Note on the matrix analysis on non-linear structures

MARCH, 1961

by

E. D. Poppleton

TECH ISCHE HOGESCHOOL

DElFT

v L.I~GTUIGBOUWO(..JNDE Michiel de Ruy'erweg 10 . DElFT

1

Z okf.1961

NOTE ON THE MATRIX ANALYSIS OF NON-LINEAR STRUCTURES

..

by

E. D. Poppleton

The author is indebted to Dr. G. N. Patterson, Director of the Institute, for providing the opportunity to carry out the investiga-tion reported herein, and to the Defence Research Board of Canada for financial assistance.

SUMMARY

The matrix method of Argyris and Kelsey is applied to the analysis of a structure in which members have a stress-strain curve consisting of two linear segments. Application of the analysis to a

simple box-beam indicates that. with a suitable choice of material pro-perties. the stress distribution is in reasonable agreement with that found using the more realistic Ramberg-Osgood relationship.

I.

Il. IIl. IV. V. TABLE OF CONTENTS NOTATION INTRODUCTION

THE MATRIX EQUATIONS OF ARGYRIS AND KELSEY USE OF THE EQUATIONS WHEN STRESS-STRAIN CURVE IS NON-LINEAR

3. 1 The IdeaHsed Stress-Strain Curve

3. 2 Derivation of the Initial Displacement Matrix 3. 3 Method of Solution of the Equations

SOME CALCULATED RESULTS CONCLUDING REMARKS APPENDIX A APPENDIX B 1 2 3 3 4 7 11 12 14 20

A a b, bo' b1i .1,,\ I C D, Do d E Et F f G H I L m n p p, Q q R (ii) NOTATION

cross sectional area of flange length of flange

matrices defined on page 2 . matrix defined on page 2. matrices defined on page 2 width of sheet

Young's modulus

slope of second linear segment of LS stress -strain curve

flexibility matrix giving displacement of external loads.

matrix of the flexibilities of the unassembled structural members.

shear modulus

column matrix of initial displacements unit matrix

matrix having elements

1

matrix defined by Equations 3, 4, 5. E/Et - 1

'Ramberg - Osgood exponent column matrix having elements p numbers defined on p. 10

square matrix having elements q numbers defined on p. 10

r S t v

x

Subscripts L y 0.7 Superscripts (1) (2) Abbrevia tions L-S R-O

column matrix of displacements of external loads column matrix of generalized loads in members. sheet thickness

column matrix of generalized displacements of members.

column matrix of generalized loads in redundant

members.

fraction of length of flange to which yield point has penetrated

strain

tEl

CTyor

initial strain

column matrix defined by Equations 3, 4, 5 stress

or m ean value of

conditions when stress-strain curve is linear. conditions at yield point in L-S stress strain curve conditions at intersection of R-O curve with the line

cr

=

o

.

7 E

é

.

conditions at ligp.tly loaded end of flange conditions at highly loaded end of flange prime indicates transpose of matrix.

Refers to linear segment stress-strain curve Refers to Ramberg-Osgood stress-strain curve

t"

.

'

.{.

(1)

1. INTRODUCTION

This note presents a, simple method of estimating the effect of a non-linear stress-strain relationship upon the behaviour of a structure. The method is based on the ideas of Argyris and Kelsey, (Ref. 1), and the state of stress in a structural member with a non-linear stress-strain curve is identified with that in a corresponding linearly elastic member which has an initial displacement or "lack of fit". This "lack of fit" is a function of applied stress, in the present application, but, if the non-linearity is due to yielding of the material, then this initial displacement

--..."

is, in fact, the permanent set which the member would suffer, on unload-ing. Recognition of this fact makes it possible to analyse structures in which loc al unloading takes place, during the application of the external load system.

The equation giving the stress distribution in the structure is the same as that given in Ref. 2, but a different point of view is used in the derivation. In Ref. 2, the flexibility matrix is considered to be a function of the stress distribution and this tends to mask the interpretation of the terms in the equation and leads Denke to restrict his analysis to cases where no unloading takes place in a member subsequent to yielding. The equation is used in Ref. 2, in conjunction with the Ramberg-Osgood stress-strain curve, and the Newton-Raphson method is used to obtain an iterative solution. It is shown here, however, th at a correction term is necessary in the solution given in Ref. 2, for the case where the structure contains members carrying linearly varying axial loads. This correction can be quite large in typical cases, and tends to make the calculation more

cumbersome.

