Archive of Applied Mechanics 63 (1993) 253-260 Archive of

Applied Mechanics �9 Springer-Verlag 1993

Snapthrough instability of a stochastic a yon Mises truss

S. A. Ramu and R. Ganesan, Bangalore

Summary: Avon Mises truss with stochastically varying material properties is investigated for snapthrough instability. The variability of the snap-through load is calculated analytically as a function of the material property variability represented as a stochastic process. The bounds are established which are independent of the knowledge of the complete description of correlation structure which is seldom possible using the experimental data. Two processes are considered to represent the material property variability and the results are presented graphically.

Durchschlagsinstabilitfit eines stochastischen von Mises Fachwerkes

Obersieht: Ein von Mises Fachwerk mit stochastisch verteilten Materialeigenschaften wird beziiglich der Durchschlagsinstabi- lit/it untersucht. Die Spannbreite der Durchschlagslast wird analytisch als Funktion der Spannbreite der Materialeigenschaf- ten berechnet, die stochastisch verteilt angenommen werden. Eine explizite Gesamtbeschreibung der Struktur ist bei Benutzung experimenteller Daten selten m6glich. Deshalb werden Grenzen ftir die Durchschlagskraft entwickelt, die vonder Kenntnis dieser Gesamtbeschreibung unabhfingig sind. Zwei Grenzf/ille werden betrachtet, um die Spannbreite der Materialeigenschaften darzustellen. Die Ergebnisse werden grafisch dargestellt.

1 Introduction

The importance of reckoning with the variability of mechanical properties of engineering materials in structural analysis and design needs hardly to be emphasized. The most popular materials like reinforced concrete in civil engineering and composites in aerospace engineering are perfect examples of materials with property variability. A witness to the recognition of this fact is the recent spurt in research activity in the engineering mechanics of stochastic fields [1-14]. The pr imary focus is currently centered on the response variability due to variability of material properties which are interpreted as stochastic fields. In the area of elastic stability investigations are principally concerned with deterministic structures under stochastic loading [15-18]. Ar iara tnam [15] considered the behaviour of a column when subjected to r andom loading and a deterministic truss subjected to a r andom snapthrough loading. Ar iara tnam and Xie [16] considered the behaviour of a shallow deterministic arch subjected to stochastic loading.

In this paper the snapthrough instability of a pinjointed truss representing a shallow arch is carried out. The elastic property of the material of the members of the truss is assigned a stochastic description so as to account for variability along the member length.

2 Basic equations

The problem of snap-buckling of shallow trusses and arches has been the subject of investigation of many researchers [19, 20].

The truss in Fig. 1 consists of two identical members hinged to each other at the crown and to rigid abutments at their other end. If the angle ~ is small, it is well known that the truss will snapthrough at a load

W = W m a x = 211 - - c 0 s 2 / 3 ( 0 ~ ) ] 3 /2 A E . (1)

18"

254 Archive of Applied Mechanics 63 (1993)

///• W

/ / / / Z . / / / / / / / - / 7

~ - -~ / " / ~ . . . . _ _ ~ . Snapped posi{ion

Fig. 1. Snapthrough of a shallow stochastic truss

In the above, it is assumed that all the quantities: A = area of cross section of each member, E = modulus of elasticity of material of member ~ -- the angle of the truss, and hence Wmax the critical load are constant deterministic quantities.

Suppose, on the other hand, the modulus of elasticity E of the material of the members is variable. Let the spatially variable modulus of elasticity E(x) be represented as a one dimensional, homogeneous, univariate stochastic field. This can be written as

1 1 - [1 + f ( x ) ] , (2)

E(x) Eo where: x is the coordinate measured along the member axis, and f(x) is a homogeneous one dimensional stochastic field with mean zero and with very small fluctuations.

It is to be noted that 1/Eo = {1/E(x)), where ( . ) denotes expectation, and (0r(x)) = 0;f(x) <d in the sense that 0C2(x)) ~d.

Further, the variance of 1/E(x) is (1) /{, 1;) ,: Var E -~ = E o [ l + f ( x ) ] - E00 - Eo 2'

where o-}: is the variance off(x). Hence the coefficient of variation of 1/E(x) is

2 2 :/Eo - (3)

mean

It is easy to derive the load-deformation relation as

U = 2 s i n O AEo L + I f(~) d~ 0

where: u is the total shortening of the member, and O is the inclination of the deformed bar axis with the horizontal.

