution at Opening

Figure 2

Ql ID Q)

Jt p ID

er

I I I I I I I

E I I I poundy Ipoundst strain

Idealized Stress-Strain Curve for structural Steel

Figure 3

---------- -----

95

MM unperforated beam no interaction

10~----~==~==~L---------------------~

~

unperforated beam including interaction

10 vVp

Interaction Curve - Unperforated Beam

Figure 4

-- -----------

96

deflection Pure Bending

(a)

~ o

r-I

deflection Bending with Shear

(b)

Load-Deflection Curves

Figure 5

97

1O~------------------------------------------~

l I

unperforated beamshy

shy

I r perforated beam

no interaction

----------- perforated beam l including interaction I I I I I I I I I

I I

Interaction Curve - Perforated Beam

Figure 6

i

98

1 CD (2) TIT

L

( a)

b

d2

(b)

Hinge Locations and Cross-section of Member

Figure 7

99

( a)

(S r CSyr+

tltS ltl direct shear direct shear

Stress Distribution at CD Stress Distribution at CD (for reversal in reinforcing)

Case I - Low Shear

+

CSyr ltS shy

direct shear

Stress Distribution at CD

Case 11 - High Shear

(b)

Stress Distributions and Resultant Forces

Figure 8

100

low highshear shear

I Ml4p k--~_____U_k-----LS_~_(_U_+-=q-)_____-L (u+q)kl s~s

I I r I I

I I I 1 I10

~-int eraction II curve 11

11

11 I I 11 I I

- I - I Iapproximate ~

interaction curve------ - 1

11~I11

i I I

I (d-2h-2t)I_~ I

d-2t 11

(1 -~) q

Interaction Curve Showing Low and High Shear Regions

Figure 9

101

14D38 7 x lof Opening

6 bull77611 x ~middotReinforcing 1 0513

03~

CH 125 1412700

0313 =--9F 0375 =Jb

approximate method ---l------ Case 11

04 vVp

Interaction Curve - 14WF38 Nominal

Figure 10

102

14WF38 7 x 10r Opening 6776 05132tbull

x e 3middot

Reinforcing

1412 11

10

approximate method Case I

04


Figure 11

103

14WF38 7 x 10f Opening

0513No Reinforcing

MMp

10

-----approximate method for unreinforoed opening

04 VVp


Figure 12

104

l4WF38 7 x l4~ Opening1pound x tReinforcing

Iapproximate method Case 11

I I I

04

Interaction Curve - l4WF38 Nominal

Figure 13

105

14WF38 9X 13- Opening lmiddotfx p Reinforcing

0513

1412

~-approximate method ~ Case IIIB


04


Figure 14

106

10WF2l

5 x sf Opening 034(iIf x iWReinforcing t~l0240

CH h

55Q 125 990 025Q 0250~

approximate method ~ Case IlIA

approximate method I Case I

I I I

04

Interaction Curve - lOWF2l Nominal

Figure 15

107

43 11 39 18 0 (

I

I I i lA

26

I

111 99 I- middot1 t

22 2-S 45 J

) C

r I

2A

13 ) lA I ~

I)II45 35 ( ()

I

I

I If 2B

f 16~ 9 25 I2t~ 21-+1 1 1 i

30 11 24 27 0- 10- ro- )

I ~ I I

20

2119YlO 151( 21~fI211~ - rr I I

Test Beam Dimensions and

Dial Gauge Looations

Figure 16

108

tI

2D

22tl 231 45 tlI

( l (

I

3A

3119 Yt 11~ 34ft J3n~

45 25 J 35 ) (~ i-

I

I 1 3B

H 25 ft2ft5 cI 16 J2++ I I1 I

t 0- 23 45 J0- 0- (

I 4A

~5 ( 32 J211~ 2 YYt -IIII


Dial Gauge Locations

Figure 17

log

45 ~ 25 5S ) ~ (

4B

2 crYI 15 25 J-I2++ Irr

L J30 24 27 ~lt) (P-L

I 5A

2tYY 13 26211 1 I

22 6

6A

i



Figure 18

--

110

If JL

a (1) p 02 ~

Cfl

tlOO ~ r-I 0 0 Q)

m ~ ~

Q SoOM

r-I Xl m ~ (1) p cD H

r-shy

~ ishy fO I- It

111

o N

-()

112

~ --- ~

~

~~ ~~

II I of

rf

A tshyshy I

1

ID I ~ I 0I middotrt

a1 +gt

- -l~ 0 ()

-t I H CJ

4gt 4gt bO ~

~ ~ Cl -rt

fiI

~ a1 ~

+gt CJI

~ amp ~ ~ ~ --- ~ I

II s I

~CJ~ 4t ~

I~ i ~

o

i-~ ~ - l

II I +1 I Ij

~I r

I tf~ft N

- shy

stress 8 1

lower Yield-lt ~i == 11 static yield-shyI l- 2 1 2 ~

( a)

F F-CL F-E [(4 ~

W-H- - shy

strain ( b)

Tensile Coupons Stress-Strain Curve from Instron Testing Machine

Figure 22 Figure 23 I- I- vJ

I

114

lA 2A 2B

20 2D 3A

3B 4A 4B

5A 6A

FIGURE 24

115

2B 20 2D 3A 3B 4At 4B

Variable Moment to Shear Ratio

20 5A

Variable Opening Depth to Beam Depth

FIGURE 25

116

2A 4A 2B 4B

Variable Opening Length to Depth

2B 3B

Variable Reinforoing Size

FIGURE 26

bullbull

117

Notes Shear force shown in kipsbott of flange bull Elastic o Yielded

r ~ 3

o

top of reinf

boundaryof opening bott of reinf~ ioiiIo ~_ 10 o 10 ~ ~ - Dending Stress Shear Stress

Ca) Stress Distribution at High Moment Edge of Opening - 20

~ bull

~ ~ boundary ~ of opening~

-0 I) 10 40 o to lO ~ Sending Stress Shear stress

(b) Stress Distribution at Low Moment Edge of Opening - 20

Figure 27

118

boundaryof opening

-70 0 __ tQ

Bending Stress Shear Stress

Ca) Stress Distribution at Oenterline of Opening - 20

-1iCgt

~I--=-shyoo---~-=-~ogt---- ---0 - ____ A~Sshy-

middot------------e----- ~~~ ~_______~-----Lr---t-o~p~of flangeo

oE

Cb) Bending Stress Distribution Across Flange at High Moment Edge of Opening - 20

Figure 28

119

-40

-lt0 lt----- shylow

moment edge gtgt bull

lt 0

high54 boundary of opening moment

edge ltgtgt

to

Ca) Bending Stress Distribution Across Length of Opening at Top of Flange - 20

high moment

edge

o

Cb) Bending Stress Distribution Across Length of Opening at Top of Reinforcing - 20

Figure 29

120

boundary---~~~~~~~~___ of Opening

o10 D


(a) Stress Distribution at High Moment Edge of Opening - 3B

ibull

boundaryof opening

10 Igt lO


(b) Stress Distribution at Low Moment Edge of Opening - 3B

Figure 30

bull bull bull bull

121

11 9 ~ oS j j ~

boundaryof opening

-lt10 -20 lt) -I-Lo Q lO 10


(a) stress Distribution at High Moment Edge of Opening - 2D

boundaryof openin

-40 -20 0 ZO o 10 ZQ 30


(b) Stress Distribution at High Moment Edge of Opening - 4B

Figure 31

122

Test Beam - lA

O6in relative defleotion between opening ends

Figure 32

Test Beam - 2A

O6in relative deflection between opening ends

Figure 33

123

lateral buckling - highmoment edge of opening413k-- shy

Test Beam - 2B


Figure 34

bull first strain hardening --~ complete yielding - low

moment edge of opening-J- -- --shy

J complete yielding - high moment edge I of opening I ----first yielding

Test Beam - 20


Figure 35

124

---- complete yielding - high moment edgeof opening

first yielding

Test Beam - 2D


Figure 36

Test Beam - 3A


Figure 37

125

k ---shy497 ---~ --1shy complete yieldingI J

---- first yielding

O6in

Figure 38

O6in

Figure 39

- high moment edge of opening

Test Beam - 3B

relative deflection between opening ends

Test Beam - 4A


126

430k--shy

k445--shy

---f-----shy ----complete yielding shy

I ----first yielding

06in

Figure 40

O6in

Figure 41

high moment edgeof opening

Test Beam - 4B

relative defleetion between opening ends

Test Beam - 5A


127

k45 o-~--

Test Beam - 6A


Figure 42

128

10 shy

10WF21

5t x ampf Opening If x t Reinforcing

~~approximate interaction ~ nominal dimensions and

~ ~

+~ ~ ~

I interaction curve ---I Imeasured dimensions and yields I

I I

04

Interaction Curves - Test Beam lA

Figure 43

curve yields

129

14WF38 7 x lot Opening

bull SIt x a Reinforcing

~-----approximate interaction curve nominal dimensions and yields

interaction curve-------- I

measured dimensions and yields

bull

Interaction Curves - Test Beams 2A and 2B

Figure 44

130

l4WF38 f x 10i Opening 11 x middotf Reinforcing

10 - ~approximate interaction curve nominal dimensions and yields

interaction curve----~ measured dimensions and yields

04

Interaction Curves - Test Beams 2C and 2D

Figure 45

10

131

l4WF38 7middotx lof Opening 2f x ~uReinforoing

approximate interaction curve nominal dimensions and yields

~

1 +3A

I interaction curve -------J


04

Interaction Curves - Test Beam 3A

Figure 46

10

132

14WF38 7 x lOi~ Opening2f x ~ Reinforcing

~ ~approximate interaction curve ~~ nominal dimensions and yields

~ ~ ~ ~ ~

I I

interaction curve ------I measured dimensions and yields

04

Interaction Ourves - Test Beam 3B

Figure 47

10

133

14WF38 H

7 x 14 Opening 1t x Reinforcing

-~

~~approximate interaction curve ~ nominal dimensions and yields

~ ~ +4B

~ ~ ~

~ interaction curve-----l measured dimensions and yields I

I I

04


Figure 48

134

14WF38 9 x 13t~ Opening lix ~uReinforcing

10 --~ ~~approximate interaction curve

nominal dimensions and yields

~ ~

SAl

interaction curve measured dimensions and yields

04 vVp


Figure 49

10

135

14WF38 7 x lof Opening No Reinforcing

r----approximate interaction curve nominal dimensions and yields

+ 6A

----interaction curve measured dimensions and yields

04


Figure 50

136

shear (kips)

70

60

---- shear corresponding to experimental failure

50

40

30

20

laterally unsupported length = 54 inohes10

40 80 120 160 200 240

lateral deflection (in x 10-3 )

Lateral Deflection at High Moment Edge of Opening - 20

Figure 51

137

shear (kips)

70

60

------------shear oorresponding to experimental failure

50

40

30

20

10

4 8 12 16 20 24 28

strain differenoe at top flange (inin x 10-3)

Flange Buokling at High Moment Edge of Opening - 20

Figure 52

138

D f

P --IL-=====bull--Jf--P + dP

Shear Stresses at End of Reinforcing

Figure 53

139

l4WF38 7 x lot

Opening

3shy10 f ~--2f x e

Reinforcing

If x ~ Reinforcing

+3A +2A

No Reinforcing + 6A

04

Variable Reinforcing Size

Figure 54

140

14WF38 7 x lof Opening If xl-Reinforcing

+ 2B

+20

+2A

+2D

04 vVp


Figure 55

141

14WF38 7x 10f Opening2i x iReinforcing

10 ---------- shy

+3B

+3A

04 vVp


Figure 56

142

14WF38 7x 14~Opening 1f x f Reinforcing

+ 4B

+ 4A

04 vVp


Figure 57

143

14WF38 1thx imiddotReinforCing

7 x lot Opening

7 x 14 Opening---

04

Variable Aspect Ratio

Figure 58

144shy

14WF38

------7middot x 10i Openinglift x ~~Reinforeing

9 x 13f Opening +20 Ii x i Reinforoing

+5A

04


Figure 59

--

r b --1 t l=Ii= W

d

u q

Beam Size ~ d b t w 2a 2h u q c x R bull lA 10WF21 43 989 578 0324

11

0244 875 550 N

0260 0253 2720 N

400 316

2A 14WF38 22 1413 677 0496 0326 1046 700 0383 0381 2813 300 58

tI tI If If If tI tf tI It n tI It2B 45

ff 11 It2C 30 1422 668 0490 0305 1045 0267 0368 2720 n tI n tt tI tI n n tI2D 17 If tI n3A 22 1413 677 0496 0326 1046 0383 0381 5313 600

It 11 It If3B 45 1422 668 0490 0305 1045 0329 0361 5230

tf tI 11 tI fI tI tI4A 22 0340 0368 2720 300 If It tI ff If tI If It n4B 45 n n tI tt f1 tI5A 30 1344 901 0305 0371 3770 425

11If fI6A 22 1045 700 shy - - shy Table 1 -I

foo J1

146

Ave Test Coupon Desig- Testing Lower Static Lower Beam

Beam Location nation Machine Yield Yield Yield Series lA web W Riehle 573 - 573 lA

nflange F 362 - 431 nF 435 shy

F-E It 462 shyItreinf R 370 370

2B web W Instron 417 405 413 2A2B W 11 3A412 398

flange F Riehle 374 - 388 F-CL Instron 382 371 F-E It 397 395

reinf Rl Riehle 434 - 437 2D web W Instron 567 550 558 2C2D

rtW 540 527 3B4A 4B5Aflange F 490 474 446 6AItF-CL 418 406

F-E 478 shyIt 2C2Dreinf R6 398 384 398 4A4B

3A web W 11 411 shyfIflange F 398 379

reinf R2 Riehle 440 shyfI3B reinf R4 330 - 331 3B

R4 11 332 shyR4 Instron 332 shy

4A web W 565 549 flange F 11 487 476

F-CL 406 398 nF-E 429 415 5A reinf R5 tI 437 429 444 5A

R5 450 422 It6A web W 559 545 Itflange F 463 450 fIF-CL 408 402

F-E ff 438 425

Tensile Test Results

Table 2

147

Beam 20 2D 3B 4BLocation Top edge upper fl - BM edge 450K 575K 433lC 300

Bott upper reinf - LIvI edge 450 500 400 317 ~ top upper fl - BM edge 483 600 367 317 Web - ~ open 483 500 517 450 Web - BM edge 517 550 400 350 Web - LM end of upper reinf 517 - - -Bott upper reinf - BM edge 550 550 500 350 Top upper reinf - BM edge 567 800 467 433 ~ web - between LM ends reinf 583 - - shyWeb - LM edge 600 - 583 shy~ top upper fl - LM edge 633 - 600 shyTop upper reinf - LM edge 633 - 467 500 Bott edge upper fl - BM edge 667 - 517 383 Complete yield - BM edge - 567 625 483 383 Complete yield - LM edge 633 - 600 shy

(a) Shear Values Corresponding to Yielding

Location Beam 2C 2D 3B 4B

Top edge upper fl - BM edge 667k 775

k 567k 483k

Bott upper reinf - LM edge - 800 583 -~ top upper fl - BM edge - 775 533 -Web - ~ open 633 700 583 -Web - EM edge - 750 - -Bott upper reinf - BM edge 667 775 - -

(b) Shear Values Corresponding to Strain Hardening

Table 3

- - - - - - -

- - - - - - - - -

- -

BeamLocation Upper fl - BM edge Lower fl - BM edge Lower LM corner open Upper LM corner open Lower BM corner open Upper ID~ corner open Web above open Web below open Low Uif corner to end reinf Up n~ corner to end reinf Low IDH corner to end reinf Up rIM corner to end reinf Between ends reinf LM end Between ends reinf HM end Upper reinf - BM edge open Lower reinf - BM edge open Upper reinf - ilA edge open Lower reinf - LM edge open Lateral buckling - BM edge Web buckling at open Tearing - lower LM corner Tearing - upper BM corner

lA 2A 2B 20 2D 3A 3B ~ lI Ilt v 162 500 400 583 700 525 433

180 - - - 725 - 517 191 550 417 617 600 550 567 191 575 442 567 575 600 433

- 575 467 567 575 600 433

- 575 - - 600 600 shy202 625 475 600 600 700 533

- 625 475 600 625 700 583

- - - 617 700 - 550

- - - 600 675 675 550

- - - - - 675 shy

- 600 458 650 675 650 583

- - - - 725 - -- - - - - 725 -- - - - - 675 -- - - - - - -- - - - - 675 shy- - 483 - - - shy

- 700 517 703 825 775 shy- - - - 825 - 600

4A 4B Ilt Ilt

450 333 500 417 450 367 450 366 500 383 450 417 550 483 600 483 550 400 550 433 625 shy625 shy575 467 575 shy500 shy550 417

- 450 500 383

--675 513

5A le

350 500 400 400 433 483 433 467 500 417

--

500

--

467 500

---

517

-

6A

400 533

-367 367 383 433 533

-----------

550

--

Shear Values Corresponding to Whitewash Flaking

J-ITable 4 ~ 00

149

~hear a1 Beam Size MV [Failure Vp Exp VVp Theor VV_ Diff

lA 10WF21 43 180K 746K 0241 0235 + 26 2A 14WF38 22 532 1L021 0520 0466 +116 2B 11 45 413 0404 0363 +113 2C If 30 548 1301 0421 0403 + 45

u n2D 17 670 0515 0420 +226 n3A 22 575 1021 0562 0466 +206 tt3B 45 497 1301 0382 0345 +107 11 n4A 22 572 0440 0360 +222 n4B 45 430 0330 0295 +119 If It5A 30 445 0342 0296 +155 116A 22 450 0346 0271 +277

Correlation Between Experiment and Theory

Table 5

~hear at Approximate Beam Size MV Failure Exp VV Theor VV DiffVp p p

lA 10WF21 43 180K 746 K

0241 0254 - 51 2A 14VlF38 22 532 1021 0520 0503 + 34

It2B 45 413 0404 0344 +175 If2C 30 548 1301 0421 0414 + 17

It2D 17 670 0515 0505 + 20 3A 22 575 1021 0562 0505 +103

n3B 45 497 1301 0382 0377 + 13 4A tI 22 572 n 0440 0463 - 50

It It 03094B 45 430 0330 + 68 5A 30 445 tr 0342 0353 - 31 6A 22 450 tt 0346 0257 +346

Correlation Between Experiment and Approximate Theory

Table 6

150

Shear at Shear at Shear at Beam MV Failure Full Yield Dif Full Yield Dif

Exp H M Edge L M Edge k k k2C 30 548 567 + 35 633 +155

2D 17 670 625 - 67 - shy3B 45 497 483 - 28 600 +207 4B 45 430 383 -109 - shy

Correlation Between Experimental Failure and Complete Yielding

Table 7

Theor Shear at Shear at Beam MV Shear at Full Yield Dif Full Yield Diff

Failure H M Edge L M Edge

2C 30n 525k 567k + 80 633k +206 2D 17 546 625 +145 - shy3B 45 449 483 + 76 600 +336 4B 45 384 383 - 03 - shy

Correlation Between Theoretical Failure and Complete Yielding

Table 8

151

APPENDIX A

Computer Program for Interaction Curve

A computer program was developed to generate an intershy

action curve for a specified beam opening and reinforcing

size according to the method described in Chapter Ill The

program was wri tten in Fortran IV and can be used on any

standard digital computer that makes use of this language

The input for the program should be of the form given in

the following table

Data Number Items on of Cards Data Cards

Number of beams 1 NB

Dimensions of beams NB dbtw

Yield Stresses NB CS111 S~ lts

Number of openings per beam NB NE

Sizes of openings NH ah

Number of reinforcing sizes per opening NH NR

Sizes of reinforcing NR uqc

A listing of the computer program is given on the following

pages

152

C THIS PROGRA~ CO~PUTES ThE INTERACTION CURVE RELATING MOMENT AND C SHEAR VALUES WHICH CAUSE FAILURE OF A WICE FLANGE bEAM CONTAINING C A REINFORCED RECTANGULAR OPENING AT ThE MIDOEPTH C

REAL KlK2MPMMPKllK12K21 t K22 581 FORMAT(5FIO3) 584 FORMAT(3FIO3) 582 FOR~AT(2FIO3) 583 FORMA1(I5 681 FORMAT(lINTERACTION

lOt 0 B T 682 FORMAT( F133F63 683 FORMAT(O SIGF 684 FORMAT( 5F93

BETwEEN MOMENT AND SHEARtw)

SIGw SIGR VP Mpt)

605 FORMAT(O OPENING LENGTH middotF73middot HEIGHT =F73) 606 FORMAT(O MMP VVP SIG 601 FORMAT( middot5FI04 608 FORMATOV IS TOO LARGE SO SIGMA BECCMES 609 FORMATtOHIGH SHEAR SOLUTION) 611 FORMAT(O U QC) 612 FORMATe 3F63) 613 FORMAT(O STRESS REVERSAL IN REINFORCING 614 FORMATOSTRESS REVERSAL IN CLEAR WEe 615 FORMAT(O STRESS REVERSAL IN WEB STUB) b16 FORMAT(OLOW SHEAR SClUTIONt) 618 FORMATtO MAxIMU~ VVP Fb3)

c READ(S583)NB 00 200 I=lNB RE~D(5581tOBTW

READ(5584)SIGFSIGWSIGR VP=W(0~2T)SIGWSQRT(3) MP=BTSIGF(D-T bullbullW(D-2Tl2SIGW4 WRITE(6681) WRITEI6682)DtB t TW WRITE(6683) WRITE(~684)SIGFSIGWSlGRVPtMP

REAO(SS83)NH DO 200 ~=lNH

RE~D(5582JAH

REAC5583) NR A2=2A H2=2H WRITE(6605)A2H2 VVPM=(O-2T-2H)0-2T) WRITEt6618) VVPM CO 200 J=lNR READ(5584) UQC

Kl K2~)

COMPLEXJ

153

WRITE(661U WRITE(661Z) UQC WRITE(6606) VINCR=OO V=OO S=OZ-H-T AW=SW Af=BT AR=(C-WjQ R=U+Q

C C LOW SHEAR SOLUTION C C STRESS REVERSAL IN WEB STUB C

WRITE(66161 WRITE(661S)

101 V=V+VINCR XX=$IGWZ-O7SV2AW2 IF(XX) 201]00300

300 SIG=SQRT(XX) ALPHA=ARSIGR(AFSIGf BETA=AWSIGCAfSIGf) Cl=T+SBETA C2=-(D-2HJ C3=VAAFSIGF) C4=C22-40ClC3 IF(C4408399399

399 K21=(-C2+$QRT(C4)(2Cl K22=(-C2-SQRTCC4raquo)(2Clt Kll=K21BETA KI2=K22BETA IF(K21)330301301

301 IFK21-10)30230233C 302 IFCKll1330304304 304 IFKllS-U)320320330 330 IF(K22)408)31331 331 IF(K22-10)33233Z40a 332 IF(KIZ)408333333 333 IF(K12S-UJ32132140S 320 K2=K21

Kl=Kl1 GO TO 322

321 K2K22 Kl=K12

322 FY=AFSIGF(S+T2)+AWSIGSS(1-2Kl2)+ARSIGR(U+Q2) F=AFSIGF+AWSIG(1-2Kl+ARSIGR M=2FY+20HF-VA

154

MMP=MMP VVP=VVP WRITE(6607)MMPVVPSIGKlKZ VINCR=VP500 GO TO 101

C C STRESS REVERSAL IN REINFORCING C

408 WRITE(6613 GO TO 409

902 V=Vl 701 V=V+VINCR

XX=SIGW2-015V2AW2 IF(XX9011CQ100

100 SIG=SQRTXX) 409 Rl=(C-W)SIGR

R2=AWSIG+SRl Cl=T+SAFSIGFRZ C2=2SURIR2-(D-2HJ C3=VA(AFSIGF)+U2Rl(AFSIGFraquo)(SR1RZ-1) C4=C22-40C1C3 IF(C4)418403403

403 K21=(-C2+SQRT(C4)(Z0Cl KZ2=-C2-SQRT(C4)(Z0Cl) Kll=AFSIGFK21R2+URlR2 K12=AFSIGFK22R2+URIR2 IF(K21)430401401

401 IF(K21-10)4C240243C 402 IF(K11S-U)430404404 404 IF(Kl1S-R)42042043C 430 IFK22)418431431 431 IF(K22-10)432432418 432 IF(K12S-U418433433 433 [FtK12S-R)421421418 420 K2=K21

