chapter 6 theoretical models for flexure, deflection and...

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CHAPTER 6

THEORETICAL MODELS FOR FLEXURE,

DEFLECTION AND DUCTILITY

6.1 GENERAL

Experimental investigation on Prefabricated Cage Reinforced

Concrete beams has been reported in Chapter 5. The behaviour of these

beams under increasing loads upto failure, modes of failure, load –

deformation response and ductile characteristics were discussed in

detail. In this chapter, an attempt is made to derive theoretical models

using the test results of previous chapter.

6.2 MATERIAL PROPERTIES

PCRC beams consist of two construction materials, each with

its own unique constitutive and strength properties and relationships.

Those considered herein are the concrete and perforated cold-formed

steel sheeting. The properties assumed for each are outlined in the

following sections.

6.2.1 Constitutive Behaviour of Concrete

Since the tensile strength of concrete is very low, its

contribution in the tension zone is neglected. The stress distribution of

156

concrete in the compression zone and the maximum strain were found

from the tests on concrete cylinders confined with prefabricated cage.

In compression, the uniaxial behaviour of concrete is found

experimentally for both confined and unconfined concrete cylinders.

Usually, the result obtained from the prediction equation is then

multiplied by a capacity reduction factor or factor of safety to reduce the

probability of failure within acceptable levels. In this investigation, it is

found from test results of confined cylinders (Chapter 3) that, the

strength of concrete due to Prefabricated Cage confinement is enhanced

by 1.2 to 1.8 times. This enhancement due to partial confinement is

taken in the following chapters as a partial safety factor. Hence, a factor

of safety is not added in the analysis.

For ultimate strength analysis that deals with the material

failure of concrete, the stress-strain relationship is assumed to have an

idealised stress distribution as shown in Figure 6.1, where the tensile

strength of the concrete is assumed to be zero.

6.2.2 Constitutive Behaviour of Cold-Formed Steel

The primary purpose of the sheet element in a composite beam

is to carry tensile stresses. Cold formed steel is elastic for a certain

region, and then its stress strain relationship becomes nonlinear as the

onset of plasticity is developed. The idealized stress-strain

representation, which was used to develop the model, is shown in

Figure 6.1.

157

Figure 6.1 Stress Strain Behaviour of Steel and Concrete

6.3 DEVELOPMENT OF MODEL FOR FLEXURAL

STRENGTH

6.3.1 Design Methodology

It has been shown that linear elastic analysis can be applied at

working or service loads to control both serviceability limit states such

as deflection and crack widths and the ultimate limit state of fatigue.

Linear elastic analysis can also be applied at ultimate loads to determine

the onset of buckling, or when the yield first occurs. However, in order

to determine the maximum possible strength of a composite beam, the

nonlinear techniques such as rigid plastic analysis (Deric Oehlers 1993,

Deric Oehlers and Mark Bradford 1995) is used.

The rigid plastic strength is, therefore, an upper bound to the

strength of the composite beam and requires that all other modes of

failure do not occur prior to this strength being achieved. It is, therefore,

fy or fck

STRESS

STRAIN

158

necessary here to ensure that premature failure through the following

modes does not occur:

a) Local buckling of steel elements

b) Failure of concrete due to concentrated dowel loads

imposed in it by the shear connectors

Furthermore, it is also necessary to ensure that the composite

beam has a sufficient rotational capacity to attain ultimate strength.

6.3.2 Assumptions Made in Developing the Analytical Model

In deriving expressions for flexural strength of a Prefabricated

Cage Reinforced Concrete beam using rigid plastic analysis, the

following five major assumptions are made.

1. Plane sections remain plane even after bending.

2. The stress-strain curve for cold formed sheet is the same

both in tension and compression.

3. Tensile strength of concrete is neglected.

4. Compressive stress distribution is represented by a

rectangular stress block.

5. The steel in the compression zone is neglected in the

moment of resistance.

The analysis and design of structures in flexure are based on

the satisfaction of the three basic sets of relationships, which must be

159

applied, to any problem of structural mechanics or stress analysis. They

are:

(a) Material properties (i.e., constitutive behaviour).

