elasticity theory and design of flexible supports for

TN-6’7-9 Knut Skarpaas March 1967

further disclosed without approval

ELASTICITY THEORY AND DESIGN OF FLEXIBLE SUPPORTS

FOR STANFORD TWO-MILE LINEAR ACCELERATOR

ACKNOWLEDGEMENT

The author wishes to express his appreciation to his advisor, Professor

James N. Cbodier, for his guidance and helpful criticism in the presentation of

this thesis.

The author also wishes tc extend his appreciation to Stanford Linear

Accelerator Center for its help in providing test facilities and computer time,

and for typing the final manuscript.

. . . - 111 -

TABLE OF CONTENTS PAGE

INTRODUCTION . . . . . . . . . 0 . . . . . 0 . . . . . . . o . . 2

DESIGNCRITERIA... . . . . . . . a . . . . . . . . . DO... 9

GENERAL ASSUMPTIONS. . . . o . . . . . . . . . . 0 . . . . . . 13

DEVELOPMENT OF FORMULAS FOR THE COMPUTER PROGRAM

Description of the Flexible Support Assembly . . . . ,, . . . . . . 15

Vertical Deformation of Tilted Plate Column . . . 0 0 0 . . ., . ., 18

Shortening of the Flexible Sheets Caused by Axial Deflection ,, . 0 . 20

Derivation of Internal Load P . . . . 0 . D D . D . 0 o D 0 0 . . 24

Listing of the Dependent Variables . D . . . 0 . , , . 0 D . ., . . 25

Operating Range of the Flexible Support Assembly . 0 . 0 0 . 0 D . 27

Basic Dimensions and Material Properties of the Flexible Support

Assembly D . . 0 . . . . 0 . . . . 0 . . . . . 0 0 . . . . . . 29

DESIGN FROM COMPUTER DATA

Choice of Design Based on Computer Data . . . 0 D . . . . 0 . . 33

Stiffness Variations Caused by Change of Initial Interference . . . . 42

Change of Stiffness by External Load . . . . . . . . . 0 . D 0 . . 42

TESTS ON FLEXIBLE SUPPORT ASSEMBLIES

Test Objectives and Primary Test Results a 0 . . D ., D . . . . . 47

Test Assemblies ........................ 50

Test Apparatus and Procedure .................. 51

The Effective Interference in the Flexible Support Assembly . . . . 54

The Equivalent Sheetlength in the Flexible Support Assembly . o . . 56

Discussion of Test Results o . . 0 . . D . . . 0 ., 0 . . , . . . 56

- iv -

.

PAGE BIBLIOGRAPHY.... . . . . . *..a0 . . . . . . . . . . . 0 68

APPENDIX A

Computer Program for the Flexible apport Assembly . . 0 . . . . 70

APPENDIX B

Tabulated Data for the Optimum Design Chosen From

ComputerData . . . . . . . . . . . . . . . . . . . o . o . . 73

APPENDIX C

Data to Determine the Equivalent Sheetlength . . . , . . . . . . 0 75

-v-

L I

I

LIST OF FIGURES

FIGURE PAGE

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

Typical Cross Section of Stanford University Two-Mile

Linear Accelerator ........... .......... 5

Ten-Foot Accelerator Section Mounted on Aluminum Extrusion . . 6

Forty-Foot Accelerator Section. ............... 7

Design Loads and Axial Displacements at the Free

Downstream End of the Forty-Foot Girder ........... 11

Allowable Transverse Displacements ............. 11

Scale Cross Section of Flexible Support Assembly ....... 16

Flexible Support Assembly Prior to Assembly ......... 17

Flexible Support Assembly After Assembly .......... 17

Flexible Support Assembly with External Load ......... 17

Flexible Support Assembly Deflected Axially ......... 17

Free Body Diagram of the Plate Column ............ 19

The Bearing Surfaces .................... 19

Free Body Diagram for the Top Half of the Flexible

Support Assembly ..................... 22

Test Fixture ........................ 52

Test Stand ......................... 53

Transition Zone of the Flexible Sheets ............ 76

- vi -

GRAPHS

LIST OF GRAPHS

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

Deflection Versus Sideload, for Supports Which Have

Plate Columns With l/2-Inch End Radii , . . . . . . . . .

Deflection Versus Overall Decrease of Height, for Supports

Which Have Plate Columns With l/2-Inch End Radii. . . . .

Deflection Versus Sideload, for Supports Which Have Plate

Columns With 3/8-Inch End Radii . . . . . . . . . . . .


Which Have Plate Columns With 3/8-&h End Radii . , . .

Deflection Versus Sideload, for Supports Which Have Plate

Columns With l/4-Inch End Radii . . . . . . . . . . . .


Which Have Plate Columns With l/4-Inch End Radii . . . . .

Deflection Versus Sideload, for Different Values of the

Initially Introduced Interference 0 . . . . . . . . . . . .

Deflection Versus Axial Stiffness, for Several External Loads

Deflection Versus Overall Decrease of Height Plotted

From Primary Test Data . . . . . . . . . . . . . . . .

The Effect of Surface Roughness on one of the Earlier

Aluminum Test Assemblies . . . . . . . . . . . . . . .

Deflection Versus Sideload for the Test Assembly Which Had

a 3-Jnch-Long Plate column With 3/8-Inch End Radii. The

Effective Interference of This Assembly was .0029 Inch . . .

PAGE

. . 34

D . 35

. l 36

. l 38

. . 40

. . 41

. . 43

. . 45

. . 48

. . 59

. . 60

- vii -

GRAPHS PAGE 12. Deflection Versus Sideload for the Test Assembly Which

Had a 2-l/2-Inch-Long Plate Column With 3/8-Inch End

Radii. The Effective Interference for This Assembly was

. 0045Inch . . . . . . . . . . . . . . . . . . . . . . . . 61

13. Deflection Versus Sideload for the Test Assembly Which Had

14.

a 2-Inch-Long Plate Column with 3/8-Inch End Radii. The

Effective Interference in This Assembly was .0046 Inch . . . , 62

Deflection Versus Sideload for the Test Assembly Which Had

a 1-l/2-Inch-Long Plate Column With 3/8-Inch End Radii. The

Effective Interference for This Assembly was .0045 Inch . . . 63

15. Deflection Versus Sideload for the Test Assembly Which

Had a 2 -l/2-Inch-Long Plate Column With 3/8-Inch End

Radii. The Effective Interference for This Assembly was

. 0012 Inch . . . . . . . . . . . . , . . . . . . . . . . . 64

. . . - Vlll-

LIST OF TABLES TABLE PAGE

I. Effective Interference for the Test Units . . . . . . . . . . 0 . 55

II . Tabulated Data for the Optimum Design Chosen From

Computer Data . . . . . . . . . . . . . . . . . . . . . . . 73

III. Table to Determine the Width of the Constant Thickness Strip . . . 76

INTRODUCTION

INTRODUCTION

The Stanford University two-mile linear accelerator is a high energy,

20-40 billion electron volts, research tool used for particle physics. Figure I

shows the major components in a typical cross section. The accelerator pipe

which contains the beam of accelerating particles is a 10, OOO-foot long four-

inch diameter copper pipe. An underground tunnel which has a ten foot by

eleven foot cross section, is the housing for this accelerator pipe, its adjust-

able support and alignment system, and its power feeding waveguides.

The four-inch diameter accelerator pipe is fabricated in ten-foot sec-

tions, which are so weak that they cannot support their own weight. These

ten-foot sections are therefore, immediately after fabrication, permanently

mounted on ten-foot aluminum extrusions, see Figure 2. These extrusions

are designed to protect the accelerator sections and their input and output

waveguides during processing and in the final installation.

Four of these ten-foot accelerator sections, each on an aluminum ex-

trusion are then aligned and mounted on forty-foot aluminum support girders,

see Figure 3. These girders all contain diffraction alignment targets which

are used in the continuous alignment of the accelerator. Finally the forty-foot

accelerator sections are mounted into the accelerator housing on remotely ad-

justable supports, and connected to the power feeding waveguides.

Even though the electron beam can be steered through the accelerator

when the structure is not perfectly aligned, it is desirable, both for the ease

of operation, and for the preservation of good geometrical qualities of the

beam, to keep the accelerator as straight as possible. A maximum transverse

-2-

deviation of .040 inch from a true straight line has been given for the align-

ment of the axis of the accelerator pipe over the two-mile length. After the

initial alignment, the accelerator may move outside of the alignment tolerance

periodically due to earth motions in the region of the accelerator; these mo-

tions may be caused by rebound, fill settlement, moisture variations or deeper

geological phenomena, During the necessary realignment relative motion be-

tween the klystron gallery and the accelerator pipe takes place. These rela-

tive motions introduce variable reactions through the waveguides to the accel-

erator pipe at four locations on each forty-foot girder. Since realignment

only is possible at each end of these forty-foot girders, adequate stiffness

must be provided throughout the girders and by the individual accelerator pipe

supports, to keep transverse deformation within set limits.