K l=K 11 GO TO 422

421 K2=K22 Kl=K12

422 FY=AFSIGFtS+T2)+AWSIGSS(1-2Kl2) 1 +(C-W)SIGR(U2+UC+SQ2-(KlS)2)

F=AFSIGF+AWSIG(I-2Kl)+(C-WJSIGR(2U+Q-2KlS) M=20FY+20HF-VA ~MP=MMP

VVP=VVP IF(MMP 418423423

423 WRITE(6601)MMPVVPSIGKlK2 VINCR=VP500

155

GO TO 701 C C STRESS REVERSAL IN CLEAR WEB C

418 WRITE(6614) GO TO 419


XX=SIGW2-075V2AW2 IF(XX)921800800

800 SIG=SQRT XX) 419 Cl=AFSIGFS(AWSIG)+T

CZ=-O+2H-2SARSIGR(AWSIG) C3=VACAFSIGF)+2U+Q2)ARSIGR(AFSIGF)

1 +S(ARSIGR2fAFSIGFAWSIG C4= C224 ClC3 IF(C4)428410410

410 KZl=(-C2+SQRTCC4)(ZCl K22=(-C2-SQRTCC4)ZC1) Kll=AFSIGFK21(AWSIG)-ARSIGR(A~SIG)

K12~AFSIGFK22(AWSIG)-ARSIGRAWSIG)

IF(K21)S30501501 501 IFK21-10)502502530 50Z IF(KllS-RS30504504 S04 IF(Kll-l0S20520510 530 IFtKZ2)428531531 531IF(K22-10)532532428 532 IF(K12S-R428533533 533 IF(K12-10)521S21428 520 1lt2=K21

Kl=Kl1 GO TO 522

521 1lt2=K2Z Kl=K12

522 FY=AFSIGF($+T5+AWSIGS(i-2bullbullKl212-ARSIGR(U+Q2) F=AFSIGF+AWSIG(1-2Kl-ARSIGR M=2FY+2HF-VA MMP=MMP VVP=VVP IF (MM P) 428523523

523 WRITE(6601)MMPVVPSIGKlK2 VINCR=VP 500 GO TO 801

c C HIGH SHEAR SOLUTION C

428 WRITE(oo09) GO TO 352

156

2 V=Vl 4 V=V+V INCR

XX=SIGW2-015V2AW2 IF(XXJI0235D350

350 SIG=SQRHXX) 352 AlPHA=ARSIGR(AfSIGf

BETA=AWSIG(AfSIGf) 429 02=-10-BETA-AlPHA

03=05VA(AFSlGFT)+AlPHA+8ETA+O5(AlPHA+8ETA)2+OS8ETAST 1 +AlPHA(U+Q2IT-CD-2H(AlPHA+8ETA)O5T

04=D22-403 IFCD4)102351351

351 K21=(-D2+SQRTID4raquo2 K22=(-D2-SQRT(04112 Kll=1O+8ETA+AlPHA-K21 K12=10+BETA+AlPHA-K22 IFCK21630601601

601 IF(K21-10602~0263C

602 IF(Kll)630604604 604 IFIKll-10)62062D63C 630IFCK22)102631o31 631 IFIK22-10)632632102 632 IF(K12)102633633 633 IF(K12-10)621621lC2 620 K K21

Kl=Kll GO TO 622

621 K2=K22 Kl=K12

622 FYPT=Kl(D-2H)-TKl2-(S+ST) FY=AFSIGFFYPT-AWSIGS2-ARSIGRlU+Q2bullbull F=AFSIGfC2KI-l-AWSIG-ARSIGR M=20FY+20HF-VA MMP=MMP VVP=VVP IF(MMP) 102623623

623 WRITE(6607)MMPVVPSIGKlK2 VINCR=VPIOO GO TO 4

102 V I=V-VINCR VINCR=VINCRIOO VIMIN=VPI00DOO IF(VINCR-VIMIN) 19922

901 Vl=V-VINCR VINCR=VINCR200 VIMIN=VPIOOOO IF(VINCR-VIMIN)199902902

c

C

157

921 V1=V-VINCR VINCR=VINCRI200 VIMIN=VPI10000 IF(VINCR-VIMIN)199922922

199 CONTINUE GO TO 200

201 CONTINUE WRITE(b608)

200 CONTINUE RETURN END

158

APPENDIX B

Plasticity Relationships

In the elastic range stresses were computed from measured

strains by the usual elastic theory The values used for the

modulus of elasticity E and the shear modulus Gtwere 29OOOksi

and 11200ksi respectively When local yielding had occurred

according to the von Mises criterionas stated by equation (2)

Chapter 11 stresses were then computed by the plasticity

relations given by Hill(8) and reduced to the form given by

Bower(16) These relations are stated here for easy reference

Plasticity relations for a Prandtl-Reuss solid yield

dE = des _ ~(d)+ 2G dt (a)xT3Gt~xy

where E x is strain in the longitudinal direction and l$ xy

is the shearing strain Eliminating ~ by use of the von

Mises criteria gives

(b)

Since explicit integration of equation (b) would be

extremely difficult the equation is diVided by dE-x and the

derivatives d tSd E and d 6 yld poundx are replaced byx x

159

(c)

( d)

where~i and ~XYi are the bending and shearing stresses

respectively corresponding to ~ and ~f and ~xYf are

these stresses corresponding to poundx + 6 (x

Substituting equations (c) and (d) into equation (b)

gives

A~x - B6~ + AtS cs = AY l (e)

f

where (f)

and ( g)

The shear stress corresponding to a change in strain is given

by

2 2 ~ =Jcsy - CS f (h)

f [3

In order to determine whether strain hardening had occurred

an equivalent strain Xp was computed from

160

where lIE Xl ~yp Ezp and xyp are the plastic components of

strain and are given by

~ xp = poundx-t ( j)

Eyp = E yy (S

- i (k)

E (1)poundzp = z -~ (m)xyp = 6xy -

~

~

The constant volume condition was used to calculate ~ zp

(n)

The equivalent strain was compared to a reference strain

~ pr computed from the value for ~p for the onset of strain

hardening for a tensile test The strain hardening strain

was taken as euro x and it was assumed that E = poundoz For yy

A36 steel the strain hardening strain was taken as 115 ~yIE

and ~y was determined from a tensile test of a coupon from

the web of the test beam Strain hardening was considered

resent J f 6 gt or J P degpr-

The above relationships were included in a paper

presented by Bower(16) in 1968

161

BIBLIOGRAPHY

1 Muskhelishvili NISome Basic Problems of the Mathematical Theory of ElasticityU 2nd Edition P Noordhoff Itd 1963

2 HelIer SR Jr Brock JS and Bart R The itresses Around a Rectangular Opening with Rounded Corners in a Beam Subjected to Bending with Shear Proceedings Vol 1 Fourth U National Congress on Applied Mechanics Berkeley Calif 1962

3 Bower JE ItElastic Stresses Around Holes in WideshyFlange Beams Journal of the Structural Division ASCE Vol 92 No ST2 Proc Paper 4773April 1966

4 BowerJE Experimental Stresses in Wide-Flange Beams with Holes tl Journal of the Structural Division ASCE Vol 92 No ST5 Proc Paper 4945October 1966

5 So WC The Stresses Around Large Circular Openingsin the Webs of Wide-Flange Beams MEng Thesis McGill University Aug 1963

6 Chen IC tiThe Stresses Around Two Large Openings in the Webs of Wide-Flange Beams MEng Thesis McGill University Aug 1967

7 Segner EP Jr Reinforcement Requirements for Girder Web Openings Journal of the Structural Division ASCE Vol 90 No ST3 Proc Paper3919 June 1964

8 HillR The Mathematical Theory of PlastiCityClarendon Press Oxford University Press London 1950

9 Handbook of Steel Construction Canadian Institute of Steel Construction OntariO 1967

10 ltSpecifications for the DeSign Fabrication and Erection of Structural Steel for Buildings AISC 1963

11 Basler K Strength of Plate Girders in Shear Journal of the Structural Division ASCE Vol 87 No ST7 Proc Paper 2967 October 1961

162

12 Worley WJ Inelastic Behavior of Aluminum AlloyI-Beams With Web Cutouts University of Illinois Engineering Experimental Station Bulletin No 448 April 1958

13 Redwood RG and McCutcheon J 0 Beam Tests with Unreinforced Web Openings Journal of the Structural Division ASCE Vol 94 No ST1 Proc Paper5706 Jan 1968

14 nCommentary of Plastic DeSign in Steel ASCE Manual of Engineering Practice No 41 1961

15 Cheng SY flAn Investigation of Large Extended Openingsin the Webs of Wide-Flange Beams MBng Thesis McGill University Aug 1966

16 Bower JE Ultimate Strength of Beams with RectangularHoles Journal of the Structural Division ASCE Vol 94 No ST6 Proc Paper 5982 June 1968

17 Redwood RG Plastic Behavior and Design of Beams with Web Openings ff Proceedings of the Canadian Structural Engineering Conference Canadian Institute of Steel Construction Toronto Feb 1968

18 Redwood RG The Strength of Steel Beams with Unreinshyforced Web Holes Civil Engineering and PubliC Works Review Vol 64 No 755 London June 1969

19 Redwood RG Ultimate Strength Design of Beams with Multiple Openings Proceedings of the Structural Engineering Conference American Society of Civil Engineers Pittsburgh October 1968

ii

SUMMARY

Results are presented for tests to destruction performed

as part of this study on one IOWF21 and ten 14WF38 beams

each containing a single rectangular opening in the web The

openings in all but one of the beams were reinforced with

straight horizontal reinforcing plates welded above and below

the opening on each side of the web The variables investishy

gated include moment to shear ratiO opening depth to beam

depth ratiO opening length to depth ratio and reinforcing

size

An ultimate strength analysis is offered based on failure

by development of a four hinge mechanism the hinges occurring

at cross-sections at the edges of the opening A simple to

use approximate method of solution is also offered and a proshy

cedure for design is suggested

The experimental results show the theory to be reasonably

accurate at high moment to shear ratios but conservative at

high values of shear The approximate method is less conshy

servative in this region

iii

ACKNOVlLEDGEMENTS


















iv

NOTATION


Af area of flange


Aw area of web

b width of flange


d depth of beam






G shear modulus









x

v

p applied load






shear force

plastic shear force

w thickness of web












vi








~ normal stress

yield stress




shear stress





vii

LIST OF FIGURES

Page




















viii

Page




















ix

Page
























x

Page




xi

LIST OF TABLES

Page










SUJn1[ARY

ACKNOWLEDGEMENTS

NOTATION

LIST OF FIGURES

LIST OF TABLES

TABLE OF CONTENTS

Page

ii

iii

iv

vii

xi











31 Assumptions 27

















63 other Factors 72




73 Failure Loads 80



81 Conclusions 85



FIGURES 93

TABLES 145



BIBLIOGRAPHY 161

1

CHAPTER I

INTRODUCTION






















2


be increased













stresses


stresses








3


























4














given in Figure lb











5


























6


























7

--~~

















the problem

8

CHAPTER 11























9





is the yield stress
















10





















11





Figure 4





















12






hardening range(ll)







is reduced to

(4)



v = w( d-2t-2h) $-3





13




















(Vv) = d-2t-2h (6)P max d-2t




14





this task











increase in load









15






















decreases




16


























17
























18

























19

























20

























was offered by Chen

21





















the case




22








this effect

















23
























24

























25
























26









27

CHAPTER III


31 Assumptions








7b




dictated









28

























29







forcing is attached









section (2) gives






30


yields

(8)



(9)




stresses

Fl = Af ~yf + SW(1-2~)G + Ar ~yr (10)









31


calculated from

t2 _ r- 2 3(V)2 ~ - yw-4sw (12)
















given by

32




f yf





mined from

(17)













33








(13) and (18) yield


- (d-2h)]





34












(22)


(24)


(25)

35











become





(28)

(29)

36





















37







is written as

d-2t-2h (6)= d-2t











38






















(30)

39




(31)


given by

A =2 ( 32) r 13















40














reached

41

CHAPTER IV

APPROXIMATE METHOD

41 General Remarks





















42






(33)





Xf


by

(35 )




43




(36)











IS = If f)Af 3 r ( 37 ) - (s2 + 4a2 )

T





44



becomes

M (39 )M= p AW

1 + 4A f


(40)













by equation (38)


45





- AW(l 2h) flA (41)r - T -a-~




ityas given by

V 2hr = (I-a-) (42) p



Ar 1 - A

M fr= A p 1 + w







46












rewri tten as

M A (45 ) M=

p 1 + w4Af






47







previous section

















(43) then becomes

48

A 2h l1 - ~(l-a)J~

M ( 46)~= A 1 + W


aw 1 - 73Af

= ---1-shyA 1 + w
















49








this capacity
















(49a)

(50)

50

A q (5la )


Mp Aw 1 + 1iA

f


simplified to

(49)

(51)







Case I


if


given by

V (1 _ ~h) (42)Vp =

51


is Ar

1- ri- = Af (43)

p 1 + w 41f


Case 11



by

(40)


acting is

1 1 + w



Case Ill



52



follows

Case IlIA

For (48)


Ar 1 - I - ~

~ = __f_~_ ill A

P 1+ w4If


(38)

Case IIIB

For A2 gt (aw) 2

+ (~)2w2 (48)r 3 ~



A1 + w

4Af

53


+l (50)s

(49)




















54







these reduoe to

M AW 2h 1

1 - 2If(l-or~ r= A (52)

p 1 + w4If

v (2h)JTr = I-d~ p






given by

111 Z - wh2 (54)r = z

p




55

MM = 1 (55) p





56

CHAPTER V























57
























requires that

58

wh2 gt- (56)2h + 2u + q




















codes



59
























60

























61

























62
























63

















frame







64

























65
















the test beams

66

CHAPTER VI























67

























68

























69

























70


























71
























72






Figure 31












63 Other Factors






73

























74








shear


















75
























76



77

CHAPTER VII

ANALYSIS OF RESULTS






















78
























79

























80











73 Failure Loads













81

























82
























83













in Figure 54












84















85

CHAPTER VIII


81 Conclusions



drawn
















lateral buckling


86














opening






distributions





87

























88







force

















89

























go
























91

























92













93

1)

) )J + 1 +1 I I

I

( a) (b)

I + --+--~+ -f--- -shy

I (c) (d)

II I I

I I j+shy

) 1)I l J I

(e) ( f)


Figure 1

94




Figure 2

Ql ID Q)

Jt p ID

er

I I I I I I I



Figure 3

---------- -----

95


10~----~==~==~L---------------------~

~


10 vVp


Figure 4

-- -----------

96


(a)

~ o

r-I


(b)


Figure 5

97

1O~------------------------------------------~

l I


shy

I r perforated beam

no interaction


I I


Figure 6

i

98

1 CD (2) TIT

L

( a)

b

d2

(b)


Figure 7

99

( a)

(S r CSyr+



Case I - Low Shear

+

CSyr ltS shy

direct shear



(b)


Figure 8

100

low highshear shear


I I r I I

I I I 1 I10


11

11 I I 11 I I



11~I11

i I I

I (d-2h-2t)I_~ I

d-2t 11

(1 -~) q


Figure 9

101



03~

CH 125 1412700

0313 =--9F 0375 =Jb


04 vVp


Figure 10

102


x e 3middot

Reinforcing

1412 11

10


04


Figure 11

103


0513No Reinforcing

MMp

10


04 VVp


Figure 12

104



I I I

04


Figure 13

105


0513

1412



04


Figure 14

106

10WF2l


CH h

55Q 125 990 025Q 0250~



I I I

04


Figure 15

107

43 11 39 18 0 (

I

I I i lA

26

I

111 99 I- middot1 t

22 2-S 45 J

) C

r I

2A

13 ) lA I ~

I)II45 35 ( ()

I

I

I If 2B

f 16~ 9 25 I2t~ 21-+1 1 1 i

30 11 24 27 0- 10- ro- )

I ~ I I

20

2119YlO 151( 21~fI211~ - rr I I



Figure 16

108

tI

2D

22tl 231 45 tlI

( l (

I

3A

3119 Yt 11~ 34ft J3n~

45 25 J 35 ) (~ i-

I

I 1 3B

H 25 ft2ft5 cI 16 J2++ I I1 I

t 0- 23 45 J0- 0- (

I 4A

~5 ( 32 J211~ 2 YYt -IIII



Figure 17

log

45 ~ 25 5S ) ~ (

4B

2 crYI 15 25 J-I2++ Irr

L J30 24 27 ~lt) (P-L

I 5A

2tYY 13 26211 1 I

22 6

6A

i



Figure 18

--

110

If JL

a (1) p 02 ~

Cfl

tlOO ~ r-I 0 0 Q)

m ~ ~

Q SoOM


r-shy

~ ishy fO I- It

111

o N

-()

112

~ --- ~

~

~~ ~~

II I of

rf

A tshyshy I

1


a1 +gt

- -l~ 0 ()

-t I H CJ

4gt 4gt bO ~

~ ~ Cl -rt

fiI

~ a1 ~

+gt CJI

~ amp ~ ~ ~ --- ~ I

II s I

~CJ~ 4t ~

I~ i ~

o

i-~ ~ - l

II I +1 I Ij

~I r

I tf~ft N

- shy

stress 8 1


( a)

F F-CL F-E [(4 ~

W-H- - shy

strain ( b)



I

114

lA 2A 2B

20 2D 3A

3B 4A 4B

5A 6A

FIGURE 24

115

2B 20 2D 3A 3B 4At 4B


20 5A


FIGURE 25

116

2A 4A 2B 4B


2B 3B


FIGURE 26

bullbull

117


r ~ 3

o

top of reinf



~ bull




Figure 27

118

boundaryof opening

-70 0 __ tQ



-1iCgt

~I--=-shyoo---~-=-~ogt---- ---0 - ____ A~Sshy-


oE


Figure 28

119

-40

-lt0 lt----- shylow


lt 0


edge ltgtgt

to


high moment

edge

o


Figure 29

120


o10 D



ibull

boundaryof opening

10 Igt lO



Figure 30

bull bull bull bull

121

11 9 ~ oS j j ~

boundaryof opening

-lt10 -20 lt) -I-Lo Q lO 10



boundaryof openin

-40 -20 0 ZO o 10 ZQ 30



Figure 31

122

Test Beam - lA


Figure 32

Test Beam - 2A


Figure 33

123


Test Beam - 2B


Figure 34




Test Beam - 20


Figure 35

124


first yielding

Test Beam - 2D


Figure 36

Test Beam - 3A


Figure 37

125


---- first yielding

O6in

Figure 38

O6in

Figure 39


Test Beam - 3B


Test Beam - 4A


126

430k--shy

k445--shy



06in

Figure 40

O6in

Figure 41


Test Beam - 4B


Test Beam - 5A


127

k45 o-~--

Test Beam - 6A


Figure 42

128

10 shy

10WF21



~ ~

+~ ~ ~


I I

04


Figure 43

curve yields

129






bull


Figure 44

130




04


Figure 45

10

131



~

1 +3A



04


Figure 46

10

132



~ ~ ~ ~ ~

I I


04


Figure 47

10

133

14WF38 H


-~


~ ~ +4B

~ ~ ~


I I

04


Figure 48

134




~ ~

SAl


04 vVp


Figure 49

10

135



+ 6A


04


Figure 50

136

shear (kips)

70

60


50

40

30

20


40 80 120 160 200 240



Figure 51

137

shear (kips)

70

60


50

40

30

20

10

4 8 12 16 20 24 28



Figure 52

138

D f



Figure 53

139

l4WF38 7 x lot

Opening

3shy10 f ~--2f x e

Reinforcing

If x ~ Reinforcing

+3A +2A

No Reinforcing + 6A

04


Figure 54

140


+ 2B

+20

+2A

+2D

04 vVp


Figure 55

141


10 ---------- shy

+3B

+3A

04 vVp


Figure 56

142


+ 4B

+ 4A

04 vVp


Figure 57

143


7 x lot Opening

7 x 14 Opening---

04


Figure 58

144shy

14WF38



+5A

04


Figure 59

--

r b --1 t l=Ii= W

d

u q


11

0244 875 550 N

0260 0253 2720 N

400 316

2A 14WF38 22 1413 677 0496 0326 1046 700 0383 0381 2813 300 58



It 11 It If3B 45 1422 668 0490 0305 1045 0329 0361 5230



foo J1

146













4A web W 565 549 flange F 11 487 476



F-E ff 438 425


Table 2

147






k 567k 483k



Table 3

- - - - - - -

- - - - - - - - -

- -


lA 2A 2B 20 2D 3A 3B ~ lI Ilt v 162 500 400 583 700 525 433

180 - - - 725 - 517 191 550 417 617 600 550 567 191 575 442 567 575 600 433

- 575 467 567 575 600 433

- 575 - - 600 600 shy202 625 475 600 600 700 533

- 625 475 600 625 700 583

- - - 617 700 - 550

- - - 600 675 675 550

- - - - - 675 shy

- 600 458 650 675 650 583

- - - - 725 - -- - - - - 725 -- - - - - 675 -- - - - - - -- - - - - 675 shy- - 483 - - - shy

- 700 517 703 825 775 shy- - - - 825 - 600

4A 4B Ilt Ilt

450 333 500 417 450 367 450 366 500 383 450 417 550 483 600 483 550 400 550 433 625 shy625 shy575 467 575 shy500 shy550 417

- 450 500 383

--675 513

5A le

350 500 400 400 433 483 433 467 500 417

--

500

--

467 500

---

517

-

6A

400 533

-367 367 383 433 533

-----------

550

--


J-ITable 4 ~ 00

149


lA 10WF21 43 180K 746K 0241 0235 + 26 2A 14WF38 22 532 1L021 0520 0466 +116 2B 11 45 413 0404 0363 +113 2C If 30 548 1301 0421 0403 + 45



Table 5


lA 10WF21 43 180K 746 K

0241 0254 - 51 2A 14VlF38 22 532 1021 0520 0503 + 34

It2B 45 413 0404 0344 +175 If2C 30 548 1301 0421 0414 + 17

It2D 17 670 0515 0505 + 20 3A 22 575 1021 0562 0505 +103

n3B 45 497 1301 0382 0377 + 13 4A tI 22 572 n 0440 0463 - 50

It It 03094B 45 430 0330 + 68 5A 30 445 tr 0342 0353 - 31 6A 22 450 tt 0346 0257 +346


Table 6

150



2D 17 670 625 - 67 - shy3B 45 497 483 - 28 600 +207 4B 45 430 383 -109 - shy


Table 7



2C 30n 525k 567k + 80 633k +206 2D 17 546 625 +145 - shy3B 45 449 483 + 76 600 +336 4B 45 384 383 - 03 - shy


Table 8

151

APPENDIX A








the following table










pages

152





SIGw SIGR VP Mpt)





RE~D(5582JAH


Kl K2~)

COMPLEXJ

153








Kl=Kl1 GO TO 322

321 K2K22 Kl=K12


154



408 WRITE(6613 GO TO 409







K l=K 11 GO TO 422

421 K2=K22 Kl=K12





155


418 WRITE(6614) GO TO 419


XX=SIGW2-075V2AW2 IF(XX)921800800







Kl=Kl1 GO TO 522

521 1lt2=K2Z Kl=K12





156

2 V=Vl 4 V=V+V INCR





04=D22-403 IFCD4)102351351


601 IF(K21-10602~0263C


Kl=Kll GO TO 622

621 K2=K22 Kl=K12





c

C

157





158

APPENDIX B















(b)




159

(c)

( d)





gives


f

where (f)

and ( g)


by

2 2 ~ =Jcsy - CS f (h)

f [3



160




Eyp = E yy (S

- i (k)


~

~


(n)











161

BIBLIOGRAPHY












162









iii

ACKNOVlLEDGEMENTS


















iv

NOTATION


Af area of flange


Aw area of web

b width of flange


d depth of beam






G shear modulus









x

v

p applied load






shear force

plastic shear force

w thickness of web












vi








~ normal stress

yield stress




shear stress





vii

LIST OF FIGURES

Page




















viii

Page




















ix

Page
























x

Page




xi

LIST OF TABLES

Page










SUJn1[ARY

ACKNOWLEDGEMENTS

NOTATION

LIST OF FIGURES

LIST OF TABLES

TABLE OF CONTENTS

Page

ii

iii

iv

vii

xi











31 Assumptions 27

















63 other Factors 72




73 Failure Loads 80



81 Conclusions 85



FIGURES 93

TABLES 145



BIBLIOGRAPHY 161

1

CHAPTER I

INTRODUCTION






















2


be increased













stresses


stresses








3


























4














given in Figure lb











5


























6


























7

--~~

















the problem

8

CHAPTER 11























9





is the yield stress
















10





















11





Figure 4





















12






hardening range(ll)







is reduced to

(4)



v = w( d-2t-2h) $-3





13




















(Vv) = d-2t-2h (6)P max d-2t




14





this task











increase in load









15






















decreases




16


























17
























18

























19

























20

























was offered by Chen

21





















the case




22








this effect

















23
























24

























25
























26









27

CHAPTER III


31 Assumptions








7b




dictated









28

























29







forcing is attached









section (2) gives






30


yields

(8)