In this note, the use of a simple stress-strain curve having two linear segments is investigated and it is found that a very simple iterative procedure can be used to solve the equation for the stress distri-bution in the structure. Moreover, in simple examples, it is found that

the properties of the assumed stress-strain curve can be matched to a Ramberg-Osgood curve to give acceptable agreement in the stress distri-bution.

The present discus sion is limited to structures which can be idealised as an assembly of flanges (having ,constant cross-sectional area between structural .nodes) and rectangular sheets carrying constant shear flows along their edges. The axialloads in the' flanges may thus vary linearly between nodes and pin-jointed frame-works are, of course,

included in the analysis. An example of this latter application is given in Appendix B.

IT. THE MATRIX EQUA TIONS OF ARGYRIS AND KELSEY

The notation of Ref. 1 wil! be used here as far as possible

. and we describe the stress distribution in the structure by means of a

column matrix of generalized loads CS). The elements of this column will

include

(a) two-row sub-columns giving the force at each end of flanges

in which the load varies linearly along the length,

(b) scalar numbers giving the force in each pin-jointed bar

carrying constant load,

(c) scalar numbers giving the shear force on an edge of each

shear web. Frequently it may be convenient to consider as a structural

member a combination of simple structural elements as indicated in Ref. 1.

In this event, the corresponding element in S would be a sub-column,

containing combinations of elements of the type listed above.

The external forces and generalized loads in members (or

combinations of members) chosen as redundancies are represented by

columns Rand X respectively, and it is shown in Ref. 1 that the matrix

equation giving the generalised loads in the members is

where S = _{b o R}

+

bI X ~ bI D -1 bI H 1 = _{b o R - bI D} -1 _{Do R - bI D} -1 _{bI H} 1 = bR - cH D = b

_l

fbI ( =. fl DL f of Ref. 2) D ₀ = _{bi fbo}

H

=

column of generalized initial displacements (due to

thermal effects or lack of fit) (equivalent to DL ~ F

of Ref. 2).

b

=

b _o - bI D- l D ₀

c -1 fl of Ref. 2)

f = diagonal matrix of flexibilities of the unassembled

members (

=-

DL of Ref. 2)

The matrices b o and bI can be readily written down from

equilibrium equations, if the redundancies are judiciously chosen as

indicated in Ref. 1. Typical diagonal elements of f are

- - - -- - - -- -

-(a) a sub-matrix a 6EA

(3 )

[~

with constant cross-sectional area A,

~

J'

for a flange of length a

(b) a for a pin-jointed bar of length a and constant cross-EA

sectional area A,

(c) d for a rectangular shear panel (a x d) carrying a shear Gat

force along the edge a.

The generalized displacements of the members are given by the matrix equation v = fS

+

H and the displacements of the external load points are given by

r

=

FR

+

bI H

where

F

=

bI f b - DI D- 1 D

o 0 0 0

lil. USE OF THE EQUATIONS WHEN STRESS-STRAIN CURVE IS NON-LINEAR

3.1 The Idealised Stress'TIStrain Curve

In order to use the equations listed in II, in the case where some of the members in the structure are loaded beyond the proportional limit (to condition A for example in Fig. 3), we state that the load condition is identical with th at obtaining in a linear member having an initial strain

Eo

=

é. ... -

cr;., /

E.. This initia! strain is a real permanent set in the case of a member which would unload along the path AB, but if the loading proeess 4.s. reversible (as might be the case with a buckled member) th en the

initial strain has no physical significance as such.

In order to use this idea we require a value of the initial strain appropriate to a given stress and we shall consider two stress-strain curves, one consisting of two linear segments (referred to sub-sequently as L-S) and the Ramberg-Osgood (R-O) stress-strain curve.

The equation to the former is

= a-

cr

=-

<T/qy

L

I

( j

7/

where m

=

E - 1 , Et

=

tangent modulus ,and the Ramberg-Osgood Et

curve is

t.. -

E

f:../~.7

₌

q=-

+

3/,

~

l-1

where

q=

=-

CT

Iv;.,

We have therefore the following expressions for the initial strain

Vvt

(CF -

I ) (L-S)

(R-O)

3.2 Derivation of the Initial Displacement Matrix

At this stage, we shall restrict the analysis to structures having three types of

members:-a) Rectangular sheets carrying shear only,

b) Pin-ended bars carrying constant axial load only,

c) Constant area flanges carrying a linearly varying axial load only.

