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A2136.1 (REV. 6.61)

https://ntrs.nasa.gov/search.jsp?R=19660010158 2018-07-08T07:39:55+00:00Z

) t )O lT110 . GD/A-DDG-64-021 DATE 31 Auguet 1964

WO.OFPACLS 1 8 + i i

SECONDARY STRESSES

I N TRUSSES

GD/A -DDG-64-021

e CONTENTS

1 eo INTRODUCTION

2.0 SECONDARY STRESS CONCEPT

3 0 0 FIXED J O I N T TRUSS EXAMPLE

3 0 1 Manual So lu t ion

3,2 7090/4 Program Solu t ion

3.3 S t a t i c Check of Program Output

4 0 0 LIST OF REFERENCES AND BIBLIOGRAPHY

1

ILLUSTRATIONS

T i t l e

Fixed Truss-Jo in t Relaxed

2 Fixed Truss-Jo in t Loaded and Unloaded

3 Example - Fixed J o i n t T r u s s

4 Bar Forces and S t r e s s I n t e n s i t i e s

5 Moment D i s t r i b u t i o n on Example T r u s s

6 F ina l J o i n t Moments

7 F ina l Axial Loads

NO - TABLES

T i t l e - I Geometric P r o p e r t i e s

I1 Computation of Angle Changes

0 111 Computation of I n i t i a l End Moments

Page

1

2

4

5

14

17

18

Page

2

3

4

5

10

11

12

Page

ii

GD/A -DDG-64 -021

1 a 0 INTRODUCTION

A widely used method f o r the a n a l y s i s of two-dimensional f i x e d j o i n t

t r u s s e s is presented i n References 1-4, and is commonly c a l l e d t h e method

of secondary stresses. In t h i s approach, t h e t r u s s is f i r s t analyzed un-

de r t h e assumption t h a t a l l members meeting a t each j o i n t a r e connected

by f r i c t i o n l e s s pins . Next, i t is assumed t h a t t h e j o i n t s of t h e s t r u c -

t u r e are "lockedfv i.e., that a l l members meeting a t a j o i n t r o t a t e through

t h e same angle . Fu r the r assumptions a re then made: ( a ) t h a t t h e a x i a l

( or, "primaryft) stresses previouely computed have n e g l i g i b l e e f f e c t upon

member s t resses from bending, and ( b ) , t h a t t h e bending (or, ttsecondarytt)

s tresses have n e g l i g i b l e e f fec t i n modifying t h e primary stresses. Thus,

from a computation of j o i n t displacements followed by a moment d i s t r i b u -

t i o n ? t h e secondary s t r e s s e s may be found.

stresses may be i t e r a t e d , and then i t is seen t h a t success ive i t e r a t i o n s

This c a l c u l a t i o n of secondary 0

converge i n an a l t e r n a t i n g manner t o some t r u e so lu t ion . The method of

secondary stresses obv ia t e s t h e need f o r so lv ing a l a r g e s e t of s i s u l t a -

neous equat ions; b u t t h i s is o f f e e t by t h e l abor of t h e computation of

j o i n t d e f l e c t i o n s and moment d i s t r i b u t i o n . Moreover, i t seems imprac t ica l

t o apply t h i s technique t o a three dimensional s t r u c t u r e .

The a v a i l a b i l i t y of l a r g e computers has made i t f e a s i b l e t o ana lyze

a t r u s s (two or t h r e e dimensional) by d i r e c t a p p l i c a t i o n of t h e energy

theorems of s t r u c t u r q l ana lys i s . For i l l u s t r a t i o n , t h e following sample

problem (from R e f . 2) is so lved by t h e method of secondary stresses and - J b $ . < * A 7r2"

by a d i r e c t method, w h i c h has been programmed and is a v a i l a b l e t o a l l

S t r e s s Groups a t Ast ronaut ics (Program No. 2785, Report No. ERR-AN-206).

O F o r an unsymmetrical loading condi t ion, t h e moment d i s t r i b u t i o n must be modified t o account f o r s idemay .

1 See Ref. 2, page 461 f o r d e t a i l s .

II)

GD/A -Dffi-64-021

2.0 SECONDARY STRESS CONCEPT

For a n adequate understanding of t h e method of secondary o t reoeeo

it is important that t h e rechanior r e s u l t i n g from t h e assumptions be

explained.