(9)




stresses

Fl = Af ~yf + SW(1-2~)G + Ar ~yr (10)









31


calculated from

t2 _ r- 2 3(V)2 ~ - yw-4sw (12)
















given by

32




f yf





mined from

(17)













33








(13) and (18) yield


- (d-2h)]





34












(22)


(24)


(25)

35











become





(28)

(29)

36





















37







is written as

d-2t-2h (6)= d-2t











38






















(30)

39




(31)


given by

A =2 ( 32) r 13















40














reached

41

CHAPTER IV

APPROXIMATE METHOD

41 General Remarks





















42






(33)





Xf


by

(35 )




43




(36)











IS = If f)Af 3 r ( 37 ) - (s2 + 4a2 )

T





44



becomes

M (39 )M= p AW

1 + 4A f


(40)













by equation (38)


45





- AW(l 2h) flA (41)r - T -a-~




ityas given by

V 2hr = (I-a-) (42) p



Ar 1 - A

M fr= A p 1 + w







46












rewri tten as

M A (45 ) M=

p 1 + w4Af






47







previous section

















(43) then becomes

48

A 2h l1 - ~(l-a)J~

M ( 46)~= A 1 + W


aw 1 - 73Af

= ---1-shyA 1 + w
















49








this capacity
















(49a)

(50)

50

A q (5la )


Mp Aw 1 + 1iA

f


simplified to

(49)

(51)







Case I


if


given by

V (1 _ ~h) (42)Vp =

51


is Ar

1- ri- = Af (43)

p 1 + w 41f


Case 11



by

(40)


acting is

1 1 + w



Case Ill



52



follows

Case IlIA

For (48)


Ar 1 - I - ~

~ = __f_~_ ill A

P 1+ w4If


(38)

Case IIIB

For A2 gt (aw) 2

+ (~)2w2 (48)r 3 ~



A1 + w

4Af

53


+l (50)s

(49)




















54







these reduoe to

M AW 2h 1

1 - 2If(l-or~ r= A (52)

p 1 + w4If

v (2h)JTr = I-d~ p






given by

111 Z - wh2 (54)r = z

p




55

MM = 1 (55) p





56

CHAPTER V























57
























requires that

58

wh2 gt- (56)2h + 2u + q




















codes



59
























60

























61

























62
























63

















frame







64

























65
















the test beams

66

CHAPTER VI























67

























68

























69

























70


























71
























72






Figure 31












63 Other Factors






73

























74








shear


















75
























76



77

CHAPTER VII

ANALYSIS OF RESULTS






















78
























79

























80











73 Failure Loads













81

























82
























83













in Figure 54












84















85

CHAPTER VIII


81 Conclusions



drawn
















lateral buckling


86














opening






distributions





87

























88







force

















89

























go
























91

























92













93

1)

) )J + 1 +1 I I

I

( a) (b)

I + --+--~+ -f--- -shy

I (c) (d)

II I I

I I j+shy

) 1)I l J I

(e) ( f)


Figure 1

94




Figure 2

Ql ID Q)

Jt p ID

er

I I I I I I I



Figure 3

---------- -----

95


10~----~==~==~L---------------------~

~


10 vVp


Figure 4

-- -----------

96


(a)

~ o

r-I


(b)


Figure 5

97

1O~------------------------------------------~

l I


shy

I r perforated beam

no interaction


I I


Figure 6

i

98

1 CD (2) TIT

L

( a)

b

d2

(b)


Figure 7

99

( a)

(S r CSyr+



Case I - Low Shear

+

CSyr ltS shy

direct shear



(b)


Figure 8

100

low highshear shear


I I r I I

I I I 1 I10


11

11 I I 11 I I



11~I11

i I I

I (d-2h-2t)I_~ I

d-2t 11

(1 -~) q


Figure 9

101



03~

CH 125 1412700

0313 =--9F 0375 =Jb


04 vVp


Figure 10

102


x e 3middot

Reinforcing

1412 11

10


04


Figure 11

103


0513No Reinforcing

MMp

10


04 VVp


Figure 12

104



I I I

04


Figure 13

105


0513

1412



04


Figure 14

106

10WF2l


CH h

55Q 125 990 025Q 0250~



I I I

04


Figure 15

107

43 11 39 18 0 (

I

I I i lA

26

I

111 99 I- middot1 t

22 2-S 45 J

) C

r I

2A

13 ) lA I ~

I)II45 35 ( ()

I

I

I If 2B

f 16~ 9 25 I2t~ 21-+1 1 1 i

30 11 24 27 0- 10- ro- )

I ~ I I

20

2119YlO 151( 21~fI211~ - rr I I



Figure 16

108

tI

2D

22tl 231 45 tlI

( l (

I

3A

3119 Yt 11~ 34ft J3n~

45 25 J 35 ) (~ i-

I

I 1 3B

H 25 ft2ft5 cI 16 J2++ I I1 I

t 0- 23 45 J0- 0- (

I 4A

~5 ( 32 J211~ 2 YYt -IIII



Figure 17

log

45 ~ 25 5S ) ~ (

4B

2 crYI 15 25 J-I2++ Irr

L J30 24 27 ~lt) (P-L

I 5A

2tYY 13 26211 1 I

22 6

6A

i



Figure 18

--

110

If JL

a (1) p 02 ~

Cfl

tlOO ~ r-I 0 0 Q)

m ~ ~

Q SoOM


r-shy

~ ishy fO I- It

111

o N

-()

112

~ --- ~

~

~~ ~~

II I of

rf

A tshyshy I

1


a1 +gt

- -l~ 0 ()

-t I H CJ

4gt 4gt bO ~

~ ~ Cl -rt

fiI

~ a1 ~

+gt CJI

~ amp ~ ~ ~ --- ~ I

II s I

~CJ~ 4t ~

I~ i ~

o

i-~ ~ - l

II I +1 I Ij

~I r

I tf~ft N

- shy

stress 8 1


( a)

F F-CL F-E [(4 ~

W-H- - shy

strain ( b)



I

114

lA 2A 2B

20 2D 3A

3B 4A 4B

5A 6A

FIGURE 24

115

2B 20 2D 3A 3B 4At 4B


20 5A


FIGURE 25

116

2A 4A 2B 4B


2B 3B


FIGURE 26

bullbull

117


r ~ 3

o

top of reinf



~ bull




Figure 27

118

boundaryof opening

-70 0 __ tQ



-1iCgt

~I--=-shyoo---~-=-~ogt---- ---0 - ____ A~Sshy-


oE


Figure 28

119

-40

-lt0 lt----- shylow


lt 0


edge ltgtgt

to


high moment

edge

o


Figure 29

120


o10 D



ibull

boundaryof opening

10 Igt lO



Figure 30

bull bull bull bull

121

11 9 ~ oS j j ~

boundaryof opening

-lt10 -20 lt) -I-Lo Q lO 10



boundaryof openin

-40 -20 0 ZO o 10 ZQ 30



Figure 31

122

Test Beam - lA


Figure 32

Test Beam - 2A


Figure 33

123


Test Beam - 2B


Figure 34




Test Beam - 20


Figure 35

124


first yielding

Test Beam - 2D


Figure 36

Test Beam - 3A


Figure 37

125


---- first yielding

O6in

Figure 38

O6in

Figure 39


Test Beam - 3B


Test Beam - 4A


126

430k--shy

k445--shy



06in

Figure 40

O6in

Figure 41


Test Beam - 4B


Test Beam - 5A


127

k45 o-~--

Test Beam - 6A


Figure 42

128

10 shy

10WF21



~ ~

+~ ~ ~


I I

04


Figure 43

curve yields

129






bull


Figure 44

130




04


Figure 45

10

131



~

1 +3A



04


Figure 46

10

132



~ ~ ~ ~ ~

I I


04


Figure 47

10

133

14WF38 H


-~


~ ~ +4B

~ ~ ~


I I

04


Figure 48

134




~ ~

SAl


04 vVp


Figure 49

10

135



+ 6A


04


Figure 50

136

shear (kips)

70

60


50

40

30

20


40 80 120 160 200 240



Figure 51

137

shear (kips)

70

60


50

40

30

20

10

4 8 12 16 20 24 28



Figure 52

138

D f



Figure 53

139

l4WF38 7 x lot

Opening

3shy10 f ~--2f x e

Reinforcing

If x ~ Reinforcing

+3A +2A

No Reinforcing + 6A

04


Figure 54

140


+ 2B

+20

+2A

+2D

04 vVp


Figure 55

141


10 ---------- shy

+3B

+3A

04 vVp


Figure 56

142


+ 4B

+ 4A

04 vVp


Figure 57

143


7 x lot Opening

7 x 14 Opening---

04


Figure 58

144shy

14WF38



+5A

04


Figure 59

--

r b --1 t l=Ii= W

d

u q


11

0244 875 550 N

0260 0253 2720 N

400 316

2A 14WF38 22 1413 677 0496 0326 1046 700 0383 0381 2813 300 58



It 11 It If3B 45 1422 668 0490 0305 1045 0329 0361 5230



foo J1

146













4A web W 565 549 flange F 11 487 476



F-E ff 438 425


Table 2

147






k 567k 483k



Table 3

- - - - - - -

- - - - - - - - -

- -


lA 2A 2B 20 2D 3A 3B ~ lI Ilt v 162 500 400 583 700 525 433

180 - - - 725 - 517 191 550 417 617 600 550 567 191 575 442 567 575 600 433

- 575 467 567 575 600 433

- 575 - - 600 600 shy202 625 475 600 600 700 533

- 625 475 600 625 700 583

- - - 617 700 - 550

- - - 600 675 675 550

- - - - - 675 shy

- 600 458 650 675 650 583

- - - - 725 - -- - - - - 725 -- - - - - 675 -- - - - - - -- - - - - 675 shy- - 483 - - - shy

- 700 517 703 825 775 shy- - - - 825 - 600

4A 4B Ilt Ilt

450 333 500 417 450 367 450 366 500 383 450 417 550 483 600 483 550 400 550 433 625 shy625 shy575 467 575 shy500 shy550 417

- 450 500 383

--675 513

5A le

350 500 400 400 433 483 433 467 500 417

--

500

--

467 500

---

517

-

6A

400 533

-367 367 383 433 533

-----------

550

--


J-ITable 4 ~ 00

149


lA 10WF21 43 180K 746K 0241 0235 + 26 2A 14WF38 22 532 1L021 0520 0466 +116 2B 11 45 413 0404 0363 +113 2C If 30 548 1301 0421 0403 + 45



Table 5


lA 10WF21 43 180K 746 K

0241 0254 - 51 2A 14VlF38 22 532 1021 0520 0503 + 34

It2B 45 413 0404 0344 +175 If2C 30 548 1301 0421 0414 + 17

It2D 17 670 0515 0505 + 20 3A 22 575 1021 0562 0505 +103

n3B 45 497 1301 0382 0377 + 13 4A tI 22 572 n 0440 0463 - 50

It It 03094B 45 430 0330 + 68 5A 30 445 tr 0342 0353 - 31 6A 22 450 tt 0346 0257 +346


Table 6

150



2D 17 670 625 - 67 - shy3B 45 497 483 - 28 600 +207 4B 45 430 383 -109 - shy


Table 7



2C 30n 525k 567k + 80 633k +206 2D 17 546 625 +145 - shy3B 45 449 483 + 76 600 +336 4B 45 384 383 - 03 - shy


Table 8

151

APPENDIX A








the following table










pages

152





SIGw SIGR VP Mpt)





RE~D(5582JAH


Kl K2~)

COMPLEXJ

153








Kl=Kl1 GO TO 322

321 K2K22 Kl=K12


154



408 WRITE(6613 GO TO 409







K l=K 11 GO TO 422

421 K2=K22 Kl=K12





155


418 WRITE(6614) GO TO 419


XX=SIGW2-075V2AW2 IF(XX)921800800







Kl=Kl1 GO TO 522

521 1lt2=K2Z Kl=K12





156

2 V=Vl 4 V=V+V INCR





04=D22-403 IFCD4)102351351


601 IF(K21-10602~0263C


Kl=Kll GO TO 622

621 K2=K22 Kl=K12





c

C

157





158

APPENDIX B















(b)




159

(c)

( d)





gives


f

where (f)

and ( g)


by

2 2 ~ =Jcsy - CS f (h)

f [3



160




Eyp = E yy (S

- i (k)


~

~


(n)











161

BIBLIOGRAPHY












162









iv

NOTATION


Af area of flange


Aw area of web

b width of flange


d depth of beam






G shear modulus









x

v

p applied load






shear force

plastic shear force

w thickness of web












vi








~ normal stress

yield stress




shear stress





vii

LIST OF FIGURES

Page




















viii

Page




















ix

Page
























x

Page




xi

LIST OF TABLES

Page










SUJn1[ARY

ACKNOWLEDGEMENTS

NOTATION

LIST OF FIGURES

LIST OF TABLES

TABLE OF CONTENTS

Page

ii

iii

iv

vii

xi











31 Assumptions 27

















63 other Factors 72




73 Failure Loads 80



81 Conclusions 85



FIGURES 93

TABLES 145



BIBLIOGRAPHY 161

1

CHAPTER I

INTRODUCTION






















2


be increased













stresses


stresses








3


























4














given in Figure lb











5


























6


























7

--~~

















the problem

8

CHAPTER 11























9





is the yield stress
















10





















11





Figure 4





















12






hardening range(ll)







is reduced to

(4)



v = w( d-2t-2h) $-3





13




















(Vv) = d-2t-2h (6)P max d-2t




14





this task











increase in load









15






















decreases




16


























17
























18

























19

























20

























was offered by Chen

21





















the case




22








this effect

















23
























24

























25
























26









27

CHAPTER III


31 Assumptions








7b




dictated









28

























29







forcing is attached









section (2) gives






30


yields

(8)



(9)




stresses

Fl = Af ~yf + SW(1-2~)G + Ar ~yr (10)









31


calculated from

t2 _ r- 2 3(V)2 ~ - yw-4sw (12)
















given by

32




f yf





mined from

(17)













33








(13) and (18) yield


- (d-2h)]





34












(22)


(24)


(25)

35











become





(28)

(29)

36





















37







is written as

d-2t-2h (6)= d-2t











38






















(30)

39




(31)


given by

A =2 ( 32) r 13















40














reached

41

CHAPTER IV

APPROXIMATE METHOD

41 General Remarks





















42






(33)





Xf


by

(35 )




43




(36)











IS = If f)Af 3 r ( 37 ) - (s2 + 4a2 )

T





44



becomes

M (39 )M= p AW

1 + 4A f


(40)













by equation (38)


45





- AW(l 2h) flA (41)r - T -a-~




ityas given by

V 2hr = (I-a-) (42) p



Ar 1 - A

M fr= A p 1 + w







46












rewri tten as

M A (45 ) M=

p 1 + w4Af






47







previous section

















(43) then becomes

48

A 2h l1 - ~(l-a)J~

M ( 46)~= A 1 + W


aw 1 - 73Af

= ---1-shyA 1 + w
















49








this capacity
















(49a)

(50)

50

A q (5la )


Mp Aw 1 + 1iA

f


simplified to

(49)

(51)







Case I


if


given by

V (1 _ ~h) (42)Vp =

51


is Ar

1- ri- = Af (43)

p 1 + w 41f


Case 11



by

(40)


acting is

1 1 + w



Case Ill



52



follows

Case IlIA

For (48)


Ar 1 - I - ~

~ = __f_~_ ill A

P 1+ w4If


(38)

Case IIIB

For A2 gt (aw) 2

+ (~)2w2 (48)r 3 ~



A1 + w

4Af

53


+l (50)s

(49)




















54







these reduoe to

M AW 2h 1

1 - 2If(l-or~ r= A (52)

p 1 + w4If

v (2h)JTr = I-d~ p






given by

111 Z - wh2 (54)r = z

p




55

MM = 1 (55) p





56

CHAPTER V























57
























requires that

58

wh2 gt- (56)2h + 2u + q




















codes



59
























60

























61

























62
























63

















frame







64

























65
















the test beams

66

CHAPTER VI























67

























68

























69

























70


























71
























72






Figure 31












63 Other Factors






73

























74








shear


















75
























76



77

CHAPTER VII

ANALYSIS OF RESULTS






















78
























79

























80











73 Failure Loads













81

























82
























83













in Figure 54












84















85

CHAPTER VIII


81 Conclusions



drawn
















lateral buckling


86














opening






distributions





87

























88







force

















89

























go
























91

























92













93

1)

) )J + 1 +1 I I

I

( a) (b)

I + --+--~+ -f--- -shy

I (c) (d)

II I I

I I j+shy

) 1)I l J I

(e) ( f)


Figure 1

94




Figure 2

Ql ID Q)

Jt p ID

er

I I I I I I I



Figure 3

---------- -----

95


10~----~==~==~L---------------------~

~


10 vVp


Figure 4

-- -----------

96


(a)

~ o

r-I


(b)


Figure 5

97

1O~------------------------------------------~

l I


shy

I r perforated beam

no interaction


I I


Figure 6

i

98

1 CD (2) TIT

L

( a)

b

d2

(b)


Figure 7

99

( a)

(S r CSyr+



Case I - Low Shear

+

CSyr ltS shy

direct shear



(b)


Figure 8

100

low highshear shear


I I r I I

I I I 1 I10


11

11 I I 11 I I



11~I11

i I I

I (d-2h-2t)I_~ I

d-2t 11

(1 -~) q


Figure 9

101



03~

CH 125 1412700

0313 =--9F 0375 =Jb


04 vVp


Figure 10

102


x e 3middot

Reinforcing

1412 11

10


04


Figure 11

103


0513No Reinforcing

MMp

10


04 VVp


Figure 12

104



I I I

04


Figure 13

105


0513

1412



04


Figure 14

106

10WF2l


CH h

55Q 125 990 025Q 0250~



I I I

04


Figure 15

107

43 11 39 18 0 (

I

I I i lA

26

I

111 99 I- middot1 t

22 2-S 45 J

) C

r I

2A

13 ) lA I ~

I)II45 35 ( ()

I

I

I If 2B

f 16~ 9 25 I2t~ 21-+1 1 1 i

30 11 24 27 0- 10- ro- )

I ~ I I

20

2119YlO 151( 21~fI211~ - rr I I



Figure 16

108

tI

2D

22tl 231 45 tlI

( l (

I

3A

3119 Yt 11~ 34ft J3n~

45 25 J 35 ) (~ i-

I

I 1 3B

H 25 ft2ft5 cI 16 J2++ I I1 I

t 0- 23 45 J0- 0- (

I 4A

~5 ( 32 J211~ 2 YYt -IIII



Figure 17

log

45 ~ 25 5S ) ~ (

4B

2 crYI 15 25 J-I2++ Irr

L J30 24 27 ~lt) (P-L

I 5A

2tYY 13 26211 1 I

22 6

6A

i



Figure 18

--

110

If JL

a (1) p 02 ~

Cfl

tlOO ~ r-I 0 0 Q)

m ~ ~

Q SoOM


r-shy

~ ishy fO I- It

111

o N

-()

112

~ --- ~

~

~~ ~~

II I of

rf

A tshyshy I

1


a1 +gt

- -l~ 0 ()

-t I H CJ

4gt 4gt bO ~

~ ~ Cl -rt

fiI

~ a1 ~

+gt CJI

~ amp ~ ~ ~ --- ~ I

II s I

~CJ~ 4t ~

I~ i ~

o

i-~ ~ - l

II I +1 I Ij

~I r

I tf~ft N

- shy

stress 8 1


( a)

F F-CL F-E [(4 ~

W-H- - shy

strain ( b)



I

114

lA 2A 2B

20 2D 3A

3B 4A 4B

5A 6A

FIGURE 24

115

2B 20 2D 3A 3B 4At 4B


20 5A


FIGURE 25

116

2A 4A 2B 4B


2B 3B


FIGURE 26

bullbull

117


r ~ 3

o

top of reinf



~ bull




Figure 27

118

boundaryof opening

-70 0 __ tQ



-1iCgt

~I--=-shyoo---~-=-~ogt---- ---0 - ____ A~Sshy-


oE


Figure 28

119

-40

-lt0 lt----- shylow


lt 0


edge ltgtgt

to


high moment

edge

o


Figure 29

120


o10 D



ibull

boundaryof opening

10 Igt lO



Figure 30

bull bull bull bull

121

11 9 ~ oS j j ~

boundaryof opening

-lt10 -20 lt) -I-Lo Q lO 10



boundaryof openin

-40 -20 0 ZO o 10 ZQ 30



Figure 31

122

Test Beam - lA


Figure 32

Test Beam - 2A


Figure 33

123


Test Beam - 2B


Figure 34




Test Beam - 20


Figure 35

124


first yielding

Test Beam - 2D


Figure 36

Test Beam - 3A


Figure 37

125


---- first yielding

O6in

Figure 38

O6in

Figure 39


Test Beam - 3B


Test Beam - 4A


126

430k--shy

k445--shy



06in

Figure 40

O6in

Figure 41


Test Beam - 4B


Test Beam - 5A


127

k45 o-~--

Test Beam - 6A


Figure 42

128

10 shy

10WF21



~ ~

+~ ~ ~


I I

04


Figure 43

curve yields

129






bull


Figure 44

130




04


Figure 45

10

131



~

1 +3A



04


Figure 46

10

132



~ ~ ~ ~ ~

I I


04


Figure 47

10

133

14WF38 H


-~


~ ~ +4B

~ ~ ~


I I

04


Figure 48

134




~ ~

SAl


04 vVp


Figure 49

10

135



+ 6A


04


Figure 50

136

shear (kips)

70

60


50

40

30

20


40 80 120 160 200 240



Figure 51

137

shear (kips)

70

60


50

40

30

20

10

4 8 12 16 20 24 28



Figure 52

138

D f



Figure 53

139

l4WF38 7 x lot

Opening

3shy10 f ~--2f x e

Reinforcing

If x ~ Reinforcing

+3A +2A

No Reinforcing + 6A

04


Figure 54

140


+ 2B

+20

+2A

+2D

04 vVp


Figure 55

141


10 ---------- shy

+3B

+3A

04 vVp


Figure 56

142


+ 4B

+ 4A

04 vVp


Figure 57

143


7 x lot Opening

7 x 14 Opening---

04


Figure 58

144shy

14WF38



+5A

04


Figure 59

--

r b --1 t l=Ii= W

d

u q


11

0244 875 550 N

0260 0253 2720 N

400 316

2A 14WF38 22 1413 677 0496 0326 1046 700 0383 0381 2813 300 58



It 11 It If3B 45 1422 668 0490 0305 1045 0329 0361 5230



foo J1

146













4A web W 565 549 flange F 11 487 476



F-E ff 438 425


Table 2

147






k 567k 483k



Table 3

- - - - - - -

- - - - - - - - -

- -


lA 2A 2B 20 2D 3A 3B ~ lI Ilt v 162 500 400 583 700 525 433

180 - - - 725 - 517 191 550 417 617 600 550 567 191 575 442 567 575 600 433

- 575 467 567 575 600 433

- 575 - - 600 600 shy202 625 475 600 600 700 533

- 625 475 600 625 700 583

- - - 617 700 - 550

- - - 600 675 675 550

- - - - - 675 shy

- 600 458 650 675 650 583

- - - - 725 - -- - - - - 725 -- - - - - 675 -- - - - - - -- - - - - 675 shy- - 483 - - - shy

- 700 517 703 825 775 shy- - - - 825 - 600

4A 4B Ilt Ilt

450 333 500 417 450 367 450 366 500 383 450 417 550 483 600 483 550 400 550 433 625 shy625 shy575 467 575 shy500 shy550 417

- 450 500 383

--675 513

5A le

350 500 400 400 433 483 433 467 500 417

--

500

--

467 500

---

517

-

6A

400 533

-367 367 383 433 533

-----------

550

--


J-ITable 4 ~ 00

149


lA 10WF21 43 180K 746K 0241 0235 + 26 2A 14WF38 22 532 1L021 0520 0466 +116 2B 11 45 413 0404 0363 +113 2C If 30 548 1301 0421 0403 + 45



