Rigidity reliability analysis of structural systems

Comprfers d Srrucrures Vol. 48, No. 3, pp. 54lMlO. Printed in Great Britain.

RIGIDITY

1993 0

0045-7949193 %.OO + 0.00 1993 Fwgmott Press Ltd

RELIABILITY ANALYSIS OF STRUCTURAL SYSTEMS YONG CANG ZHANG

Aircraft Engineering Department, Northwestern Polytechnical University, Xi’an, Shasnxl Province,

People’s Republic of China (Received 17 Jumuzry 1992) Abstract-A

method is presented for computing the load carrying capacity of structural systems subjected to rigidity requirements by introducing the conception of a ‘general&d element’ into the incremental load approach, so that the reliability analysis range of structural systems is extended to both strength failure

and rigidity failure circumstances. The principle of the method is clear and simple. Some illustrative examples are given to clarify the method and its application.

1. INTBODUCI’ION

random loads and subjected to certain deformation limitations at prescribed nodes in prescribed directions, may fail to perform its function in either of the following ways:

Randomness is widespread in nature, and structural engineering is no exception to this. The design parameters involved in a structure, such as external load, component strength, etc., are uncertain, therefore the load carrying capacity of the structure is uncertain. A realistic assessment of a structure’s performance, in the light of uncertainty, may be stated only in terms of probability. The probability for a structure to resist external load to carry out its designed functions is the measurement of its reliability. In this field, a great deal of exciting progress and remarkable research achievements have been made, but most works are focused on the aspect of strength and failure of structural systems. According to their functional requirements, structures may have several types of failure, such as statical strength failure, rigidity failure, fatigue failure, etc. For those structures which must possess a certain rigidity, rigidity failure is sometimes more severe than strength failure. For example, at some point in an airplane the structure stiffness must be sufficient to meet the need of aerodynamics or vibration frequency. So far, very few studies on rigidity reliability are found in the literature. Without doubt, the development of this research is significant for the extension of both theory and application of structural reliability. The author of this paper presents a method to compute the structural capacity for rigidity failure, thus the reliability can be analysed for the structures with both strength and rigidity requirements. Some truss examples are used to illustrate the application of the method.

1. Strength failure: a certain number of elements fail, then the structure hecomes a mechanism and loses its load carrying ability. 2. Rigidity failure: the prescribed deformation exceeds the allowable limit even though the structure can still carry a load. 1. Generalized element concept and incremental loading approach Assume that a structure consists of n ductile elements (Fig. l), and is acted on by one load L. Along a failure path (i.e. a failure mode), m elements failed in succession, but the structure has not yet collapsed. Before the next element fails, the prescribed deformation 6, that is often expressed by displacement, first reaches its allowable limit. With the incremental loading approach [I], the structure is incrementally loaded until collapse. Summing all the load increments gives the structure’s ultimate load or capacity. Let the first load increment S, just cause the first element’s failure, S, the second, and S,,, the mth. The incremental loading equations are then

2. THE LOAD CARRYING

CAPACITY OF STRUCTURAL SYSTEMS SURJECI’ED TO RIGIDITY LJMITATIONS

A structural system which is composed of n elements, acted on by one or more concentrated 505

all

0

...

0

SI

a2,

a,

...

0

s2

...

.., ...

a,,

a,

. . . a_

> I (1)

... &II

where Ri is the strength of the ith element, sj thejth load increment, and au the internal force of the ith element as S, = 1. While loading the structure, the prescribed displacement 6 is also growing (Fig. 2), thus

YONG CANG ZHANG

506

Combining

4’

a11

R2

41

a22 . . .

...

...

...

RIB8

a,,

a,

. . . a_

PI

fi

h

*** fm fm+l

=

Dispbcunau

-

(1) and (4a) yields 0

...

0

0

s,

0

0

s2

...

... 0

Sm+l

(5)

Fig. 1. Idealized failure behaviour of a ductile element.

corresponding to load increments S,, S,, . . . , S,,,, there are displacement increments 6,, a,, . . . ,a,. Define

then A is the jth displacement increment caused by unit Sj. Its reciprocal, l/‘, has a geometrical interpretation as the slope of the jth line segment in Fig. 2 and a physical interpretation as the structure stiffness during the jth load increment period. At the end of the mth load increment, the displacement value is 6 =

5 sj= f j-1

AS,.

(3)

j-l

The (m + 1)th load increment S,,,, , causes the displacement to just reach its allowable limit. It gives

j-1

which is the incremental loading equation for rigidity failure. Comparing (5) with the incremental loading equations for strength failure [1], one can find that the difference is only from the last load increment, which is determined by allowable displacement limit in rigidity failure or by the (m + 1)th element strength in strength failure. If we consider 6 as a ‘generalized element’, the allowable displacement limit as its ‘strength’ and the displacement value as its ‘load’, eqn (5) is same in form as that of strength failure, so the theory and method for strength failure analysis can be introduced to the rigidity failure analysis. Let j=l,2 ,..., m+l, theneqn

= 4

0

. . .

0

8

a21

022

...

0

s2

. . .

. . .