(b) Equilibrium of forces and

(c) Compatibility of strains

Making use of the above assumptions and the three basic sets

of relationship, the equations are developed. Analytical models have

been developed based on the models proposed by Deric Oehlers (1993)

for Profiled Composite beams and Hussain et. al. (1995) for Thin

Walled Composite beams.

6.3.3 Rigid Plastic Analysis

The PCRC beam is assumed to exhibit full interaction and

hence, there is no slip across the steel-concrete interface due to the bond

offered by the slots available in the prefabricated cage. The same strain

distribution will exist in sheeting and concrete with neutral axis of both

steel and concrete section, and coincident with each other when full

interaction is developed.

Based on the above assumption, the PCRC beams were

analysed using the rigid plastic analysis, and equations were developed

for the theoretical moment of resistance of three different profiles of

prefabricated cage used in the experimental study. The theoretical

results were validated with the experimental results and tabulated in

Table 6.1.

160

6.3.3.1 Theoretical Moment of Resistance of PCRC Beams with

Profile I

The stress diagram of concrete and sheet are given in

Figure 6.2.

Figure 6.2 Rigid Plastic Analysis of PCRC Beam with Profile I

Fundamental Equations

(a) Axial Strengths

Axial strengths of the various components were calculated

using the following expressions:

Axial Tensile force of steel sheet T= T1 + T2

T =2fyts(bt+dt) (6.1)

Axial compressive force of concrete C = f ckB Nc (6.2)

161

(b) Neutral Axis

Considering the steel sheets that are at yield, the equation for

Nc can be derived as:

Depth of Neutral axis in concrete componentBf

TNck

c . (6.3)

(c) Ultimate Moment

Taking moment of all the forces about the top fibre of the

beam, the moment capacity (Mth) of the PCRC beams can be determined

from the expression:

2)(

)(22

21cck

ttsythBNfdddbtfM (6.4)


Profile II

The stress diagram of concrete and sheet are given in Figure 6.3.

Figure 6.3 Rigid Plastic Analysis of PCRC Beam with Profile II

162


(a) Axial Strengths



Axial Tensile force of steel sheet T =fyts(2bt1+bt2+2dt) (6.5)

Axial compressive force of concrete C = f ckB Nc (6.6)

(b) Neutral Axis




TNck

c . (6.7)

(c) Ultimate Moment




2)(

2)2(2

2121cck

tttsythBNfdddbbtfM (6.8)


Profile III

The stress diagram of concrete and sheet are given in Figure 6.4.

163

Figure 6.4 Rigid Plastic Analysis of PCRC Beam with Profile III


(a) Axial Strengths



Axial Tensile force of steel sheet T=T1+T2+T3

T = 2fyts(bt1+bt2+dt) (6.9)

Axial compressive force of concrete C= f ckB Nc (6.10)

(b) Neutral Axis




TNck

c . (6.11)

164

(c) Ultimate Moment




2)(

22

32211cck

tttsythBNfdddbdbtfM (6.12)

Table 6.1 Comparison of Experimental and Theoretical Results

BeamSeries Profile Experimental

Moment (kNm)Theoretical

Moment (kNm) Mexp/Mtheo

A1I

9.144 7.995 1.14A2 10.478 10.494 1.00A3 13.335 13.635 0.98B1

I9.779 8.064 1.21

B2 10.922 10.619 1.03B3 13.716 13.856 0.99C1

I10.16 8.106 1.25

C2 11.43 10.694 1.07C3 13.97 13.989 1.00D1

II9.589 8.130 1.18

D2 10.795 10.679 1.01D3 13.653 13.886 0.98E1

II10.351 8.187 1.26

E2 11.240 10.781 1.04E3 13.970 14.067 0.99F1

II10.795 8.249 1.31

F2 12.002 10.892 1.10F3 14.605 14.263 1.02

165

Table 6.1 (Continued)