The elements in the initial displacement matrix correspond-ing to the first two types of member are easily written down.

For a sheet carrying a dimensionless shear stress 0- and having an edge length d, we have

or H

=

dt.

_o =

~

W\

l~-l)

-=-

1

Z

Gt

-.l c :::: 3

cl.

c:::fó'7 ~ - V\ H

=

O\c:,...o _ _

cr

=

I

Gr-and for a uniform pin-jointed bar of length a,

or where (L-S) (1) (R-O) (L-S) (2) (R-O)

(5)

The value of H for a tapered bar is also readily derived, when the shape of the bar is known.

For the flange, we require a generalized displacement, due to initial strain, consistent with the generalized displac~unents~ due..to stress. This will be represented by a two-element column for each flange, the

elements giving the displacement at each end of the flange and we find these using the unit load theorem as in Ref. 1.

Let us consider first the case of a flange of length a, having an L-S stress-strain curve on which the yield point has penetrated a dis-tance tXa.. from the more highly loaded end. Since the load in the flange is distributed linearly, the initial strain will be distributed linearly also. To find the displacement at the more highly loaded end we apply a unit load and hold the flange in equilibrium by a shear flow

thus:-I:

~ ~ _:~_~ ~~I 1--~2)

:J .... ,

S

(stress due to unit load) x (initial strain) d (volume)

( )

îCX:G\.

J

~)

-x:

-:>C H 2

=

D.

1.

[CI-IX)+

~ ~o

o(ä..da:

o

which gives

_

~

o...iX

(3

-<X.)

Z:~)

oU\., Similarly H(1) = ö.. Jo

[0( -

~J t.~) ~O-

cl ~

Hence we have H where

than

, ; When the stresses at both end~ of the flange are greater

V

y , then we may use the formula given in Ref. land we have

{~)

- CÀ:EG>

L~ ~J ~~I,J

=-

[-e]{Z}

(4)

where

{~7

:

Turning now to the Ramberg-Osgood stress-strain

relation-ship, we now have an initial strain distribution which is not linear, since

it is the stress which varies linearly along the length of the flange. Hence

we have

cJ=

where x is measured from the lightly loaded end, and consequently

~o

= :;

(JE'

[èT

6J

+

(\J.l~)_qm J~]vt

Using the unit load theorem again we have

where

o

..!

~ <::Jo" Î -E

[~J

{Z1

(5)

In passing, it may be noted that the total elongation of the

(7)

where

cr

=

~

(0-(1)+

~)

)

=

mean stress in flange. In the analysis of Denke (Ref. 2), the elongation of a flange is taken to be

~ A. q-~,

o-Vl

,

and it appears that serious errors might arise due to the neglect of the correction terms. The value of ~ is generally quite large, and the first correction term is not negligible, even with moderately smal! stress gradients in the flange.

Given the stress distribution in a structure, therefore, it is now possible to write down the initial displacement matrix in the form

H=L~

where L is a diagonal matrix having as elements the,l defined in Equations 1 to 5. ~ is a column whose elements are the functions of stress and yield point penetration shown in the same equations . With the L-S stress-strain curve these matrices, of course, contain non-zero elements only in the locations corresponding to members in which yield-ing has taken place.

3. 3 Method of Solution of the Equations

The matrix equation giving the stress distribution in the structure m ay now be written

S

=

bR - cLL.

(6) where the column SL gives the distribution of generalized loads in the corresponding linear structure.

. For solution it is convenient to pre-multiply by [Sy'J -1

or lSo. 7J -1 , depending on the stress-strain relations being used, where

Sy

= load in member at yield and SO.7 = load in member when stress is

0-

₀. 7.

Equation 6 then becomes

(L-S)

(R-O)

The latter equation may now be compared with that of Ref. 2, which in the notation of that p~per is

Y=YL-C'<P Y

(7a)

On making the necessary change in notation this becomes

(8)

It should be remembered that this equation is in terms of .the mean stresses in the flanges whereas, in Equation 6, the load in a flan ge is described by a two element column giving the stresses at the two ends of the flange . . The two equations are, of course, equivalent when the structure consists only of constant load bars, but not when it contains flanges.