. . .

. . .

4

41

R2 Rm+l

%,I,1

%,I,2

***

am+h+l

Ill

Sm+l (5a)

or in matrix form as

or

{RI = MW.

(5W

Solving (5b) for load increments results in {S} = [A]-‘(R).

Fig. 2. Load-displacement curve of rigidity failure.

(6)

Rigidity reliability analysis of structural systems The load carrying rigidity failure is

capacity of the structure m+l ‘a = j;, ‘J.

for

507

K failure components. ability is

The structural

failure prob-

Pr= P(M,
(7)

(12)

and the structural reliability is The terms of matrix A are computed by structure

analysis in which the related physical parameters are used at their mean values, such as mean section areas of bars, mean Young’s modulus of material, mean load, etc., thus Rs is a linear function of R,s and [6] R,=C,R,+C,R,f...

+G&+G+J~l~

(8)

w=I-Pf

U2a)

In general, correlativity between failure modes exists, so it is very difficult to obtain an accurate solution from (12). Usually P, is evaluated by lower and upper bounds. Taking account of all mode-l failure and all mode-2 joint failure probabilities, Ditlevsen presented second-order bounds [3] as

where Cis are constants. Suppose R, to be a normal variabie, [S] to be constant, then, by probability knowledge, R8 is also a normal variable.

p,~p,+~*~i-~~pij,o)

K R PfG C Pi- C max Pij, ia* i
(13)

2. Strength failure and rigidity failure For the rare case when m elements in a structure fail, the prescribed displacement would exactly reach its limit, then the structure capacity is still determined by rigidity failure Rd=J+5’j. _

(9) The reliability index of the ith failure mode is

If the structure becomes a mechanism after m elements and fails, but the ~spla~ent has not reached its limit during the mth load increment, the structure capacity will be determined by the strength failure R,=is,. J-1

The structural capacity is determined as mentioned above, then the safety margin corresponding to each failure mode can be expressed as

i=l,2,...,K

00)

in which, R,, is the structure capacity in the ith failure mode and K is the number of sign&ant failure modes. The faihrre probability of the structure for the ith failure mode is Pf = P(M, < 0).

Therefore Pi=@(-pi)

Pa)

3. RELIABILITY ANALYSIS OF STRUCTURAL SY!nEMS

h$=R,-L,

Pij = (Mi < ofMj < O), where Pi = P(h!fi c O), i,j=1,2 ,..., K, i#j. From probability considerations, if Rsi and L are normally distributed, Mi is normally distributed, too

(11)

The fake of the total structural system may be thought of as the failure of a series of systems with

Pij =

f(Mi, Mjl wi

dM_

(14)

in which Pri can be computed quite accurately by numerical integration, f(M,, M,) is the joint probability density function of Mi and Mj with parameters PM,, gMrPPM,’ 0~ and Mimi, Mjl* 4. ILLUSTRATIVE

Exhale

EXAMPLES

1. ~i~e-~ar truss (Fig. 3)

The structure is acted on at node 1 by a horizontal concentrated load S, a nomal variable. The mean and the coefficient of variation of S are 18 and 0.10, respectively (i.e. ps = 18, Ys = 0.10). All elements are ductile and their strengths are normal variables with flRI+ = 20 (tension), ,uRi = 15 (compression), V, = 0.05. The element’s strength is uncorrelated with each other, also with load S, i.e. p[R,, Rj] = 0, p[Ri, S] = 0. A displacement limit [S] = 0.55 is imposed on node 2 in the horizontal direction. Young% modulus ..~~ ~~ E = 100.

?iONG

CANG i%.ANG M,=0.7071R;+0.7071R;-S Ms=0.7071R;+R;-S M4=0.7071R;+R,+R;-S M2=0.7071R;+ 14.3678-S

Fig. 4. Failure tree for a five-bar truss.

ively. For simplicity and clarity, all mode-l and mode-2 joint failure probabilities are expressed as matrix P as ElementareaA,

0.033823

=A,=-=A,=i.O

Fig. 3. Five-bar truss. [‘I The failure analysis of the strucmre is shown in Table 1. The criterion method [4] is applied to enumerate the significant failure modes. Element 4 would fail first, and the second failure may be generalized element 6 or element 2 or element 3, and any second element failure would result in the failure of the structure. For failure sequence 4-6, the corresponding incremental loading equations are as follows:

Ri

I[

=

0.048014

0.03f010

0.044836

0.044068 0.048362

L0.044836

9

0.044068 0.042319 0.048362 1 (18)

where 0.0-‘3823 = 0.0003823, Piis the term of row i and column i, Pij is that of row i and column j. By (13), the failure probability of the structure system is 0.034031 < P,g o.034737

WI

0.7888

sym.

(19)

and by (12a), the reliability of the structure system is

[ @I = 0.01693

0.g35263 < 1 G 0.935969.