Table 5


lA 10WF21 43 180K 746 K

0241 0254 - 51 2A 14VlF38 22 532 1021 0520 0503 + 34

It2B 45 413 0404 0344 +175 If2C 30 548 1301 0421 0414 + 17

It2D 17 670 0515 0505 + 20 3A 22 575 1021 0562 0505 +103

n3B 45 497 1301 0382 0377 + 13 4A tI 22 572 n 0440 0463 - 50

It It 03094B 45 430 0330 + 68 5A 30 445 tr 0342 0353 - 31 6A 22 450 tt 0346 0257 +346


Table 6

150



2D 17 670 625 - 67 - shy3B 45 497 483 - 28 600 +207 4B 45 430 383 -109 - shy


Table 7



2C 30n 525k 567k + 80 633k +206 2D 17 546 625 +145 - shy3B 45 449 483 + 76 600 +336 4B 45 384 383 - 03 - shy


Table 8

151

APPENDIX A








the following table










pages

152





SIGw SIGR VP Mpt)





RE~D(5582JAH


Kl K2~)

COMPLEXJ

153








Kl=Kl1 GO TO 322

321 K2K22 Kl=K12


154



408 WRITE(6613 GO TO 409







K l=K 11 GO TO 422

421 K2=K22 Kl=K12





155


418 WRITE(6614) GO TO 419


XX=SIGW2-075V2AW2 IF(XX)921800800







Kl=Kl1 GO TO 522

521 1lt2=K2Z Kl=K12





156

2 V=Vl 4 V=V+V INCR





04=D22-403 IFCD4)102351351


601 IF(K21-10602~0263C


Kl=Kll GO TO 622

621 K2=K22 Kl=K12





c

C

157





158

APPENDIX B















(b)




159

(c)

( d)





gives


f

where (f)

and ( g)


by

2 2 ~ =Jcsy - CS f (h)

f [3



160




Eyp = E yy (S

- i (k)


~

~


(n)











161

BIBLIOGRAPHY












162









x

v

p applied load






shear force

plastic shear force

w thickness of web












vi








~ normal stress

yield stress




shear stress





vii

LIST OF FIGURES

Page




















viii

Page




















ix

Page
























x

Page




xi

LIST OF TABLES

Page










SUJn1[ARY

ACKNOWLEDGEMENTS

NOTATION

LIST OF FIGURES

LIST OF TABLES

TABLE OF CONTENTS

Page

ii

iii

iv

vii

xi











31 Assumptions 27

















63 other Factors 72




73 Failure Loads 80



81 Conclusions 85



FIGURES 93

TABLES 145



BIBLIOGRAPHY 161

1

CHAPTER I

INTRODUCTION






















2


be increased













stresses


stresses








3


























4














given in Figure lb











5


























6


























7

--~~

















the problem

8

CHAPTER 11























9





is the yield stress
















10





















11





Figure 4





















12






hardening range(ll)







is reduced to

(4)



v = w( d-2t-2h) $-3





13




















(Vv) = d-2t-2h (6)P max d-2t




14





this task











increase in load









15






















decreases




16


























17
























18

























19

























20

























was offered by Chen

21





















the case




22








this effect

















23
























24

























25
























26









27

CHAPTER III


31 Assumptions








7b




dictated









28

























29







forcing is attached









section (2) gives






30


yields

(8)



(9)




stresses

Fl = Af ~yf + SW(1-2~)G + Ar ~yr (10)









31


calculated from

t2 _ r- 2 3(V)2 ~ - yw-4sw (12)
















given by

32




f yf





mined from

(17)













33








(13) and (18) yield


- (d-2h)]





34












(22)


(24)


(25)

35











become





(28)

(29)

36





















37







is written as

d-2t-2h (6)= d-2t











38






















(30)

39




(31)


given by

A =2 ( 32) r 13















40














reached

41

CHAPTER IV

APPROXIMATE METHOD

41 General Remarks





















42






(33)





Xf


by

(35 )




43




(36)











IS = If f)Af 3 r ( 37 ) - (s2 + 4a2 )

T





44



becomes

M (39 )M= p AW

1 + 4A f


(40)













by equation (38)


45





- AW(l 2h) flA (41)r - T -a-~




ityas given by

V 2hr = (I-a-) (42) p



Ar 1 - A

M fr= A p 1 + w







46












rewri tten as

M A (45 ) M=

p 1 + w4Af






47







previous section

















(43) then becomes

48

A 2h l1 - ~(l-a)J~

M ( 46)~= A 1 + W


aw 1 - 73Af

= ---1-shyA 1 + w
















49








this capacity
















(49a)

(50)

50

A q (5la )


Mp Aw 1 + 1iA

f


simplified to

(49)

(51)







Case I


if


given by

V (1 _ ~h) (42)Vp =

51


is Ar

1- ri- = Af (43)

p 1 + w 41f


Case 11



by

(40)


acting is

1 1 + w



Case Ill



52



follows

Case IlIA

For (48)


Ar 1 - I - ~

~ = __f_~_ ill A

P 1+ w4If


(38)

Case IIIB

For A2 gt (aw) 2

+ (~)2w2 (48)r 3 ~



A1 + w

4Af

53


+l (50)s

(49)




















54







these reduoe to

M AW 2h 1

1 - 2If(l-or~ r= A (52)

p 1 + w4If

v (2h)JTr = I-d~ p






given by

111 Z - wh2 (54)r = z

p




55

MM = 1 (55) p





56

CHAPTER V























57
























requires that

58

wh2 gt- (56)2h + 2u + q




















codes



59
























60

























61

























62
























63

















frame







64

























65
















the test beams

66

CHAPTER VI























67

























68

























69

























70


























71
























72






Figure 31












63 Other Factors






73

























74








shear


















75
























76



77

CHAPTER VII

ANALYSIS OF RESULTS






















78
























79

























80











73 Failure Loads













81

























82
























83













in Figure 54












84















85

CHAPTER VIII


81 Conclusions



drawn
















lateral buckling


86














opening






distributions





87

























88







force

















89

























go
























91

























92













93

1)

) )J + 1 +1 I I

I

( a) (b)

I + --+--~+ -f--- -shy

I (c) (d)

II I I

I I j+shy

) 1)I l J I

(e) ( f)


Figure 1

94




Figure 2

Ql ID Q)

Jt p ID

er

I I I I I I I



Figure 3

---------- -----

95


10~----~==~==~L---------------------~

~


10 vVp


Figure 4

-- -----------

96


(a)

~ o

r-I


(b)


Figure 5

97

1O~------------------------------------------~

l I


shy

I r perforated beam

no interaction


I I


Figure 6

i

98

1 CD (2) TIT

L

( a)

b

d2

(b)


Figure 7

99

( a)

(S r CSyr+



Case I - Low Shear

+

CSyr ltS shy

direct shear



(b)


Figure 8

100

low highshear shear


I I r I I

I I I 1 I10


11

11 I I 11 I I



11~I11

i I I

I (d-2h-2t)I_~ I

d-2t 11

(1 -~) q


Figure 9

101



03~

CH 125 1412700

0313 =--9F 0375 =Jb


04 vVp


Figure 10

102


x e 3middot

Reinforcing

1412 11

10


04


Figure 11

103


0513No Reinforcing

MMp

10


04 VVp


Figure 12

104



I I I

04


Figure 13

105


0513

1412



04


Figure 14

106

10WF2l


CH h

55Q 125 990 025Q 0250~



I I I

04


Figure 15

107

43 11 39 18 0 (

I

I I i lA

26

I

111 99 I- middot1 t

22 2-S 45 J

) C

r I

2A

13 ) lA I ~

I)II45 35 ( ()

I

I

I If 2B

f 16~ 9 25 I2t~ 21-+1 1 1 i

30 11 24 27 0- 10- ro- )

I ~ I I

20

2119YlO 151( 21~fI211~ - rr I I



Figure 16

108

tI

2D

22tl 231 45 tlI

( l (

I

3A

3119 Yt 11~ 34ft J3n~

45 25 J 35 ) (~ i-

I

I 1 3B

H 25 ft2ft5 cI 16 J2++ I I1 I

t 0- 23 45 J0- 0- (

I 4A

~5 ( 32 J211~ 2 YYt -IIII



Figure 17

log

45 ~ 25 5S ) ~ (

4B

2 crYI 15 25 J-I2++ Irr

L J30 24 27 ~lt) (P-L

I 5A

2tYY 13 26211 1 I

22 6

6A

i



Figure 18

--

110

If JL

a (1) p 02 ~

Cfl

tlOO ~ r-I 0 0 Q)

m ~ ~

Q SoOM


r-shy

~ ishy fO I- It

111

o N

-()

112

~ --- ~

~

~~ ~~

II I of

rf

A tshyshy I

1


a1 +gt

- -l~ 0 ()

-t I H CJ

4gt 4gt bO ~

~ ~ Cl -rt

fiI

~ a1 ~

+gt CJI

~ amp ~ ~ ~ --- ~ I

II s I

~CJ~ 4t ~

I~ i ~

o

i-~ ~ - l

II I +1 I Ij

~I r

I tf~ft N

- shy

stress 8 1


( a)

F F-CL F-E [(4 ~

W-H- - shy

strain ( b)



I

114

lA 2A 2B

20 2D 3A

3B 4A 4B

5A 6A

FIGURE 24

115

2B 20 2D 3A 3B 4At 4B


20 5A


FIGURE 25

116

2A 4A 2B 4B


2B 3B


FIGURE 26

bullbull

117


r ~ 3

o

top of reinf



~ bull




Figure 27

118

boundaryof opening

-70 0 __ tQ



-1iCgt

~I--=-shyoo---~-=-~ogt---- ---0 - ____ A~Sshy-


oE


Figure 28

119

-40

-lt0 lt----- shylow


lt 0


edge ltgtgt

to


high moment

edge

o


Figure 29

120


o10 D



ibull

boundaryof opening

10 Igt lO



Figure 30

bull bull bull bull

121

11 9 ~ oS j j ~

boundaryof opening

-lt10 -20 lt) -I-Lo Q lO 10



boundaryof openin

-40 -20 0 ZO o 10 ZQ 30



Figure 31

122

Test Beam - lA


Figure 32

Test Beam - 2A


Figure 33

123


Test Beam - 2B


Figure 34




Test Beam - 20


Figure 35

124


first yielding

Test Beam - 2D


Figure 36

Test Beam - 3A


Figure 37

125


---- first yielding

O6in

Figure 38

O6in

Figure 39


Test Beam - 3B


Test Beam - 4A


126

430k--shy

k445--shy



06in

Figure 40

O6in

Figure 41


Test Beam - 4B


Test Beam - 5A


127

k45 o-~--

Test Beam - 6A


Figure 42

128

10 shy

10WF21



~ ~

+~ ~ ~


I I

04


Figure 43

curve yields

129






bull


Figure 44

130




04


Figure 45

10

131



~

1 +3A



04


Figure 46

10

132



~ ~ ~ ~ ~

I I


04


Figure 47

10

133

14WF38 H


-~


~ ~ +4B

~ ~ ~


I I

04


Figure 48

134




~ ~

SAl


04 vVp


Figure 49

10

135



+ 6A


04


Figure 50

136

shear (kips)

70

60


50

40

30

20


40 80 120 160 200 240



Figure 51

137

shear (kips)

70

60


50

40

30

20

10

4 8 12 16 20 24 28



Figure 52

138

D f



Figure 53

139

l4WF38 7 x lot

Opening

3shy10 f ~--2f x e

Reinforcing

If x ~ Reinforcing

+3A +2A

No Reinforcing + 6A

04


Figure 54

140


+ 2B

+20

+2A

+2D

04 vVp


Figure 55

141


10 ---------- shy

+3B

+3A

04 vVp


Figure 56

142


+ 4B

+ 4A

04 vVp


Figure 57

143


7 x lot Opening

7 x 14 Opening---

04


Figure 58

144shy

14WF38



+5A

04


Figure 59

--

r b --1 t l=Ii= W

d

u q


11

0244 875 550 N

0260 0253 2720 N

400 316

2A 14WF38 22 1413 677 0496 0326 1046 700 0383 0381 2813 300 58



It 11 It If3B 45 1422 668 0490 0305 1045 0329 0361 5230



foo J1

146













4A web W 565 549 flange F 11 487 476



F-E ff 438 425


Table 2

147






k 567k 483k



Table 3

- - - - - - -

- - - - - - - - -

- -


lA 2A 2B 20 2D 3A 3B ~ lI Ilt v 162 500 400 583 700 525 433

180 - - - 725 - 517 191 550 417 617 600 550 567 191 575 442 567 575 600 433

- 575 467 567 575 600 433

- 575 - - 600 600 shy202 625 475 600 600 700 533

- 625 475 600 625 700 583

- - - 617 700 - 550

- - - 600 675 675 550

- - - - - 675 shy

- 600 458 650 675 650 583

- - - - 725 - -- - - - - 725 -- - - - - 675 -- - - - - - -- - - - - 675 shy- - 483 - - - shy

- 700 517 703 825 775 shy- - - - 825 - 600

4A 4B Ilt Ilt

450 333 500 417 450 367 450 366 500 383 450 417 550 483 600 483 550 400 550 433 625 shy625 shy575 467 575 shy500 shy550 417

- 450 500 383

--675 513

5A le

350 500 400 400 433 483 433 467 500 417

--

500

--

467 500

---

517

-

6A

400 533

-367 367 383 433 533

-----------

550

--


J-ITable 4 ~ 00

149


lA 10WF21 43 180K 746K 0241 0235 + 26 2A 14WF38 22 532 1L021 0520 0466 +116 2B 11 45 413 0404 0363 +113 2C If 30 548 1301 0421 0403 + 45



Table 5


lA 10WF21 43 180K 746 K

0241 0254 - 51 2A 14VlF38 22 532 1021 0520 0503 + 34

It2B 45 413 0404 0344 +175 If2C 30 548 1301 0421 0414 + 17

It2D 17 670 0515 0505 + 20 3A 22 575 1021 0562 0505 +103

n3B 45 497 1301 0382 0377 + 13 4A tI 22 572 n 0440 0463 - 50

It It 03094B 45 430 0330 + 68 5A 30 445 tr 0342 0353 - 31 6A 22 450 tt 0346 0257 +346


Table 6

150



2D 17 670 625 - 67 - shy3B 45 497 483 - 28 600 +207 4B 45 430 383 -109 - shy


Table 7



2C 30n 525k 567k + 80 633k +206 2D 17 546 625 +145 - shy3B 45 449 483 + 76 600 +336 4B 45 384 383 - 03 - shy


Table 8

151

APPENDIX A








the following table










pages

152





SIGw SIGR VP Mpt)





RE~D(5582JAH


Kl K2~)

COMPLEXJ

153








Kl=Kl1 GO TO 322

321 K2K22 Kl=K12


154



408 WRITE(6613 GO TO 409







K l=K 11 GO TO 422

421 K2=K22 Kl=K12





155


418 WRITE(6614) GO TO 419


XX=SIGW2-075V2AW2 IF(XX)921800800







Kl=Kl1 GO TO 522

521 1lt2=K2Z Kl=K12





156

2 V=Vl 4 V=V+V INCR





04=D22-403 IFCD4)102351351


601 IF(K21-10602~0263C


Kl=Kll GO TO 622

621 K2=K22 Kl=K12





c

C

157





158

APPENDIX B















(b)




159

(c)

( d)





gives


f

where (f)

and ( g)


by

2 2 ~ =Jcsy - CS f (h)

f [3



160




Eyp = E yy (S

- i (k)


~

~


(n)











161

BIBLIOGRAPHY












162









vi








~ normal stress

yield stress




shear stress





vii

LIST OF FIGURES

Page




















viii

Page




















ix

Page
























x

Page




xi

LIST OF TABLES

Page










SUJn1[ARY

ACKNOWLEDGEMENTS

NOTATION

LIST OF FIGURES

LIST OF TABLES

TABLE OF CONTENTS

Page

ii

iii

iv

vii

xi











31 Assumptions 27

















63 other Factors 72




73 Failure Loads 80



81 Conclusions 85



FIGURES 93

TABLES 145



BIBLIOGRAPHY 161

1

CHAPTER I

INTRODUCTION






















2


be increased













stresses


stresses








3


























4














given in Figure lb











5


























6


























7

--~~

















the problem

8

CHAPTER 11























9





is the yield stress
















10





















11





Figure 4





















12






hardening range(ll)







is reduced to

(4)



v = w( d-2t-2h) $-3





13




















(Vv) = d-2t-2h (6)P max d-2t




14





this task











increase in load









15






















decreases




16


























17
























18

























19

























20

























was offered by Chen

21





















the case




22








this effect

















23
























24

























25
























26









27

CHAPTER III


31 Assumptions








7b




dictated









28

























29







forcing is attached









section (2) gives






30


yields

(8)



(9)




stresses

Fl = Af ~yf + SW(1-2~)G + Ar ~yr (10)









31


calculated from

t2 _ r- 2 3(V)2 ~ - yw-4sw (12)
















given by

32




f yf





mined from

(17)













33








(13) and (18) yield


- (d-2h)]





34












(22)


(24)


(25)

35











become





(28)

(29)

36





















37







is written as

d-2t-2h (6)= d-2t











38






















(30)

39




(31)


given by

A =2 ( 32) r 13















40














reached

41

CHAPTER IV

APPROXIMATE METHOD

41 General Remarks





















42






(33)





Xf


by

(35 )




43




(36)











IS = If f)Af 3 r ( 37 ) - (s2 + 4a2 )

T





44



becomes

M (39 )M= p AW

1 + 4A f


(40)













by equation (38)


45





- AW(l 2h) flA (41)r - T -a-~




ityas given by

V 2hr = (I-a-) (42) p



Ar 1 - A

M fr= A p 1 + w







46












rewri tten as

M A (45 ) M=

p 1 + w4Af






47







previous section

















(43) then becomes

48

A 2h l1 - ~(l-a)J~

M ( 46)~= A 1 + W


aw 1 - 73Af

= ---1-shyA 1 + w
















49








this capacity
















(49a)

(50)

50

A q (5la )


Mp Aw 1 + 1iA

f


simplified to

(49)

(51)







Case I


if


given by

V (1 _ ~h) (42)Vp =

51


is Ar

1- ri- = Af (43)

p 1 + w 41f


Case 11



by

(40)


acting is

1 1 + w



Case Ill



52



follows

Case IlIA

For (48)


Ar 1 - I - ~

~ = __f_~_ ill A

P 1+ w4If


(38)

Case IIIB

For A2 gt (aw) 2

+ (~)2w2 (48)r 3 ~



A1 + w

4Af

53


+l (50)s

(49)




















54







these reduoe to

M AW 2h 1

1 - 2If(l-or~ r= A (52)

p 1 + w4If

v (2h)JTr = I-d~ p






given by

111 Z - wh2 (54)r = z

p




55

MM = 1 (55) p





56

CHAPTER V























57
























requires that

58

wh2 gt- (56)2h + 2u + q




















codes



59
























60

























61

























62
























63

















frame







64

























65
















the test beams

66

CHAPTER VI























67

























68

























69

























70


























71
























72






Figure 31












63 Other Factors






73

























74








shear


















75
























76



77

CHAPTER VII

ANALYSIS OF RESULTS






















78
























79

























80











73 Failure Loads













81

























82
























83













in Figure 54












84















85

CHAPTER VIII


81 Conclusions



drawn
















lateral buckling


86














opening






distributions





87

























88







force

















89

























go
























91

























92













93

1)

) )J + 1 +1 I I

I

( a) (b)

I + --+--~+ -f--- -shy

I (c) (d)

II I I

I I j+shy

) 1)I l J I

(e) ( f)


Figure 1

94




Figure 2

Ql ID Q)

Jt p ID

er

I I I I I I I



Figure 3

---------- -----

95


10~----~==~==~L---------------------~

~


10 vVp


Figure 4

-- -----------

96


(a)

~ o

r-I


(b)


Figure 5

97

1O~------------------------------------------~

l I


shy

I r perforated beam

no interaction


I I


Figure 6

i

98

1 CD (2) TIT

L

( a)

b

d2

(b)


Figure 7

99

( a)

(S r CSyr+



Case I - Low Shear

+

CSyr ltS shy

direct shear



(b)


Figure 8

100

low highshear shear


I I r I I

I I I 1 I10


11

11 I I 11 I I



11~I11

i I I

I (d-2h-2t)I_~ I

d-2t 11

(1 -~) q


Figure 9

101



03~

CH 125 1412700

0313 =--9F 0375 =Jb


04 vVp


Figure 10

102


x e 3middot

Reinforcing

1412 11

10


04


Figure 11

103


0513No Reinforcing

MMp

10


04 VVp


Figure 12

104



I I I

04


Figure 13

105


0513

1412



04


Figure 14

106

10WF2l


CH h

55Q 125 990 025Q 0250~



I I I

04


Figure 15

107

43 11 39 18 0 (

I

I I i lA

26

I

111 99 I- middot1 t

22 2-S 45 J

) C

r I

2A

13 ) lA I ~

I)II45 35 ( ()

I

I

I If 2B

f 16~ 9 25 I2t~ 21-+1 1 1 i

30 11 24 27 0- 10- ro- )

I ~ I I

20

2119YlO 151( 21~fI211~ - rr I I



Figure 16

108

tI

2D

22tl 231 45 tlI

( l (

I

3A

3119 Yt 11~ 34ft J3n~

45 25 J 35 ) (~ i-

I

I 1 3B

H 25 ft2ft5 cI 16 J2++ I I1 I

t 0- 23 45 J0- 0- (

I 4A

~5 ( 32 J211~ 2 YYt -IIII



Figure 17

log

45 ~ 25 5S ) ~ (

4B

2 crYI 15 25 J-I2++ Irr

L J30 24 27 ~lt) (P-L

I 5A

2tYY 13 26211 1 I

22 6

6A

i



Figure 18

--

110

If JL

a (1) p 02 ~

Cfl

tlOO ~ r-I 0 0 Q)

m ~ ~

Q SoOM


r-shy

~ ishy fO I- It

111

o N

-()

112

~ --- ~

~

~~ ~~

II I of

rf

A tshyshy I

1


a1 +gt

- -l~ 0 ()

-t I H CJ

4gt 4gt bO ~

~ ~ Cl -rt

fiI

~ a1 ~

+gt CJI

~ amp ~ ~ ~ --- ~ I

II s I

~CJ~ 4t ~

I~ i ~

o

i-~ ~ - l

II I +1 I Ij

~I r

I tf~ft N

- shy

stress 8 1


( a)

F F-CL F-E [(4 ~

W-H- - shy

strain ( b)



I

114

lA 2A 2B

20 2D 3A

3B 4A 4B

5A 6A

FIGURE 24

115

2B 20 2D 3A 3B 4At 4B


20 5A


FIGURE 25

116

2A 4A 2B 4B


2B 3B


FIGURE 26

bullbull

117


r ~ 3

o

top of reinf



~ bull




Figure 27

118

boundaryof opening

-70 0 __ tQ



-1iCgt

~I--=-shyoo---~-=-~ogt---- ---0 - ____ A~Sshy-


oE


Figure 28

119

-40

-lt0 lt----- shylow


lt 0


edge ltgtgt

to


high moment

edge

o


Figure 29

120


o10 D



ibull

boundaryof opening

10 Igt lO



Figure 30

bull bull bull bull

121

11 9 ~ oS j j ~

boundaryof opening

-lt10 -20 lt) -I-Lo Q lO 10



boundaryof openin

-40 -20 0 ZO o 10 ZQ 30



Figure 31

122

Test Beam - lA


Figure 32

Test Beam - 2A


Figure 33

123


Test Beam - 2B


Figure 34




Test Beam - 20


Figure 35

124


first yielding

Test Beam - 2D


Figure 36

Test Beam - 3A


Figure 37

125


---- first yielding

O6in

Figure 38

O6in

Figure 39


Test Beam - 3B


Test Beam - 4A


126

430k--shy

k445--shy



06in

Figure 40

O6in

Figure 41


Test Beam - 4B


Test Beam - 5A


127

k45 o-~--

Test Beam - 6A


Figure 42

128

10 shy

10WF21



~ ~

+~ ~ ~


I I

04


Figure 43

curve yields

129






bull


Figure 44

130




04


Figure 45

10

131



~

1 +3A



04


Figure 46

10

132



~ ~ ~ ~ ~

I I


04


Figure 47

10

133

14WF38 H


-~


~ ~ +4B

~ ~ ~


I I

04


Figure 48

134




~ ~

SAl


04 vVp


Figure 49

10

135



+ 6A


04


Figure 50

136

shear (kips)

