The objective is to demonstrate discrete extremum methods of structural analysis as a tool for structural system reliability evaluation. Specifically, linear and multiobjective linear programming models for analysis of rigid plastic frames under proportional and multiparametric loadings, respectively, are considered. Kinematic and static approaches for analysis form a primal-dual pair in each of these models and have a polyhedral format. Duality relations link extreme points and hyperplanes of these polyhedra and lead naturally to dual methods for system reliability evaluation

Nafday, Avinash M.

English

NASA Technical Reports Server

N91-10308
PROBABILISTIC STRUCTURAL ANALYSIS
BY EXTREMUM METHODS
AVINASH M. NAFDAY
EQE Engineering, Inc.
Costa Mesa, California
103
https://ntrs.nasa.gov/search.jsp?R=19910000995 2020-03-19T20:17:26+00:00Z
STRUCTURAL MODELS FOR SYSTEM RELIABILITY EVALUATION
Evaluation of structural system reliability is based on the determination of
probabilistic response of structures. The study of probabilistic behavior of
structural systems requires an underlying structural analysis model. The current
system reliability analysis procedures utilize the structural analysis methods
basically developed with a deterministic point of view. The applicability of these
methods when dealing with a random situation needs examination. It may be that
methods of structural analysis that are quite suitable for deterministic analysis
are not as suitable for probabilistic analysis and vice versa. Development of
structural system models especially suited for random state variables may be more
efficient and likely to lead to important insights into random structural behavior.
A recent National Science Foundation Workshop on Structural System Reliability held
at the University of Colorado, Boulder [I] also emphasized the need for the
development of structural analysis models from the probabilistic viewpoint.
The objective of this paper is to demonstrate discrete extremum methods of
structural analysis as a tool for structural system reliability evaluation.
Specifically, linear and multiobjective linear programming models for analysis of
rigid plastic frames under proportional and multiparametric loadings, respectively,
are considered. Kinematic and static approaches for analysis form a primal-dual
pair in each of these models and have a polyhedral format. Duality relations link
extreme points and hyperplanes of these polyhedra and lead naturally to dual
methods for system reliability evaluation.
DIRECT METHOD
Formulation of the global limit state surface
Computation of probability of random variables
having an outcome in the safe set
INDIRECT METHOD
Identification of all failure modes
Computation of the probability of failure of
individual modes
Evaluation of system reliability from modal
probabilities
104
LIMITATIONS OF CURRENT APPROACHES
It is known that the structural problems can be analyzed by idealizing these
mathematically as either continuous or discrete models, leading respectively to the
solution of differential or algebraic equations/inequalities. The underlying
solution principles can be based on the solution of simultaneous equations or the
use of extremum principles. In the sixties, Schmit and coworkers [2] briefly
explored extremum methods for the deterministic structural analysis problems but
found them to be not competitive with solution methods for simultaneous equations.
However, in the use of simultaneous equations procedure for probabilistic
structural analysis, one has to solve the structural analysis problem repeatedly
for different realizations of random variables and this is computionally costly.
Use of extremum principles, on the other hand, elucidates the mathematical
structure of the problem corresponding to various random realizations of state
variables. This structure is extremely coherent with a definite pattern about the
solutions of the problem. An understanding of such patterns leads one to gain
important insights into response under random variables without analyzing the
structure for all such combinations. This coupled with use of recent computational
advances in algorithms [3] and vector processing of information on supercomputers
are likely to make these methods extremely attractive for use in probabilistic
analysis. For example, recent research [4] shows extremum methods to be ideally
suited for structural analysis required in the system reliability assessment of
structures with rigid plastic material behavior.
m STRUCTURAL ANALYSIS FOR RELIABILITY EVALUATION
Discrete/Continuous models
Classical methods, elastic/plastic
Extremum methods
RESPONSE PATTERNS
- Polyhedral response regions
- Other response regions
• ADVANTAGES OF EXTREMUM METHODS
105
EXTREMUM METHODS FOR STRUCTURAL ANALYSIS
Structural analysis requires both the calculation of the distribution of forces
throughout the structure and the displaced state of the system under the action of
applied loads. One of the fundamental features of structural analysis is the
possibility of using either forces or displacements as basic variables, with the
respective approaches referred to as static and kinematic methods. The algebraic