BeamSeries Profile Experimental

Moment (kNm)Theoretical

Moment (kNm) Mexp/Mtheo

G1III

20.574 16.398 1.25G2 21.908 18.323 1.20G3 20.193 17.77 1.14H1

III21.717 16.694 1.30

H2 21.717 18.555 1.17H3 22.860 17.99 1.27I1

III23.813 16.880 1.41

I2 25.146 18.746 1.34I3 20.955 18.17 1.15J1

I18.860 16.635 1.13

J2 19.050 17.080 1.12J3 18.860 17.359 1.09K1

I18.288 16.793 1.09

K2 20.955 17.240 1.22K3 18.479 17.523 1.05L1

I18.479 16.919 1.09

L2 18.479 17.367 1.06L3 17.526 17.654 0.99

6.4 DEVELOPMENT OF MODEL FOR DEFLECTION

6.4.1 General

Composite beam is usually designed first for ultimate limit

states. Its behaviour in service must then be checked. For a simply

supported beam, the most critical serviceability limit state is usually

excessive deflection, which can govern the design when unpropped

construction is used.

166

For a composite beam with complete shear interaction between

the steel and the concrete components, there is no relative slip at the

steel/concrete interface. In practice, complete shear interaction is

assumed to coincide with the full shear connection. It may not always

be possible or necessary to have full shear connection.

While under-estimated deflection may result in serviceability

problem, an over estimated deflection would result in a rejection of a

design, which would have adverse financial effects on the project.

Therefore, accurate prediction of deflection in a reasonable effort

becomes increasingly important in a building design.

6.4.2 Assumptions

The following assumptions are made for analysis of deflection.

1. The behaviour of the beam is linearly elastic.

2. The curvature and vertical deflection of both the sheet and

concrete components are the same as those of the beam.

3. Deflections are small and shear deformation in the sheet,

and concrete components are neglected.

4. Provision of slots in the Prefabricated Cage provides

composite action between the sheet and concrete

components along with chemical bond.

5. Slip is negligible at the steel – concrete interface when the

loading is at service stage.

167

Figure 6.5 Transformed Section

6.4.3 Development of Model

The midspan deflection ( ) for the two point loading is

expressed as

)43(48

22 aLEI

Pa (6.13)

As shear span, a = L/4 in this study, Equation (6.13) can be

rewritten as

EIPL

76811 3

(6.14)

168

To determine the effective moment of inertia, transformed area

method is adopted. Figure 6.5 shows the transformed section.

cpptppgT AmAmAA )1()1( (6.15)

)2d

tD(Dd)2t

D(Db

)2dt(Dd)

2t(Db[1)tp2(m

2DBD.c.yTA

tsct

sct

cscc

sccs

(6.16)

Solving the above equation yields cy .

ty is found from ct yDy

Then the transformed moment of inertia is calculated as

follows:

])d

tty(tddt

)t

Dty(tbtb

)d

tcy(tdcdt)

tDcy(tb

tb)[pm(tBycBy

uI

tsst

tsscst

st

cssc

sscsc

sc

23

23

223

212212

212

3

21212

3

3

3

3

(6.17)

GtensionGcomp IIpmtycyBuI )1()33(

3 (6.18)

where modular ratio of steel sheet (mp) is,

cEpE

pm

169

The moment of inertia thus calculated is for an uncracked

transformed section. In the same way, the moment of inertia for cracked

transformed section is calculated as

33 tybII uc (6.19)

From the literature (Max Porter 1984) it is understood that the

effective moment of inertia for composite steel deck slabs can be used

as,

2uIcI

effI (6.20)

This recommendation for composite deck slabs is adopted

here. Hence, the effective moment of inertia for PCRC beam is given

by

)(1uIcIeffI (6.21)

Where, is a reduction factor.