In accordance with the work of Section 3. 2, Equation 8 should contain a correction term thus

(9)

~~- (f~)

in which sum of shear for:ces applied to flange by adjacent sheets

ër-

=

mean axial load in flange

The Newton - Raphson solution used in Ref. 2 is now somewhat more complicated to apply since the matrix

which must be inverted at each iteration is no longer diagonal.

An investigation was made of behaviour of the correction term in the above solution and it was found that to a fair degree of approxi -mation Equation 7(b) could be written

where L is now a matrix in which diagonal elements corresponding to a flange are given by

and

.

~a...CJö.,

/0'39 ï

E:

LO'77

-30..

CJö·,

L' ·

0 0 1

E

1·33

0.04J

O'

Ir

if if (11)

(9)

The relationship between the true values of the end dis-placements and those given by the above are shown in Fig. land it will be seen that the results are nearly always within about 10% of the exact value, except when n is small. At this expense in accuracy, the matrix corresponding to Eq. 10 may be preserved in diagonal form.

The equation appropriate to the L-S stress -strain diagram can be solved by means of a very simple iterative procedure, however.

This consists in holding the initial displacements constant in all except one member, or a small group of members, and determining the corres-ponding stress distribution. The displacements of th is member or group of members are then held constant while those of another member or group of members are allowed to vary, and another approximation to the stress distribution is found. The process is continued until convergence is satisfactory and each iteration involves, at most, the solution of a linear matrix equation or the solution of a scalar cubic equation for the value of Q( in a single partially yielded flange.

To illustrate the procedure let us rewrite Equation 7(a) showing the three possible elements in

Lt

and L viz.

- (1)

m

₁₎ -1 _{~I ~,CJt,0}

CT

₁

₌

_1L

Sy c

-

0 0 0 0 ~EI - (2) - (2) Cl,

"",\]j,

cr

1

~L

0 _bE., 0 0 0 0

--

_cr-

-

_{--- (Ï)} (1)

_o

_{0-." ~ ...}

_o;.l

_{~Z.~lo;, 0} 2

C72L

0

-

_{~ E::L} -~

_&e.

0 - (2) _{Cf2~2) 0}

_o

a~.!-~L o~~~ 0 ₀

0"2

bEl.. :3

e.2.

-

--

-

-

-CJ3

CJ3L

0 0 0

o

Q,~~ 0 E'3

It should be noted that the column ~ will gene rally have a large number of zero elements corresponding to members which remain elastic. Due to the non-linearity of the elements inZ corresponding to a partially yielded flange, it is convenient to deal with such members "

~

singly and, as an illustration let us suppose that the calculation has pro-ceeded to the point where we wish to determine

Gï

j, while holding all

other initial displacements at the values determined by the previous iteration. The top two equations give the value of

cr,- ,

in the form

J

l

2 - (2) ]

q12

0<.1

«(]1 -1)

q22 0(. 1 (3 - <Xl)

(éJ\

(2) - 1)

which may be simplified to give the following scalar Equation for

tX

1

r(P2 - PLl L\P2 - 1

J

+31/p2-P1)

lC

P 2 - 1 since

IX,

=-+

( P2 -

PIJ

Al

+

<'x1 - 1 =0 P2 - 1

Having determined

Xl

from Equation 12, the stresses are found from

(::2 - 1)

0-+

1 )

=

1

(12)

. Calculation of the stresses in the other two types of member

is simply accornplished, since the corresponding elements in ~ are

linear. Consequently we can write

cr/I)

₌

_{P3 q33}

_V

(l)

2 - 1

Cl:

(2)

2 P4

CT

2 (2) - 1

(11)

or

0-

=

P -

Q

{cr -

Ir

where the order of the matrices is appropriate to the number of yielded members of this type.

The solution of th is equation is sim ply

(13)

It is possible that the load in a yielded member may decrease on increasing the externalloads above a certain value. In this event, the value of H appropriate to the maximum value of the load in the member may be assumed to be the permanent set in the member, and th is is held constant in subsequent calculations, and the member is allowed to unload linearly.

IV. SOME CALCULATED RESULTS

Some simple calculations have been made, firstly to check to what extent results obtained using of the L-S stress-strain curve differ from those obtained using the more realistic R. -0 curve, and secondly to make a preliminary assessment of the rapidity of convergence. The details of the calculations are given in the Appendices.

