Table Of ContentLecture Notes in
Engineering
Edited by C. A. Brebbia and S. A. Orszag
at'1
~.
IFIP 33
P. Thoft-Christensen (Editor)
Reliability and Optimization
of Structural Systems
Proceedings of the First IFIP WG 7.5 Working
Conference
Aalborg, Denmark, May 6-8,1987
Spri nger-Verlag
Berlin Heidelberg New York
London Paris Tokyo
Series Editors
C. A. Brebbia . S. A. Orszag
Consulting Editors
J. Argyris . K.-J. Bathe· A. S. Cakmak· J. Connor· R. McCrory
C. S. Desai· K.-P. Holz . F. A. Leckie· G. Pinder· A. R. S. Pont
J. H. Seinfeld . P. Silvester· P. Spanos· W. Wunderlich· S. Yip
Editor
P. Thoft-Christensen
Institute of Building Technology
and Structural Engineering
The University of Aalborg
Sohngardsholmvej 57
DK-9000 Aalborg
Denmark
ISBN-13:978-3-540-18570-3 e-ISBN-13:978-3-642-83279-6
001: 10.1007/978-3-642-83279-6
Library of Congress Cataloging-in-Publication Data
IFIP WG 7.5 Working Conference (1st: 1987 : Aalborg, Denmark)
Reliability and optimization of structural systems:
proceedings of the First IFIP WG 7.5 Working Conference,
Aalborg, Denmark, May 6-8, 1987 / P. Thoft-Christensen, editor.
p. cm. - (Lecture notes in engineering; 33)
Includes indexes.
ISBN-13:978-3-540-18570-3 (U.S.)
1. Structural design - Mathematical models - Congresses.
2. Mathematical optimization - Congresses.
3. Reliability (Engineering) - Congresses.
I. Thoft-Christensen, Palle,
II. Title.
III. Series.
TA658.2.1381987 624.1'771 - dc 19 87-28646
This work is subject to copyright. All rights are reserved, whether the whole or part of the material
is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation,
broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication
of this publication or parts thereof is only permitted under the provisions of the German Copyright
Law of September 9, 1965, in its version of June 24, 1985, and a copyright fee must always be paid.
Violations fall under the prosecution act of the German Copyright Law.
© Springer-Verlag Berlin, Heidelberg 1987
216113020-543210
PREFACE
The Proceedings contain 28 papers presented at the I st Working Conference on »Reliability
and Optimization of Structural Systems», Aalborg, Denmark, May 6 -8, 1987. The Working
Conference was organized by the IFIP Working Group 7.5. The Proceedings also include 4
papers which were submitted, but for various reasons not presented at the Working Confer
ence. The Working Conference was attended by 50 participants from 18 countries.
The Conference was the first scientific meeting of the new IFIP Working Group 7.5 on
»Reliability and Optimization of Structural Systems». The purpose of the Working Group
7.5 is
to promote modern structural system optimization and reliability theory,
to advance international cooperation in the field of structural system optimization and
reliability theory,
to stimulate research, development and application of structural system optimization
and reliability theory,
• to further the dissemination and exchange of information on reliability and optimiza
tion of structural system optimization and reliability theory,
to encourage education in structural system optimization and reliability theory.
At present the members of the Working Group are:
A. H. S. Ang, USA M. Grimmelt, Germany F. R.
G. Augusti, Italy N. C. Lind, Canada
M. J. Baker, United Kingdom H. O. Madsen, Norway
P. Bjerager, Denmark F. Moses, USA
A. Der Kiureghian, USA Y. Murotsu, Japan
O. Ditlevsen, Denmark R. Rackwitz, Germany F. R.
M. R. Gorman, USA P. Thoft-Christensen, Denmark (Chairman)
Members of the organizing committee are:
M. J. Baker, United Kingdom
H. O. Madsen, Norway
Y. Murotsu, Japan
R. Rackwitz, Germany F. R.
P. Thoft-Christensen, Denmark (Conference Director)
The Conference was financially supported by
IFIP
DANFIP
The University of Aalborg.