The values of were calculated from experimental results and

are tabulated in Table 6.2. Using the regression analysis of experimental

results, the factor can be expressed in terms of percentage of steel ( )

and yield strength of steel (fy). The derived expression for is given

below,

= 5.646 0.735 (6.22)

170

Using this effective moment of inertia, the final deflection

becomes

effEIWl

eff3

.76811 (6.23)

6.4.4 Application to Simply Supported PCRC Beams

The analysis for deflection considers the effect of loading on

the serviceability aspects of the composite beam. Therefore, for the

purpose of comparison between the measured data and the theoretical

models, only measured data upto the onset of observable nonlinear

behaviour is included. The onset of observable nonlinear behaviour is

based upon change in the slope of the measured load – deflection plots.

The aforementioned procedure is applied in order to carry out a

parametric study about deflection of simply supported beams. Using the

proposed equation, the behaviour of PCRC beam in deflection was

arrived and shown in Figure 6.6 - 6.41. The test results from the

Chapter 5 were used to compare the theoretical predictions. The results

are in good agreement under service load.

Sectional properties used for this investigation is presented in

Table 6.2. Using the reduction factor deflection at various load levels

were calculated and plotted in Figure 6.6 through 6.41. The theoretical

deflection at service load (Pu/1.5) is tabulated against corresponding

experimental deflection in Table 6.2.

171

Table 6.2 Comparison of Experimental and Theoretical deflectionof PCRC Beam Series

Sl.No

BeamSeries

Ps(kN)

fck(N/mm2)

fy(N/mm2) exp theo

exp @Ps (mm)

the @ Ps(mm)

1 A1 24.0 22.75 245 4.12 4.13 2.75 2.622 A2 27.5 22.75 262 3.83 3.60 3.60 2.743 A3 35.0 22.75 279 2.93 2.94 3.18 2.564 B1 25.7 27.86 245 4.19 4.13 3.70 3.425 B2 28.7 27.86 262 3.51 3.60 3.35 2.616 B3 36.0 27.86 279 2.76 2.94 4.21 3.627 C1 26.7 32.23 245 4.18 4.13 3.06 3.118 C2 30.0 32.23 262 3.51 3.60 3.70 3.239 C3 36.7 32.23 279 2.75 2.94 3.60 3.0010 D1 25.2 23.05 245 4.2 4.13 3.07 3.3411 D2 28.3 23.05 262 3.6 3.60 3.86 3.3812 D3 35.8 23.05 279 2.88 2.94 3.94 3.0613 E1 27.2 27.21 245 4.11 4.13 3.60 3.4014 E2 29.5 27.21 262 3.76 3.60 3.20 3.4515 E3 36.7 27.21 279 2.7 2.94 4.20 3.3716 F1 28.3 33.78 245 4.17 4.13 4.53 3.8317 F2 31.5 33.78 262 3.64 3.60 4.12 3.5518 F3 38.3 33.78 279 2.96 2.94 3.25 3.5019 G1 54.0 33.10 245 2.85 2.57 4.37 3.4320 G2 57.5 33.10 262 2.56 2.39 2.93 3.1621 G3 53.0 33.10 279 1.92 1.79 3.00 2.1622 H1 57.0 38.80 245 2.69 2.57 3.90 3.5323 H2 57.0 38.80 262 2.53 2.39 3.34 2.9624 H3 60.0 38.80 279 1.88 1.79 3.17 2.3525 I1 62.5 45.20 245 2.53 2.57 3.78 3.2726 I2 66.0 45.20 262 2.35 2.39 3.60 3.2927 I3 55.0 45.20 279 1.75 1.79 2.51 2.1328 J1 49.5 32.80 397 2.54 2.67 3.30 3.8129 J2 50.0 32.80 402 2.68 2.72 4.10 3.8230 J3 49.5 32.80 404 2.52 2.71 2.90 3.4231 K1 48.0 38.30 397 2.71 2.67 4.28 3.9732 K2 55.0 38.30 402 2.78 2.72 5.20 4.8033 K3 48.5 38.30 404 2.57 2.71 4.20 3.8934 L1 48.5 44.20 397 2.32 2.67 2.80 3.0535 L2 48.5 44.20 402 2.84 2.72 3.80 3.8736 L3 46.0 44.20 404 2.81 2.71 4.20 3.94