In addition to the transverse alignment tolerance described above,

there is also an axial location tolerance. This tolerance which applies to the

input end of each ten-foot section, is o 010 inch, and must be maintained during

operation of the accelerator. This location accuracy is achieved by mounting

the input end of the first of the four accelerator sections rigidly to the forty-

foot girder. At this end the girder is in turn rigidly tied to the ground. The

final location of the successive three accelerator sections on each girder is

accomplished by mounting them rigidly end to end, and by controlling their

temperature to within three degrees Fahrenheit.

While the accelerator pipe temperature is held nearly constant, the

forty-foot aluminum support girder temperature varies with the ambient accel-

erator housing temperature. Rigid mounting of the copper accelerator pipe to

-3 -

the aluminum girder would therefore cause high thermal stresses in the accel-

erator pipe, and bend the forty-foot section beyond the alignment tolerances.

To minimize the stresses in the accelerator pipe, and the bending of the forty-

foot section, flexible support assemblies with low axial stiffness are used to

mount the pipe to the girder.

Therefore, the flexible support assemblies which mount the accelerator

pipe to the girder must have high transverse stiffness to carry transverse loads

from the waveguides with minimum transverse deformations. These supports

must also have low axial stiffness to minimize the stresses in the accelerator

pipe, and sufficient operating range to allow the total thermal differential

expansion.

-4 -

30 feet of fill , i’.;~ffjj Settlement introd&%

reactions at t h&‘*& accelerotor.:,k

FtG.1 --TYPICAL CROSS SECTION OF STANFORD UNIVERSITY TWO-MILE LINEAR ACCELERATOR

in

- 5 -

ortY-foot 8.lm-jnm Support girder

Alumi ntlm ,.--L

iffraction alignmc,nt

FIGURE 3 "OR~-JWr ACCELFRATok SECTION

DESIGN CRITERIA

DESIGN CRITERIA

The supports which mount the accelerator pipes to the ten-foot alumi-

num extrusions must satisfy requirements related to size, load carrying ca-

pacity, axial flexibility, maximum transverse deformations, endurance and

safety.

The space available for these supports between the accelerator pipe

and the aluminum extrusion is 5-l/2 inches high and 8-3/4 inches wide, as

shown on Figure 5. The largest loads which any support might carry during

operating conditions are shown on Figure 4 as a horizontal load Q,, a vertical

load Qz and a moment M about the accelerator axis. Figure 4 also shows, by

dotted outline, the axial relative displacement Y which occurs between the end

of the forty-foot aluminum girder and the above mounted accelerator pipe dur-

ing adverse temperature conditions. Of the maximum transverse deviation of

0 040 inch, from a true straight line, which was required for the alignment of

the accelerator pipe, approximately one quarter has been allowed through the

supports which join the pipe to the aluminum extrusions. The direction and

maximum values of these transverse displacements are also shown on Figure 5.

Specific design criteria referring to Figures 4 and 5 are listed below.

1. When the accelerator pipe is displaced axially .275 inch, under op-

erating conditions, no combination of the loads shown on Figure 4

shall cause total transverse displacement exceeding the maximum

displacements shown on Figure 5.

2. When the accelerator pipe is displaced axially .375 inch, during

transportation or other temporary condition, no combination of the

loads shown on Figure 4 shall cause total transverse displacements

-9-

exceeding twice the maximum displacements shown on Figure 5.

3. The maximum accumulated axial thrust from all supports used on

one forty-foot girder should preferably not exceed 750 pounds dur-

ing accelerator operating conditions 0

4. The supports shall be capable of withstanding shock loads of two

times the design load without major permanent deformations.

5. After the supports have been cycled 600 times with full design load,

between plus and minus ,375 inch, they shall still meet criteria 1,

2, 3, and4.

6. The supports shall require no maintenance.

7. The support assembly shall be made from non-magnetic materials.

- 10 -

Q, = f 500 pounds

M = f 7600 inch pounds /ffJ$A

Accelerator

Yis relative axial displacement between accelerator pipe and

FIGURE 4

DESIGN LOADS AND AXUL DISPUCmmS AT

THE FREE DOWXSTREiAM END OF THE FORTY-FOOT GIRDER

f.010 inch

2.0002 radian t.008 inch

FIGURE 5

ALLOWABLE TRANSVERSE DISPLSLCEMENTS

- 11 -

GENERAL ASSUMPTIONS

GENERAL ASSUMPTIONS

In the analysis which follows, unless otherwise stated, it will be as-

sumed that:

(a)

O-4

(cl

(d)

(e)

(9

(g)

(h)

0)

The assembly is in its final installation, mounted in such a way

that no relative rotation can take place between the upper and the

lower mounting block during axial deflection of the assembly.

The effective column tilting angle a! (radians) is always smaller

than the static coefficient of friction between the plate column and

the mounting blocks.

The dimension Ls represents the equivalent length of the sheet

which takes into account the transition radii at both ends of the

sheet, see Appendix C.

Shortening of the plate column caused by bending is disregarded.

Localized deformations of rolling surfaces are disregarded.

Rolling friction forces are disregarded.

The mounting blocks are considered rigid.

The elements of the assembly are made of materials which follow

Hooke’s Law.

The curvature of each sheet is given by:

1 M d2y - =-= r EI dx2

From assumption (a) it can be proved that the top and bottom halves of

both sheets have identical shapes, and are subjected to identical loadings. A

free body diagram of the top half of the flexible support assembly can therefore

be drawn as shown on Figure 13.

- 13 -

DEVELOPMENT OF FORMULAS FOR THE COMPUTER PROGRAM

.

DESCRIPTION OF THE FLEXIBLE SUPPORT ASSEMBLY

The support design which is used in the Stanford Linear Accelerator is

shown on Figure 6. This riveted assembly consists of two flexible sheets, an

upper and a lower mounting block, and a plate column. The plate column,

which is rounded at both ends, is inserted with a compressive preload to insure

that an external tensile load -Q will not cause separation between the plate col-

umn and the mounting blocks. Four schematic cross sections of the assembly

are shown on Figures 7, 8, 9 and 10. For clarification there is a subscript s

on all dimensions which refer to the flexible sheets, while all plate column

dimensions carry a subscript c.

Figure 7 displays several cross-sectional dimensions and the interfer-

ence Its prior to assembly. The assembling operation is performed by shrink-

ing the plate column in liquid nitrogen before installation. When the plate col-

umn expands in place an internal load PO develops as shown on Figure 8. This

internal load is carried as a compressive load on the plate column and as a

tensile load PO/2 in each sheet,,

The flexible assembly will after it has been installed carry an external

load Q while it is deflected axially Y as shown on Figure 10. During this deflec-

tion the plate column rolls on its ends and the sheets bend. The overall height

of the assembly usually decreases, and this decrease of height is at any time

the same for both the tilted plate column and the curved sheets.

The analysis which follows uses this phenomena of consistent deforma-

tions to develop an expression for the variable internal load P, and other de-

sign variables such as the decrease of vertical height, the maximum stresses

in the sheets and the axial stiffness.

- 15 -

Q pounds per inch

Plate column

Flexible sheet

Lower mounting block/ -

L I

F pounds per inch

-.375 inch maximum

axial deflection

t------- F pounds per inch

Q pounds per inch

FIGURE 6

SCALE CROSS SECTION OF FLEXIBLE SUPPORT ASSEMBLY

- 16 -

c

-l-r z- -

-J-l- FIGURE 7

FLEXIBU SUPPORT ASSEMBLY PRIOR TO ASSEMBLY

7 L s

_1

FIGURE 8

FLEXIBLE SUPPORT ASSEMBLY AFTER ASSEMBLY

FIGURE 9

FLEXIBLF: SUP~OR'I' ASSEMBLY W ITH EXTERNAL LOAD

FIGURE 10

FLEXIBLF, SUPPORT ASSEMBLY DEFLECTED AXIALLY

- 17 -

VERTICAL DEFORMATION OF TILTED PLATE COLUMN

The total decrease of vertical height of the plate column is determined

by its final angular position p, and the compressive load it carries in this posi-

tion, see Figure 11.