relationship of the static and kinematic approaches are the mathematical transpose
of each other, a feature known as the static-kinematic duality. If the structure
is statically determinate, the number of equations is same as the number of
variables, and the forces and displacements can be found easily from the solution
of system of algebraic equations. For statically indeterminate structures,
additional conditions reflecting the material constitutive relations must be
introduced.
As an example, for redundant frame structures, partial satisfaction of structural
constraints generates a subspace in R n containing the solutions of interest and the
final solution can be reached by some optimality criterion. The optimal solution
gives the result corresponding to the solution by traditional methods. The power
of the extremum methods, however, is that all the suboptimal solutions may also be
obtained from the model, and these suboptimal solutions correspond to various
random realizations of the variables. This set of available solutions has a rich
underlying mathematical structure and such patterns have recently been studied for
rigid plastic frames under proportional and multiparametric loading [4].
\
\ I \
//1 l i_,,_
/ I \ I
106
RIGID PLASTIC FRAMES WITH PROPORTIONAL LOADING
The problem of limit analysis of frames in which the plastic behavior is activated
by a single stress resultant (such as flexure) may be formulated in terms of a
Linear Programming (LP) model [5]. Plastic hinges are assumed to form at a set of
discrete locations (j = I, 2 ..... J) and the plastic moment capacity at the jth
section is denoted by Mpj. Models formulated from dual structural consideration of
static and kinematic prznciples have been shown to be a primal-dual pair in the LP
format. The variables are M_ = moment at section j, Mj = Mj + - Mj - where Mj +
and M_ " represent the positive and negative values of moments; 0; = rotation at
sectign j; tk = a coefficient indicating the contribution of the _th elementary
mode to collapse, t k = t k + - t k - and %+, %= collapse load factors for the
kinematic and static LP's respectively. The parameters areekj- = hinge rotation of
elementary mechanism k at joint j, ek = external work associated with elementary
mechanism k, M = number of equations of equilibrium and Mpj = member capacity at
section j.
KINEMATIC LP STATIC LP
;_= Min j[_l"= Mpj (0;+ 0 i)]
;k= Max _"
M J
-- ° -"
- Okjs.t. _+ Oi k=lZ; tk s.t. j=l_ Ok j M! _," ek
j=l, . ., J k=l, -, M
+
M t k ek = 1 -Mpj <_Mj _< + MpjkN1
Oj+,Oj" > 0
j=l, -, J
107
DUALITY RELATIONS
A study of the geometric structure of the primal and dual models shows the
constraint regions for both to be polyhedral. The failure modes of the frame are
given by the extreme points of the kinematic constraint region and the facets of
the static constraint region. Duality transformations of LP actually map extreme
points of one model to the hyperplanes of the other and vice versa. More
generally, l-dimensional subspaces of the model in Rn are linked to (n-l)
dimensional subspaces of the other model.
For primal, the random material properties specified by vector M__ occur only in
the objective function and these determine the failure mode of the frame since
the solution must surely belong to at least one of the extreme points.
Therefore, unlike other procedures that require repeated solutions of structural
model, one just needs to explore the polyhedral region for identifying the
solution for a different (random) MD vectors. Similarly, the vector M_ occurs
only in the right hand side of static LP model and failure modes (facets) can be
generated from the dual variables.
DUALITY
- Static/Kinematic
- Mathematical programming
II PHYSICAL INTERPRETATION
Limit states/Failure modes
Hyperplanes/Extreme points
Relationships
108
RIGID PLASTIC FRAMES WITH MULTIPARAMETRIC LOADING
Proportional loading indicates a system of concentrated loads, each of which is
proportional to a parameter, _. However, the actual loading on the structures
may not satisfy the restriction of proportional loading. It is necessary, in
such cases, to consider the independent variation of load factor parameters for
the various loads acting on the frame. A static multiobjective linear
programming (MOLP) model has recently been formulated. Q (q = i, 2 ...... Q)
denote independent load parameters and Ckj and Dkq are the constant coefficients.
Unlike scalar optimization problems, the vector optimal solutions are not
completely ordered and there is no unique 'optimal' solution. The notion of an
optimal solution is replaced by the concept of weak noninferior solution.
The geometrical structure of the MOLP static model shows that it has two
different associated polyhedra instead of just one, as in LP models. These
polyhedral regions are defined by the feasible regions of the MOLP model in the
objective (load) space and decision (basic variable) space, respectively. The
two polyhedral feasible regions have frontiers made up of only polyhedral facets.
It has been shown [6] that maximal facets of these polyhedra correspond to the
failure modes of the structure and union of all maximal facets gives the global
limit surface for the frame.
STATIC MOLP MODEL
Max _= MaX{_l, " "'%'Q }T