70

60


50

40

30

20


40 80 120 160 200 240



Figure 51

137

shear (kips)

70

60


50

40

30

20

10

4 8 12 16 20 24 28



Figure 52

138

D f



Figure 53

139

l4WF38 7 x lot

Opening

3shy10 f ~--2f x e

Reinforcing

If x ~ Reinforcing

+3A +2A

No Reinforcing + 6A

04


Figure 54

140


+ 2B

+20

+2A

+2D

04 vVp


Figure 55

141


10 ---------- shy

+3B

+3A

04 vVp


Figure 56

142


+ 4B

+ 4A

04 vVp


Figure 57

143


7 x lot Opening

7 x 14 Opening---

04


Figure 58

144shy

14WF38



+5A

04


Figure 59

--

r b --1 t l=Ii= W

d

u q


11

0244 875 550 N

0260 0253 2720 N

400 316

2A 14WF38 22 1413 677 0496 0326 1046 700 0383 0381 2813 300 58



It 11 It If3B 45 1422 668 0490 0305 1045 0329 0361 5230



foo J1

146













4A web W 565 549 flange F 11 487 476



F-E ff 438 425


Table 2

147






k 567k 483k



Table 3

- - - - - - -

- - - - - - - - -

- -


lA 2A 2B 20 2D 3A 3B ~ lI Ilt v 162 500 400 583 700 525 433

180 - - - 725 - 517 191 550 417 617 600 550 567 191 575 442 567 575 600 433

- 575 467 567 575 600 433

- 575 - - 600 600 shy202 625 475 600 600 700 533

- 625 475 600 625 700 583

- - - 617 700 - 550

- - - 600 675 675 550

- - - - - 675 shy

- 600 458 650 675 650 583

- - - - 725 - -- - - - - 725 -- - - - - 675 -- - - - - - -- - - - - 675 shy- - 483 - - - shy

- 700 517 703 825 775 shy- - - - 825 - 600

4A 4B Ilt Ilt

450 333 500 417 450 367 450 366 500 383 450 417 550 483 600 483 550 400 550 433 625 shy625 shy575 467 575 shy500 shy550 417

- 450 500 383

--675 513

5A le

350 500 400 400 433 483 433 467 500 417

--

500

--

467 500

---

517

-

6A

400 533

-367 367 383 433 533

-----------

550

--


J-ITable 4 ~ 00

149


lA 10WF21 43 180K 746K 0241 0235 + 26 2A 14WF38 22 532 1L021 0520 0466 +116 2B 11 45 413 0404 0363 +113 2C If 30 548 1301 0421 0403 + 45



Table 5


lA 10WF21 43 180K 746 K

0241 0254 - 51 2A 14VlF38 22 532 1021 0520 0503 + 34

It2B 45 413 0404 0344 +175 If2C 30 548 1301 0421 0414 + 17

It2D 17 670 0515 0505 + 20 3A 22 575 1021 0562 0505 +103

n3B 45 497 1301 0382 0377 + 13 4A tI 22 572 n 0440 0463 - 50

It It 03094B 45 430 0330 + 68 5A 30 445 tr 0342 0353 - 31 6A 22 450 tt 0346 0257 +346


Table 6

150



2D 17 670 625 - 67 - shy3B 45 497 483 - 28 600 +207 4B 45 430 383 -109 - shy


Table 7



2C 30n 525k 567k + 80 633k +206 2D 17 546 625 +145 - shy3B 45 449 483 + 76 600 +336 4B 45 384 383 - 03 - shy


Table 8

151

APPENDIX A








the following table










pages

152





SIGw SIGR VP Mpt)





RE~D(5582JAH


Kl K2~)

COMPLEXJ

153








Kl=Kl1 GO TO 322

321 K2K22 Kl=K12


154



408 WRITE(6613 GO TO 409







K l=K 11 GO TO 422

421 K2=K22 Kl=K12





155


418 WRITE(6614) GO TO 419


XX=SIGW2-075V2AW2 IF(XX)921800800







Kl=Kl1 GO TO 522

521 1lt2=K2Z Kl=K12





156

2 V=Vl 4 V=V+V INCR





04=D22-403 IFCD4)102351351


601 IF(K21-10602~0263C


Kl=Kll GO TO 622

621 K2=K22 Kl=K12





c

C

157





158

APPENDIX B















(b)




159

(c)

( d)





gives


f

where (f)

and ( g)


by

2 2 ~ =Jcsy - CS f (h)

f [3



160




Eyp = E yy (S

- i (k)


~

~


(n)











161

BIBLIOGRAPHY












162









vii

LIST OF FIGURES

Page




















viii

Page




















ix

Page
























x

Page




xi

LIST OF TABLES

Page










SUJn1[ARY

ACKNOWLEDGEMENTS

NOTATION

LIST OF FIGURES

LIST OF TABLES

TABLE OF CONTENTS

Page

ii

iii

iv

vii

xi











31 Assumptions 27

















63 other Factors 72




73 Failure Loads 80



81 Conclusions 85



FIGURES 93

TABLES 145



BIBLIOGRAPHY 161

1

CHAPTER I

INTRODUCTION






















2


be increased













stresses


stresses








3


























4














given in Figure lb











5


























6


























7

--~~

















the problem

8

CHAPTER 11























9





is the yield stress
















10





















11





Figure 4





















12






hardening range(ll)







is reduced to

(4)



v = w( d-2t-2h) $-3





13




















(Vv) = d-2t-2h (6)P max d-2t




14





this task











increase in load









15






















decreases




16


























17
























18

























19

























20

























was offered by Chen

21





















the case




22








this effect

















23
























24

























25
























26









27

CHAPTER III


31 Assumptions








7b




dictated









28

























29







forcing is attached









section (2) gives






30


yields

(8)



(9)




stresses

Fl = Af ~yf + SW(1-2~)G + Ar ~yr (10)









31


calculated from

t2 _ r- 2 3(V)2 ~ - yw-4sw (12)
















given by

32




f yf





mined from

(17)













33








(13) and (18) yield


- (d-2h)]





34












(22)


(24)


(25)

35











become





(28)

(29)

36





















37







is written as

d-2t-2h (6)= d-2t











38






















(30)

39




(31)


given by

A =2 ( 32) r 13















40














reached

41

CHAPTER IV

APPROXIMATE METHOD

41 General Remarks





















42






(33)





Xf


by

(35 )




43




(36)











IS = If f)Af 3 r ( 37 ) - (s2 + 4a2 )

T





44



becomes

M (39 )M= p AW

1 + 4A f


(40)













by equation (38)


45





- AW(l 2h) flA (41)r - T -a-~




ityas given by

V 2hr = (I-a-) (42) p



Ar 1 - A

M fr= A p 1 + w







46












rewri tten as

M A (45 ) M=

p 1 + w4Af






47







previous section

















(43) then becomes

48

A 2h l1 - ~(l-a)J~

M ( 46)~= A 1 + W


aw 1 - 73Af

= ---1-shyA 1 + w
















49








this capacity
















(49a)

(50)

50

A q (5la )


Mp Aw 1 + 1iA

f


simplified to

(49)

(51)







Case I


if


given by

V (1 _ ~h) (42)Vp =

51


is Ar

1- ri- = Af (43)

p 1 + w 41f


Case 11



by

(40)


acting is

1 1 + w



Case Ill



52



follows

Case IlIA

For (48)


Ar 1 - I - ~

~ = __f_~_ ill A

P 1+ w4If


(38)

Case IIIB

For A2 gt (aw) 2

+ (~)2w2 (48)r 3 ~



A1 + w

4Af

53


+l (50)s

(49)




















54







these reduoe to

M AW 2h 1

1 - 2If(l-or~ r= A (52)

p 1 + w4If

v (2h)JTr = I-d~ p






given by

111 Z - wh2 (54)r = z

p




55

MM = 1 (55) p





56

CHAPTER V























57
























requires that

58

wh2 gt- (56)2h + 2u + q




















codes



59
























60

























61

























62
























63

















frame







64

























65
















the test beams

66

CHAPTER VI























67

























68

























69

























70


























71
























72






Figure 31












63 Other Factors






73

























74








shear


















75
























76



77

CHAPTER VII

ANALYSIS OF RESULTS






















78
























79

























80











73 Failure Loads













81

























82
























83













in Figure 54












84















85

CHAPTER VIII


81 Conclusions



drawn
















lateral buckling


86














opening






distributions





87

























88







force

















89

























go
























91

























92













93

1)

) )J + 1 +1 I I

I

( a) (b)

I + --+--~+ -f--- -shy

I (c) (d)

II I I

I I j+shy

) 1)I l J I

(e) ( f)


Figure 1

94




Figure 2

Ql ID Q)

Jt p ID

er

I I I I I I I



Figure 3

---------- -----

95


10~----~==~==~L---------------------~

~


10 vVp


Figure 4

-- -----------

96


(a)

~ o

r-I


(b)


Figure 5

97

1O~------------------------------------------~

l I


shy

I r perforated beam

no interaction


I I


Figure 6

i

98

1 CD (2) TIT

L

( a)

b

d2

(b)


Figure 7

99

( a)

(S r CSyr+



Case I - Low Shear

+

CSyr ltS shy

direct shear



(b)


Figure 8

100

low highshear shear


I I r I I

I I I 1 I10


11

11 I I 11 I I



11~I11

i I I

I (d-2h-2t)I_~ I

d-2t 11

(1 -~) q


Figure 9

101



03~

CH 125 1412700

0313 =--9F 0375 =Jb


04 vVp


Figure 10

102


x e 3middot

Reinforcing

1412 11

10


04


Figure 11

103


0513No Reinforcing

MMp

10


04 VVp


Figure 12

104



I I I

04


Figure 13

105


0513

1412



04


Figure 14

106

10WF2l


CH h

55Q 125 990 025Q 0250~



I I I

04


Figure 15

107

43 11 39 18 0 (

I

I I i lA

26

I

111 99 I- middot1 t

22 2-S 45 J

) C

r I

2A

13 ) lA I ~

I)II45 35 ( ()

I

I

I If 2B

f 16~ 9 25 I2t~ 21-+1 1 1 i

30 11 24 27 0- 10- ro- )

I ~ I I

20

2119YlO 151( 21~fI211~ - rr I I



Figure 16

108

tI

2D

22tl 231 45 tlI

( l (

I

3A

3119 Yt 11~ 34ft J3n~

45 25 J 35 ) (~ i-

I

I 1 3B

H 25 ft2ft5 cI 16 J2++ I I1 I

t 0- 23 45 J0- 0- (

I 4A

~5 ( 32 J211~ 2 YYt -IIII



Figure 17

log

45 ~ 25 5S ) ~ (

4B

2 crYI 15 25 J-I2++ Irr

L J30 24 27 ~lt) (P-L

I 5A

2tYY 13 26211 1 I

22 6

6A

i



Figure 18

--

110

If JL

a (1) p 02 ~

Cfl

tlOO ~ r-I 0 0 Q)

m ~ ~

Q SoOM


r-shy

~ ishy fO I- It

111

o N

-()

112

~ --- ~

~

~~ ~~

II I of

rf

A tshyshy I

1


a1 +gt

- -l~ 0 ()

-t I H CJ

4gt 4gt bO ~

~ ~ Cl -rt

fiI

~ a1 ~

+gt CJI

~ amp ~ ~ ~ --- ~ I

II s I

~CJ~ 4t ~

I~ i ~

o

i-~ ~ - l

II I +1 I Ij

~I r

I tf~ft N

- shy

stress 8 1


( a)

F F-CL F-E [(4 ~

W-H- - shy

strain ( b)



I

114

lA 2A 2B

20 2D 3A

3B 4A 4B

5A 6A

FIGURE 24

115

2B 20 2D 3A 3B 4At 4B


20 5A


FIGURE 25

116

2A 4A 2B 4B


2B 3B


FIGURE 26

bullbull

117


r ~ 3

o

top of reinf



~ bull




Figure 27

118

boundaryof opening

-70 0 __ tQ



-1iCgt

~I--=-shyoo---~-=-~ogt---- ---0 - ____ A~Sshy-


oE


Figure 28

119

-40

-lt0 lt----- shylow


lt 0


edge ltgtgt

to


high moment

edge

o


Figure 29

120


o10 D



ibull

boundaryof opening

10 Igt lO



Figure 30

bull bull bull bull

121

11 9 ~ oS j j ~

boundaryof opening

-lt10 -20 lt) -I-Lo Q lO 10



boundaryof openin

-40 -20 0 ZO o 10 ZQ 30



Figure 31

122

Test Beam - lA


Figure 32

Test Beam - 2A


Figure 33

123


Test Beam - 2B


Figure 34




Test Beam - 20


Figure 35

124


first yielding

Test Beam - 2D


Figure 36

Test Beam - 3A


Figure 37

125


---- first yielding

O6in

Figure 38

O6in

Figure 39


Test Beam - 3B


Test Beam - 4A


126

430k--shy

k445--shy



06in

Figure 40

O6in

Figure 41


Test Beam - 4B


Test Beam - 5A


127

k45 o-~--

Test Beam - 6A


Figure 42

128

10 shy

10WF21



~ ~

+~ ~ ~


I I

04


Figure 43

curve yields

129






bull


Figure 44

130




04


Figure 45

10

131



~

1 +3A



04


Figure 46

10

132



~ ~ ~ ~ ~

I I


04


Figure 47

10

133

14WF38 H


-~


~ ~ +4B

~ ~ ~


I I

04


Figure 48

134




~ ~

SAl


04 vVp


Figure 49

10

135



+ 6A


04


Figure 50

136

shear (kips)

70

60


50

40

30

20


40 80 120 160 200 240



Figure 51

137

shear (kips)

70

60


50

40

30

20

10

4 8 12 16 20 24 28



Figure 52

138

D f



Figure 53

139

l4WF38 7 x lot

Opening

3shy10 f ~--2f x e

Reinforcing

If x ~ Reinforcing

+3A +2A

No Reinforcing + 6A

04


Figure 54

140


+ 2B

+20

+2A

+2D

04 vVp


Figure 55

141


10 ---------- shy

+3B

+3A

04 vVp


Figure 56

142


+ 4B

+ 4A

04 vVp


Figure 57

143


7 x lot Opening

7 x 14 Opening---

04


Figure 58

144shy

14WF38



+5A

04


Figure 59

--

r b --1 t l=Ii= W

d

u q


11

0244 875 550 N

0260 0253 2720 N

400 316

2A 14WF38 22 1413 677 0496 0326 1046 700 0383 0381 2813 300 58



It 11 It If3B 45 1422 668 0490 0305 1045 0329 0361 5230



foo J1

146













4A web W 565 549 flange F 11 487 476



F-E ff 438 425


Table 2

147






k 567k 483k



Table 3

- - - - - - -

- - - - - - - - -

- -


lA 2A 2B 20 2D 3A 3B ~ lI Ilt v 162 500 400 583 700 525 433

180 - - - 725 - 517 191 550 417 617 600 550 567 191 575 442 567 575 600 433

- 575 467 567 575 600 433

- 575 - - 600 600 shy202 625 475 600 600 700 533

- 625 475 600 625 700 583

- - - 617 700 - 550

- - - 600 675 675 550

- - - - - 675 shy

- 600 458 650 675 650 583

- - - - 725 - -- - - - - 725 -- - - - - 675 -- - - - - - -- - - - - 675 shy- - 483 - - - shy

- 700 517 703 825 775 shy- - - - 825 - 600

4A 4B Ilt Ilt

450 333 500 417 450 367 450 366 500 383 450 417 550 483 600 483 550 400 550 433 625 shy625 shy575 467 575 shy500 shy550 417

- 450 500 383

--675 513

5A le

350 500 400 400 433 483 433 467 500 417

--

500

--

467 500

---

517

-

6A

400 533

-367 367 383 433 533

-----------

550

--


J-ITable 4 ~ 00

149


lA 10WF21 43 180K 746K 0241 0235 + 26 2A 14WF38 22 532 1L021 0520 0466 +116 2B 11 45 413 0404 0363 +113 2C If 30 548 1301 0421 0403 + 45



Table 5


lA 10WF21 43 180K 746 K

0241 0254 - 51 2A 14VlF38 22 532 1021 0520 0503 + 34

It2B 45 413 0404 0344 +175 If2C 30 548 1301 0421 0414 + 17

It2D 17 670 0515 0505 + 20 3A 22 575 1021 0562 0505 +103

n3B 45 497 1301 0382 0377 + 13 4A tI 22 572 n 0440 0463 - 50

It It 03094B 45 430 0330 + 68 5A 30 445 tr 0342 0353 - 31 6A 22 450 tt 0346 0257 +346


Table 6

150



2D 17 670 625 - 67 - shy3B 45 497 483 - 28 600 +207 4B 45 430 383 -109 - shy


Table 7



2C 30n 525k 567k + 80 633k +206 2D 17 546 625 +145 - shy3B 45 449 483 + 76 600 +336 4B 45 384 383 - 03 - shy


Table 8

151

APPENDIX A








the following table










pages

152





SIGw SIGR VP Mpt)





RE~D(5582JAH


Kl K2~)

COMPLEXJ

153








Kl=Kl1 GO TO 322

321 K2K22 Kl=K12


154



408 WRITE(6613 GO TO 409







K l=K 11 GO TO 422

421 K2=K22 Kl=K12





155


418 WRITE(6614) GO TO 419


XX=SIGW2-075V2AW2 IF(XX)921800800







Kl=Kl1 GO TO 522

521 1lt2=K2Z Kl=K12





156

2 V=Vl 4 V=V+V INCR





04=D22-403 IFCD4)102351351


601 IF(K21-10602~0263C


Kl=Kll GO TO 622

621 K2=K22 Kl=K12





c

C

157





158

APPENDIX B















(b)




159

(c)

( d)





gives


f

where (f)

and ( g)


by

2 2 ~ =Jcsy - CS f (h)

f [3



160




Eyp = E yy (S

- i (k)


~

~


(n)











161

BIBLIOGRAPHY












162









viii

Page




















ix

Page
























x

Page




xi

LIST OF TABLES

Page










SUJn1[ARY

ACKNOWLEDGEMENTS

NOTATION

LIST OF FIGURES

LIST OF TABLES

TABLE OF CONTENTS

Page

ii

iii

iv

vii

xi











31 Assumptions 27

















63 other Factors 72




73 Failure Loads 80



81 Conclusions 85



FIGURES 93

TABLES 145



BIBLIOGRAPHY 161

1

CHAPTER I

INTRODUCTION






















2


be increased













stresses


stresses








3


























4














given in Figure lb











5


























6


























7

--~~

















the problem

8

CHAPTER 11























9





is the yield stress
















10





















11





Figure 4





















12






hardening range(ll)







is reduced to

(4)



v = w( d-2t-2h) $-3





13




















(Vv) = d-2t-2h (6)P max d-2t




14





this task











increase in load









15






















decreases




16


























17
























18

























19

























20

























was offered by Chen

21





















the case




22








this effect

















23
























24

























25
























26









27

CHAPTER III


31 Assumptions








7b




dictated









28

























29







forcing is attached









section (2) gives






30


yields

(8)



(9)




stresses

Fl = Af ~yf + SW(1-2~)G + Ar ~yr (10)









31


calculated from

t2 _ r- 2 3(V)2 ~ - yw-4sw (12)
















given by

32




f yf





mined from

(17)













33








(13) and (18) yield


- (d-2h)]





34












(22)


(24)


(25)

35











become





(28)

(29)

36





















37







is written as

d-2t-2h (6)= d-2t











38






















(30)

39




(31)


given by

A =2 ( 32) r 13















40














reached

41

CHAPTER IV

APPROXIMATE METHOD

41 General Remarks





















42






(33)





Xf


by

(35 )




43




(36)











IS = If f)Af 3 r ( 37 ) - (s2 + 4a2 )

T





44



becomes

M (39 )M= p AW

1 + 4A f


(40)













by equation (38)


45





- AW(l 2h) flA (41)r - T -a-~




ityas given by

V 2hr = (I-a-) (42) p



Ar 1 - A

M fr= A p 1 + w







46












rewri tten as

M A (45 ) M=

p 1 + w4Af






47







previous section

















(43) then becomes

48

A 2h l1 - ~(l-a)J~

M ( 46)~= A 1 + W


aw 1 - 73Af

= ---1-shyA 1 + w
















49








this capacity
















(49a)

(50)

50

A q (5la )


Mp Aw 1 + 1iA

f


simplified to

(49)

(51)







Case I


if


given by

V (1 _ ~h) (42)Vp =

51


is Ar

1- ri- = Af (43)

p 1 + w 41f


Case 11



by

(40)


acting is

1 1 + w



Case Ill



52



follows

Case IlIA

For (48)


Ar 1 - I - ~

~ = __f_~_ ill A

P 1+ w4If


(38)

Case IIIB

For A2 gt (aw) 2

+ (~)2w2 (48)r 3 ~



A1 + w

4Af

53


+l (50)s

(49)




















54







these reduoe to

M AW 2h 1

1 - 2If(l-or~ r= A (52)

p 1 + w4If

v (2h)JTr = I-d~ p






given by

111 Z - wh2 (54)r = z

p




55

MM = 1 (55) p





56

CHAPTER V























57
























requires that

58

wh2 gt- (56)2h + 2u + q




















codes



59
























60

























61

























62
























63

















frame







64

























65
















the test beams

66

CHAPTER VI























67

























68

























69

























70


























71
























72






Figure 31












63 Other Factors






73

























74








shear


















75
























76



77

CHAPTER VII

ANALYSIS OF RESULTS






















78
























79

























80











73 Failure Loads













81

























82
























83













in Figure 54












84















85

CHAPTER VIII


81 Conclusions



drawn
















lateral buckling


86














opening






distributions





87

























88







force

















89

























go
























91

























92













93

1)

) )J + 1 +1 I I

I

( a) (b)

I + --+--~+ -f--- -shy

I (c) (d)

II I I

I I j+shy

) 1)I l J I

(e) ( f)


Figure 1

94




Figure 2

Ql ID Q)

Jt p ID

er

I I I I I I I



Figure 3

---------- -----

95


10~----~==~==~L---------------------~

~


10 vVp


Figure 4

-- -----------

96


(a)

~ o

r-I


(b)


Figure 5

97

1O~------------------------------------------~

l I


shy

I r perforated beam

no interaction


I I


Figure 6

i

98

1 CD (2) TIT

L

( a)

b

d2

(b)


Figure 7

99

( a)

(S r CSyr+



Case I - Low Shear

+

CSyr ltS shy

direct shear



(b)


Figure 8

100

low highshear shear


I I r I I

I I I 1 I10


11

11 I I 11 I I



11~I11

i I I

I (d-2h-2t)I_~ I

d-2t 11

(1 -~) q


Figure 9

101



03~

CH 125 1412700

0313 =--9F 0375 =Jb


04 vVp


Figure 10

102


x e 3middot

Reinforcing

1412 11

10


04


Figure 11

103


0513No Reinforcing

MMp

10


04 VVp


Figure 12

104



I I I

04


Figure 13

105


0513

1412



04


Figure 14

106

10WF2l


CH h

55Q 125 990 025Q 0250~



I I I

04


Figure 15

107

43 11 39 18 0 (

I

I I i lA

26

I

111 99 I- middot1 t

22 2-S 45 J

) C

r I

2A

13 ) lA I ~

I)II45 35 ( ()

I

I

I If 2B

f 16~ 9 25 I2t~ 21-+1 1 1 i

30 11 24 27 0- 10- ro- )

I ~ I I

20

2119YlO 151( 21~fI211~ - rr I I



Figure 16

108

tI

2D

22tl 231 45 tlI

( l (

I

3A

3119 Yt 11~ 34ft J3n~

45 25 J 35 ) (~ i-

I

I 1 3B

H 25 ft2ft5 cI 16 J2++ I I1 I

t 0- 23 45 J0- 0- (

I 4A

~5 ( 32 J211~ 2 YYt -IIII



Figure 17

log

45 ~ 25 5S ) ~ (

4B

2 crYI 15 25 J-I2++ Irr

L J30 24 27 ~lt) (P-L

I 5A

2tYY 13 26211 1 I

22 6

6A

i



Figure 18

--

110

If JL

a (1) p 02 ~

Cfl

tlOO ~ r-I 0 0 Q)

m ~ ~

Q SoOM


r-shy

~ ishy fO I- It

111

o N

-()