Or ig ina l i n t e r s e c t i o n of c e n t r o i d s

F igure 1 - Fixed Trues - J o i n t Relaxed

0 The j o i n t i n F igure 1 i 6 taken ars a t y p i c a l f i x e d j o i n t i n a t rues .

I n i t i a l l y t h e c e n t r o i d s of a l l the members framing i n t o t h e unloaded

j o i n t of t he s t r u c t u r e are a6sured t o i n t e r s e c t . Allowing f o r p in j o i n t

a c t i o n a t t h e j o i n t , the member8 rill d e f l e c t t o t h e shaded p o s i t i o n and

t h e i r c e n t r o i d s t o t h e doubly dashed c e n t e r l i n e pos i t i on . I f t h e j o i n t

is t r u l y f i x e d t h e r e l a t i v e angles between each of t h e members can no t

0 @ I n f i g u r e s 1 and 2 t h e r o t a t i o n of t h e gueset is not shown.

2

CD/A -DDG-64-021

change. Only t h e e n t i r e j o i n t can r o t a t e t o an equi l ibr ium pos i t i on . ' I )

Hence, a f te r the a x i a l loads i n t h e members have been c a l c u l a t e d from

t h e p in j o i n t a n a l y s i s , t h e c o n t i n u i t y a t the I*fixedt1 j o i n t must be r e -

s to red . To accomplish t h i s , t h e r e l a t i v e end d e f l e c t i o n o r r o t a t i o n of

each member must be computed. From t h e s e displacements , f i xed end mo-

ments a r e c a l c u l a t e d and t h e members a r e allowed t o r o t a t e e l a s t i c a l l y

u n t i l t h e j o i n t c o n t i n u i t y is res tored .

p o s i t i o n s of t h e f i x e d j o i n t are shown i n F igure 2.

The i n i t i a l 'and f i n a l d e f l e c t e d

I

Figure 2 - Fixed Truso - J o i n t Unloaded and Loaded

3

GD/A -DDG-64-021

3 0 0 FIXED JOINT TRUSS EXAMPLE

The example so lved by t h e "secondary s t r e s s 1 v method and by Astro-

n a u t i c s Program Noo 2785 is shown i n Figure 3. All j o i n t s a r e r i g i d and

a l l ex terna l load is a p p l i e d a t t h e j o i n t s . The s t r u c t u r e is as su red t o

be geometr ical ly and phys ica l ly symmetric about t h e v e r t i c a l member C c.

With t h i s cond i t ion r e a l i z e d , one-half of t h e t r u s s may be analyzed a f t e r

p r o p e r l y cons t r a in ing t h e s t r u c t u r e a t C , c , and a.

1

336"

4 a t 300lf = 1200" 4

Figure 3 - Example - Fixed J o i n t Truss

4

cc

aB

Bc

GD/A-DDG-64-021

33 6 11044 7809 0 235

45004 27.68 960.9 2 134

300 26.55 922 o 7 3.0'76

3.1 Manual Solution

B -332 c -12.50 D

279

C d D

I 279

Figure 4 - Bar Forces and Streee Intensities

TABLE I Geometric Properties

5

The primary ba r fo rces (p in j o i n t analysim) are shorn i n t h e l i n e

diagram i n Figure 4; t r u s s element geometric (and s e c t i o n ) p r o p e r t i e e

are presented i n Table I. The next s t e p is t o compute t h e ?! angle8

f o r ob ta in ing t h e f ixed end moments (FEM's). The6e 9 angle6 w i l l be

c a l c u l a t e d by t h e b a r cha in method shorn i n Reference 2.

where:

t a n g l e oppomite

Bo IC ang l8 oppos i te

=: ang l8 oppos i te

s i d g

s ide

s i d e

6

GD/A-DDG-64-021

The 9 angle. must b e computed i n o rde r t o c a l c u l a t e t h e f i x e d end

moments f o r the'moment d i s t r i b u t i o n . The a n g l e 9 is t h e r o t a t i o n of

t h e chord j o i n i n g t h e ends of t h e e l a s t i c curve r e f e r r e d t o t h e o r i g i n a l

d i r e c t i o n of t h e member.