The decrease of vertical height of the plate column, which is caused by

the tilting from vertical to an angular position P, is called Act . An expression

for this deformation in terms of the parameters shown on Figure 11 is given by

A Ct = Lc (1 - cos p) (1)

To determine the decrease of vertical height of the plate column which

is caused by the compressive load we also refer to Figure 11. Since P is the

total instantaneous vertical tension in the two sheets, the plate column will be

compressed by a vertical load (P + Q). The couple of opposing loads, each of

the value (P + Q), produce a moment on the plate column. This moment is bal-

anced by two friction forces (P + Q) tan Q which automatically develop to bring

the plate column to equilibrium. The resultant G is given by

G = (P + Q)/COS a! (2)

The effective length of the plate column is the distance between the points of

contact and is given by the following expression

2Rcos a!+Lccos (P -a) (3)

The effective cross section of the plate column is given by

Tc/cos (P - a) (4)

The shortening of the plate column between the contact points D and E is called

‘DE’ This deformation is expressed below by the average compressive strain ec

- 18 -

R c -

T- L

I-

7 c

y

FIGURE 11

FREE BODY DIAGRAM OF THE PIKCE COLUMN

FIGURE 12

THE BEARING SURFACES

- 19 -

of the plate column and its effective length

‘DE = cC[2Rcosa + Lccos (P - CY,~ (5)

Using the notation EC for the modulus of elasticity for the column material and

expressions (2) and (4) for the force and effective cross section respectively,

one can now write

S = P + Q cos (P - 01) DE cos a! TCEC

[2R cos a! + Lc cos (P - a)3 (6)

The vertical component of SDE , which is caused by the compressive load in the

column is called Act and is given by the following expression

A = (P + Q) cos (P - a) cc TcEc

[2Rcosa! + Lc cos (hi)] (7)

The expression for the total decrease of vertical height of the plate col-

umn, which is caused by tilting and compression, is called AC and is given by

the following expressions

or

Ac=Act + Act

Ac=Lc(l-co@) I- (P+Q)cos ‘-Ol) TcEc

[2Rcos~~+L~~os@-~~j) (8)

SHORTENING OF THE FLEXIBLE SHEETS CAUSED BY AXIAL DEFLECTION

The total decrease of vertical height of the flexible sheets is determined

by the axial deflection of the assembly Y, and the vertical tensile load P/2

which each sheet must sustain at this deflection.

To determine the shortening caused by the bending of the sheets, their

- 20 -

deflection curve must be known. However, because of symmetry, only one

half of one sheet need be considered. Figure 13 shows a free body diagram for

one half sheet. When the coordinate axis, deformation and loads are taken as

shown on this figure, the bending moment at the cross section mn is

M=F+(P+Q)tana x P -- 2 2y

The differential equation of the deflection curve is then

-F+(P+Q)tana x 2

Using the notations

P 2EI

= k2

and

F f (P + Q) tan a! 2EI = k2C

(9)

(10)

(11)

The ordinary differential equation can be rewritten as

Y” -k2y = -Ck2x

And the general solution of the deflection curve is then

y = A cash kx f B sinh kx - Cx

With the integration constants A and B determined from the boundary conditions,

the deflection curve and its first derivatives are’

1 kH cash 2

(12)

(13)

- 21 -

\\\\\\

1 m

-

-1 I

a

y"=-c k cash =

sinh kx (14) 2

The decrease of vertical height of the flexible assembly A,sb, which is

caused by bending of the sheets, can now be calculated from

H/2 A sb = 2

f + y12 dx

0

Using the notation

s = cosi~ The expression for the shortening of the flexible sheets is

H/2 A sb = J

C2 (1 - S cash k~)~ dx 0

Performing this integration, one obtains

A sb

(15)

(16)

For the stretching of the sheets one makes use of the knowledge that the

average tensile strain is constant along the sheet under the action of the load

P/2 as long as the deflection is small. This elongation Ast caused by the ten-

sile load can therefore be expressed as

pLS A =ZTE st s s

(17)

where Es is the modulus of elasticity of the sheets.

The total decrease of vertical height of the flexible sheets is now given

by the sum of expressions (16) and (17)

- 23 -

‘s=%b -Ast

or

n,=c2 2

F - F si.nhk++ &- (sinhkH+kH) I

pLS - 2TsEs

(18)

DERIVATION OF INTERNAL LOAD P

An internal load P develops during the assembly of the flexible supports.

This internal load is uniquely determined by ,0, the tilting angle of the plate col-

umn, and the initial interference Its. To develop an expression for the load P,

one may use the following equation of consistent deformations for the assembly

When the expressions (8)

AC and As, respectively,

I cs = AC - As

and (18) are substituted into the above equation for

and solved for P, one obtains

P = i Its - Lc(l -cosp)+c2 $ - y sinh

[ F i-‘& (sinhkH+kH) 1

Q -T cos (P - a) c c [

2R cos CY f Lc cos (p - a)

‘I/

LS

2TsEs + cos (P - al

TcEc [2 R cos CY + Lc cos (p - cz)]

(19)

Equation (19) expresses the load P in terms of several constants and vari-

ables . In this case Its, Lc, Ls , R, Tc and Ts are geometrical constants

while Es and EC are material constants. Of the variables which are

- 24 -

(Y , p, k, C , S and H, any one can be chosen as the only independent variable.

The tilting angle of the plate column p was chosen in this case.

LISTING OF THE DEPENDENT VARIABLES

The total axial deflection of the assembly Y is determined from Figures

11 and 12 which display the significant parameters for the rolling action

Y = 2j3R + Lcsin/3

To get an expression for tan Q! one will again use Figure 11

Y-2RP tan Q! = 2R -I- Lc cos /I

(20)

(21)

The notation k2 was introduced to solve the differential equation of the deflection

curve as

k2 =P 2Es1s

The notation S was introduced earlier as

1 s = kH cash 2

(22)

(23)

An expression for the axial force F is obtained from the deflection curve, Equa-

tion (l2), by substituting for C and setting x = H/2

P* Y F =

H 2s kH - (PfQ) tan Q

-- k sinh yj- (24)

- 25 -

The notation C was introduced earlier as

c _ F -I- (P f Q) tan (Y -- 2EsIsk2 ’

(25)

Expression (18) and Figure 7 are used to obtain an equation for the vertical height

of the flexible sheets

H = Ls - As

or after substitution

pLS H = Ls + 2TsEs $ (sinh kH + kH)-

1 (26)

The dependent variables listed above are all required to determine the internal

load P. However, expressions for both the axial sideload F and its associated

axial deflection Y are among these variables. These variables are therefore

all which are required to determine the axial load versus deflection curve for the

flexible support assembly. One will note that the first three equations, (20)

through (22), are explicitly expressed in terms of the independent variable p and

earlier defined dependent variables. The four remaining variables, however, Equa-

tions(23) through (26), are interrelated and cannot be solved independently. No

further attempt has been made to express each of these variables in terms of

p only.

Several sideload versus deflection curves for different load conditions

and different geometrical shapes of the support assembly are required to check

conformance between the expressions developed and experimental results. Be-

cause some of the expressions developed are rather complex and a large amount

- 26 -

of data is needed, a computer program has been written to provide the data ,,see

Appendix A . Since several widths of the flexible support assembly are used for

the accelerator, this computer program calculates all the variables for a one-mch-

wide assembly. The variables are calculated using a recycling procedure. During

each cycle the computer will give p an increment and use the latest computed

value of all other variables to compute new values of these variables. By decrease

of the increment size any desired computer accuracy can be achieved. For our

accuracy requirements a typical increment size of .0005 radian is used. This

computer program is written in FORTRAN IV language for the International

Business Machine’s digital computer number 7090.

OPERATING RANGE OF THE FLEXIBLE SUPPORT ASSEMBLY

The available operating range of each support assembly is limited by the

mechanical properties of the materials of which it is built, as well as by the

shape of the parts in the assembly. Listed below are expressions for some of

the mechanical properties which can limit the operating range of this assembly.

The compressive load on the plate column is given by Equation (2) as

(P + Q)/cos (Y , and this load should never buckle the plate column. The problem

of “Buckling of a Bar with Rounded Ends” has been solved and the critical loads

tabulated in the book, Theory of Elastic Stability, Stephen P. Timoshenko and

James M. Gere. 1 All the tabulated critical loads in this book range from one

times to four times the critical load for a pinned end bar. Since no particular

end radius or length of plate column has been chosen for our design, a conserv-

ative approach would be to compare the compressive load on the plate column

- 27 -

to the critical buckling load for a pinned end plate. Calling the ratio between the

critical load for the pinned end condition and the actual load on the plate column

SF (safety factor), one can write

SF = 7r2 EcIc cos CY/ (P + Q) l (Lc + 2R)2 (27)

No attempt has been made to develop a formula for the buckling of the plate

column in a tilted position, since the final design will give a great margin of

safety from a chosen value of SF of greater than three.