The first structure considered was the simple box beam illustrated in Fig. 2. This has already been analysed in Ref. 3 and it was found that, with the load distribution

R3

=

_{4R 1}

the load at the inboard end of the corner flanges was considerably higher than that in the central stringer. This mem ber would thus yield first and an anlysis of the post-yield behaviour was made for the following stress-strain

parameters:-1) L-S E/Et = 20, 40 2) R-O V\. = 9, 12.5, 20.

The results are shown in Fig. 4, in which the abscissae Rl are values of Rl as a fraction of the value of Rl at which

cr-

3 (2)

=

1 and the curves are plotted on the assum ption that rj y =

cr-

0.7' It will be seen that the curves corresponding to L-S and R-O are qualitatively very similar and, in particular, the difference between the shape of the curves in the region 0.8 L.. o-L..l. 05 is much less marked than on the stress-strain curves. It seems that it is possible to represent the behaviour of the member over quite a large load range by identifying a val:ue of E/Et

with a value of Vl and possibly modifying the ratio ~/

VOo

7 slightly.

For example, if we assume

r:::r

y / 0-0.7 = 1 then the curves for E/Et = 20

and E/Et

=

40 represent quite weU the behaviour of the curves for n

=

12.5

and n

=

20, at least up to

fT

=

1. 25. This would be close to the ultimate

strength of a high strength alloy such as 7075. The corresponding L-S

and R-O curves for the former case are compared in Fig. 3.

Another comparison between the result of using L-S and

R-O curves is obtained by analysing Truss III of Ref. 4. In this work the

stress-strain curves of the mernbers were measured and they appear to be well represented by two straight lines joined py a rounded 'knee', as shown inset in Fig. 5. The corresponding L-S curve was taken to be that formed by

projecting the linear parts of the measured curve to define

\Ty

and using

measured values of E and Et. It should be noted that the compression

member bowed during the test and finally buckled, rather than yielded, so that the measured value of E for the member was considerably lower than that of the material itself.

The results are shown in Fig. 5 where the present analysis

is compared with the iterative calculation of Ref. 4, which takes account of "

the round 'knee' in the stress strain curve. The two methods give very nearly the same result although the present method gives curves with three linear segments, the first crank occurring whem Member (2) "yields" and the second when Mernber (3) "buckles". The rneasured results agree well with the calculated values for the two tension members, but there is some discrepancy in the case of the compression mernber, when a large amount of bowing may have influenced the strain gauge readings.

FinaUy, some further calculations 'were done for the beam

of Fig. 2 for the case where R3'

= -

1. 3 Rl, R2

=

Rl' Here the yielding

at the root is somewhat more widespread and the results shown in Figs.

6 and 7 emphasize sorne of the interesting features of non-linear behaviour.

Fig. 6 shows how rapidly a mernber will 'shed' load to a neighbouring

elastic mem ber once the yield is reached locally, even though Fig. 7 shows that the yield point does not penetrate very deeply. The shedding of load causes a large increase in the shear in the adjacent panel and, in the case

considered here, the shear in the panel

(0-0,)

is increased by ab out 90%

when the load ratio is 1. 24. Also noteworthy is the very rapid increase in

the displacement of the end af ter the central stringer has yielded. (Fig. 7). V. CONCL UDING REMARKS

The use of the method of Argyris and Kelsey has been shown to yield a very simple solution for a structure in which the stress strain

curves for the members can be approximated by two linear segments. The "

solution is a 'relaxation' procedure in which, at each iteration, a member or group of members is allowed to deform in a non-linear manner while

(13)

the linear stress is near the yield stress are involved in the calculation and hence the order of the matrix equations may be quite small even in a large structure. In any event, the computer pro~ram required is very simple involving either the solution of a scalar cubic equation or a linear matrix equation at each iteration. The complexity of the solution thus compares favourably with that suggested by Denke where the Rarnberg-Osgood relationship is assumed and the resulting non-linear matrix equation solved by the Newton-Raphson method. The order of the

matrices involved in the latter procedure will probably be larger than in the present method since members carrying stresses less tha!l"cr- 0; 7 will still be included. However, the calculations of the present work, albeit for a very simple structure, seem to indicate that, with a proper choice of material properties, the L-S calculations will give a good approxi-mation to the R-O results. In any event, the difference between the L-S and R-Q are not likely to be of great significance bearing in mind the

structural idealisations which have to be made, perforce, in order to make any analysis a practicable proposition.