curve has r o t a t e d clockwirre from its o r i g i n a l d i r e c t i o n . S ince t h e mea-

b e r Cc i 6 assumed t o remain v e r t i c a l a f t e r loading , t h e f i n a l p o s i t i o n

of a l l t h e j o i n t s may be determined by a l g e b r a i c a l l y summing t h e EA#'#

which g ives t h e 9 f o r each member. The EAd's are c a l c u l a t e d as i n

Table 11. The *a are ca l cu la t ed a8 shown i n Table I1 and a r e then

used t o c a l c u l a t e t h e i n i t i a l end moment ( s e e Table 111).

fol lowing Table 111, presen t s the moment d i s t r i b u t i o n i te ra t ionr r .

comparison of t h e f i n a l i t e r a t i o n and t h e computer program (No. 2785)

r e s u l t s i m p resented i n f i g u r e s 6 and 7 .

* ie p o s i t i v e when t h e chord of t h e e l a o t i c

Figure 5 ,

A

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GD/A-DDG-64-02 1

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GD/A-DDG-64-02 1

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= -10'020 Bc

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TABLE I11 Computation of I n i t i a l End Moments

cc 0 0

Bc 00292 +19*30 -33 0 8 r

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. GD/A -DDG-64-021

(E'OT- )"OK-

11

GD/A - D E - 6 4 -02 1

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80.5

The c a l c u l a t i o n performed above show6 t h e s h e a r s induced by the f i x e d

end moments and t h e r e o u l t a n t secondary ax ia l load8 induced by t h e f i x e d

condi t ion .

of t h e p in j o i n t axial loads is necessary.

These ax ia l loads a r e s u f f i c i e n t l y s ~ l l so that no c o r r e c t i o n

The approximete moments obtained by t h e f i r e t t r i a l moment d i s t r i b u -

t i o n a r e aomewhat h igher than those obtained from t h e d i r e c t approach of

t h e program.

d i s t r i b u t i o n w i l l g ive 80.10 a l t e r n a t i n g anower about t h e t r u e oolu t ion .

I f s e v e r a l c y c l e s are c a r r i e d out i t would be apparent that t h e approxi-

mate approach converges t o t h e answer obtained by d i r e c t a p p l i c a t i o n of

t h e Energy Theoreur.

At3 shown i n Reference 3 and elsewhere, each c y c l e of moment

13

GD/A-DDG-64-021

* 3.2 7090/4 Program Solution

The fol lowing two pages present pr in t out of the problem input data

and a summary of the essential output.

14

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GD/A -DDG- 64 -02 1

e 303 STATIC CHECK OF PROGRAM OUTPUT

A = O

Vert ica l Reaction a t Node 1

R = (2 Direction)

(2 Direction)

= (-2.790 x lo-’) R = 2.79 x 10*1bs.

( 10’9

S t a t i c Reaction (from Fig. 3 )

= 279 Kips ’

Conc 1 us ions :

1. Def lect ions 8 a t i s f y the boundary condit ions.

2. Reaction8 and external loads s a t i s f y the equations of s t a t i c e .

3. Solution is correct .

17

v *

a 4.0 LIST OF REFERENCES AND BIBLIOGRAPHY

1. Parce l ; J. I., and Moorman, R.B.B., "Analysis of S t a t i c a l l y

Inde termina te S t ruc tures t1 , John Wiley and Sona, Inc. 1955

2. Norr i s , C. H., and Wilbur, J. Bo, "Elementary S t r u c t u r a l

Analysis", McGraw-Hill, 1960

3. Costa , J. J., ItAdvanced S t r u c t u r a l Analyeis** Edwards Brothere ,

Inc., Ann Arbor, 1953

4. Maugh, L. C., " S t a t i c a l l y Inde termina te S t ruc tures" , John Wiley

and Sons, 1946

Valania, K. C., and Lloyd, J. R., "A Mntric Method for the

Analysi8 of Complex Space frame^^^, General ~namico /A8t ronau t i c6

Report No. ERR-AN-206, 31 September 1962, Rev. 31 January 1964,

5.

64 p.

10