The maximum stress in the sheets cs is caused by a bending moment

and a tensile load. The maximum bending moment Msmax occurs at the top and

at the bottom of each sheet, and is obtained from Equation (14) by substituting

x =H/2. M smax = E I y” ss H/2 = Es= Is l C- k*S. sinhk+ (28)

At the top and bottom of the sheets there is a transition zone where the sheets

increase in thickness. Using the notations K and Kb for the stress concentra- t

tion factors for tensile stress and bending stress respectively, the maximum

stress in the sheets is expressed as

P M smax . TS

53 = Kt q + Kb Is 2

The axial stiffness of the assembly is expressed as

K dF =- Y dy

(29)

(30)

This value is approximated by the computer by dividing the incremental change

of the force F by the incremental increase of the axial displacement Y.

- 28 -

The overall height of the assembly changes with the axial deflection Y.

This change of height can be expressed in terms of the vertical deformation of

the sheets or that of the plate column. Using the notation DH for this total change

of height, and referring to expression (26), one can write the following expression

DH = Ho - H (31)

where H is the initial value of H. 0

The contact stress between the plate column and the mounting blocks is

given by the following equation2 as cconact

P+Q -

Ocontac t = .798 2 l- urn E - E

m C 1 (32)

where trm = Poisson’s ratio of the mounting block

Em = modulus of elasticity of the mounting block.

During axial deflection of the support assembly the plate column rolls on

its ends. To ensure that sufficient rolling surface is available on the plate column,

the following expression is given for the remaining rolling surface, Rsurf,

beyond the line of contact

R surf = R(Y- P)

where y = arc sin Tc/2R (see Figure 12).

BASIC DIMENSIONS AND MATERIAL PROPERTIES OF THE FLEXIBLE SUPPORT ASSEMBLY

(33)

It is advantageous to make the flexible support assembly as tall as pos-

sible, so its size is determined by the available space. The overall height of the

- 29 -

assembly is five and one-half inches and is shown in this size on Figure 6. The

flexible sheets, which are of the same height, are machined down to o 045 -inch

thickness at the middle, leaving a three-quarter -inch wide rim at either end for

riveting to the upper and lower mounting blocks. A relatively large axial deflec-

tion, of .375 inch, is required of this assembly. The sheet material must there-

fore have a low modulus of elasticity and high tensile strength. Aluminum of 7075

alloy and T6 temper provides an easily machinable material which meets these

requirements. The mounting blocks are made from three-quarter -inch thick plate

of the same material. When 7075-T6 aluminum is used the tensile working stress

in this design must be limited to 66,000 psi, and the maximum contact pressure

to 107,000 psi, as specified by the Aluminum Company of America. 3 The value

for the modulus of elasticity is also specified by the same company, and is

10.4 l lo6 psi.

The maximum stress in the sheets is expressed by formula (29). In this

formula the stress concentration factors Kt and Kb were used for the tensile

stress and the bending stress, respectively. Values for these stress concentration

factors were taken from R. E. Peterson’s book on Stress Concentration Design

Factors.4 Although the actual values were both approximately 1.02, the more

conservative value 1.05 was used for these concentration factors in the computer

program.

The plate column material must also have a high tensile strength; but since

this part is not allowed to buckle, a higher modulus of elasticity is desirable.

Since only nonmagnetic materials can be used, and aluminum is excluded because

it might cold-weld to the adjoining parts, half hard phosphor bronze strip has been

- 30 -

chosen, This material has a modulus of elasticity of 15.9 l lo6 psi and can

tolerate a contact pressure as high as 118,000 psi. Phosphor bronze is an ex-

pensive material which is available only in certain thicknesses; its thickness in

the assembly has therefore been chosen as only one-eighth of an inch. The oper-

ational requirements of the plate column are: that it provides sufficient rolling

surface at its ends; that it can sustain the contact pressure; and finally, that it

does not buckle.

The plate column is inserted with a compressive preload to insure that an

external tensile load Q will not cause separation between the plate column and

the mounting blocks. The maximum size, which this tensile had Q can attain,

is reached if an 8-3/4-inch long assembly is loaded as shown in Figure 4. A

moment of 7600 in&pounds, and a tensile load Qz, a total of 500 pounds, will

cause a maximum tensile line load Q to reach a value of 653 pounds per inch.

To guarantee that no internal separation fakes place with this external tensile load

an interference of .004 inch is used.

- 31 -

DESIGN FROM COMPUTER DATA

CHOICE OF DESIGN BASED ON COMPUTER DATA

When the specified material properties and basic dimensions are used for

the flexible support assembly, the performance of this assembly can still be

greatly altered by varying the total plate column length and its end radii. A com-

puter program is therefore used to tabulate the variables which must be monitored

during flexing of the assembly, see Appendix A. For each of the three different

end radii, l/2, 3/8 and l/4 inch, a set of five tables has been prepared. One

table in each set pertains to an assembly with one particular plate column length,

these five plate column lengths being 4, 3, 2-l/2, 2 and l-1/2 inches. Although

the tables contain several mechanical properties which must be monitored, only

two graphs have been produced from each set of five tables. The two variables

chosen are the sideload F and the decrease of overall height DH, both of which

are plotted against the axial deflection Y. Since the final optimum design has a

2-l/2-inch long plate column with 3/8-inch end radii, Table II representing this

assembly is included in Appendix B.

Near the bottom of Table II, the incremental computer stepsize of the

independent variable p is shown as D 0005 radian, while the second tabulated

value of this variable is .0075 radian. This means that for this table the com-

puter recalculates all variables in fifteen steps before the next line of variables

is printed. When this stepsize is used by the computer, the expected accuracy

of all the tabulated variables is better than two and one-half percent.

The variable F is plotted against the variable Y on Graph 3 as a solid

curve to a deflection of .374. At this deflection the maximum sheet stress has

exceeded the allowable 66,000 psi and the curve is therefore dotted from then on.

- 33 -

GRAPH 1 DEFLECTION V-ERR-ITS --,-nm,- ,-,fin

-- ..- VJ o~u~umu, FOR SUPPORTS WH.CCH HAVE PLATE COLUMNS WITH l/2-INCH END RADII

" 34-

21

20

19

18

17

16

15

14

13

l.2

11

10

9

8

7

6

5

4

3

2

1

0

Axial deflection of assembly, Y inches

GRAPH2

DEFLECTION VERSUS OVERALL DECREASE OF HEIGHT, FOR SUPPORTS WHICH HAVE PLATE COLUMNS WITH l/2-INCH END RADII

- 35 -


GRAPH3

DEFLECTION VERSUS SIDELOAD, FOR SUPPORTS WHICH HAVE PLATE COLUMNS WITH 3/B-INCfI END RADII

- 36 -

The curve is not terminated because the assembly has not ceased to function.

Examining the remaining variables at this maximum deflection, one notes that

the internal load P has increased to a value of 679.61 pounds. A value for P of

653 pounds was estimated earlier as a desirable minimum. This assures that

separation between the plate column and the mounting blocks will not take place

during load&g. The plate column has a safety factor of 6.80 against buckling when

compared with a plate which is pinned at the ends. The maximum contact pressure

between the column ends and the mounting blocks is only 63,860 psi, which is well

below the allowable 10’7,000 psi. The decrease of overall height DH is plotted

versus the deflection Y on Graph 4, and this curve is also dotted beyond the

acceptable operating range. Finally the last column in the table shows that there

is another .0074 inch of rolling surface remaining on the plate column beyond the

line of contact at this maximum deflection.

All the remaining curves on Graphs 3 and 4 are also plotted from com-

puter data. The operating range for assemblies with plate column lengths of 4,

3 and 2-l/2 inches is limited by the maximum sheet stress. The operating

range of the remaining units is limited by the available rolling surface. Sufficient

rolling surface can, however, be provided for these units, but there are another

two limitations to the operating range of these units. For the units with a plate

column length of 2 and l-1/2 inches, the internal load P decreases sharply

with increasing deflection Y. The remaining internal load P at a deflection of

.375 inch is insufficient to prevent separation between plate columns and mount-

ing blocks at high loads. To fulfill the design criteria numbers 1 and 2, the

adopted design shall give a maximum decrease of overall height of . 010 inch

- 37 -


GRAPH4

DEFLEC!I'ION VERSUS OVERAI;L DECREASE OF HEIGHT, FOR SUPPORTS WHICH HAVE PLATE COLUMNS WITH 3/8-INCH END FAD11

- 38 -

for a .275-inch deflection and a maximum decrease of .020 inch for a .375-inch

deflection. Only the assemblies with plate columns of 4- and 3-inch total lengths

meet these design criteria, but they do not have the operating range. The remain-

ing assembly with the 2 -l/2-inch long plate column is therefore the only choice,

although it allows a decrease of vertical height of .0105 inch’ at a .275-inch

deflection.