I would like to thank the organizing committee members for their valuable help in organ
izing the Working Conference, and all the authors for preparing papers for the Proceedings.
Special thanks to Mrs. Kirsten Aakjrer, University of Aalborg, for her efficient work as Con
ference Secretary before, during and after the institute.
August 1987 P. Thoft-Christensen
CONTENTS
On the Application of a Nonlinear Finite Element Formulation in Structural Systems Reliability
Amdahl. J .. Leira. B. • Wu. Yu-Lin
Fatigue Life Estimation under Random Overloads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Arone. R.
Application to Marine Structures of Asymptotic Vector Process Methods 31
Cazzulo. R.
Reliability Analysis of Discrete Dynamic Systems under Non-Stationary random Ex- . . . . . . 45
citations
Chmielewski. T.
Reliability of Partly Damaged Structures 67
Costa. F. Vasco
Uncontrolled Unreliable Process with Explicit or Implicit Breakdowns and Mixed. . . . . . . . . 77
Executive Times·
Dimilrov. B. N.. Kolev. N. V. . Petrov. P. G.
Reliability Computations for Rigid Plastic Frames with General Yield Conditions. . . . . . . . . 91
Dillevsen. O.
Parallel Systems of Series Subsystems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
Egeland. T .. Tvedt. L.
Range-Mean-Pair Exceedances in Stationary Gaussian Processes. . . . . . . . . . . . . . . . . . . . . . . 119
Ford. D. G.
A Sampling Distribution for System Reliability Assessment· . . . . . . . . . . . . . . . . . . . . . . . . . 141
Fu. G. . Moses. F.
Comparison of Numerical Schemes for the Multinormal Integral· 157
Gol/wilzer. S .• Rackwilz. R.
Reliability of Fiber Bundles under Random Time-Dependent Loads. . . . . . . . . . . . . . . . . . . . 175
Grigoriu. M.
A Practical Application of Structural System Reliability Analysis. . . . . . . . . . . . . . . . . . . . . . 183
Guenard. Y .. Lebas. G.
Outcrossing Formulation for Redundant Structural Systems under Fatigue. . . . . . . . . . . . . . 199
Guers. F.. Dolinski. K.. Rackwitz. R .
• Not presented at the Conference.
v
Optimal Bridge Design by Geometric Programming. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
Gupta. N. C. Dos. Paul. H.. Yu. C. H.
An Application of Fuzzy Linear and Nonlinear Programming to Structural Optimization 233
Koyama. K .• Kamiya. Y.
On the Calibration of ARMA Processes for Simulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
s..
Krenk. Clausen. J.
Level Four Optimization for Structural Glass Design 259
Lind. N. C.
Contribution to the Identification of Dominant Failure Modes in Structural Systems. . . . . . 275
Murotsu. Y .. Matsuzaka. S.
Reliability Estimates by Quadratic Approximation of the Limit State Surface. . . . . . . . . . . . 287
Naess. A.
Failure Mode Enumeration for System Reliability Assessment by Optimization Algorithms. . 297
Nalday. A. M. . Corotis. R. B.
Reliability Analysis of Hysteretic Multi-Storey Frames under Random Excitation. . . . . . . . . 307
Nielsen. S. R. K.. Mork. K. J .• Tholt-Christensen. P.
System Reliability Models for Bridge Structures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329
Nowak. A. s.. Lind. N. C.
Modelling of the Strain Softening in the Beta-Unzipping Method. . . . . . . . . . . . . . . . . . . . . . 339
Paczkowski. W.
Probabilistic Fracture Mechanics Applied to the Reliability Assessment of Pipes in a PWR. . . 355
Schmidt. T. . Schomburg. U.