Ps = Assumed Service load taken as ultimate load/1.5

172

Figure 6.6 P- of A1 Series Figure 6.7 P- of A2 Series

Figure 6.8 P- of A3 Series Figure 6.9 P- of B1 Series

Figure 6.10 P- of B2 Series Figure 6.11 P- of B3 Series

0

10

20

30

40

50

60

0 50 100Deflection in mm

Exp

theo

0

5

10

15

20

25

30

35

40

0 20 40 60 80Deflection in mm

Exp

theo

05

1015202530354045


Exp

theo

173

Figure 6.12 P- of C1 Series Figure 6.13 P- of C2 Series

Figure 6.14 P- of C3 Series Figure 6.15 P- of D1 Series

Figure 6.16 P- of D2 Series Figure 6.17 P- of D3 Series

05

1015202530354045

0 20 40 60Deflection in mm

Exptheo

0

10

20

30

40

50


Exp

theo

0

10

20

30

40

50

60


Exp

theo

174

Figure 6.18 P- of E1 Series Figure 6.19 P- of E2 Series

Figure 6.20 P- of E3 Series Figure 6.21 P- of F1 Series

Figure 6.22 P- of F2 Series Figure 6.23 P- of F3 Series

0

10

20

30

40

50


Exp

theo

0

10

20

30

40

50

60


Exp

theo

05

1015202530354045


Exp

theo

0

10

20

30

40

50

60


Exp

theo

175

Figure 6.24 P- of G1 Series Figure 6.25 P- of G2 Series

Figure 6.26 P- of G3 Series Figure 6.27 P- of H1 Series

Figure 6.28 P- of H2 Series Figure 6.29 P- of H3 Series

0

20

40

60

80

100


Exptheo

0102030405060708090


Exptheo

176

Figure 6.30 P- of I1 Series Figure 6.31 P- of I2 Series

Figure 6.32 P- of I3 Series Figure 6.33 P- of J1 Series

Figure 6.34 P- of J2 Series Figure 6.35 P- of J3 Series

0102030405060708090


ExpTheo

0

10

20

30

40

50

60

70

80


Exp

Theo

177

Figure 6.36 P- of K1 Series Figure 6.37 P- of K2 Series

Figure 6.38 P- of K3 Series Figure 6.39 P- of L1 Series

Figure 6.40 P- of L2 Series Figure 6.41 P- of L3 Series

0

10

20

30

40

50

60

70

80


Exp

Theo

0

10

20

30

40

50

60

70

80

0 20 40 60Deflection in mm

Exp

Theo

178

6.5 DEVELOPMENT OF MODEL FOR DUCTILITY

6.5.1 General

A structural member is to be designed with sufficient ductility

capacity to avoid brittle failure in flexure and ensure a ductile behaviour,

especially for seismic design. The current philosophy of seismic design

of moment-resisting reinforced concrete frames is based on the

formation of plastic hinges at the critical sections of the frame under the

effect of substantial load reversals in the inelastic range. The ability of

the plastic hinges to undergo several cycles of inelastic deformations

without significant loss in its strength capacity is usually assessed in

terms of the available ductility of the particular section.

The ductility capacity of a section can be expressed in the form

of curvature ductility factor ( ). The moment-curvature analysis

performed under monotonically increasing load represents only the first

quarter-cycle of the actual hysteretic behaviour of the plastic hinge on

rotation under the earthquake loading. Therefore, of a section

calculated under such an assumption is considered a theoretical estimate

of the actual ductility that can be supplied by the section when subjected

to an actual earthquake loading.

However, the theoretical estimation of under monotonic

loading is widely used as an appropriate indicator of the adequacy of the

earthquake resistant design of RC members. It is even used for

prediction of the damage level in frames under earthquake loading.

Although, the actual response of the frame subjected to severe

earthquakes is complex and involves large uncertainties, prediction of

179

damage is usually made in terms of ductility demand on individual

members.