The curves on Graphs 1 and 2 are plotted from computed data for assem-

blies with plate columns of the same lengths as on Graphs 3 and 4 but with 1/2-

inch end radii. Graphs 1 and 2 show that all these assemblies meet the design

criteria but none has the required operating range.

Graphs 5 and 6 display the behavior of assemblies with plate columns of

the same five lengths but with l/4 -inch end radii. However, on these graphs the

two curves representing the assemblies with the two shortest plate columns are

terminated. This indicates that these assemblies have ceased to operate, since

the internal load P has vanished. The loss of the internal load P results in the

plate column tklling out. The most promising assembly from these two graphs

is the 3-inch plate column assembly, but it does not have the required operating

range.

The optimum design chosen from all six graphs is the assembly with the

2 -l/2 -inch long plate column and the 3/8-inch end radii. The performance of

this assembly is therefore investigated further.

- 39 -

L is the total length of the plate column in inches.

;

id radii on all these plate columns are l/4 inch,

j E

Available operating range -------- Operating range exceeded

U.IW 0.200 0.300


GRAPH5

DEFLECTION VERSUS SIDELOAD, FOR SUPPORTS WHICH HAVE PLATE COLUMNS WITH ~/~-INCH END RADII

- 40 -

21

20

19

18

17

16

15

14

13

12

11

10

9

8


GRAPH6

DEFLECTION VERSUS OVERALL DECREASE OF HEIGHT, FOR SUPPORTS WHICH HAVE PLATE COLUMNS WITH l/b-INCH END RADII

- 41-

STIFFNESS VARIATIONS CAUSED BY CHANGE OF INITIAL INTERFERENCE

All the curves which have been plotted previously have been for assem-

blies which have an initial interference Its of .004 inch. This value of Its is

chosen to insure sufficient internal load P. But when these units are mass pro-

duced a variation of this initial interference must be tolerated. Graph 7 has there-

fore been prepared to show how assemblies with the three different plate column

lengths 4, 2 -l/2 and 1 -l/2 inches, all with the 3/8 -inch end radii, behave when

the interference is altered by plus or minus . 002 inch. The assembly with the

4 -inch long column stiffens when the interference is increased, while the assembly

with the l-l/2-inch long plate column turns unstable, and requires an external

force to return it to zero deflection. The assembly with the 2 -l/2-inch long plate

column is practically unaffected by this rather large change of interference. This

is a desirable characteristic. Even with greatly relaxed production tolerances

on the plate column and the adjoining parts, one can expect to produce assemblies

with uniform stiffness.

CHANGE OF STIFFNESS BY EXTERNAL LOAD

Assume that a moment about the accelerator axis will cause a line load

Q, which varies linearly along the 8-3/4-inch long assembly. Then, if an

assembly is loaded simultaneously by a moment of 7600 inch-pounds and a 500

pound tensile load, the line load on this assembly will vary from a maximum

tensile line load of 653 pounds per inch at one end, to a compressive line load

of 539 pounds per inch at the other end. The average of these line loads is a

tensile load of 57 pounds per inch. Assemblies with plate columns of 2-l/2-inch

- 42 -


GRAPH7

DEFLECTION VERSUS SIDELOAD FOR DIFFERENT VALUES OF THE INITIALLY INTRODUCED INTERFERENCE .

- 43 -

lengths, and 3/8-inch end radii, which are loaded with these three different loads,

will have axial stiffnesses as shown on Graph 8. As seen from the curves on this

graph, the size of the stiffnesses vary with the load, but for one particular load

they remain almost constant throughout the working range of the assembly. The

stiffness for the average line load Q = - 5’7 pounds per inch, is very close to

the average of the stiffnesses for the maximum and the minimum loading; this

indicates that the stiffness varies linearly with the external line load Q. From

this characteristic it follows that the overall stiffness of the assembly is not

changed by an external moment about the accelerator axis. An assembly which

carries a compressive line load of 134 pounds per inch, has a maximum stiffuess

of .07 pound per inch. This indicates that it is possible to make assemblies

which have nearly zero stiffness throughout their entire operating range.

- 44-

Axial deflection of Bssembly, Y inches

GRAPH8

DEFLECTION VERSUS AXIAL STIFFNESS FOR SEVERAL EXTEXNAL LOADS

- 45 -

I .

TESTS ON FLEXIBLE SUPPORT ASSEMBLIES

TEST OBJECTIVES AND PRIMARY TEST RESULTS

The purpose of the tests on the flexible support assemblies was,primarily,

to determine that this design satisfied all the design criteria, and, secondly, to

verify experimentally some of the characteristics which were predicted by the

computer program. Only a brief summary of the primary test procedure and

results is given here. Greater attention is later given to the secondary tests.

The initial tests which took place early in 1964 were performed on one of

the first mass produced assemblies of ten-foot accelerator pipes and aluminum

extrusions. Figure 2 shows one of these completed assemblies. During these

tests only the input end assembly was loaded by a moment of 7600 inch-pounds

about the accelerator axis, and by a vertical load of 300 pounds, while the other

three support assemblies on the same ten-foot section carried the remaining

weight of the ten-foot accelerator pipe. With this loading, the ten-foot assembly

was first given an endurance test which consisted of cycling the accelerator pipe

axially between plus and minus .375 inch deflection 600 times. Then a test

was made to prove that the assembly could withstand earthquake type shock loads.

That test consisted of loading the input end assembly statically with a 26,600

inch-pounds moment about the accelerator axis, and a vertical load of 800 pounds.

During the next two tests, deflections of the input end assembly were recorded, the

first of these tests showed that the loading used during the cycling test produced a

,006-inch lateral deflection, and that this deflection was not influenced by axial

deflection of the accelerator pipe. The last test produced data for plotting of

Graph 9, which shows the decrease of height DH, versus the axial deflection Y.

Since the test data closely fit a smooth curve, this indicated that very little

- 47 -


GRAPH9

DEFLECTION VERSUS OVERALL DECREZ3E OF HEIGHT PLOTTED FROM PRIMARY TEST DATA

- 48 -

destructive wear of the rolling surfaces was caused by the life test and the

high static load test. A decrease of height of .0103 inch was shown for an

axial deflection of .275 in&and a .0188 inch decrease for a .375 inch deflec-

tion. Both these deformations were acceptable when compared with the original

design criteria on page 8. The conclusion from all these primary tests was

therefore that this design was acceptable. Since during these tests all of the

four flexible support assemblies in the ten-foot assembly were loaded differently,

no effort was made to compare overall test results with computed data.

- 49 -

TEST ASSEMBLIES

Special test assemblies were made for the remaining tests. Since five-

inch-wide flexible sheets were available from the last production of flexible sup-

port assemblies, all test units were made five inches wide. The shape of these

assemblies is as shown on Figure 6. Initially nine pairs of test assemblies were

made, each pair with a different aperture to enable testing of nine different column

lengths. However, the roughness of the rolling surfaces, which were generated

using a commercially available concave milling cutter, made these assemblies

unusable for further tests. Finally four new test assemblies were made. In

these units five-inch-wide flexible aluminum sheets were used. The aluminum

mounting blocks and plate columns were replaced by hardened steel parts which

were ground to a microf inish of better than 8 RMS and also to a high dimensional

accuracy.

Assembly of the plate column into the completed riveted assembly is

usually performed by first shrinking the plate column in liquid nitrogen and then

letting it expand in place. But on the final four test assemblies four half-inch

long SR-4 foil strain gages were mounted on each assembly so that the tensile

stress in the sheets could be monitored with a Baldwin Lima-Hamilton strain

indicator. The distances between the mounting blocks were also monitored

during the assembly of these units. It was then found that to achieve a certain

tensile stress in the sheets the mounting blocks had to be separated a distance

exceeding the precalculated distance by a factor of two andone-halftothree. Since

disassembly and reassembly indicated that no measurable permanent deformation

had taken place, one could conclude from these observations that this riveted

assembly was more flexible than what.the computer program would indicate.

- 50 -

TEST APPARATUS AND PROCEDURE

The test assemblies were tested one assembly at a time in the test fixture

shown on Figure 14. This fixture was primarily made to minimize the rotation

of the top mounting block of the assembly during testing, but it also served for

mounting the required dial displacement gages. During the testing this fixture was

mounted on the table of a Bridgeport milling machine, and the top plate was con-

nected to the head stock of the mill through a push-pull force gage made especially

for these tests. Deflecting the test assembly was then performed by cranking the

table back and forth, see Figure 15. This arrangement was chosen to give the

necessary freedom of loading. With this system, either a push or a pull could

be applied to the assembly at any deflection. As shown on Figure 10, a moment

ME was required to keep the top mounting block from rotating. This moment

was here provided by a vertical force from the roller supports. But this vertical

force also produced a vertical load on the test unit, and in turn changed the stiff-

ness of the assembly being tested. Each unit was therefore always test,ed twice,

first by pushing it, then after it was turned around, by pulling it. Since the

change of the force from one arrangement to the other seldom amounted to more

than ten percent of the total reading, the average value was used for plotting of

curves.