Theoretic Information Approach to Identification and Signal Processing. . . . . . . . . . . . . . . . 373
Sobczyk. K.
Integrated Reliability-Based Optimal Design of Structures. . . . . . . . . . . . . . . . . . . . . . . . . . . 385
Sorensen. J. D. . Tholt-Christensen. P.
On Some Graph-Theoretic Concepts and Techniques Applicable in the Reliability Analysis. . 399
of Structural Systems·
Vulpe. A .• Carausu. A.
Reliability of Ideal Plastic Systems Based on Lower-Bound Theorem. . . . . . . . . . . . . . . . . . . 417
Madsen. H. 0 .. Bjerager. P.
• Not presented at the Conference.
VI
Structural System Reliability Analysis Using Multi-Dimensional Limit State Criteria. . . . . . . 433
Turner. R. c.. Baker. M. J.
Sensitivity Measures in Structural Reliability Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459
Bjerager. P. . Krenk. S.
Index of authors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471
Subject index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472
ON THE APPLICATION OF A NONLINEAR FINITE ELEMENT FORMULATION
IN STRUCTURAL SYSTEMS RELIABILITY
J. Amdahl & B. Leira
SINTEF, Division of Structural Engineering, Trondheim, Norway
and
Yu-Lin Wu
Division of Marine Structures
The Norwegian Institute of Technology, Trondheim, Norway
ABSTRACT
An attempt is made to combine methods for advanced progressive collapse analysis
with a probabilistic formulation of load and resistance variables. A brief de
scription is first given of a nonlinear computer program intended for progres
sive collapse analysis of space frame structures. The basic idea behind the
program is to use only one finite element per physical element in the structure.
Plastic interaction functions for stress resultants serve as local failure func
tions of the system. The basic variables are the external load parameters and
the yield stresses of the members. A dominant failure path is found by load in
crementation, where the direction in the load space is governed by the current
smallest distance to the failure surface. The associated failure probability
and bounds for system reliability are found by means of first order reliability
methods. The use of the method is illustrated in numerical examples.
1 INTRODUCTION
Design of a structure according to a specific code commonly involves checking of
individual structural members. However, most structures are redundant in the
sense that collapse of the first single element merely causes the load to be
redistributed. The system structural reliability may then be far different from
individual member reliability. Efficient and accurate methods for evaluation of
the system reliability will then be needed to form the basis for development of
more rational codes.
System structural reliability has been subject to considerable research effort
during the last decade. Two mainstreams of analysis can be idenfified. The
failure mode approach is based on ways in which the structure can fail, whereas
the stable configuration approach is based on ways the structure can survive.
2
The probabilistic analysis corresponding to the former involves computation of
the probability content of a union of intersections between sets. The latter
approach leads correspondingly to the intersection of unions, see e.g. Refs. [1,
2, 3, 4].
Estimation of system reliability on the basis of system failure modes has been
considered by several authors. Simple but most useful bounds were derived by
Cornell, [5], applying the failure probabilities of each mode separately. As
these bounds frequently are too wide, closer bounds obtained by taking correla
tion between failure modes into account were introduced e.g. by Ditlevsen, [6],
Ang and Ma, [7], Stevenson and Moses, [8],and Vanmarkcke, [9]. The bounds pre
sented in [6] have been extensively applied in the literature. Madsen, [10],
discusses first vs. second order reliability analysis of series structural
systems based on these bounds.
Prior to evaluation of system failure probability, identification of the failure
modes must be performed. Commonly, the most dominant modes in a stochastic sense
are sought, as the total number of modes can be very large. An incremental
method for this purpose was presented by Moses et. aI, [11, 12, 13, 14]. A
method based on a secant stiffness formulation, also including a probabilistic
algorithm, was introduced by Murotsu et. al. [15, 16, 17, 18, 19, 20]. Applica
tion of this method to reliability analysis of offshore structures has been con
ducted by Crohas et. al., [21, 22]. Guenard, [23], also studied offshore struc
tures by a similar method.