APPENDIX A

Analysis of a Simple Box-Beam A. 1 Basic Structural Matrices

As a simple example of the procedure we consider the box-beam illustrated in Fig. 2. This was analysed in Ref. 3, using panels 7, 8 9 as the redundancies,

only one quarter of the structure. b ₀

=

0 8 8 16 16 24 0 0 0 0 0 0 8 8 8 0 0 0 D

=

a 6 AE 0 0 0 0 0 0 8 0 8 0 16 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8 0 8 8 0 0 0 0 0 0

~

4.25

18.00 6.00 18.00 22.25

B.OO

6. 0fl

6.00 10. 25

and, due to the symmetry, cc:msidering The following matrices were evaluated.

bI

=

0 0 0 -1 0 0 -1 0 0 -1 -1 0 -1 -1 0 -1 -1 -1 0 0 0 1 0 0 1 0 0 1 1 t() 1 1 0 1 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 D

= -

~Q08

o 6AE 160 64 96 88 40

2~

24 16

a f = -6AE 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 (15) 2 1 1 2 O. 9375 0.9375 O. 9375 6.25 6.25 6.25

and from these the Flexibility matrix is readily derived F

=

bI f b - DI D- 1 D o 0 0 0 a [

= -

596 6 AE 318 106 A. 2 Solution of Equations 318 215 81 106J ₈₁ 53

Consider first the load system,used in Ref. 1, where R3

=

4R1' R2

=

3R1· This gives the generalized forc'e in Mernber (3)

S3 =

l-

18. 15]

Rl 70.76

and, since this is by far the most highly loaded member, we shall first of all consider that only this member yields.

The matrix c appears in fuU in Ref. 3 and the ·elerne.nts appropriate to Mernber (3) are

•

6 AE

So Eq. 7 (a) can be written

rëT;(1»)

=

[ls.15

l

Rl/py - 3:2

L

Q;

(2) 70. 76J

=l-

o.

256

1

Rl 1. 000

J

l

-2.18

O.

90 O. 90][!X'32(cr3 (2) - 1)

J

- (2) 4. 18 cX3 (3

-~)(

q

3 - 1) (2)

where Rl

=

value of Rl as a multiple of value of Rl when

(J'"

3

=

1, and we have assumed E IEt

=

40.

As an example, consider the case when

ih

~l. 5. : Bl,lbstituting for the values of p, and q in ~quation (12) yields

-2.78

rX

+

18.23 ()( '2

+

2. 24

eX. -

1 = 0 the required solution of which is

rX

=

0.183 and hence

= [ : : : : : ]

For this simple case, Equation ,'Z(b) becomes

-

L-O.

256 ] Rl -

-.!..

x 8 0. 7 1. 000 -AE x

~

a<T"0.7 [210 3752a 7 E

87

=

fo·

256l

RCrO. 144

h

·

000

J

LO.060 _ ... /

87J2.

402

(17)

in which the terms involving

CT

₃(1) in ~ have been omitted since they will be of the order of (0. 256)~ and hence very smal!. It was _ found most convenient to solve the above ~uation for a fixed value of

.

0-

₃(2)

and determine the app~priate value~of

03

(1) and Rl. For example, for the case

vt

= 20, (T3(2) = 1. 1, \T3 (1) = 0.3 was assumed as a first approximation and this yie!ded Rl

=

1. 215, 3(1)

=

0.284.

A further iteration gave the final. solution Ri

=

1. 2f2, 0"'3(1)

=

0.284

The results of calculations of the value of

cr-

3 (2) for several different values of

Vl ,

and E/Et are shown in Fig. 4.