To be able to vary the line load Q on the test assembly a loading frame

was used. The maximum load which could be applied on the five-inch-wide as-

sembly, was limited to 540 pounds since at this load the milling machine tended

to tip over. Therefore, the maximum line load applied to any assembly was 108

pounds per inch.

- 51 -

Dial displacement

f gages

Top plate G --f

I a

I Assembly to be tested

Push-pull force gage

Roller support \

FIGURF 14

TESTFIXTURE

I CIzz

I, I

- .--__

.._--- -__

_ - -.

_ I

- 53

-

THE EFFECTIVE INTERFERENCE IN THE FLEXIBLE SUPPORT ASSEMBLY

One of the assumptions made during the writing of the computer program

was: that the deformations in the mounting blocks, and in the riveted joints

between the mounting blocks and the sheets were small enough to be ignored.

During the installation of the plate columns into the assemblies it was found that

other deformations existed,because the certain interference did not produce nearly

the expected tensile stress in the sheets. Since it was beyond the scope of this

paper to investigate the deformations other than those in the sheets and the plate

columns, the expression, “effective interference, ” was introduced. This ef-

fective interference would raise the initial pure tensile stresses in the sheets

to their actual measured values if only the sheets and the plate columns were

deforming. The tensile stresses were then measured during the installation of

the plate columns into the five test assemblies. These stresses were tabulated

in Table I together with the actual changes of spacing between the mounting

blocks and the calculated effective interferences. The effective interferences

rather than the actual interferences were finally used in the computer program

for the five test conditions.

- 54 -

TABLE I EFFECTIVE INTERFERENCE FOR THE TEST UNITS

Plate Average tensile stress Effective Change of column in the flexible sheets interference spacing between length after installation mounting blocks

inch psi inch inch

3 7,654 .0029 .0087

2% 12,022 too45 .0133

2 12,558 .0046 .0118

1; 12,428 .0045 .0109

2; 3,206 0 0012 .0040

- 55 -

THE EQUIVALENT SHEETLENGTH IN THE FLEXIBLE SUPPORT ASSEMBLY

Flexible sheets for the flexible support assemblies were produced with

transition radii at the top and the bottom where they changed from .045-inch

thickness into thicker sections, see Figure 13. The transition zones were each

.187 -inch wide, and located in the areas of the sheets where the highest bending

moments occurred. A certain amount of rotation and deflection would therefore

take place in these zones. Since the effect of these transition zones was not ac-

counted for in the computer program, it was required to determine what equivalent

length of the constant thickness sheets should be used to approximate the bending

of the sheets in the assemblies. A strip of .045-inch thickness and .106-inch

width was found to produce the same edge rotation as an actual transition zone

when exposed to a constant bending moment, see Appendix C. The double of the

width of this strip was therefore added to the constant thickness sheet length of

3.625 inches, producing an equivalent sheet length of 3.838 inches. The width

of the constant thickness strips was calculated to produce the same rotation as

the actual transition zone,rather than deflection of the actual transition zone. This

choice was made because it was assumed that the overall deflection of the assembly

was influenced more by the rotation than the deflection in these zones.

DISCUSSION OF TEST RESULTS

An optimum flexible support assembly design was chosen from the

Graphs 1 through 6. That assembly had 4 inch long flexible sheets, and a

&l/%-inch long plate column with 3/8-inch end radii. It was assembled with

.004-inch interference.

- 56 -

The first assemblies which were built as shown on Figure 6, were ex-

pected to perform like the calculated optimum design. However, after the pri-

mary tests, it was discovered that these assemblies had greater axial stiffness

than predicted, but otherwise satisfied all requirements. This greater axial stiff-

ness would cause larger axial forces to be transmitted through the accelerator

pipe during adverse temperature conditions in the accelerator housing. An

estimate was made of the accumulated axial thrust in the accelerator pipe from

all the flexible support assemblies used on one forty-foot girder. The maximum

thrust, during accelerator operating conditions, was found to be approximately

600 pounds. The exact size of this accumulated thrust depended not only on the

temperature conditions in the accelerator housing, but also on how much vertical

load was transmitted to the accelerator pipe through the waveguides. Since the

axial thrust did not exceed the 750 pounds allowed in the design criteria,the de-

sign of these first flexible supports was chosen for the accelerator.

During the research for this paper the discrepancy between the calculated

and the actual stiffness of the flexible support assemblies was investigated further.

It was then found that by introducing the value for the effective interference and

the value for the equivalent sheet length, the comE;uter program could better

predict test results.

All the test assemblies were made with flexible sheets left over from the

last production run of flexible support assemblies. These flexible sheets were

found to have an equivalent sheet length of 3.838 inches. This equivalent sheet

length was used in the computer program to tabulate data for the test assemblies.

The effective interference was determined from measurements of the

average tensile stress produced during installation of the plate column into the

- 57 -

support assembly. These effective interferences were therefore different for

each test assembly and were listed in Table I.

The push-pull force gage which was initially used on the old aluminum

test assemblies was recalibrated prior to the tests on the four final test as-

semblies. The data from these last tests were accurate to within two percent.

As mentioned earlier each test assembly was tested twice, first by pushing on it,

and then after it was turned around, by pulling on it. The average value of the

pushing sideload and the pulling sideload was then used for plotting one curve.

Earlier tests had shown that assemblies did not always behave symmetrically,

so the average sideloads required to deflect the assemblies both to the left and

to the right were plotted on Graphs 10 through 15.

The curves onGraph 10 were plotted using data from a test on one of the

old aluminum test assemblies. This assembly was made with aluminum mounting

blocks and an aluminum plate column. This graph was included to show the ef-

fect of surface irregularities on the performance of an assembly. These surface

irregularities were so small that they were not detected before the curves from

the test data were plotted. As a result of this discovery new test units were made

with steel plate columns and mounting blocks.

The new test assemblies were first tested with no external load, and then

with a total external load of 520 pounds. This load amounted to a compressive

line load of 108 pounds per inch width of the assembly. The results from these

tests were shown on Graphs 11 through 15. Curves from computer data,

which predicted the performance of the test assemblies under these load condi-

tions were also included on each graph. As seen from the graphs a reasonable

agreement between the predicted and actual results was achieved. For all these

tables, the equivalent sheetlengths and the effective interferences were used.

- 58 -

M

a3 -3

W

@.I

0 r-l

a.3 l-l

4

spunod J

psoxap-p puxaqxg

- 59 -

16

I2

10

8

6

4

2

0 3

Axial deflection to the left

GRAPH 11

Axial deflection to the right

DEFLECTION VERSUS SIDELOAD FOR THE TEST ASSEMBLY WHICH HAD A 3-INCH-LONG PLATE COLUMN WITH 3/8-INCH END RADII. THE EFFECTIVE INTERFERENCE OF THIS ASSEMBLY WAS .OO29 INCH.

16

2

0

GRAPH12

DEFIJXTION VERSUS SIDELOAD FOR THE-TEST ASSEMBLY WHICH HAD A 2-l/2-INCH-LONG PLATE COLUMN WITH 3/8-INCH END RADII. THE EFFECTIVE INTERFEmCE FOR THIS ASSEMEiLY WAS .0045 INCH.

j pi

16

10

a

6

4

-6

Axial deflection to the left Axial deflection to the right

GRAPH13

DEFLECTION VERSUS SIDELOAD FOR THE TEST ASSEMBLY WHICH HAD A 2-INCH-LONG PLclTE COLUMN WITH 3/8-INCH END RADII. THE EFFECTIVE INTERFERF,NCE FOR THIS ASSEMBLY WAS -0046 INCH.

- 63 -

Axial deflection to the left

GRAPH 15

Axial deflection to the right

DEFLECTION VERSUS SIDELOAD FOR THE TEST ASSEMBLY WHICH HAD A 2-l/2-ITJCH-LONG PLATE COLUMN WITH 3/8-INCH END RADII. THE EFFECTIVE INTERFERENCE FOR THIS ASSEMBLY WAS .0012 INCH.

When the external compressive load was applied to the test assemblies

with the 2 and 1-l/2-inch long plate columns, a shift in the no sideload position

was recorded. This shift occurred because the plate columns were not properly

aligned in the assembly prior to testing. No such shifting occurred during the

tests on the other assemblies.