J Q
s.t. Z; Ckj Mj - _ Dkq %,q = 0
i=1 q=l
k= 1,.. ., M
- +
-Mpj < Mj = < Mpj
j=l_ -, J
109
RESPONSE REGIONS
These response patterns for frames define the limits of variation of random
variables and once such patterns are generated for a given structure, different
solutions corresponding to any random vector must belong to the defined regions.
Therefore, by considering powerful algorithmic methods developed in mathematical
and computer science literature for extremum problems, alternative structural
responses can be predicted without reanalysis of the structure. Often, it is
possible to further simplify the procedure in some cases, based on the
decomposition of parametric space. These procedures decompose the parametric
space into mutually exclusive (non-overlapping) and collectively exhaustive
subdomains corresponding to various failure modes [7]. This enables one to
replace the consideration of an infinite number of parameter combinations with a
finite number of parametric regions, which are also polyhedral. Multiparametric
procedures lead to partitioning of both the load and basic variable space. All
these procedures do not in any way depend on the probabilistic information. The
probability considerations can be subsequently introduced to evaluate structural
system reliability. This facilitates investigation of different loading
conditions and probabilistic assumptions since reliability evaluations can be
obtained without any further structural analyses.
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i ] /.
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110 OF POOR :'_"_"_ '
SYSTEM RELIABILITY EVALUATION
System reliability evaluation of frames for ultimate collapse is simplified by
the use of structural responses generated by extremum procedures. For example,
in the static case, a method has been proposed [8] which replaces the safe
polyhedral response region of the frame with an analytically tractable region of
equivalent volume, where the term volume is to be interpreted in a broad sense,
since the volume may be of different dimensions and order. Reliability can be
computed from the properties of the substituted region, which can be a
parallelotope, hypersphere, hyperellipsoid, or any other suitable form. Use of
hyperspherical equivalent region leads to the expression for structural system
reliability in terms of the chi-square distribution.
System reliability evaluation of frames for ultimate collapse by the kinematic
approach requires the enumeration of the failure modes, calculation of the
probability of failure for each mode and then computation of the overall
reliability by suitable aggregation. A simulation approach that exploits the
special structure of the kinematic model has been proposed [9]. Load and
resistance proportionalities are determined by each simulation, and the
associated failure mode is identified as an extreme point of the LP model. The
procedure gains its efficiency from the fact that every simulation derives an
associated failure condition and its probability which are then combined into a
system reliability.
SYSTEM RELIABILITY- STATIC APPROACH
Replacement of response regions by an
equivalent region
Concept of equivalence
Hypersphere, parallelotope, hyperellipsoid
SYSTEM RELIABILITY- KINEMATIC APPROACH
- Generation of failure modes
Simulation procedure
111
CONCLUSIONS
Extremum methods of structural analysis offer significant promise for advances in
the analysis of random structural systems and their reliability assessment. The
complexity of the physical problem and the randomness of the variables makes the
solution otherwise intractable. Fortunately, the mathematical nature of the
problem lends itself to mathematical programming formulations and use of powerful
algorithmic procedures. This has been illustrated by consideration of rigid
plastic frames subject to collapse by flexural action. Linear and multiobjective
linear programming models were discussed for structural systems analysis under
proportional and multiparametric loading, respectively. Duality relations
between the static and kinematic approaches for each of these models and their
response patterns lead naturally to alternative methods for system reliability
evaluation.
Author's ongoing research aims to demonstrate the use of extremum methods for the
reliability analysis of different structural systems for varying material
behavior, structural dynamics problems and stability analysis. It is planned to
explore structural behavior patterns with the objective of gaining insights into
random structural behavior, dual relationships of patterns from static and
kinematic considerations, causes of redundancy and the feasibility of using
insights for the development of simplified and efficient computational methods
for structural reanalysis and system reliability evaluation.
• PROMISE OF EXTREMUM METHODS
EXTENSIONS
- Other structures
Material behavior models
Structural dynamics problems
Stability analysis
• REDUNDANCY
• STRUCTURAL REANALYSIS
• SIMPLIFIED METHODS FOR SYSTEM RELIABILITY
112
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Probabilistic structural analysis by extremum methods

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