Improving the ductility of the materials generally leads to an

improvement in the ductility of the section. Based on a closer

examination of published literature, ductility of concrete is being

improved at present by confining it using steel binders, as ties in

compression members and as closely spaced stirrups in beams. The

ductility of structures can be improved by adding compression steel in

the concrete section or by confining the compression zone which leads

to an improvement in the ductility of the material. The concrete confined

in such a way is called confined concrete or ductile concrete (Park and

Dai Tuitong 1988, Park and Paulay 1975).

In PCRC beams, compression zone is partially confined by the

perforated steel sheet. This aids in confinement, which helps to increase

ductility factor.

6.5.2 Methods of Defining Ductility

The term ductility is generally quantitatively described by a

parameter called “ductility factor” or “ductility ratio”.

The most common measures of ductility are:

The inelastic deformation u y.

Cross sectional ductility in terms of Ductility factor

180

Two types of ductility factors were employed in this study:

a) Displacement ductility factor (µ = u / y).

where,

uis the displacement at ultimate load and

y is the displacement at which tension steel yields.

b) Curvature ductility factor (µØ = Øu / Øy)

where,

Øuis the curvature corresponding to u, and

Øy is the curvature at which tension steel yields.

The energy absorbed by the element or structure as given

by the area under the force deformation diagram.

Rotation capacity

Limiting value of neutral axis depth.

6.5.2.1 Method: 1 Inelastic Deformation

The length of an inelastic zone formed in the load – deflection

response is usually defined as inelastic deformation. The idealized

inelastic zone in a beam is shown in Figure 6.42. This is calculated from

experimental results and is presented in Table 5.5.

181

Figure 6.42 Idealized load – Deformation Curve

6.5.2.2 Method:2 Cross Sectional Ductility of PCRC Beam

The ductility of beams may be defined in terms of the

behaviour of individual cross-sections or the behaviour of entire beams.

The cross sectional ductility is widely used as a measure of beam

ductility represented as the ductility factor.

6.5.2.3 Calculation of Yield and Ultimate Curvatures

A typical PCRC beam section with its corresponding strain

distribution at the yield and ultimate stages are shown in Figures 6.43

and 6.44.

As all the beams exhibit nearly full shear connection at

ultimate, for the calculation of the ductility factor, slip in these beams is

ignored. The curvature calculations are given by the following

equations:

182

a) At yield stage

Figure 6.43 Stress-strain Diagram at Yield Stage

Yield Curvaturey

yy N

(6.24)

where Ny can be determined by assuming the linear stress distribution

and satisfying the equilibrium condition of forces. This leads to:

The neutral axis (Ny) at the yield state is obtained from the

equilibrium condition which is given in Equation (6.25).

TC

For Profile I,

ttsyyck bdtfBNf 2

ck

ttysy Bf

bdftN

)(2 (6.25)

183

For Profile II,

fckBNy= fyts(2bt1+bt2+2dt)

ck

tttysy Bf

bbdftN

)22( 21 (6.26)

For Profile III,

fckBNy= 2fyts(bt1+bt2+dt)

ck

tttysy Bf

bbdftN

)(2 21 (6.27)

where,

ckf - cube compression strength at 28th day in MPa

yf - yield stress of steel sheet in MPa

At ultimate stage

Ultimate Curvatureu

uu N

(6.28)

where,

u = Concrete compressive strain at crushing of concrete or

at ultimate moment

Nu = Depth of compression zone at ultimate, can be

determined by satisfying equilibrium equation of forces.