Graphs 11, 12 and 15 showed that the assemblies with 3-inch and

&l/&inch long plate columns had lower axial stiffness then predicted by the com-

puter. Graphs 13 and 14 showed that the assemblies with shorter plate

columns had higher axial stiffness than predicted. These differences between

the predicted and achieved results should be expected, considering the earlier

finding from tests, that the initial interference produced a smaller internal load

P than was calculated. This point is discussed further below.

i Computer data showed that the assembly with the 3-inch long plate column

had a higher axial stiffness than the assembly with the 2-l/2-inch long plate

column, see Graphs 11 and 12. It was shown in the computer data that the

internal load P increased faster in the assembly with the 3-inch plate column

than in the assembly with the 2-l/2-inch column when these were deflected

axially. Therefore, a faster increasing internal load P is associated with a

stiffer assembly.

The calculated internal load PO was proportional to the initial interference.

The calculated variable load P was dependent on the interference in any tilted

position. However, in testing, it was found that the initial interference caused a

much smaller internal load PO than had been calculated. Prom this it was as-

sumed that a variation in this interference, such as that caused by tilting from one

position to another, would also cause a smaller change in the internal load P than

had been calculated. Therefore, if the internal load in the test assembly with the

- 65 -

3-inch long plate column did not increase as fast as the computed data showed,

one would expect lower axial stiffness than predicted by the computer. This

conclusion was supported by the test data which was plotted on Graph 11. A

similar argument could be made to conclude that test assemblies with shorter

plate columns should have higher axial stiffness than predicted by the computer.

The last graph, number 15, showed the test results on a test assembly

with a 2-l/2-inch long plate column with 3/8-inch end radii. These dimensions

were also used in the flexible support assemblies for the Stanford Linear Ac-

celerator. The test assembly had steel mounting blocks and a steel plate column

instead of aluminum mounting blocks and a phosphor bronze plate column as in

the design for the accelerator. This alternate choice of materials should not

make a significant difference between the test assembly and the accelerator as-

semblies. The test assembly and computer data based on the test assembly

could therefore represent all the assemblies used for the accelerator.

The actual interference for the test assembly was .004 inch and the ef-

fective interference was found to be .0012 inch. The actual interference in the

mass produced assemblies for the accelerator ranged from .0047 inch to .0063

inch. One would therefore expect the effective interferences for these units to

stay between . 001 and .002 inch. The remaining curves on Graph 15, which

were based on computer data for the test assembly, showed that the external

sideload did not change significantly when the effective interference varied

between .OOl and .002 inch. The mass produced assemblies would therefore

have uniform stiffness, and each react similarly to loading along the two-mile

act elerator .

- 66 -

According to computer data, the maximum stress in the flexible sheets

of the accelerator assemblies would, because of the variation of the effective

interference, range from 62,000 to 72,000 psi, at the full .375-inch axial de-

flection. This stress would occur only during the most adverse environmental

conditions and was therefore considered acceptable. The axial stiffness for the

test assembly was 44 pounds per inch for no external line load. For an external

line load of 108 pounds per inch, the axial stiffness was 14 pounds per inch.

Since the assemblies for the accelerator carried line loads of approximately

10 pounds per inch, the expected stiffness for these assemblies would be approxi-

mately 41 pounds per inch. With this axial stiffness the maximum accumulated

axial thrust from all the flexible support assemblies on one forty-foot girder,

would be smaller than the 750 pounds allowed in the design criteria.

The decrease of the vertical height of the assemblies was recorded

during all the tests. This decrease of height gave continuous information on

how well the plate columns had been aligned during installation. However, the

test data corresponded so well with the calculated data, that no curves were

prepared to show this variable.

- 67 -

BIBLIOGRAPHY

1. Stephen P. Timoshenko and James M. Gere. Theory of Elastic Stability.

(New York: McGraw-Hill Book Company, 1961), p. 59.

2. A. Foppl. Technische Mechanik. 4th ed. , Vol. 5, p. 350 .

3. Aluminum Company of America. Alcoa Structural Handbook - A Design

Manual for Aluminum D (Pittsburg, Pennsylvania, 1960), pp. 48-49.

4. R. E. Peterson. Stress Concentration Design Factors. (New York: John

Wiley & Sons, Inc. , 1953), ppO 28 and 31.

- 68 -

APPENDIX A

COMPUTER PROGRAM FOR THE FLEXIBLE

SUPPORT ASSEMBLY

I

0” I

SLAC SKARPA 343 9091 0 DATE 02/28/67 TIME A - EFN SOURCE STATEMENT - IFNISJ -

C C C C C C

99 100

101

102

103

STIFFNESS OF FLEXIBLE SUPPORT ASSEMBLY WITH EXTERNAL LOAD P LBS PER INCH SHEET DIMENSIONS ARE LENGTH LS, THICKNESS TS, ELASTIC MODULUS ES COLUMN DIMENSIONS ARE LENGTH LC, RADIUS R, TXIWNESS TC, ELASTIC UOD EC PRELOAD IS CAUSED BY INITIAL INTERFERENCE CALLED ICS

DIMENSION TITLE1261 REAL LSO,LS,LC,MSJTAX,K,iCS,KT,KB,KY,f4SLOP SINHIXJ = .5 l ( EXPIXJ - EXPI-XJJ COSHIXJ = .5 l I EXPIXJ + EXPl-XJJ READl5,lOOJTiTLE FORJ4ATI13Ab/13Abl READl5,1DlJCJ,ES,EC,LS,TS,LC.R,TClltS1L,M,N,NN FORMAT(3F10.0,bFb.4/4IlOJ WRiTETbrlOZJTITLE FORl4ATllHl,13Ab/13Ab////J WRITE1brl03lQ,ES,EC,LS,TS,LC,R,TCIICS FORHATT25X,13HEXTERNAL LOAD,i4X,27HSHEET MODULUS OF ELASTiCiTY,7X,

12BHCOLUMN MODULUS OF fLASTiCITY/27X,BHL6S/INCH,2BX,3HPSI,32X,3NPSi 2/ 3F35.0///llX,12HSHEET LENGTH,7X.l5HSHEET THICKNESS,bX.l3HCDLUMN 3LENGTH17X,13HCDLUHN RADIUS,bX,1bHCOLUf4N THICKNESS,ZX,12HINTERFEREN 4tE/14X~4H1NCH~lbX,4HINCH~lbX,4HiNCH,lbX,4HiNCH~lbX,4HINCH,lbX,4HIN 5CH/ bF20.4////3

WRITEfb,104J 104 FORMAT ISXBHSLOPE OF.4X,?HAXiAL rbX.BHEXTERNAL.4X.BHINTERNA

lL,2X,bHCDLUHN,bX,7HHAXIMUM,bX,5HAXIAL,BX,BHDECREASE,5X~7HMAXiMU~,3 2X.7HREJ4AIN./5X,BHCOLUMN B,4X,lOHDEFLECTION,3X,BHSiDELOAD,4X~7HLOAD 3 P ,3X,bHSAFETY,bX,5HSHEET,BX;9HSTiFFNESS,4X~lOHOF DVERALL,3X,7HCO 4NTACT,3X,7HROLLJNG/5X.7HRADIANS15X1BHY INCHES.SX.5HF LBSe7X,3HLBS, 57X,bHFACTDR,bX,bHSTRESS,7X,BHLBS/INCHISX,9HHEiGHT DH,4X,BHPRESSURE 6,2X,7HSURFACE//J

B = .O c = .o F = .O J = 0 Y = .o TANA = s = 1:: K 5 1.0 KT = 1.05 KB = 1.05 vs = .32 vc = .29 vn = vc EM = EC PO =IICS-Pcl2.~R+LCJ/lTC~ECJJ/ILS/(2.rTSIESJ+I2.~R+LCJ/ITC~ECJJ LSD = LS + LS*PO/I2.*TS*ESJ CONST = 3.14159+~2+EC*TC~+3/lil.-VC~~2J~l2.~lLC+2..RJ.~2J COEFF =.798*SQRT1,5/lR *Ill.-VJW=2J/EM+ll.-VC+c2)/Et))) GAMMA = ATANllTC/12.*RJJ/I1.-lTC/l2.*RJJ**2JJ

LSO :O 107-I = L , Jd , N 8 = FLOAT(I) . .OOOl

19 P = lICS-LC~l1.-COSIBlJ+C~~2~l.5~H-2.~S~SINHl.5~K~H~/KtS~~2~ 1.25*ISINHlK*HJ+K*HJ/KJ -P*COSlB-ATANITANAJJ+l2.*R~COS(ATINITANAJJ

2 3 b 7 9

11

lb

la

19

21 22

1.605 SEC. PAGE 1

SLAC SKARPA 343 Cl091 0 DATE 02/28/6? TIME 1.605 SEC. PAGE 2 A - EFN SOURCE STATEMENT - IFNlSl -

27 28 29 30 31

34

37

43 44

53

57

61

62

63

2tLC*COSTB-ATAN(TANA)))/1TC*EC~~/LLS/(2.*TS*ES~+ COSLB-ATANITANA)) 3,(2.,R+COS(ATAN(TANA))+LC*COS(B-ATANITANA)))/~TC~EC~)