112

~ --- ~

~

~~ ~~

II I of

rf

A tshyshy I

1


a1 +gt

- -l~ 0 ()

-t I H CJ

4gt 4gt bO ~

~ ~ Cl -rt

fiI

~ a1 ~

+gt CJI

~ amp ~ ~ ~ --- ~ I

II s I

~CJ~ 4t ~

I~ i ~

o

i-~ ~ - l

II I +1 I Ij

~I r

I tf~ft N

- shy

stress 8 1


( a)

F F-CL F-E [(4 ~

W-H- - shy

strain ( b)



I

114

lA 2A 2B

20 2D 3A

3B 4A 4B

5A 6A

FIGURE 24

115

2B 20 2D 3A 3B 4At 4B


20 5A


FIGURE 25

116

2A 4A 2B 4B


2B 3B


FIGURE 26

bullbull

117


r ~ 3

o

top of reinf



~ bull




Figure 27

118

boundaryof opening

-70 0 __ tQ



-1iCgt

~I--=-shyoo---~-=-~ogt---- ---0 - ____ A~Sshy-


oE


Figure 28

119

-40

-lt0 lt----- shylow


lt 0


edge ltgtgt

to


high moment

edge

o


Figure 29

120


o10 D



ibull

boundaryof opening

10 Igt lO



Figure 30

bull bull bull bull

121

11 9 ~ oS j j ~

boundaryof opening

-lt10 -20 lt) -I-Lo Q lO 10



boundaryof openin

-40 -20 0 ZO o 10 ZQ 30



Figure 31

122

Test Beam - lA


Figure 32

Test Beam - 2A


Figure 33

123


Test Beam - 2B


Figure 34




Test Beam - 20


Figure 35

124


first yielding

Test Beam - 2D


Figure 36

Test Beam - 3A


Figure 37

125


---- first yielding

O6in

Figure 38

O6in

Figure 39


Test Beam - 3B


Test Beam - 4A


126

430k--shy

k445--shy



06in

Figure 40

O6in

Figure 41


Test Beam - 4B


Test Beam - 5A


127

k45 o-~--

Test Beam - 6A


Figure 42

128

10 shy

10WF21



~ ~

+~ ~ ~


I I

04


Figure 43

curve yields

129






bull


Figure 44

130




04


Figure 45

10

131



~

1 +3A



04


Figure 46

10

132



~ ~ ~ ~ ~

I I


04


Figure 47

10

133

14WF38 H


-~


~ ~ +4B

~ ~ ~


I I

04


Figure 48

134




~ ~

SAl


04 vVp


Figure 49

10

135



+ 6A


04


Figure 50

136

shear (kips)

70

60


50

40

30

20


40 80 120 160 200 240



Figure 51

137

shear (kips)

70

60


50

40

30

20

10

4 8 12 16 20 24 28



Figure 52

138

D f



Figure 53

139

l4WF38 7 x lot

Opening

3shy10 f ~--2f x e

Reinforcing

If x ~ Reinforcing

+3A +2A

No Reinforcing + 6A

04


Figure 54

140


+ 2B

+20

+2A

+2D

04 vVp


Figure 55

141


10 ---------- shy

+3B

+3A

04 vVp


Figure 56

142


+ 4B

+ 4A

04 vVp


Figure 57

143


7 x lot Opening

7 x 14 Opening---

04


Figure 58

144shy

14WF38



+5A

04


Figure 59

--

r b --1 t l=Ii= W

d

u q


11

0244 875 550 N

0260 0253 2720 N

400 316

2A 14WF38 22 1413 677 0496 0326 1046 700 0383 0381 2813 300 58



It 11 It If3B 45 1422 668 0490 0305 1045 0329 0361 5230



foo J1

146













4A web W 565 549 flange F 11 487 476



F-E ff 438 425


Table 2

147






k 567k 483k



Table 3

- - - - - - -

- - - - - - - - -

- -


lA 2A 2B 20 2D 3A 3B ~ lI Ilt v 162 500 400 583 700 525 433

180 - - - 725 - 517 191 550 417 617 600 550 567 191 575 442 567 575 600 433

- 575 467 567 575 600 433

- 575 - - 600 600 shy202 625 475 600 600 700 533

- 625 475 600 625 700 583

- - - 617 700 - 550

- - - 600 675 675 550

- - - - - 675 shy

- 600 458 650 675 650 583

- - - - 725 - -- - - - - 725 -- - - - - 675 -- - - - - - -- - - - - 675 shy- - 483 - - - shy

- 700 517 703 825 775 shy- - - - 825 - 600

4A 4B Ilt Ilt

450 333 500 417 450 367 450 366 500 383 450 417 550 483 600 483 550 400 550 433 625 shy625 shy575 467 575 shy500 shy550 417

- 450 500 383

--675 513

5A le

350 500 400 400 433 483 433 467 500 417

--

500

--

467 500

---

517

-

6A

400 533

-367 367 383 433 533

-----------

550

--


J-ITable 4 ~ 00

149


lA 10WF21 43 180K 746K 0241 0235 + 26 2A 14WF38 22 532 1L021 0520 0466 +116 2B 11 45 413 0404 0363 +113 2C If 30 548 1301 0421 0403 + 45



Table 5


lA 10WF21 43 180K 746 K

0241 0254 - 51 2A 14VlF38 22 532 1021 0520 0503 + 34

It2B 45 413 0404 0344 +175 If2C 30 548 1301 0421 0414 + 17

It2D 17 670 0515 0505 + 20 3A 22 575 1021 0562 0505 +103

n3B 45 497 1301 0382 0377 + 13 4A tI 22 572 n 0440 0463 - 50

It It 03094B 45 430 0330 + 68 5A 30 445 tr 0342 0353 - 31 6A 22 450 tt 0346 0257 +346


Table 6

150



2D 17 670 625 - 67 - shy3B 45 497 483 - 28 600 +207 4B 45 430 383 -109 - shy


Table 7



2C 30n 525k 567k + 80 633k +206 2D 17 546 625 +145 - shy3B 45 449 483 + 76 600 +336 4B 45 384 383 - 03 - shy


Table 8

151

APPENDIX A








the following table










pages

152





SIGw SIGR VP Mpt)





RE~D(5582JAH


Kl K2~)

COMPLEXJ

153








Kl=Kl1 GO TO 322

321 K2K22 Kl=K12


154



408 WRITE(6613 GO TO 409







K l=K 11 GO TO 422

421 K2=K22 Kl=K12





155


418 WRITE(6614) GO TO 419


XX=SIGW2-075V2AW2 IF(XX)921800800







Kl=Kl1 GO TO 522

521 1lt2=K2Z Kl=K12





156

2 V=Vl 4 V=V+V INCR





04=D22-403 IFCD4)102351351


601 IF(K21-10602~0263C


Kl=Kll GO TO 622

621 K2=K22 Kl=K12





c

C

157





158

APPENDIX B















(b)




159

(c)

( d)





gives


f

where (f)

and ( g)


by

2 2 ~ =Jcsy - CS f (h)

f [3



160




Eyp = E yy (S

- i (k)


~

~


(n)











161

BIBLIOGRAPHY












162









ix

Page
























x

Page




xi

LIST OF TABLES

Page










SUJn1[ARY

ACKNOWLEDGEMENTS

NOTATION

LIST OF FIGURES

LIST OF TABLES

TABLE OF CONTENTS

Page

ii

iii

iv

vii

xi











31 Assumptions 27

















63 other Factors 72




73 Failure Loads 80



81 Conclusions 85



FIGURES 93

TABLES 145



BIBLIOGRAPHY 161

1

CHAPTER I

INTRODUCTION






















2


be increased













stresses


stresses








3


























4














given in Figure lb











5


























6


























7

--~~

















the problem

8

CHAPTER 11























9





is the yield stress
















10





















11





Figure 4





















12






hardening range(ll)







is reduced to

(4)



v = w( d-2t-2h) $-3





13




















(Vv) = d-2t-2h (6)P max d-2t




14





this task











increase in load









15






















decreases




16


























17
























18

























19

























20

























was offered by Chen

21





















the case




22








this effect

















23
























24

























25
























26









27

CHAPTER III


31 Assumptions








7b




dictated









28

























29







forcing is attached









section (2) gives






30


yields

(8)



(9)




stresses

Fl = Af ~yf + SW(1-2~)G + Ar ~yr (10)









31


calculated from

t2 _ r- 2 3(V)2 ~ - yw-4sw (12)
















given by

32




f yf





mined from

(17)













33








(13) and (18) yield


- (d-2h)]





34












(22)


(24)


(25)

35











become





(28)

(29)

36





















37







is written as

d-2t-2h (6)= d-2t











38






















(30)

39




(31)


given by

A =2 ( 32) r 13















40














reached

41

CHAPTER IV

APPROXIMATE METHOD

41 General Remarks





















42






(33)





Xf


by

(35 )




43




(36)











IS = If f)Af 3 r ( 37 ) - (s2 + 4a2 )

T





44



becomes

M (39 )M= p AW

1 + 4A f


(40)













by equation (38)


45





- AW(l 2h) flA (41)r - T -a-~




ityas given by

V 2hr = (I-a-) (42) p



Ar 1 - A

M fr= A p 1 + w







46












rewri tten as

M A (45 ) M=

p 1 + w4Af






47







previous section

















(43) then becomes

48

A 2h l1 - ~(l-a)J~

M ( 46)~= A 1 + W


aw 1 - 73Af

= ---1-shyA 1 + w
















49








this capacity
















(49a)

(50)

50

A q (5la )


Mp Aw 1 + 1iA

f


simplified to

(49)

(51)







Case I


if


given by

V (1 _ ~h) (42)Vp =

51


is Ar

1- ri- = Af (43)

p 1 + w 41f


Case 11



by

(40)


acting is

1 1 + w



Case Ill



52



follows

Case IlIA

For (48)


Ar 1 - I - ~

~ = __f_~_ ill A

P 1+ w4If


(38)

Case IIIB

For A2 gt (aw) 2

+ (~)2w2 (48)r 3 ~



A1 + w

4Af

53


+l (50)s

(49)




















54







these reduoe to

M AW 2h 1

1 - 2If(l-or~ r= A (52)

p 1 + w4If

v (2h)JTr = I-d~ p






given by

111 Z - wh2 (54)r = z

p




55

MM = 1 (55) p





56

CHAPTER V























57
























requires that

58

wh2 gt- (56)2h + 2u + q




















codes



59
























60

























61

























62
























63

















frame







64

























65
















the test beams

66

CHAPTER VI























67

























68

























69

























70


























71
























72






Figure 31












63 Other Factors






73

























74








shear


















75
























76



77

CHAPTER VII

ANALYSIS OF RESULTS






















78
























79

























80











73 Failure Loads













81

























82
























83













in Figure 54












84















85

CHAPTER VIII


81 Conclusions



drawn
















lateral buckling


86














opening






distributions





87

























88







force

















89

























go
























91

























92













93

1)

) )J + 1 +1 I I

I

( a) (b)

I + --+--~+ -f--- -shy

I (c) (d)

II I I

I I j+shy

) 1)I l J I

(e) ( f)


Figure 1

94




Figure 2

Ql ID Q)

Jt p ID

er

I I I I I I I



Figure 3

---------- -----

95


10~----~==~==~L---------------------~

~


10 vVp


Figure 4

-- -----------

96


(a)

~ o

r-I


(b)


Figure 5

97

1O~------------------------------------------~

l I


shy

I r perforated beam

no interaction


I I


Figure 6

i

98

1 CD (2) TIT

L

( a)

b

d2

(b)


Figure 7

99

( a)

(S r CSyr+



Case I - Low Shear

+

CSyr ltS shy

direct shear



(b)


Figure 8

100

low highshear shear


I I r I I

I I I 1 I10


11

11 I I 11 I I



11~I11

i I I

I (d-2h-2t)I_~ I

d-2t 11

(1 -~) q


Figure 9

101



03~

CH 125 1412700

0313 =--9F 0375 =Jb


04 vVp


Figure 10

102


x e 3middot

Reinforcing

1412 11

10


04


Figure 11

103


0513No Reinforcing

MMp

10


04 VVp


Figure 12

104



I I I

04


Figure 13

105


0513

1412



04


Figure 14

106

10WF2l


CH h

55Q 125 990 025Q 0250~



I I I

04


Figure 15

107

43 11 39 18 0 (

I

I I i lA

26

I

111 99 I- middot1 t

22 2-S 45 J

) C

r I

2A

13 ) lA I ~

I)II45 35 ( ()

I

I

I If 2B

f 16~ 9 25 I2t~ 21-+1 1 1 i

30 11 24 27 0- 10- ro- )

I ~ I I

20

2119YlO 151( 21~fI211~ - rr I I



Figure 16

108

tI

2D

22tl 231 45 tlI

( l (

I

3A

3119 Yt 11~ 34ft J3n~

45 25 J 35 ) (~ i-

I

I 1 3B

H 25 ft2ft5 cI 16 J2++ I I1 I

t 0- 23 45 J0- 0- (

I 4A

~5 ( 32 J211~ 2 YYt -IIII



Figure 17

log

45 ~ 25 5S ) ~ (

4B

2 crYI 15 25 J-I2++ Irr

L J30 24 27 ~lt) (P-L

I 5A

2tYY 13 26211 1 I

22 6

6A

i



Figure 18

--

110

If JL

a (1) p 02 ~

Cfl

tlOO ~ r-I 0 0 Q)

m ~ ~

Q SoOM


r-shy

~ ishy fO I- It

111

o N

-()

112

~ --- ~

~

~~ ~~

II I of

rf

A tshyshy I

1


a1 +gt

- -l~ 0 ()

-t I H CJ

4gt 4gt bO ~

~ ~ Cl -rt

fiI

~ a1 ~

+gt CJI

~ amp ~ ~ ~ --- ~ I

II s I

~CJ~ 4t ~

I~ i ~

o

i-~ ~ - l

II I +1 I Ij

~I r

I tf~ft N

- shy

stress 8 1


( a)

F F-CL F-E [(4 ~

W-H- - shy

strain ( b)



I

114

lA 2A 2B

20 2D 3A

3B 4A 4B

5A 6A

FIGURE 24

115

2B 20 2D 3A 3B 4At 4B


20 5A


FIGURE 25

116

2A 4A 2B 4B


2B 3B


FIGURE 26

bullbull

117


r ~ 3

o

top of reinf



~ bull




Figure 27

118

boundaryof opening

-70 0 __ tQ



-1iCgt

~I--=-shyoo---~-=-~ogt---- ---0 - ____ A~Sshy-


oE


Figure 28

119

-40

-lt0 lt----- shylow


lt 0


edge ltgtgt

to


high moment

edge

o


Figure 29

120


o10 D



ibull

boundaryof opening

10 Igt lO



Figure 30

bull bull bull bull

121

11 9 ~ oS j j ~

boundaryof opening

-lt10 -20 lt) -I-Lo Q lO 10



boundaryof openin

-40 -20 0 ZO o 10 ZQ 30



Figure 31

122

Test Beam - lA


Figure 32

Test Beam - 2A


Figure 33

123


Test Beam - 2B


Figure 34




Test Beam - 20


Figure 35

124


first yielding

Test Beam - 2D


Figure 36

Test Beam - 3A


Figure 37

125


---- first yielding

O6in

Figure 38

O6in

Figure 39


Test Beam - 3B


Test Beam - 4A


126

430k--shy

k445--shy



06in

Figure 40

O6in

Figure 41


Test Beam - 4B


Test Beam - 5A


127

k45 o-~--

Test Beam - 6A


Figure 42

128

10 shy

10WF21



~ ~

+~ ~ ~


I I

04


Figure 43

curve yields

129






bull


Figure 44

130




04


Figure 45

10

131



~

1 +3A



04


Figure 46

10

132



~ ~ ~ ~ ~

I I


04


Figure 47

10

133

14WF38 H


-~


~ ~ +4B

~ ~ ~


I I

04


Figure 48

134




~ ~

SAl


04 vVp


Figure 49

10

135



+ 6A


04


Figure 50

136

shear (kips)

70

60


50

40

30

20


40 80 120 160 200 240



Figure 51

137

shear (kips)

70

60


50

40

30

20

10

4 8 12 16 20 24 28



Figure 52

138

D f



Figure 53

139

l4WF38 7 x lot

Opening

3shy10 f ~--2f x e

Reinforcing

If x ~ Reinforcing

+3A +2A

No Reinforcing + 6A

04


Figure 54

140


+ 2B

+20

+2A

+2D

04 vVp


Figure 55

141


10 ---------- shy

+3B

+3A

04 vVp


Figure 56

142


+ 4B

+ 4A

04 vVp


Figure 57

143


7 x lot Opening

7 x 14 Opening---

04


Figure 58

144shy

14WF38



+5A

04


Figure 59

--

r b --1 t l=Ii= W

d

u q


11

0244 875 550 N

0260 0253 2720 N

400 316

2A 14WF38 22 1413 677 0496 0326 1046 700 0383 0381 2813 300 58



It 11 It If3B 45 1422 668 0490 0305 1045 0329 0361 5230



foo J1

146













4A web W 565 549 flange F 11 487 476



F-E ff 438 425


Table 2

147






k 567k 483k



Table 3

- - - - - - -

- - - - - - - - -

- -


lA 2A 2B 20 2D 3A 3B ~ lI Ilt v 162 500 400 583 700 525 433

180 - - - 725 - 517 191 550 417 617 600 550 567 191 575 442 567 575 600 433

- 575 467 567 575 600 433

- 575 - - 600 600 shy202 625 475 600 600 700 533

- 625 475 600 625 700 583

- - - 617 700 - 550

- - - 600 675 675 550

- - - - - 675 shy

- 600 458 650 675 650 583

- - - - 725 - -- - - - - 725 -- - - - - 675 -- - - - - - -- - - - - 675 shy- - 483 - - - shy

- 700 517 703 825 775 shy- - - - 825 - 600

4A 4B Ilt Ilt

450 333 500 417 450 367 450 366 500 383 450 417 550 483 600 483 550 400 550 433 625 shy625 shy575 467 575 shy500 shy550 417

- 450 500 383

--675 513

5A le

350 500 400 400 433 483 433 467 500 417

--

500

--

467 500

---

517

-

6A

400 533

-367 367 383 433 533

-----------

550

--


J-ITable 4 ~ 00

149


lA 10WF21 43 180K 746K 0241 0235 + 26 2A 14WF38 22 532 1L021 0520 0466 +116 2B 11 45 413 0404 0363 +113 2C If 30 548 1301 0421 0403 + 45



Table 5


lA 10WF21 43 180K 746 K

0241 0254 - 51 2A 14VlF38 22 532 1021 0520 0503 + 34

It2B 45 413 0404 0344 +175 If2C 30 548 1301 0421 0414 + 17

It2D 17 670 0515 0505 + 20 3A 22 575 1021 0562 0505 +103

n3B 45 497 1301 0382 0377 + 13 4A tI 22 572 n 0440 0463 - 50

It It 03094B 45 430 0330 + 68 5A 30 445 tr 0342 0353 - 31 6A 22 450 tt 0346 0257 +346


Table 6

150



2D 17 670 625 - 67 - shy3B 45 497 483 - 28 600 +207 4B 45 430 383 -109 - shy


Table 7



2C 30n 525k 567k + 80 633k +206 2D 17 546 625 +145 - shy3B 45 449 483 + 76 600 +336 4B 45 384 383 - 03 - shy


Table 8

151

APPENDIX A








the following table










pages

152





SIGw SIGR VP Mpt)





RE~D(5582JAH


Kl K2~)

COMPLEXJ

153








Kl=Kl1 GO TO 322

321 K2K22 Kl=K12


154



408 WRITE(6613 GO TO 409







K l=K 11 GO TO 422

421 K2=K22 Kl=K12





155


418 WRITE(6614) GO TO 419


XX=SIGW2-075V2AW2 IF(XX)921800800







Kl=Kl1 GO TO 522

521 1lt2=K2Z Kl=K12





156

2 V=Vl 4 V=V+V INCR





04=D22-403 IFCD4)102351351


601 IF(K21-10602~0263C


Kl=Kll GO TO 622

621 K2=K22 Kl=K12





c

C

157





158

APPENDIX B















(b)




159

(c)

( d)





gives


f

where (f)

and ( g)


by

2 2 ~ =Jcsy - CS f (h)

f [3



160




Eyp = E yy (S

- i (k)


~

~


(n)











161

BIBLIOGRAPHY












162









x

Page




xi

LIST OF TABLES

Page










SUJn1[ARY

ACKNOWLEDGEMENTS

NOTATION

LIST OF FIGURES

LIST OF TABLES

TABLE OF CONTENTS

Page

ii

iii

iv

vii

xi











31 Assumptions 27

















63 other Factors 72




73 Failure Loads 80



81 Conclusions 85



FIGURES 93

TABLES 145



BIBLIOGRAPHY 161

1

CHAPTER I

INTRODUCTION






















2


be increased













stresses


stresses








3


























4














given in Figure lb











5


























6


























7

--~~

















the problem

8

CHAPTER 11























9





is the yield stress
















10





















11





Figure 4





















12






hardening range(ll)







is reduced to

(4)



v = w( d-2t-2h) $-3





13




















(Vv) = d-2t-2h (6)P max d-2t




14





this task











increase in load









15






















decreases




16


























17
























18

























19

























20

























was offered by Chen

21





















the case




22








this effect

















23
























24

























25
























26









27

CHAPTER III


31 Assumptions








7b




dictated









28

























29







forcing is attached









section (2) gives






30


yields

(8)



(9)




stresses

Fl = Af ~yf + SW(1-2~)G + Ar ~yr (10)









31


calculated from

t2 _ r- 2 3(V)2 ~ - yw-4sw (12)
















given by

32




f yf





mined from

(17)













33








(13) and (18) yield


- (d-2h)]





34












(22)


(24)


(25)

35











become





(28)

(29)

36





















37







is written as

d-2t-2h (6)= d-2t











38






















(30)

39




(31)


given by

A =2 ( 32) r 13















40














reached

41

CHAPTER IV

APPROXIMATE METHOD

41 General Remarks





















42






(33)





Xf


by

(35 )




43




(36)











IS = If f)Af 3 r ( 37 ) - (s2 + 4a2 )

T





44



becomes

M (39 )M= p AW

1 + 4A f


(40)













by equation (38)


45





- AW(l 2h) flA (41)r - T -a-~




ityas given by

V 2hr = (I-a-) (42) p



Ar 1 - A

M fr= A p 1 + w







46












rewri tten as

M A (45 ) M=

p 1 + w4Af






47







previous section

















(43) then becomes

48

A 2h l1 - ~(l-a)J~

M ( 46)~= A 1 + W


aw 1 - 73Af

= ---1-shyA 1 + w
















49








this capacity
















(49a)

(50)

50

A q (5la )


Mp Aw 1 + 1iA

f


simplified to

(49)

(51)







Case I


if


given by

V (1 _ ~h) (42)Vp =

51


is Ar

1- ri- = Af (43)

p 1 + w 41f


Case 11



by

(40)


acting is

1 1 + w



Case Ill



52



follows

Case IlIA

For (48)


Ar 1 - I - ~

~ = __f_~_ ill A

P 1+ w4If


(38)

Case IIIB

For A2 gt (aw) 2

+ (~)2w2 (48)r 3 ~



A1 + w

4Af

53


+l (50)s

(49)




















54







these reduoe to

M AW 2h 1

1 - 2If(l-or~ r= A (52)

p 1 + w4If

v (2h)JTr = I-d~ p






given by

111 Z - wh2 (54)r = z

p




55

MM = 1 (55) p





56

CHAPTER V























57
























requires that

58

wh2 gt- (56)2h + 2u + q




















codes



59
























60

























61

























62
























63

















frame







64

























65
















the test beams

66

CHAPTER VI























67

























68

























69

























70


























71
























72






Figure 31












63 Other Factors






73

























74








shear


















75
























76



77

CHAPTER VII

ANALYSIS OF RESULTS






















78
























79

























80











73 Failure Loads













81

























82
























83













in Figure 54












84















85

CHAPTER VIII


81 Conclusions



drawn
















lateral buckling


86














opening






distributions





87

























88







force

















89

























go
























91

























92













93

1)

) )J + 1 +1 I I

I

( a) (b)

I + --+--~+ -f--- -shy

I (c) (d)

II I I

I I j+shy

) 1)I l J I

(e) ( f)


Figure 1

94




Figure 2

Ql ID Q)

Jt p ID

er

I I I I I I I



Figure 3

---------- -----

95


10~----~==~==~L---------------------~

~


10 vVp


Figure 4

-- -----------

96


(a)