184

For Profile I,

)

= 2 ) (6.29)

For Profile II,

[ ]

[2( ) ] (6.30)

For Profile III,

)[ ]

= 2 ( ) (6.31)

where,

d= Effective depth

Ast= Area of tension reinforcement

185

Figure 6.44 Stress – Strain Diagram at Ultimate Stage

From strain compatibility conditions,

)( u

s

u

cu

NdN (6.32)

=

=

(6.33)

From equilibrium conditions,

21 TTC

styuck AfBNf

(6.34)

186

= 0 (6.35)

; ;

02 cbNaN uu (6.36)

Solving Equation (6.36), Nu can be obtained as

= ±

The addition of compression reinforcement in the form of

continuous sheet to a beam will shift the neutral axis upwards and

increase the ultimate curvature substantially, although it has a little

effect on its yield strength or yield curvature. Hence, the curvature

ductility factor is defined as,

y

u (6.37)

ssy

y

u

cu Ef

NdN

)( (6.38)

Using Equation (6.38), curvature ductility factor (CDF) for the

sections tested in Chapter 5 was evaluated and are presented in

Table 6.3. The ratio between the experimental and theoretical ductility

factor shows the accuracy of the equation.

187

Table 6.3 Theoretical and Experimental Ductility Factor

Sl.No BeamID

Neutral axis Curvature ductilityfactor

Ratio

THE

EXP

,

,Nu

(mm)Ny

(mm)Theory( T, )

Expt.( E, )

1 A1 37.03 14.93 12.19 12.93 1.062 A2 40.49 19.96 9.91 10.88 1.103 A3 44.72 26.57 8.12 8.65 1.074 B1 35.42 12.19 12.98 13.93 1.075 B2 38.76 16.30 10.61 11.03 1.046 B3 42.84 21.70 8.78 8.97 1.027 C1 34.30 10.54 13.55 14.34 1.068 C2 37.55 14.09 11.11 11.28 1.029 C3 41.53 18.76 9.24 9.58 1.04

10 D1 37.23 14.74 12.34 13.12 1.0611 D2 40.71 19.70 10.03 10.21 1.0212 D3 44.97 26.23 8.24 8.91 1.0813 E1 35.90 12.49 12.93 14.10 1.0914 E2 39.28 16.69 10.61 11.85 1.1215 E3 43.42 22.22 8.77 9.06 1.0316 F1 34.23 10.06 13.83 14.05 1.0217 F2 37.47 13.44 11.36 11.18 0.9818 F3 41.45 17.90 9.46 9.54 1.0119 G1 49.01 18.66 8.37 8.37 0.9720 G2 49.20 17.35 8.33 8.33 0.9721 G3 52.38 18.70 8.50 8.13 0.9622 H1 47.41 13.27 9.23 9.23 1.0023 H2 47.60 14.80 9.04 9.04 1.0024 H3 50.69 15.95 8.93 8.93 1.0025 I1 45.92 11.39 9.83 9.83 1.0226 I2 46.09 12.70 9.34 9.34 1.0027 I3 49.11 13.70 9.37 9.10 0.9728 J1 38.61 21.17 7.20 9.41 1.3129 J2 38.67 21.24 7.17 8.58 1.1930 J3 39.11 21.55 7.15 8.16 1.1431 K1 37.32 18.13 7.60 9.07 1.1932 K2 37.37 18.19 7.56 8.66 1.1533 K3 37.80 18.46 7.55 8.24 1.0934 L1 36.16 15.71 7.971 8.70 1.0935 L2 36.21 15.76 7.93 8.27 1.0436 L3 36.63 15.99 7.91 8.08 1.02

188

6.6 KEY FINDINGS

From the analytical investigations carried out in the present

study, the following major findings can be arrived at.

The theoretical models have been developed for flexural

strength using rigid plastic analysis. The proposed model

incorporates the enhanced compressive strength of

concrete due to confinement, and full shear connection is

assumed between perforated sheet and concrete.

The analytical expressions developed are in close

agreement with experimental results in predicting the

ultimate strength in bending.

The deflection equation is developed considering

uncracked and cracked transformed moment of inertia. A

reduction factor ( ) is introduced in the deflection

equation. An equation is also derived for in terms of

percentage of steel and yield strength of steel using

regression analysis.

Beyond service load level, theoretical deflection is started

to vary from the actual deflection.

From equilibrium forces and strain compatibility

conditions at yield stage and at ultimate stage, the

equation for curvature ductility factor is arrived. The

experimental and theoretical ductility factors are in good

agreement.

chapter 6 theoretical models for flexure, deflection and...

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