An alternative strategy for performing the branch and bound operations as de
scribed by Murotsu has been adopted by Thoft-Christensen et. al., [24]. This
so-called p-unzipping method has been applied by Baadshaug et. al., [25], for
evaluation of reliability of jacket platform structures.
Ang and Ma, [26], proposed a different method for determination of the most pro
bable failure modes by using mathematical programming based on the basic failure
modes of the structure. Klingmfiller, [27], applied a procedure based on the
principle of virtual work and mathematical programming algorithms.
It seems that realistic models for structural behaviour is difficult to incorpo
rate in the search procedcures for identification of dominant failure modes. One
possible solution would be a Monte Carlo type of approach, see e.g. [28, 29].
However, this will lead to numerous structural analyses, which constitute the
most expensive part of the reliability determination. Alternative methods have
been developed by Kam et. al., [30], for nonlinear structures with deterministic
3
strength. Similar simplified analyses based on mean values of random structural
strength parameters are described by Lin el. al., [31). Methods based on exten
sion of Moses' method have been proposed by Melchers and Tang, [32, 33]. Recent
ly, a consistent formulation has been outlined by Gollwitzer and Rackwitz, [34],
also including instabilities.
Brittle structural behaviour has been included by several of the authors refer
red to above, see e.g. Refs. [14, 16, 24, 26]. Consideration of this topic has
also been given by Giannini et.al., [35], Bjerager, [36].
Although there has been significant achievements with respect to techniques for
reliability as~essment as such, the representation of structural behaviour is
still very simple and idealized. To enhance the acceptance of reliability
analysis there is a strong need for applying structural behaviour models with
which the designers are familiar.
In this paper, a method for identification of the most significant failure modes
for nonlinear structures is outlined, based on a statistical representation of
both external load and random strength. Instability failure is represented in a
uniform way by progressive inclusion of internal hinges in the structural finite
element model. The basis of the method is an incremental formulation of the
equilibrium equations.
2 NONLINEAR STRUCTURAl. ANALYSIS FORHULATION FOR SPACE FRAHES
For a thorough description of the computer program USFOS it is referred to [37,
38]. In the following only the basic concepts are reviewed.
The main idea behind USFOS is to represent each physical element by one finite
element as shown in Figure 1. This facilitates that input models from conven
tional linear analysis can be used with minor modifications.
The basic finite element is the spatial beam element with 12 degrees of freedom.
As stress and strain measures the stress resultants and the corresponding
displacements are used.
The coupling between axial and lateral displacements is automatically taken care
of by including large deflection effects in the strain expression on local
element level (von Karman theory).
4
..fl
f--: - -~: . . ~~
plastic hinge
f.J
ve'" ve •
0 (Elastic)
1/t.
o
Non-linear material Non-linear geometry
Figure 1. U5F05 - basic concepts
In the elastic range equilibrium equations and incremental equations can be
derived from the first and second variation of the potential energy. An impor
tant feature with the method is the choice of interpolation functions. These
satisfy exactly the differential equation for a beam under axial force and
lateral bending and yield trigonometric functions for compression and exponen
tial (hyberbolic) functions for tension. This facilitates closed form solutions
for all integrations in the equilibrium and incremental equations.
The modelling of plasticity uses stress resultants as basic parameters. The
capacity of a cross-section is expressed by an interaction equation
i 1 ... 6 ( 1 )
where 0y = yield stress, 5i, denotes a stress resultant. If a cross-section
reaches the plastic state defined by Eq. (1) a hinge is introduced. Elastic and
plastic displacements are separated. The plastic deformations are concentrated
to the plastic hinges whereas the beam remains elastic between hinges. The next
step is to introduce the normality criterion for the incremental plastic dis
placements and the consistency criterion for the incremental stress resultants.
The latter criterion states that the stress state lies on the interaction sur-