Consider now the case where R3 =-1. _{3 Rl' R 2} = Rl and E/Et = 40. There are now three fairly heavily loaded .members and selecting appropriate elements of c from Ref. 3, we get the equation

- (1) CT2 = O. 184 Rl - 2.00 0.46 0.46 O. 19 -0.46 -0. 19 0"2(2) 0.805 0.46 2. 18 2. 18 O. 90 -2. 18 -0. 90

-

--

-

-([3 (1) 0.805 0.46 2.18 2. 18 O. 90 -2. 18 -0. 90 CT3(2) 1. 000 O. 19 0.90 0 .. 90 4. 18 -0. 90 -4.18

-

-

-

- -

-

0-6 (1) 0.584 -0.46 -2. 18 -2. 18 -0. 90 2. 18 O. 90

-06(2) 0.710 -0.19 -0.90 -0.90 -4. 18 0.90 4.18

This equation is now solved for various values of Rl ' as indicated in Section 3.3. In this case it was found that the yield point always remains within the member so that the elements of

L3

are

j-

d'

(0-(2) - 1) ]

L

tX.

(3 - ()( )( \7(2) - 1)

~

and the iteration proced,ure proceeds by 'relaxing' one member at a time. Values of Rl up to 1. 26 were investigated and up to this value member (2) had not yielded. It was found that with a judicious choice of first

approximation the convergence in this simple problem was quite rapid. As an example, consider the case Rl = 1. 26. Extrapolation of previous

(18)

r

~1)1

=

ro.

810

1

l

ëJ~2)

0·

08~

which give

rXb

= 0.302, and hence the equation for

ft

3 becomes

r

U3

(1)]

₌

r1.

_{094] - [2.18 0.90J}

r

_D(32 _{(0-3(2) -1)}

J

Lèr3(2)

~.

550 0.90 4.18

LtX

3 (3 -

r.i

3) (0=3(2) - 1)

Use of Equation 12 yields

tX..

3 = 0.569 and

[~:::J

=

[:"::7

0

:]

.

and hence the equation for the second approximation to ~ becomes

l

~(l)l=

rO.889]

cr~2)

Ll.

370 -r2.18

~.

90

o.

90Jfo( ~

(06(2) -

1)

J

4. 18

~6

(3 ...;

~6)( ëf

_{6 (2) - 1} This gives

tX

6 = 0.304 and

~(~:J

=

I~.

8121/

G·

0821J

-and the next equation forU3 gave 0(3 = 0.571

[~:(::]

=

~

..

:::j

Use of these results made no apparent change in

0{

6 so the final result was taken to be

r!:(lX

= 0.942 3

if3 (2) 1. 077

Vs(1)

0.812

(19)

Having found the distribution of

cr-

in the structure the matrix H can be determined and hence the delfection under the external loads may be found from

r

=

FR

+

b'H

The stresses, displacements and values of

tx:.

are shown plotted in Figs. 6 and 7 as functions of Rl .

APPENDIX B

Analysis of Truss lIl, Ref. 4 B. 1 Basic Structural Matrices

The measured stress-strain curves for the members were wel! represented by two straight lines joined by a curved 'knee' (see Fig. 5). The L-S curve was taken to be composed of the two straight lines only and this yielded the approximate material properties given below.

Length Area CTyx 10 -3 Ex 10-6 Et x 10-6

Me.mber in _{sq. in. psi psi psi}

wt.

1 67.00

o.

125 49. 5 10.5 0.518 19.3 2 23.68 0.~25 49. 5 11.0.5 0. 518 19.3 3 20.00 0.500 10.4 1. 95 O. 123 14.8

Member (1) was taken as the redundancy and the following matrices were deter.mined b o

=t

·

:

25

J

b 1

=

lo

.

l825J

b =

U·

295

U

0.581 -0.667 0. 667 -0.469 -6 D

=

72. 35 x 10 in/lb

-e

=

ll.

₃₈₂ -1. 140 _{0. 922 ]} x 104 lb/in. -1. 140

O.

941 -0.760 0.922 -0.760 0.615 B. 2 Solution of Equations

\Ti

CJ2

-<J3

Equation (7a) becOInes

=

4.768 Qx10- 5 9.400 -9.040 =[4. 768J Qx 10-5 .9.400 -9.040

-

8 x 10- 4 _{G 6. 10}

--

_tr

1 - 1 4.95 _{8 x 10- 4} 2.16 .

_\f

2 - 1 4.95 10.-4 _{1. 58} \1""3

+

1 1. 04 -11. 240

-8

3 .624 10.798 -3. 976 . 2.

35j~CT1

-

Ü

3.288 -1.944

512 -

1 -3. 152 1. 866 0-3

+

1

(21)

It should be noted that the initial displacement is

propor-tional to

(CT +

1) in Member (3), since this member is in compression.