~ o

r-I


(b)


Figure 5

97

1O~------------------------------------------~

l I


shy

I r perforated beam

no interaction


I I


Figure 6

i

98

1 CD (2) TIT

L

( a)

b

d2

(b)


Figure 7

99

( a)

(S r CSyr+



Case I - Low Shear

+

CSyr ltS shy

direct shear



(b)


Figure 8

100

low highshear shear


I I r I I

I I I 1 I10


11

11 I I 11 I I



11~I11

i I I

I (d-2h-2t)I_~ I

d-2t 11

(1 -~) q


Figure 9

101



03~

CH 125 1412700

0313 =--9F 0375 =Jb


04 vVp


Figure 10

102


x e 3middot

Reinforcing

1412 11

10


04


Figure 11

103


0513No Reinforcing

MMp

10


04 VVp


Figure 12

104



I I I

04


Figure 13

105


0513

1412



04


Figure 14

106

10WF2l


CH h

55Q 125 990 025Q 0250~



I I I

04


Figure 15

107

43 11 39 18 0 (

I

I I i lA

26

I

111 99 I- middot1 t

22 2-S 45 J

) C

r I

2A

13 ) lA I ~

I)II45 35 ( ()

I

I

I If 2B

f 16~ 9 25 I2t~ 21-+1 1 1 i

30 11 24 27 0- 10- ro- )

I ~ I I

20

2119YlO 151( 21~fI211~ - rr I I



Figure 16

108

tI

2D

22tl 231 45 tlI

( l (

I

3A

3119 Yt 11~ 34ft J3n~

45 25 J 35 ) (~ i-

I

I 1 3B

H 25 ft2ft5 cI 16 J2++ I I1 I

t 0- 23 45 J0- 0- (

I 4A

~5 ( 32 J211~ 2 YYt -IIII



Figure 17

log

45 ~ 25 5S ) ~ (

4B

2 crYI 15 25 J-I2++ Irr

L J30 24 27 ~lt) (P-L

I 5A

2tYY 13 26211 1 I

22 6

6A

i



Figure 18

--

110

If JL

a (1) p 02 ~

Cfl

tlOO ~ r-I 0 0 Q)

m ~ ~

Q SoOM


r-shy

~ ishy fO I- It

111

o N

-()

112

~ --- ~

~

~~ ~~

II I of

rf

A tshyshy I

1


a1 +gt

- -l~ 0 ()

-t I H CJ

4gt 4gt bO ~

~ ~ Cl -rt

fiI

~ a1 ~

+gt CJI

~ amp ~ ~ ~ --- ~ I

II s I

~CJ~ 4t ~

I~ i ~

o

i-~ ~ - l

II I +1 I Ij

~I r

I tf~ft N

- shy

stress 8 1


( a)

F F-CL F-E [(4 ~

W-H- - shy

strain ( b)



I

114

lA 2A 2B

20 2D 3A

3B 4A 4B

5A 6A

FIGURE 24

115

2B 20 2D 3A 3B 4At 4B


20 5A


FIGURE 25

116

2A 4A 2B 4B


2B 3B


FIGURE 26

bullbull

117


r ~ 3

o

top of reinf



~ bull




Figure 27

118

boundaryof opening

-70 0 __ tQ



-1iCgt

~I--=-shyoo---~-=-~ogt---- ---0 - ____ A~Sshy-


oE


Figure 28

119

-40

-lt0 lt----- shylow


lt 0


edge ltgtgt

to


high moment

edge

o


Figure 29

120


o10 D



ibull

boundaryof opening

10 Igt lO



Figure 30

bull bull bull bull

121

11 9 ~ oS j j ~

boundaryof opening

-lt10 -20 lt) -I-Lo Q lO 10



boundaryof openin

-40 -20 0 ZO o 10 ZQ 30



Figure 31

122

Test Beam - lA


Figure 32

Test Beam - 2A


Figure 33

123


Test Beam - 2B


Figure 34




Test Beam - 20


Figure 35

124


first yielding

Test Beam - 2D


Figure 36

Test Beam - 3A


Figure 37

125


---- first yielding

O6in

Figure 38

O6in

Figure 39


Test Beam - 3B


Test Beam - 4A


126

430k--shy

k445--shy



06in

Figure 40

O6in

Figure 41


Test Beam - 4B


Test Beam - 5A


127

k45 o-~--

Test Beam - 6A


Figure 42

128

10 shy

10WF21



~ ~

+~ ~ ~


I I

04


Figure 43

curve yields

129






bull


Figure 44

130




04


Figure 45

10

131



~

1 +3A



04


Figure 46

10

132



~ ~ ~ ~ ~

I I


04


Figure 47

10

133

14WF38 H


-~


~ ~ +4B

~ ~ ~


I I

04


Figure 48

134




~ ~

SAl


04 vVp


Figure 49

10

135



+ 6A


04


Figure 50

136

shear (kips)

70

60


50

40

30

20


40 80 120 160 200 240



Figure 51

137

shear (kips)

70

60


50

40

30

20

10

4 8 12 16 20 24 28



Figure 52

138

D f



Figure 53

139

l4WF38 7 x lot

Opening

3shy10 f ~--2f x e

Reinforcing

If x ~ Reinforcing

+3A +2A

No Reinforcing + 6A

04


Figure 54

140


+ 2B

+20

+2A

+2D

04 vVp


Figure 55

141


10 ---------- shy

+3B

+3A

04 vVp


Figure 56

142


+ 4B

+ 4A

04 vVp


Figure 57

143


7 x lot Opening

7 x 14 Opening---

04


Figure 58

144shy

14WF38



+5A

04


Figure 59

--

r b --1 t l=Ii= W

d

u q


11

0244 875 550 N

0260 0253 2720 N

400 316

2A 14WF38 22 1413 677 0496 0326 1046 700 0383 0381 2813 300 58



It 11 It If3B 45 1422 668 0490 0305 1045 0329 0361 5230



foo J1

146













4A web W 565 549 flange F 11 487 476



F-E ff 438 425


Table 2

147






k 567k 483k



Table 3

- - - - - - -

- - - - - - - - -

- -


lA 2A 2B 20 2D 3A 3B ~ lI Ilt v 162 500 400 583 700 525 433

180 - - - 725 - 517 191 550 417 617 600 550 567 191 575 442 567 575 600 433

- 575 467 567 575 600 433

- 575 - - 600 600 shy202 625 475 600 600 700 533

- 625 475 600 625 700 583

- - - 617 700 - 550

- - - 600 675 675 550

- - - - - 675 shy

- 600 458 650 675 650 583

- - - - 725 - -- - - - - 725 -- - - - - 675 -- - - - - - -- - - - - 675 shy- - 483 - - - shy

- 700 517 703 825 775 shy- - - - 825 - 600

4A 4B Ilt Ilt

450 333 500 417 450 367 450 366 500 383 450 417 550 483 600 483 550 400 550 433 625 shy625 shy575 467 575 shy500 shy550 417

- 450 500 383

--675 513

5A le

350 500 400 400 433 483 433 467 500 417

--

500

--

467 500

---

517

-

6A

400 533

-367 367 383 433 533

-----------

550

--


J-ITable 4 ~ 00

149


lA 10WF21 43 180K 746K 0241 0235 + 26 2A 14WF38 22 532 1L021 0520 0466 +116 2B 11 45 413 0404 0363 +113 2C If 30 548 1301 0421 0403 + 45



Table 5


lA 10WF21 43 180K 746 K

0241 0254 - 51 2A 14VlF38 22 532 1021 0520 0503 + 34

It2B 45 413 0404 0344 +175 If2C 30 548 1301 0421 0414 + 17

It2D 17 670 0515 0505 + 20 3A 22 575 1021 0562 0505 +103

n3B 45 497 1301 0382 0377 + 13 4A tI 22 572 n 0440 0463 - 50

It It 03094B 45 430 0330 + 68 5A 30 445 tr 0342 0353 - 31 6A 22 450 tt 0346 0257 +346


Table 6

150



2D 17 670 625 - 67 - shy3B 45 497 483 - 28 600 +207 4B 45 430 383 -109 - shy


Table 7



2C 30n 525k 567k + 80 633k +206 2D 17 546 625 +145 - shy3B 45 449 483 + 76 600 +336 4B 45 384 383 - 03 - shy


Table 8

151

APPENDIX A








the following table










pages

152





SIGw SIGR VP Mpt)





RE~D(5582JAH


Kl K2~)

COMPLEXJ

153








Kl=Kl1 GO TO 322

321 K2K22 Kl=K12


154



408 WRITE(6613 GO TO 409







K l=K 11 GO TO 422

421 K2=K22 Kl=K12





155


418 WRITE(6614) GO TO 419


XX=SIGW2-075V2AW2 IF(XX)921800800







Kl=Kl1 GO TO 522

521 1lt2=K2Z Kl=K12





156

2 V=Vl 4 V=V+V INCR





04=D22-403 IFCD4)102351351


601 IF(K21-10602~0263C


Kl=Kll GO TO 622

621 K2=K22 Kl=K12





c

C

157





158

APPENDIX B















(b)




159

(c)

( d)





gives


f

where (f)

and ( g)


by

2 2 ~ =Jcsy - CS f (h)

f [3



160




Eyp = E yy (S

- i (k)


~

~


(n)











161

BIBLIOGRAPHY












162









xi

LIST OF TABLES

Page










SUJn1[ARY

ACKNOWLEDGEMENTS

NOTATION

LIST OF FIGURES

LIST OF TABLES

TABLE OF CONTENTS

Page

ii

iii

iv

vii

xi











31 Assumptions 27

















63 other Factors 72




73 Failure Loads 80



81 Conclusions 85



FIGURES 93

TABLES 145



BIBLIOGRAPHY 161

1

CHAPTER I

INTRODUCTION






















2


be increased













stresses


stresses








3


























4














given in Figure lb











5


























6


























7

--~~

















the problem

8

CHAPTER 11























9





is the yield stress
















10





















11





Figure 4





















12






hardening range(ll)







is reduced to

(4)



v = w( d-2t-2h) $-3





13




















(Vv) = d-2t-2h (6)P max d-2t




14





this task











increase in load









15






















decreases




16


























17
























18

























19

























20

























was offered by Chen

21





















the case




22








this effect

















23
























24

























25
























26









27

CHAPTER III


31 Assumptions








7b




dictated









28

























29







forcing is attached









section (2) gives






30


yields

(8)



(9)




stresses

Fl = Af ~yf + SW(1-2~)G + Ar ~yr (10)









31


calculated from

t2 _ r- 2 3(V)2 ~ - yw-4sw (12)
















given by

32




f yf





mined from

(17)













33








(13) and (18) yield


- (d-2h)]





34












(22)


(24)


(25)

35











become





(28)

(29)

36





















37







is written as

d-2t-2h (6)= d-2t











38






















(30)

39




(31)


given by

A =2 ( 32) r 13















40














reached

41

CHAPTER IV

APPROXIMATE METHOD

41 General Remarks





















42






(33)





Xf


by

(35 )




43




(36)











IS = If f)Af 3 r ( 37 ) - (s2 + 4a2 )

T





44



becomes

M (39 )M= p AW

1 + 4A f


(40)













by equation (38)


45





- AW(l 2h) flA (41)r - T -a-~




ityas given by

V 2hr = (I-a-) (42) p



Ar 1 - A

M fr= A p 1 + w







46












rewri tten as

M A (45 ) M=

p 1 + w4Af






47







previous section

















(43) then becomes

48

A 2h l1 - ~(l-a)J~

M ( 46)~= A 1 + W


aw 1 - 73Af

= ---1-shyA 1 + w
















49








this capacity
















(49a)

(50)

50

A q (5la )


Mp Aw 1 + 1iA

f


simplified to

(49)

(51)







Case I


if


given by

V (1 _ ~h) (42)Vp =

51


is Ar

1- ri- = Af (43)

p 1 + w 41f


Case 11



by

(40)


acting is

1 1 + w



Case Ill



52



follows

Case IlIA

For (48)


Ar 1 - I - ~

~ = __f_~_ ill A

P 1+ w4If


(38)

Case IIIB

For A2 gt (aw) 2

+ (~)2w2 (48)r 3 ~



A1 + w

4Af

53


+l (50)s

(49)




















54







these reduoe to

M AW 2h 1

1 - 2If(l-or~ r= A (52)

p 1 + w4If

v (2h)JTr = I-d~ p






given by

111 Z - wh2 (54)r = z

p




55

MM = 1 (55) p





56

CHAPTER V























57
























requires that

58

wh2 gt- (56)2h + 2u + q




















codes



59
























60

























61

























62
























63

















frame







64

























65
















the test beams

66

CHAPTER VI























67

























68

























69

























70


























71
























72






Figure 31












63 Other Factors






73

























74








shear


















75
























76



77

CHAPTER VII

ANALYSIS OF RESULTS






















78
























79

























80











73 Failure Loads













81

























82
























83













in Figure 54












84















85

CHAPTER VIII


81 Conclusions



drawn
















lateral buckling


86














opening






distributions





87

























88







force

















89

























go
























91

























92













93

1)

) )J + 1 +1 I I

I

( a) (b)

I + --+--~+ -f--- -shy

I (c) (d)

II I I

I I j+shy

) 1)I l J I

(e) ( f)


Figure 1

94




Figure 2

Ql ID Q)

Jt p ID

er

I I I I I I I



Figure 3

---------- -----

95


10~----~==~==~L---------------------~

~


10 vVp


Figure 4

-- -----------

96


(a)

~ o

r-I


(b)


Figure 5

97

1O~------------------------------------------~

l I


shy

I r perforated beam

no interaction


I I


Figure 6

i

98

1 CD (2) TIT

L

( a)

b

d2

(b)


Figure 7

99

( a)

(S r CSyr+



Case I - Low Shear

+

CSyr ltS shy

direct shear



(b)


Figure 8

100

low highshear shear


I I r I I

I I I 1 I10


11

11 I I 11 I I



11~I11

i I I

I (d-2h-2t)I_~ I

d-2t 11

(1 -~) q


Figure 9

101



03~

CH 125 1412700

0313 =--9F 0375 =Jb


04 vVp


Figure 10

102


x e 3middot

Reinforcing

1412 11

10


04


Figure 11

103


0513No Reinforcing

MMp

10


04 VVp


Figure 12

104



I I I

04


Figure 13

105


0513

1412



04


Figure 14

106

10WF2l


CH h

55Q 125 990 025Q 0250~



I I I

04


Figure 15

107

43 11 39 18 0 (

I

I I i lA

26

I

111 99 I- middot1 t

22 2-S 45 J

) C

r I

2A

13 ) lA I ~

I)II45 35 ( ()

I

I

I If 2B

f 16~ 9 25 I2t~ 21-+1 1 1 i

30 11 24 27 0- 10- ro- )

I ~ I I

20

2119YlO 151( 21~fI211~ - rr I I



Figure 16

108

tI

2D

22tl 231 45 tlI

( l (

I

3A

3119 Yt 11~ 34ft J3n~

45 25 J 35 ) (~ i-

I

I 1 3B

H 25 ft2ft5 cI 16 J2++ I I1 I

t 0- 23 45 J0- 0- (

I 4A

~5 ( 32 J211~ 2 YYt -IIII



Figure 17

log

45 ~ 25 5S ) ~ (

4B

2 crYI 15 25 J-I2++ Irr

L J30 24 27 ~lt) (P-L

I 5A

2tYY 13 26211 1 I

22 6

6A

i



Figure 18

--

110

If JL

a (1) p 02 ~

Cfl

tlOO ~ r-I 0 0 Q)

m ~ ~

Q SoOM


r-shy

~ ishy fO I- It

111

o N

-()

112

~ --- ~

~

~~ ~~

II I of

rf

A tshyshy I

1


a1 +gt

- -l~ 0 ()

-t I H CJ

4gt 4gt bO ~

~ ~ Cl -rt

fiI

~ a1 ~

+gt CJI

~ amp ~ ~ ~ --- ~ I

II s I

~CJ~ 4t ~

I~ i ~

o

i-~ ~ - l

II I +1 I Ij

~I r

I tf~ft N

- shy

stress 8 1


( a)

F F-CL F-E [(4 ~

W-H- - shy

strain ( b)



I

114

lA 2A 2B

20 2D 3A

3B 4A 4B

5A 6A

FIGURE 24

115

2B 20 2D 3A 3B 4At 4B


20 5A


FIGURE 25

116

2A 4A 2B 4B


2B 3B


FIGURE 26

bullbull

117


r ~ 3

o

top of reinf



~ bull




Figure 27

118

boundaryof opening

-70 0 __ tQ



-1iCgt

~I--=-shyoo---~-=-~ogt---- ---0 - ____ A~Sshy-


oE


Figure 28

119

-40

-lt0 lt----- shylow


lt 0


edge ltgtgt

to


high moment

edge

o


Figure 29

120


o10 D



ibull

boundaryof opening

10 Igt lO



Figure 30

bull bull bull bull

121

11 9 ~ oS j j ~

boundaryof opening

-lt10 -20 lt) -I-Lo Q lO 10



boundaryof openin

-40 -20 0 ZO o 10 ZQ 30



Figure 31

122

Test Beam - lA


Figure 32

Test Beam - 2A


Figure 33

123


Test Beam - 2B


Figure 34




Test Beam - 20


Figure 35

124


first yielding

Test Beam - 2D


Figure 36

Test Beam - 3A


Figure 37

125


---- first yielding

O6in

Figure 38

O6in

Figure 39


Test Beam - 3B


Test Beam - 4A


126

430k--shy

k445--shy



06in

Figure 40

O6in

Figure 41


Test Beam - 4B


Test Beam - 5A


127

k45 o-~--

Test Beam - 6A


Figure 42

128

10 shy

10WF21



~ ~

+~ ~ ~


I I

04


Figure 43

curve yields

129






bull


Figure 44

130




04


Figure 45

10

131



~

1 +3A



04


Figure 46

10

132



~ ~ ~ ~ ~

I I


04


Figure 47

10

133

14WF38 H


-~


~ ~ +4B

~ ~ ~


I I

04


Figure 48

134




~ ~

SAl


04 vVp


Figure 49

10

135



+ 6A


04


Figure 50

136

shear (kips)

70

60


50

40

30

20


40 80 120 160 200 240



Figure 51

137

shear (kips)

70

60


50

40

30

20

10

4 8 12 16 20 24 28



Figure 52

138

D f



Figure 53

139

l4WF38 7 x lot

Opening

3shy10 f ~--2f x e

Reinforcing

If x ~ Reinforcing

+3A +2A

No Reinforcing + 6A

04


Figure 54

140


+ 2B

+20

+2A

+2D

04 vVp


Figure 55

141


10 ---------- shy

+3B

+3A

04 vVp


Figure 56

142


+ 4B

+ 4A

04 vVp


Figure 57

143


7 x lot Opening

7 x 14 Opening---

04


Figure 58

144shy

14WF38



+5A

04


Figure 59

--

r b --1 t l=Ii= W

d

u q


11

0244 875 550 N

0260 0253 2720 N

400 316

2A 14WF38 22 1413 677 0496 0326 1046 700 0383 0381 2813 300 58



It 11 It If3B 45 1422 668 0490 0305 1045 0329 0361 5230



foo J1

146













4A web W 565 549 flange F 11 487 476



F-E ff 438 425


Table 2

147






k 567k 483k



Table 3

- - - - - - -

- - - - - - - - -

- -


lA 2A 2B 20 2D 3A 3B ~ lI Ilt v 162 500 400 583 700 525 433

180 - - - 725 - 517 191 550 417 617 600 550 567 191 575 442 567 575 600 433

- 575 467 567 575 600 433

- 575 - - 600 600 shy202 625 475 600 600 700 533

- 625 475 600 625 700 583

- - - 617 700 - 550

- - - 600 675 675 550

- - - - - 675 shy

- 600 458 650 675 650 583

- - - - 725 - -- - - - - 725 -- - - - - 675 -- - - - - - -- - - - - 675 shy- - 483 - - - shy

- 700 517 703 825 775 shy- - - - 825 - 600

4A 4B Ilt Ilt

450 333 500 417 450 367 450 366 500 383 450 417 550 483 600 483 550 400 550 433 625 shy625 shy575 467 575 shy500 shy550 417

- 450 500 383

--675 513

5A le

350 500 400 400 433 483 433 467 500 417

--

500

--

467 500

---

517

-

6A

400 533

-367 367 383 433 533

-----------

550

--


J-ITable 4 ~ 00

149


lA 10WF21 43 180K 746K 0241 0235 + 26 2A 14WF38 22 532 1L021 0520 0466 +116 2B 11 45 413 0404 0363 +113 2C If 30 548 1301 0421 0403 + 45



Table 5


lA 10WF21 43 180K 746 K

0241 0254 - 51 2A 14VlF38 22 532 1021 0520 0503 + 34

It2B 45 413 0404 0344 +175 If2C 30 548 1301 0421 0414 + 17

It2D 17 670 0515 0505 + 20 3A 22 575 1021 0562 0505 +103

n3B 45 497 1301 0382 0377 + 13 4A tI 22 572 n 0440 0463 - 50

It It 03094B 45 430 0330 + 68 5A 30 445 tr 0342 0353 - 31 6A 22 450 tt 0346 0257 +346


Table 6

150



2D 17 670 625 - 67 - shy3B 45 497 483 - 28 600 +207 4B 45 430 383 -109 - shy


Table 7



2C 30n 525k 567k + 80 633k +206 2D 17 546 625 +145 - shy3B 45 449 483 + 76 600 +336 4B 45 384 383 - 03 - shy


Table 8

151

APPENDIX A








the following table










pages

152





SIGw SIGR VP Mpt)





RE~D(5582JAH


Kl K2~)

COMPLEXJ

153








Kl=Kl1 GO TO 322

321 K2K22 Kl=K12


154



408 WRITE(6613 GO TO 409







K l=K 11 GO TO 422

421 K2=K22 Kl=K12





155


418 WRITE(6614) GO TO 419


XX=SIGW2-075V2AW2 IF(XX)921800800







Kl=Kl1 GO TO 522

521 1lt2=K2Z Kl=K12





156

2 V=Vl 4 V=V+V INCR





04=D22-403 IFCD4)102351351


601 IF(K21-10602~0263C


Kl=Kll GO TO 622

621 K2=K22 Kl=K12





c

C

157





158

APPENDIX B















(b)




159

(c)

( d)





gives


f

where (f)

and ( g)


by

2 2 ~ =Jcsy - CS f (h)

f [3



160




Eyp = E yy (S

- i (k)


~

~


(n)











161

BIBLIOGRAPHY












162









SUJn1[ARY

ACKNOWLEDGEMENTS

NOTATION

LIST OF FIGURES

LIST OF TABLES

TABLE OF CONTENTS

Page

ii

iii

iv

vii

xi











31 Assumptions 27

















63 other Factors 72




73 Failure Loads 80



81 Conclusions 85



FIGURES 93

TABLES 145



BIBLIOGRAPHY 161

1

CHAPTER I

INTRODUCTION






















2


be increased













stresses


stresses








3


























4














given in Figure lb











5


























6


























7

--~~

















the problem

8

CHAPTER 11























9





is the yield stress
















10





















11





Figure 4





















12






hardening range(ll)







is reduced to

(4)



v = w( d-2t-2h) $-3





13




















(Vv) = d-2t-2h (6)P max d-2t




14





this task











increase in load









15






















decreases




16


























17
























18

























19

























20

























was offered by Chen

21





















the case




22








this effect

















23
























24

























25
























26









27

CHAPTER III


31 Assumptions








7b




dictated









28

























29







forcing is attached









section (2) gives






30


yields

(8)



(9)




stresses

Fl = Af ~yf + SW(1-2~)G + Ar ~yr (10)









31


calculated from

t2 _ r- 2 3(V)2 ~ - yw-4sw (12)
















given by

32




f yf





mined from

(17)













33








(13) and (18) yield


- (d-2h)]





34












(22)


(24)


(25)

35











become





(28)

(29)

36





















37







is written as

d-2t-2h (6)= d-2t











38






















(30)

39




(31)


given by

A =2 ( 32) r 13















40














reached

41

CHAPTER IV

APPROXIMATE METHOD

41 General Remarks





















42






(33)





Xf


by

(35 )




43




(36)











IS = If f)Af 3 r ( 37 ) - (s2 + 4a2 )

T





44



becomes

M (39 )M= p AW

1 + 4A f


(40)













by equation (38)