The length of the deformed bar l, is related to the originl length L through the relation

l cos O = L cos c~.

Using the above, the load-deformation relation can be written as

. . . . . w 1 c~ Jc~176 i L �9 (4) w AEo 1

1 + ~ f(~)d~

0

In the above w is the dimensionless load parameter.

S. A. Ramu and R. Ganesan: Snapthrough instability

W,

1 s / ---

255

Fig. 2. Structural nonlinear response using mean value of E(x)

3 Critical load

The critical level of deformat ion O at which the maximal or snapthrough value of the load is reached is given by the condi t ion for averaged problem

dw = 0, i.e., when cos O = l /cos c~. (5)

dO

The behaviour of averaged problem is shown in Fig. 2. The snapthrough load is then equal to

(/E '; 1) Wm,x = 211 -- COS 2/3 (C~)] a/2" 1 1 + ~ f (~ )d~ . (6)

0

Since the above is a stochastic quanti ty, it will be useful only when its expected value and variance are calculated.

The expected value of Wm~x

( W m a x ) ~- 211 - - COS 2/3 (~)]3/2 1 + Z f ( ~ ) . d { .

Using the binominal expansion for the denomina tor the expected value of the snapthrough load is found to be

(Wmax> = 211 - c o s 2/3 (~x)] 3/2 .

4 Calculation of variance

The variance of the snapthrough load is given by

Var ( W m a x ) = ( [ W m a x - - ( W m a x ) ] 2 )

Z f (~ ) .d~ - 211 - cos 2/3 (cO] 3/2

0

f ( ~ l d r + 1 - 2 1 + g f(~')d{ .

0

(8)

The above can be evaluated term by term.

256 Archive of Applied Mechanics 63 (1993)

Evaluation of the first term:

({/[ i ]'ti( ; , [i ] 1 2 2 3 1 2 I = 1 1 + ~ f (~ )d4 = 1 - ~ f ( { ) d { + - s f (~ )d4

0

+ ... higher order terms / .

Neglecting the higher order terms, this reduces to

3/[i; 11 I = 1 + ~5 f (~ l ) f(~2) d41 d~2

0

L L L L

' i f ' f f = 1 + ~5 (,f(r "f(42)) d4, d42 = 1 + ~5 Rsy(r - 42)de, dr (9)

0 0 0 0

where Rir(41 - 4z) is the autocorrelation function of the process f(~). Similarly, the third term in eqn. (8) can be shown to be

< / [ f ] ) 1 1 1 f f ] I I I = 2 1 + Z f(r 2 1 + ~ Rsy(r162 dg, dr �9 (10)

0 0 0

Introducing (9) and (10) in (8), the variance of the snapthrough load is found to be

L L

4 f f Var (Wmax) = ~ [1 - c o s 2/3 (e ) ] 3 Rsf(4a - - ~2) d ~ l d ~ 2 . ( 1 1 )

0 0

Using Wiener - Khinchtine relations, this can be written in terms of the power spectral density as follows:

LL i 4 ff Var (Wmax) = ~ [1 - - COS 2/3 ((~)]3

0 0 -co

Sff(co) ei~162162 ) dco d~l d42. (12)

5 Bounds for variance

For a band limited process, with upper and lower cut off frequencies + co,, - co, respectively, two cases can be considered as follows"

Case 1. When co, --* 0, the PSD function sharpens around the origin and we have 9-o0 q-to u

o'}y = j" S~rs(co) do = j" Ssi(co )dco and so, - co -~o u

SsAco) = ~}I ' ~(co.)

where 8(09.) is a direc delta function.

s. A. Ramu and R. Ganesan: Snapthrough instability 257

Case 2. For a finite power white noise (actually ideal white noise cannot be a causal one) i.e. band limited white noise,

S ~}* for I(DI <(Du, SfS((D ) = o 2(D,

0 for I(DI > co..

where So = strength of white noise. (For co, -+ oo we have ideal white noise which is of theoretical interest as the total power becomes infinite). Consider the Case 1,

-{-oo

Rr162 - 42) = ~ 5(0)) a}~-e i~'(r162 do) = a}f; for - Go < (~1 - - ~2) < -]- GO. - o o

Using this in the variance equation, after the double integration, we will get,

Var (Wm~x) = 411 -- COS 2/3 (C013 " a}f ,

o- 211 - - COS 2/3 (0{)] 3 /2 G f f Coefficient of variat ion -

m e a n 211 - - COS 2 /3 (0{)] 3 /2 ~- (TZf.