Inspection of the equations shows that member (2) yields

first and the behaviour of the member in this condition is described by

the second equation. The solution of this simple linear scalar equation

leads to 82 = O. 136 Q

+

4760 and, hence,

~

_S2

81J

=

L

_O0_. .₁₃₆ 834] Q

S3 -0.110

+[--570Q

₄₇₆₀

-3800

When Member (3) buckles the behaviour is described by the solution of the second and third equation, a simple matrix equation, viz.

r~21

=

/e.400]

Q x 10-5

L

fT

3J

t

e

.

040

the solution of which yields

r:;]

=

r~

:

~::J

Q

ls3

L-0.077

- r3. 288 [ 3. 152 - 1.

944tl-cr2 -

J

1.

866JLf

3

+

1

+

t-6360

J

5280 -4230

The loads in the members are shown plotted, as a function of Q, in Fig. 5

together with the measurem ents and calculations of Ref. 4.

( ' • r

1. Argyris,.J. H. Kelsey, S. 2. Denke, P. H. 3. Poppleton, E. D. 4. Steinbacher, F. R. Gaylord, C. N. Rey, W. K. REFERENCES

The Matrix Force Method of Structural Analysis and Some New Applications. Aeronautical Research Council, Rand M No. 3034, Feb., 1956.

The Matrix Solution of Certain Nonlinear Problems in Structural Analysis Journal of the Aeronautical Sciences, Vol. 23, No. 3 March, 1956.

Note on the Design of Redundant Structures, UTIA Technical Note. No. 36, July, 1960. Method for Analysing Indeterm inate

Structures Stress Above Proportional Limit. NACA Tech Note 2376, June 1951.

"lAPPRO 1.0 0·1 FIG. 1. 7E

_.J::L

,=

3 Cl.

cr

0.'1

(fCl>] "

r=

cr

(1)

7(1)

r= Z·S

"=4

r=

2 n=4 n 0 ₄₀ 0 20

II

10 'V -4-1.0 _{1 0 · 0} " eXACT

COMPARISON OF EXACT VALUE OF THE ELONGATION OF A FLANGE WITH THAT GIVEN BY EQUATION 11

Thickness of Upper and Lower Skin Thickness of Vertical Web

Area of Corner Flanges

=

A

Area of Central Flanges

=

2A

Flange length

=

a

Sheet width

=

d

Ribs at loaded sections infinitely stiff

= 0.03 in. = 0.05 in

=

1 sq. in.

=

2 sq. in. = 40 in.

=

20 in. 4R2.

FIG. 2. SIMPLE BOX-BEAM

4R,

1.0

0.'

0·4 0.2. o 1.2, 1·0 _CZ)

<r3

0·4 o'Zo o FIG. 4. FIG. 3 0·8 0·5 1.0

COMPARISON OF L-S AND R-O CURVES GIVING APPROXIMATELY SAME STRESS DISTRIBUTION IN BOX-BEAM. 1·2 _z·o .Ë.. n E-t 0 _{40 6} 9 \l 2.0 ~ 20

e

12. .5 1·5 2.0 2.5 RI

5 4

S

3 -3 I( 10 LBS. Zo o TRUSS III

,

4 FIG. 5.

I

MEMBER 2 YIELDS MEM8&:R LOADS

o

THE.;)RETICAL. (REF.4) " ACT U A L ( ~ E F. <4 ) PRESENT CALC. 6

•

10 -3 Q )( 10 LBS. 10

•

-3 G'""xIO G P.f.I. 4 2. 12. ~ MEMBER 3 BUCKLES

,

ASSUMED LlNEAR SEGMENTS

"'~

STRESS-STRAIN DIAGRAM FOR MEMBER 3 o &. 4 • • E I( 103 IN/IN 1+ Ui

·8 .2. o 1.0 FIG. 6. 1.10 -(2.)

<r.

L I I MEMBER 6 YIELDS ... 0":99 __

-<r.L

I I I I

,. z.

1·7 ,., 0·6 ',5 o·S 1·4 0·4 1'3 O·J "l. 0.2

'ol

0·\ 1.0 FIG. 7 MEMBER 6 YIELDS

r,=

R,'

I

I

J I I

,

I. , R,

DEFLECTION OF TIP OF BOX-BEAM AND PENETRATION OF YIELD POINT INTO FLANGE.