45





- AW(l 2h) flA (41)r - T -a-~




ityas given by

V 2hr = (I-a-) (42) p



Ar 1 - A

M fr= A p 1 + w







46












rewri tten as

M A (45 ) M=

p 1 + w4Af






47







previous section

















(43) then becomes

48

A 2h l1 - ~(l-a)J~

M ( 46)~= A 1 + W


aw 1 - 73Af

= ---1-shyA 1 + w
















49








this capacity
















(49a)

(50)

50

A q (5la )


Mp Aw 1 + 1iA

f


simplified to

(49)

(51)







Case I


if


given by

V (1 _ ~h) (42)Vp =

51


is Ar

1- ri- = Af (43)

p 1 + w 41f


Case 11



by

(40)


acting is

1 1 + w



Case Ill



52



follows

Case IlIA

For (48)


Ar 1 - I - ~

~ = __f_~_ ill A

P 1+ w4If


(38)

Case IIIB

For A2 gt (aw) 2

+ (~)2w2 (48)r 3 ~



A1 + w

4Af

53


+l (50)s

(49)




















54







these reduoe to

M AW 2h 1

1 - 2If(l-or~ r= A (52)

p 1 + w4If

v (2h)JTr = I-d~ p






given by

111 Z - wh2 (54)r = z

p




55

MM = 1 (55) p





56

CHAPTER V























57
























requires that

58

wh2 gt- (56)2h + 2u + q




















codes



59
























60

























61

























62
























63

















frame







64

























65
















the test beams

66

CHAPTER VI























67

























68

























69

























70


























71
























72






Figure 31












63 Other Factors






73

























74








shear


















75
























76



77

CHAPTER VII

ANALYSIS OF RESULTS






















78
























79

























80











73 Failure Loads













81

























82
























83













in Figure 54












84















85

CHAPTER VIII


81 Conclusions



drawn
















lateral buckling


86














opening






distributions





87

























88







force

















89

























go
























91

























92













93

1)

) )J + 1 +1 I I

I

( a) (b)

I + --+--~+ -f--- -shy

I (c) (d)

II I I

I I j+shy

) 1)I l J I

(e) ( f)


Figure 1

94




Figure 2

Ql ID Q)

Jt p ID

er

I I I I I I I



Figure 3

---------- -----

95


10~----~==~==~L---------------------~

~


10 vVp


Figure 4

-- -----------

96


(a)

~ o

r-I


(b)


Figure 5

97

1O~------------------------------------------~

l I


shy

I r perforated beam

no interaction


I I


Figure 6

i

98

1 CD (2) TIT

L

( a)

b

d2

(b)


Figure 7

99

( a)

(S r CSyr+



Case I - Low Shear

+

CSyr ltS shy

direct shear



(b)


Figure 8

100

low highshear shear


I I r I I

I I I 1 I10


11

11 I I 11 I I



11~I11

i I I

I (d-2h-2t)I_~ I

d-2t 11

(1 -~) q


Figure 9

101



03~

CH 125 1412700

0313 =--9F 0375 =Jb


04 vVp


Figure 10

102


x e 3middot

Reinforcing

1412 11

10


04


Figure 11

103


0513No Reinforcing

MMp

10


04 VVp


Figure 12

104



I I I

04


Figure 13

105


0513

1412



04


Figure 14

106

10WF2l


CH h

55Q 125 990 025Q 0250~



I I I

04


Figure 15

107

43 11 39 18 0 (

I

I I i lA

26

I

111 99 I- middot1 t

22 2-S 45 J

) C

r I

2A

13 ) lA I ~

I)II45 35 ( ()

I

I

I If 2B

f 16~ 9 25 I2t~ 21-+1 1 1 i

30 11 24 27 0- 10- ro- )

I ~ I I

20

2119YlO 151( 21~fI211~ - rr I I



Figure 16

108

tI

2D

22tl 231 45 tlI

( l (

I

3A

3119 Yt 11~ 34ft J3n~

45 25 J 35 ) (~ i-

I

I 1 3B

H 25 ft2ft5 cI 16 J2++ I I1 I

t 0- 23 45 J0- 0- (

I 4A

~5 ( 32 J211~ 2 YYt -IIII



Figure 17

log

45 ~ 25 5S ) ~ (

4B

2 crYI 15 25 J-I2++ Irr

L J30 24 27 ~lt) (P-L

I 5A

2tYY 13 26211 1 I

22 6

6A

i



Figure 18

--

110

If JL

a (1) p 02 ~

Cfl

tlOO ~ r-I 0 0 Q)

m ~ ~

Q SoOM


r-shy

~ ishy fO I- It

111

o N

-()

112

~ --- ~

~

~~ ~~

II I of

rf

A tshyshy I

1


a1 +gt

- -l~ 0 ()

-t I H CJ

4gt 4gt bO ~

~ ~ Cl -rt

fiI

~ a1 ~

+gt CJI

~ amp ~ ~ ~ --- ~ I

II s I

~CJ~ 4t ~

I~ i ~

o

i-~ ~ - l

II I +1 I Ij

~I r

I tf~ft N

- shy

stress 8 1


( a)

F F-CL F-E [(4 ~

W-H- - shy

strain ( b)



I

114

lA 2A 2B

20 2D 3A

3B 4A 4B

5A 6A

FIGURE 24

115

2B 20 2D 3A 3B 4At 4B


20 5A


FIGURE 25

116

2A 4A 2B 4B


2B 3B


FIGURE 26

bullbull

117


r ~ 3

o

top of reinf



~ bull




Figure 27

118

boundaryof opening

-70 0 __ tQ



-1iCgt

~I--=-shyoo---~-=-~ogt---- ---0 - ____ A~Sshy-


oE


Figure 28

119

-40

-lt0 lt----- shylow


lt 0


edge ltgtgt

to


high moment

edge

o


Figure 29

120


o10 D



ibull

boundaryof opening

10 Igt lO



Figure 30

bull bull bull bull

121

11 9 ~ oS j j ~

boundaryof opening

-lt10 -20 lt) -I-Lo Q lO 10



boundaryof openin

-40 -20 0 ZO o 10 ZQ 30



Figure 31

122

Test Beam - lA


Figure 32

Test Beam - 2A


Figure 33

123


Test Beam - 2B


Figure 34




Test Beam - 20


Figure 35

124


first yielding

Test Beam - 2D


Figure 36

Test Beam - 3A


Figure 37

125


---- first yielding

O6in

Figure 38

O6in

Figure 39


Test Beam - 3B


Test Beam - 4A


126

430k--shy

k445--shy



06in

Figure 40

O6in

Figure 41


Test Beam - 4B


Test Beam - 5A


127

k45 o-~--

Test Beam - 6A


Figure 42

128

10 shy

10WF21



~ ~

+~ ~ ~


I I

04


Figure 43

curve yields

129






bull


Figure 44

130




04


Figure 45

10

131



~

1 +3A



04


Figure 46

10

132



~ ~ ~ ~ ~

I I


04


Figure 47

10

133

14WF38 H


-~


~ ~ +4B

~ ~ ~


I I

04


Figure 48

134




~ ~

SAl


04 vVp


Figure 49

10

135



+ 6A


04


Figure 50

136

shear (kips)

70

60


50

40

30

20


40 80 120 160 200 240



Figure 51

137

shear (kips)

70

60


50

40

30

20

10

4 8 12 16 20 24 28



Figure 52

138

D f



Figure 53

139

l4WF38 7 x lot

Opening

3shy10 f ~--2f x e

Reinforcing

If x ~ Reinforcing

+3A +2A

No Reinforcing + 6A

04


Figure 54

140


+ 2B

+20

+2A

+2D

04 vVp


Figure 55

141


10 ---------- shy

+3B

+3A

04 vVp


Figure 56

142


+ 4B

+ 4A

04 vVp


Figure 57

143


7 x lot Opening

7 x 14 Opening---

04


Figure 58

144shy

14WF38



+5A

04


Figure 59

--

r b --1 t l=Ii= W

d

u q


11

0244 875 550 N

0260 0253 2720 N

400 316

2A 14WF38 22 1413 677 0496 0326 1046 700 0383 0381 2813 300 58



It 11 It If3B 45 1422 668 0490 0305 1045 0329 0361 5230



foo J1

146













4A web W 565 549 flange F 11 487 476



F-E ff 438 425


Table 2

147






k 567k 483k



Table 3

- - - - - - -

- - - - - - - - -

- -


lA 2A 2B 20 2D 3A 3B ~ lI Ilt v 162 500 400 583 700 525 433

180 - - - 725 - 517 191 550 417 617 600 550 567 191 575 442 567 575 600 433

- 575 467 567 575 600 433

- 575 - - 600 600 shy202 625 475 600 600 700 533

- 625 475 600 625 700 583

- - - 617 700 - 550

- - - 600 675 675 550

- - - - - 675 shy

- 600 458 650 675 650 583

- - - - 725 - -- - - - - 725 -- - - - - 675 -- - - - - - -- - - - - 675 shy- - 483 - - - shy

- 700 517 703 825 775 shy- - - - 825 - 600

4A 4B Ilt Ilt

450 333 500 417 450 367 450 366 500 383 450 417 550 483 600 483 550 400 550 433 625 shy625 shy575 467 575 shy500 shy550 417

- 450 500 383

--675 513

5A le

350 500 400 400 433 483 433 467 500 417

--

500

--

467 500

---

517

-

6A

400 533

-367 367 383 433 533

-----------

550

--


J-ITable 4 ~ 00

149


lA 10WF21 43 180K 746K 0241 0235 + 26 2A 14WF38 22 532 1L021 0520 0466 +116 2B 11 45 413 0404 0363 +113 2C If 30 548 1301 0421 0403 + 45



Table 5


lA 10WF21 43 180K 746 K

0241 0254 - 51 2A 14VlF38 22 532 1021 0520 0503 + 34

It2B 45 413 0404 0344 +175 If2C 30 548 1301 0421 0414 + 17

It2D 17 670 0515 0505 + 20 3A 22 575 1021 0562 0505 +103

n3B 45 497 1301 0382 0377 + 13 4A tI 22 572 n 0440 0463 - 50

It It 03094B 45 430 0330 + 68 5A 30 445 tr 0342 0353 - 31 6A 22 450 tt 0346 0257 +346


Table 6

150



2D 17 670 625 - 67 - shy3B 45 497 483 - 28 600 +207 4B 45 430 383 -109 - shy


Table 7



2C 30n 525k 567k + 80 633k +206 2D 17 546 625 +145 - shy3B 45 449 483 + 76 600 +336 4B 45 384 383 - 03 - shy


Table 8

151

APPENDIX A








the following table










pages

152





SIGw SIGR VP Mpt)





RE~D(5582JAH


Kl K2~)

COMPLEXJ

153








Kl=Kl1 GO TO 322

321 K2K22 Kl=K12


154



408 WRITE(6613 GO TO 409







K l=K 11 GO TO 422

421 K2=K22 Kl=K12





155


418 WRITE(6614) GO TO 419


XX=SIGW2-075V2AW2 IF(XX)921800800







Kl=Kl1 GO TO 522

521 1lt2=K2Z Kl=K12





156

2 V=Vl 4 V=V+V INCR





04=D22-403 IFCD4)102351351


601 IF(K21-10602~0263C


Kl=Kll GO TO 622

621 K2=K22 Kl=K12





c

C

157





158

APPENDIX B















(b)




159

(c)

( d)





gives


f

where (f)

and ( g)


by

2 2 ~ =Jcsy - CS f (h)

f [3



160




Eyp = E yy (S

- i (k)


~

~


(n)











161

BIBLIOGRAPHY












162

















63 other Factors 72




73 Failure Loads 80



81 Conclusions 85



FIGURES 93

TABLES 145



BIBLIOGRAPHY 161

1

CHAPTER I

INTRODUCTION






















2


be increased













stresses


stresses








3


























4














given in Figure lb











5


























6


























7

--~~

















the problem

8

CHAPTER 11























9





is the yield stress
















10





















11





Figure 4





















12






hardening range(ll)







is reduced to

(4)



v = w( d-2t-2h) $-3





13




















(Vv) = d-2t-2h (6)P max d-2t




14





this task











increase in load









15






















decreases




16


























17
























18

























19

























20

























was offered by Chen

21





















the case




22








this effect

















23
























24

























25
























26









27

CHAPTER III


31 Assumptions








7b




dictated









28

























29







forcing is attached









section (2) gives






30


yields

(8)



(9)




stresses

Fl = Af ~yf + SW(1-2~)G + Ar ~yr (10)









31


calculated from

t2 _ r- 2 3(V)2 ~ - yw-4sw (12)
















given by

32




f yf





mined from

(17)













33








(13) and (18) yield


- (d-2h)]





34












(22)


(24)


(25)

35











become





(28)

(29)

36





















37







is written as

d-2t-2h (6)= d-2t











38






















(30)

39




(31)


given by

A =2 ( 32) r 13















40














reached

41

CHAPTER IV

APPROXIMATE METHOD

41 General Remarks





















42






(33)





Xf


by

(35 )




43




(36)











IS = If f)Af 3 r ( 37 ) - (s2 + 4a2 )

T





44



becomes

M (39 )M= p AW

1 + 4A f


(40)













by equation (38)


45





- AW(l 2h) flA (41)r - T -a-~




ityas given by

V 2hr = (I-a-) (42) p



Ar 1 - A

M fr= A p 1 + w







46












rewri tten as

M A (45 ) M=

p 1 + w4Af






47







previous section

















(43) then becomes

48

A 2h l1 - ~(l-a)J~

M ( 46)~= A 1 + W


aw 1 - 73Af

= ---1-shyA 1 + w
















49








this capacity
















(49a)

(50)

50

A q (5la )


Mp Aw 1 + 1iA

f


simplified to

(49)

(51)







Case I


if


given by

V (1 _ ~h) (42)Vp =

51


is Ar

1- ri- = Af (43)

p 1 + w 41f


Case 11



by

(40)


acting is

1 1 + w



Case Ill



52



follows

Case IlIA

For (48)


Ar 1 - I - ~

~ = __f_~_ ill A

P 1+ w4If


(38)

Case IIIB

For A2 gt (aw) 2

+ (~)2w2 (48)r 3 ~



A1 + w

4Af

53


+l (50)s

(49)




















54







these reduoe to

M AW 2h 1

1 - 2If(l-or~ r= A (52)

p 1 + w4If

v (2h)JTr = I-d~ p






given by

111 Z - wh2 (54)r = z

p




55

MM = 1 (55) p





56

CHAPTER V























57
























requires that

58

wh2 gt- (56)2h + 2u + q




















codes



59
























60

























61

























62
























63

















frame







64

























65
















the test beams

66

CHAPTER VI























67

























68

























69

























70


























71
























72






Figure 31












63 Other Factors






73

























74








shear


















75
























76



77

CHAPTER VII

ANALYSIS OF RESULTS






















78
























79

























80











73 Failure Loads













81

























82
























83













in Figure 54












84















85

CHAPTER VIII


81 Conclusions



drawn
















lateral buckling


86














opening






distributions





87

























88







force

















89

























go
























91

























92













93

1)

) )J + 1 +1 I I

I

( a) (b)

I + --+--~+ -f--- -shy

I (c) (d)

II I I

I I j+shy

) 1)I l J I

(e) ( f)


Figure 1

94




Figure 2

Ql ID Q)

Jt p ID

er

I I I I I I I



Figure 3

---------- -----

95


10~----~==~==~L---------------------~

~


10 vVp


Figure 4

-- -----------

96


(a)

~ o

r-I


(b)


Figure 5

97

1O~------------------------------------------~

l I


shy

I r perforated beam

no interaction


I I


Figure 6

i

98

1 CD (2) TIT

L

( a)

b

d2

(b)


Figure 7

99

( a)

(S r CSyr+



Case I - Low Shear

+

CSyr ltS shy

direct shear



(b)


Figure 8

100

low highshear shear


I I r I I

I I I 1 I10


11

11 I I 11 I I



11~I11

i I I

I (d-2h-2t)I_~ I

d-2t 11

(1 -~) q


Figure 9

101



03~

CH 125 1412700

0313 =--9F 0375 =Jb


04 vVp


Figure 10

102


x e 3middot

Reinforcing

1412 11

10


04


Figure 11

103


0513No Reinforcing

MMp

10


04 VVp


Figure 12

104



I I I

04


Figure 13

105


0513

1412



04


Figure 14

106

10WF2l


CH h

55Q 125 990 025Q 0250~



I I I

04


Figure 15

107

43 11 39 18 0 (

I

I I i lA

26

I

111 99 I- middot1 t

22 2-S 45 J

) C

r I

2A

13 ) lA I ~

I)II45 35 ( ()

I

I

I If 2B

f 16~ 9 25 I2t~ 21-+1 1 1 i

30 11 24 27 0- 10- ro- )

I ~ I I

20

2119YlO 151( 21~fI211~ - rr I I



Figure 16

108

tI

2D

22tl 231 45 tlI

( l (

I

3A

3119 Yt 11~ 34ft J3n~

45 25 J 35 ) (~ i-

I

I 1 3B

H 25 ft2ft5 cI 16 J2++ I I1 I

t 0- 23 45 J0- 0- (

I 4A

~5 ( 32 J211~ 2 YYt -IIII



Figure 17

log

45 ~ 25 5S ) ~ (

4B

2 crYI 15 25 J-I2++ Irr

L J30 24 27 ~lt) (P-L

I 5A

2tYY 13 26211 1 I

22 6

6A

i



Figure 18

--

110

If JL

a (1) p 02 ~

Cfl

tlOO ~ r-I 0 0 Q)

m ~ ~

Q SoOM


r-shy

~ ishy fO I- It

111

o N

-()

112

~ --- ~

~

~~ ~~

II I of

rf

A tshyshy I

1


a1 +gt

- -l~ 0 ()

-t I H CJ

4gt 4gt bO ~

~ ~ Cl -rt

fiI

~ a1 ~

+gt CJI

~ amp ~ ~ ~ --- ~ I

II s I

~CJ~ 4t ~

I~ i ~

o

i-~ ~ - l

II I +1 I Ij

~I r

I tf~ft N

- shy

stress 8 1


( a)

F F-CL F-E [(4 ~

W-H- - shy

strain ( b)



I

114

lA 2A 2B

20 2D 3A

3B 4A 4B

5A 6A

FIGURE 24

115

2B 20 2D 3A 3B 4At 4B


20 5A


FIGURE 25

116

2A 4A 2B 4B


2B 3B


FIGURE 26

bullbull

117


r ~ 3

o

top of reinf



~ bull




Figure 27

118

boundaryof opening

-70 0 __ tQ



-1iCgt

~I--=-shyoo---~-=-~ogt---- ---0 - ____ A~Sshy-


oE


Figure 28

119

-40

-lt0 lt----- shylow


lt 0


edge ltgtgt

to


high moment

edge

o


Figure 29

120


o10 D



ibull

boundaryof opening

10 Igt lO



Figure 30

bull bull bull bull

121

11 9 ~ oS j j ~

boundaryof opening

-lt10 -20 lt) -I-Lo Q lO 10



boundaryof openin

-40 -20 0 ZO o 10 ZQ 30



Figure 31

122

Test Beam - lA


Figure 32

Test Beam - 2A


Figure 33

123


Test Beam - 2B


Figure 34




Test Beam - 20


Figure 35

124


first yielding

Test Beam - 2D


Figure 36

Test Beam - 3A


Figure 37

125


---- first yielding

O6in

Figure 38

O6in

Figure 39


Test Beam - 3B


Test Beam - 4A


126

430k--shy

k445--shy



06in

Figure 40

O6in

Figure 41


Test Beam - 4B


Test Beam - 5A


127

k45 o-~--

Test Beam - 6A


Figure 42

128

10 shy

10WF21



~ ~

+~ ~ ~


I I

04


Figure 43

curve yields

129






bull


Figure 44

130




04


Figure 45

10

131



~

1 +3A



04


Figure 46

10

132



~ ~ ~ ~ ~

I I


04


Figure 47

10

133

14WF38 H


-~


~ ~ +4B

~ ~ ~


I I

04


Figure 48

134




~ ~

SAl


04 vVp


Figure 49

10

135



+ 6A


04


Figure 50

136

shear (kips)

70

60


50

40

30

20


40 80 120 160 200 240



Figure 51

137

shear (kips)

70

60


50

40

30

20

10

4 8 12 16 20 24 28



Figure 52

138

D f



Figure 53

139

l4WF38 7 x lot

Opening

3shy10 f ~--2f x e

Reinforcing

If x ~ Reinforcing

+3A +2A

No Reinforcing + 6A

04


Figure 54

140


+ 2B

+20

+2A

+2D

04 vVp


Figure 55

141


10 ---------- shy

+3B

+3A

04 vVp


Figure 56

142


+ 4B

+ 4A

04 vVp


Figure 57

143


7 x lot Opening

7 x 14 Opening---

04


Figure 58

144shy

14WF38



+5A

04


Figure 59

--

r b --1 t l=Ii= W

d

u q


11

0244 875 550 N

0260 0253 2720 N

400 316

2A 14WF38 22 1413 677 0496 0326 1046 700 0383 0381 2813 300 58



It 11 It If3B 45 1422 668 0490 0305 1045 0329 0361 5230



foo J1

146













4A web W 565 549 flange F 11 487 476



F-E ff 438 425


Table 2

147






k 567k 483k



Table 3

- - - - - - -

- - - - - - - - -

- -


lA 2A 2B 20 2D 3A 3B ~ lI Ilt v 162 500 400 583 700 525 433

180 - - - 725 - 517 191 550 417 617 600 550 567 191 575 442 567 575 600 433

- 575 467 567 575 600 433

- 575 - - 600 600 shy202 625 475 600 600 700 533

- 625 475 600 625 700 583

- - - 617 700 - 550

- - - 600 675 675 550

- - - - - 675 shy

- 600 458 650 675 650 583

- - - - 725 - -- - - - - 725 -- - - - - 675 -- - - - - - -- - - - - 675 shy- - 483 - - - shy

- 700 517 703 825 775 shy- - - - 825 - 600

4A 4B Ilt Ilt

450 333 500 417 450 367 450 366 500 383 450 417 550 483 600 483 550 400 550 433 625 shy625 shy575 467 575 shy500 shy550 417

- 450 500 383

--675 513

5A le

350 500 400 400 433 483 433 467 500 417

--

500

--

467 500

---

517

-

6A

400 533

-367 367 383 433 533

-----------

550

--


J-ITable 4 ~ 00

149


lA 10WF21 43 180K 746K 0241 0235 + 26 2A 14WF38 22 532 1L021 0520 0466 +116 2B 11 45 413 0404 0363 +113 2C If 30 548 1301 0421 0403 + 45



Table 5


lA 10WF21 43 180K 746 K

0241 0254 - 51 2A 14VlF38 22 532 1021 0520 0503 + 34

It2B 45 413 0404 0344 +175 If2C 30 548 1301 0421 0414 + 17

It2D 17 670 0515 0505 + 20 3A 22 575 1021 0562 0505 +103

n3B 45 497 1301 0382 0377 + 13 4A tI 22 572 n 0440 0463 - 50

It It 03094B 45 430 0330 + 68 5A 30 445 tr 0342 0353 - 31 6A 22 450 tt 0346 0257 +346


Table 6

150



2D 17 670 625 - 67 - shy3B 45 497 483 - 28 600 +207 4B 45 430 383 -109 - shy


Table 7



2C 30n 525k 567k + 80 633k +206 2D 17 546 625 +145 - shy3B 45 449 483 + 76 600 +336 4B 45 384 383 - 03 - shy


Table 8

151

APPENDIX A








the following table










pages

152





SIGw SIGR VP Mpt)





RE~D(5582JAH


Kl K2~)

COMPLEXJ

153








Kl=Kl1 GO TO 322

321 K2K22 Kl=K12


154



408 WRITE(6613 GO TO 409







K l=K 11 GO TO 422

421 K2=K22 Kl=K12





155


418 WRITE(6614) GO TO 419


XX=SIGW2-075V2AW2 IF(XX)921800800







Kl=Kl1 GO TO 522

521 1lt2=K2Z Kl=K12





156

2 V=Vl 4 V=V+V INCR





04=D22-403 IFCD4)102351351


601 IF(K21-10602~0263C


Kl=Kll GO TO 622

621 K2=K22 Kl=K12





c

C

157





158

APPENDIX B















(b)




159

(c)

( d)





gives


f

where (f)

and ( g)


by

2 2 ~ =Jcsy - CS f (h)

f [3



160




Eyp = E yy (S

- i (k)


~

~


(n)











161

BIBLIOGRAPHY












162









ultimate strength of beams vqth by judith gloede …

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