The coefficient of variat ion of snapthrough load is the coefficient of variat ion of stochastic process f(x), describing the Young's modulus fluctuation. For Case 2,

sin cou({, - 42) Rr162 - 42) = 2So"

( ~ - 42 L L

4 f ; Var ( W m a x ) = ~ " [1 - - COS 2/3 (00] 3 (.0 u 2 S 0 '

0 0

For W.N. process, with strength So,

sin COu(~l - ~-2) RfS(~I - ~ 2 ) = (Du Lt -'9" 0 2So

(Du(4, - ~ )

sin cou(41 - ~2)

(D.(~I - ~ ) d41 d~2.

= 2rcSo- 5(41 - {2).

For finite power W.N.,

sin co.(41 - ~2) { a}f; ~1 = ~2 co. L'--,oo 2So'(Du" c o . ( 4 1 - 4 2 ) = 0; ~l 4= ~2.

Since a}I is a constant, we come to the conclusion that if f (x) is a finite power w.n.

Var (Win,x) ---' 0.

Thus, the lower bound becomes zero. However,

Var (Wmax) < 4[(1 - cos 2/3 (c0] 3 "o-}f

since,

L L

f f d41 d~2 = ([F(L) F(0)] 2) = - Rzf(41 42) 2[Rff(O) Rff(L)]. 0 0

This is the upper bound for snapthrough load variance.

258 Archive of Applied Mechanics 63 (1993)

6 Numerica l examples

Let L = 1.5 m; af f = 0.08. F r o m (12), we know,

L L

Var(Wmax)= 4(1-c~ L2 Rff(41 - - 4 2 ) d~l d42.

0 0

If

x

F(x) = S f(4) d4, 0

1 ty2ffb3foZe_blo~l, If SyI(OJ ) = ~-

f b4(b2_ -__ 342)'~ R f f(4) = @ f k, (42 + bZ) 3 J '

1 a}xb3e_bl,o E

1 b 4 RFF(4 ) = ~ a}f (~2 + b2i

Var ( W m a x ) =

1 the P S D Svr(o9) = ~ i Syf(co).

b = constant ,

411 - cos 2/3 (c0] 3 L2 [2(RFv(0)) -- (RFF(L)]

The coefficient of variat ion of Wma x is

b2L 0.08b V w m a x - - (Tff

' L 2 l / b 2 + (1.5)2

This is plot ted in Fig. 3.

td o >

"5

(D (D

0.08

0.07

0.06

0.05

0.0l,

0.03

0.02

0.01

17

16

15

%D 13 ~2

ql

10

i I I l i I I I I I I ~ I I 1

�89 �88 6 8 tt3 1'2 1l, 16 90 2 L, 6 8 10 12 16 18 Values of b b ,"

3 4

Fig. 3 and 4. 3 Variation of coefficient of variation of snapthrough load with correlation distance (Ex. 1); 4 Variation of coefficient of variation of snapthrough load with correlation distance (Ex. 2)

s. A. Ramu and R. Ganesan: Snapthrough instability 259

When b --, 0%

0.08b 0.08b Vw . . . . - - ~ - - 0 . 0 8 .

] /b 2 + (1.5) 2 b

U p p e r b o u n d of V w , m a x = 0.08. Cons ider the P S D funct ion

1 bS. c04 Sss(~176 = "e- l l

and cor respond ing au to rco re la t ion funct ion

f rom which, we get

SFF(0)) = O-}f . b5 "0)2. e_bl~o I and Rvv(4) = a}I b6(b2 - 342) 48 12 (42 + b2) 3

1 afs . x/( L2 + b2) 3 b2 - b6( b2 - 3L2)

V~v . . . . : L ' 2 ] / 3 __ ( L 2 + b 2 ) 3

As b --+ 0% L 2 ~ b 2, and so, Vw,max --+ constant , as b --+ Go. The plot is in Fig. 4.