DY = 2.*B*R+LC*SINLB) - Y 20 r =YtDY 21 TANA = ( Y -2.*R*B )/TZ.*R+LC+COSLBJI

IF(P) 108,108r22 22 K = SQRT I5.347*P/LES*LTS**31)) 23 S = 1. / COSH(.5*K*H)

OF = P*Y/LH-2.+S*SINHL.S+K+H)/K) - LP+Q)*TANA - F 24 F = F t OF 25 c = IFtLPtQ)*TANA)/P 26 H = LS t P+LS/12.*TS*ES)-C l +2+(.5+H-Z.*S*SINHi.5,K+H)/K+S++2+

~(SINH(KIH)+K+HI/(~.~K) 1 27 SF = CONST l COSLATANIJANA))/IPtQ) 28 MSHAX = ES*L JS**3/10.693)*C*K*S*SINHl.5*K*H) 29 SPSI = KJ*P/IZ.*JS)tKB +MS~AX*5.347/1JS+*2~

IFIDY) 10,10,30 30 KY = OF / DY

GO TO 31 10 KY = .O 31 OH = LSO - Ii 32 CONJP = COEFF * SQRTlP+Q) 33 RSURF = R * L GAMMA - 0 I 34 MSLOP = c * (1, - 5 1

IFLJ-I) 105,105,L07 105 WRITEL6,106)B,Y,F,P,SF,SPSI,KY~DH,CONTP,RSURF 106 FORMAT~2F12.4,5F12.2,lFl2.4,lFl2.O,lFll.4/~

J = J t NN 107 CONTINUE 108 WRITEt6,109) 109 FORMATI /// 1

BSTEP = FLOAT(N) i .OOOl WRITE{ 6,110) BSTEP

110 FORMATL6X52HTO COMPUTE THE NUMBERS LISTED IN THE TABLE ABOVE,THE/ lbX,45HCOMPUTER USED AN INCREMENTAL SJEPSIZE OF B = ,lF7.4//)

WRITEt6,lll) 111 FORMATLbX53HTHE ABOVE COMPUTATIONS WERE PERFORMED ON FE6 27THl967)

GO TO 99 END

APPENDIX B

TABLE II

TABULATED DATA FOR THE OPTIMUM DESIGN

CHOSEN FROM COMPUTER DATA

TABLE OF CALCULATED VARIABLES FOR A FLEXIBLE SUPPORT ASSEMBLY WITH TOTAL PLATE COLUMN LENGTH = 2.500 RADIUS = .375 AN0 INTERFERENCE = .004

EXTERNAL LOAD SHEET MODULUS OF ELASTICITY LBS/INCH PSI

SHEET LENGTH INCH 4.0000

SLOPE OF AXIAL EXTERNAL INTERNAL COLUMN MAXIMUM AXIAL DECREASE MAXIMUM REMAIN. COLUMN B DEFLECTION SIOELOAO LOAD P SAFETY SHEET STIFFNESS OF OVERALL CONTACT ROLLING RADIANS Y INCHES F LBS CBS FACTOR STRESS LBS/INCH HEIGHT OH PRESSURE SURFACE

0. 0. 0. 723.15 6.44 8436.73 0. 0.0015 0.0187 0.70 721.94 6.45 11507.13 37.49 0.0150 0.0375 1.41 720.54 6.46 14571.49 37.50 0.0225 0.0562 2.11 719.01 6.48 17629.90 37.52 0.0300 0.0750 2.81 717.35 6.49 20682.07 37.53 0.0375 0.0937 3.52 715.56 6.51 23727.74 37.56 0.0450 0.1125 4.22 713.67 6.52 26766.82 37.58 0.0525 0.1312 4.92 711.68 6.54 29799.31 37.61 0.0600 0.1499 5.63 709.60 6.56 32825.19 37.65 0.0675 0.1687 6.33 707.45 6.58 35844.58 37.68 0.0750 0.1874 7.04 705.25 6.59 38857.92 37.72 0.0825 0.2061 7.75 703.01 6.61 41865.40 37.77 0.0900 0.2248 8.45 700.75 6.63 44867.70 37.81 0.0975 0.2435 9.16 698.49 6.65 47865.38 37.86 0.1050 0.2622 9.87 696.24 6.67 50859.08 37.91 0.1125 0.2808 10.58 694.03 b.b9 53849.81 37.97 0.1200 0.2995 11.29 691.87 6.71 56838.36 38.03 0.1275 0.3181 12.00 689.78 6.72 59825.99 38.08 0.1350 0.3368 12.71 687.78 6.74 62813.56 38.15 0.1425 0.3554 13.42 685.87 6.76 65802.55 38.21 0.1500 0.3740 14.13 b84.09 6.77 68794.17 38.28 0.1575 0.3926 14.84 682.44 6.78 71789.68 38.35 0.1650 0.4112 15.55 680.94 6.79 74790.75 38.42 0.1725 0.4298 lb.27 679.bl 6.80 77798.63 38.49

0. 65874. 0.0637 0.0001 65819. 0.0609 0.0002 65755. 0.0580 0.0005 65685. 0.0552 0.0008 65609. 0.0524 0.0013 65528. 0.0496 0.0018 65441. 0.0468 0.0024 65349. 0.0440 0.0032 65254. 0.0412 0.0040 65155. 0.0384 0.0050 65054. 0.0355 0.0060 64950. 0.0327 0.0071 64846. 0.0299 0.0084 64741. 0.0271 0.0097 64637. 0.0243 0.0111 64534. 0.0215 0.0126 64434. 0.0187 0.0143 64336. 0.0159 0.0160 64243. 0.0130 0.0178 64154. 0.0102 0.0197 64070. 0.0074 0.0217 63993. 0.0046 0.0239 63923. 0.0018 0.0261 63860. -0.0010

0. 10400000.

SHEET THICKNESS COLUMN LENGTH COLUMN RADIUS INCH INCH INCH 0.0450 1.7500 0.3750

:OLUMN MODULUS OF ELASTICITY PSI

15900000.

COLUMN THICKNESS INTERFERENCE INCH INCH 0.1250 0.0040

TO COMPUTE THE NUHBERS LISTEO IN THE TABLE ABOVE,THE COMPUTER USE0 AN INCREMENTAL STEPSIZE OF 8 = 0.0005

THE ABOVE COMPUTATIONS WERE PERFORMED ON FEB 27TH1967

APPENDIX C

DATA TO DETERMINE THE EQUIVALENT SHEETLENGTH

DATA TO DETERMINE THE EQUIVALENT SHEETLENGTH

When the computer program was written for the flexible support assembly,

it was assumed that the flexible sheets were of constant thickness and length. The

transition zones at the ends of each sheet were not accounted for. An estimatewas

therefore made to determine the equivalent length of a constant thickness sheet,

which would produce the same total flexibility of the assembly. It was assumed

that the stiffness of the assembly was influenced more by the bending in the tran-

sition zone than the deflection in this zone. It was necessary to determine which

width of a constant thickness strip would produce the same edge rotation as the

actual transition zone, when both were bent by a constant bending moment. To

simplify calculations, the .187-inch-wide transition zone was assumed to have

the same stiffness as the total of seven strips each of constant thickness as shown

on Figure 16. The rotations through each of these sections were calculated, and

were tabulated on Table III.

The thickness t was given for any value of x by the following equation:

t = .045 + ,375 (1 - cos y)

where y = arc sin x/. 375 .,

The rotation through each of the seven short sections was

Ml e =EsIs=

12M Q

Esb (1 - v2) 7 =c.L

t3

where C is constant.

The values of t and 8 were tabulated in Table III. As seen from this table,

a strip of .106-inch width and . @&-inch thickness gave the same edge rotation as

the whole transition zone.

- 75 -

Of R/VETS

FIGURE 16 TRANSITION ZONE ON THE FLEXIBLE SHEETS

TABLE III

TABLE TO DETERMINE THE WIDTH OF THE CONSTANT THICKNESS STRIP

Strip Number Thickness Length 8 /c = L/t3

1 D 0450 . 0150 . 0164 lo4

2 . 0462 .0300 . 0304 lo*

3 . 0498 . 0300 D 0243 1O-4

4 . 0560 .0300 . 0171 10dr

5 . 0647 . 0300 . 0111 1o-4

6 0 0763 . 0525 -4 . 0118 10

7 o 187 D 3750 . 0057 1o-4

Accumulated L/t3 for strips numbered 1 through 7 = . 1168 1O-4

Equivalent strip . 045 . 106 * 1168 10 -4

- 76 -

elasticity theory and design of flexible supports for

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