(20)

Solving (15) for S, and S, and then summing them giVeS

Example 2. Ten-bar truss (Fig. 5)

This example is extracted from [2], but a horizontal displacement limit to node 2 is added here with [S] = 1.8. The load is a normal variable with ,us = 2.5 The safety margin is and V, = 0.12. The element strengths are also normal M,=R,-S=0.7071R;+l4.3678-S. (17) variables (Fig. 5) with V, = 0.05. Assume that E = 1000, P[&, Rj]= 0,and p[Ri, S]= 0. In [2] all 27 strength failure modes are enumerated Similarly, the other safety margins can be obtained. for the structure and three significant failure modes The failure tree and the safety margins corresponding are 10-4, 10-9 and 8-7. With the consideration of to each branch are shown in Fig. 4. The reliability rigidity failure for each of the 27 failure modes, four index for each failure mode is /I1 = 3.3653, significant rigidity failure modes are determined, thus & = 3.7167, & = 3.7641 and & = 3.7641, respect& = S, + S, = 0.707lR;

+ 14.3678.

(16)

Table 1. Data for example 1 Element No. 4 failed Bar

%

1

0.5578 -0.4422 -0.4422

2 3

4 5 6

-0.7888 0.6254 0.01693

&I 20 -15 -15

-1s 20 0.55

a;: $ 0.02789 0.02948 0.02948

0.05259t 0.03127 0.0260

3,

19.0162

ai, I%

3,

&

%

%

0 -1 -1

9.3927 -6.5910

0.1517f

6.5GO

2X%72

-6.5910 0 8.1073 0.228 I

0.1517t 0.1770t 0.1678t

6.5910 5.7328 5.9576

25.6072 24.7490 24.9738

0 1.4142 0.03828

0

5, is the mean strength of the ith element during the S, period. Rs is the mean available strength of the it) eleFent_during the S, period, & = & - a, 3,. R, is the mean capacity of the structure, Rs = S, + 22,. t indicates the most likely failure element.

509

Rigidity reliability analysis of structural systems Element nlmn rtmgtb Bar No. Tension Compr. Am8 20.0 10.0 1.0 l-6 7 - to 5.0 2.5 0.25

Mean

Mode

capacity

number

4.0267

1

4.1603

4

4.1603

5

4.0557 4.0371 4.492 1 4.4559 Fig. 6. Failure tree for a 10-bar truss.

The failure probability and reliability bounds of the structure system are respectively 0.055294 < P/< 0.055579

(221

0.9’4421 d R g 0.9’4706.

(231

Fig. 5. Ten-bar truss. there are seven significant failure modes for the present situation. The failure tree and the mean capacity of each branch are shown in Fig. 6. The safety margins are listed as follows: M, = M,, = 0.333313; + 0.277411; - S M2 = M,,* = 0.2805R,

+ 3.3358 - S

M, = M,o,s = 0.2820R, + 3.3507 - S I& = M,,

= 0.55471;

+ 0.5547R, - S

MS = MB,, = 0.5547R,’ + 0.554712, -S

MG=h&=0.3351R;+

1.1048-S

M, = MS = 4.4921 - S.

The reliability indices and mode failure probabilities are &=4.4256,

&=5.0889,

& = 4.9164,

& = 5.6918,

/&=5.1502,

j&=4.9164,

5. DJSCUSSION AND CONCLUSIONS

1. According to the designed functions, structural systems have different types of failure. The key problem is how to determine the load carrying capacity of the structural systems for each type of failure. The capacity for rigidity limitation can be determined by the incremental loading approach. As the generalized element concept is introduced to where the displacement limit exists, the structure reliability analysis can be made in the same way as that for the strength failure which has been greatly studied. 2. By the method presented in this paper, the analysis range of structure system reliability is extended to both strength and rigidity failure, which is especially significant to the structures with rigidity requirements. 3. In the case of the normal strength of an element, normal load and constant rigidity limit, the capacity of a structural system is the normal variable, as is the safety margin, which gives a quite convenient operation to use second-order reliability bounds as shown by the examples.

8, = 6.6403

- 0.0548 13

[PI =

0.061071

o.oWM

0.0’8876

0.0’~27

0.061303

0.061205

0.0’5299

0.0’5476

0.064414

0.06101 1

0.0’6749

0.0’4308

0.0’3537

0.064414

0.0’6286

0.0’3085

0.082786

0.0*1763

0.081763

0.0*6304

0.0i01565

0.0’*1551

0.0’0t551

0.0’01059

_0.0L01563 0.0’01565

sym.

(21)

0.0”1573_

YONGCANGZHANG

510

REFERENCES 1. F. Moses, System reliability developments in structural engineering. Srrucr. Safety 1, 3-13 (1982). 2. M. R. German, Reliability of structural systems. Report CE 79-2, Department of Civil Engineering, Case Western Reserve University, Cleveland, Ohio (1979).

3. 0. Ditlevsm, Narrow reliability bounds for structural systems. J. Sfruct. Mech. 7,453-412 (1979). 4. Y. S. Feng, Enumerating significant failure modes of a structural system by using criterion methods. Cornput. Strrcfr. 30, 1153-1157 (1988). 5. Y. C. Zhang, Fatigue reliability of structural systems under a symmetrically circulating load. Acta Aeronautica et Astronautica Sinica 9, 135-141 (1988).