7 Concluding remarks

The foregoing has d e m o n s t r a t e d the m e t h o d for calculat ing the snap th rough critical load of a shal low f rame whose m e m b e r s are m a d e of a mater ia l with var iable mater ia l p roper ty . Represen ta t ion as a s tochast ic process of the var iable mater ia l p rope r ty is thus seen to be the effective way of analysing real life structures.

The f rame considered herein is the simplest example of a s tochast ic s t ructural system. The direct der iva t ion of the analyt ical re la t ionship between the variabili t ies of the critical load and the mater ia l p r o b l e m has been possible because of the inherent simplicity of a statically de te rmina te structure. Extens ion of the m e t h o d to m o r e complex s t ructures like statically inder te rmina te f rames or plates and shells of con t inua m a y not be a s t ra igh t forward easy exercise.

References

1. Shinozuka, M.: Structural response variability. J. Eng. Mech. 6 (1987) 825-842 2. Shinozuka, M.; Leone, E.: A probabilistic model for spatial distribution of material properties. J. Eng. Fract. Mech. 8 (1976)

825-842 3. Vanmarcke, E.: Random fields- analysis and synthesis. Cambridge: MIT Press 1983 4. Shinozuka, M. (ed.): Stochastic mechanics. New York: Columbia University 1987 5. Schfiller, G. I.; Shinozuka, M. (eds.): Stochastic methods in Structural dynamics. Dordrecht: Martinus Nijhoff 1987 6. Augusti, G.; Borri, A.; Casciati, F. C.: Structural design under random uncertainties Economical return and intangible

quantities. In: ICOSSAR 81, pp. 483-494. Trondheim: Elsevier 1981 7. Soong, T. T.; Cozzarelli, F. A.: Vibration of disordered structural systems. Shock and Vibrations digest 8 (1976) 21 - 3 5 8. Vomsheidt, J.; Purkert, W.: Random eigenvalue problems. New York: Elsevier 1983 9. Boyce, W. E.: Random eigenvalue problems. In: Bharucha Reid, A. T. (ed.) Probabilistic methods in applied methematics,

Vol. 1. pp. 1-73. New York: Academic Press 1968 10. Kozin, E: Stability of flexible structures with random parameters. In: Ariaratnam, S. T.; Sch/iller, G. I.; Elishakoff, I. (eds.)

Stochastic structural dynamics pp. 173 180. New York: Elsevier 1988 11. Shinozuka, M.; Astill, C. A.: Random eigenvalue problems in structural analysis. AIAA J. 10 (1972) 456-462

260 Archive of Applied Mechanics 63 (1993)

12. Collins, D.; Thomson, W. T.: The eigenvalue problem for structural systems with statistical properties. AIAA J. 7 (1969) 642-648

13. Herrmann, G.: Determinism and uncertainty in stability. In: Leipholz, H. (ed.) Instability of continuous systems, pp. 238-247. Berlin: Springer 1971

14. Anantha Ramu, S.; Ganesan, R.: Free vibration of a stochastic beam-column using stochastic FEM. Int. J. for Computers and Strs. 41 (1991) 987-994

15. Ariaratnam, S. T.: Stability of structures under stochastic disturbances. In: Leipholz, H. (ed.) Instability of continuous systems, pp. 78-84. Berlin: Springer Verlag 1971

16. Ariaratnam, S. T.; Xie, W. C.: Dynamic Snapbuckling of Structures under stochastic loads. In: Ariaratnam, S. T.; Schneller, G. I.; Elishakoff, I. (eds.) Stochastic Structural Dynamics, pp. 1 20. New York: Elsevier 1988

17. Seide, R: Snapthrough of initially buckled beams under uniform random pressure. In: Elishakoff, I.; Lyon, R. H. (eds.) Random vibration - status and recent developments. New York: Elsevier 1986

18. Pi, H. N.; Ariaratnam, S. T.; Lennox, W. C.: First passage time for the snapthrough of shell type structures. J. Sound and Vibration. 14 (1971) 375-384

19. Biezeno, C. B.: Uber eine Stabilit/itsfrage beim gelenkig gelagerten schwachgekrfimmten Stab. Proc. Acad. Sci. Amst. 32 (1929) 990-994

20. Bazant, S. E; Cedolin, L.: Stability of structures. New York: Oxford University Press 1991

Received January 29, 1992

Prof. S. A. Ramu Dr. R. Ganesan Indian Institute of Science Department of Civil Engineering Bangalore 560012 India