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Title
Finite element response sensitivity analysis using force-based frame models
Permalink
https://escholarship.org/uc/item/5fz9h6pc
Journal
INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, 59(13)
Authors
Conte, Joel P
Barbato, Michele
Spacone, Enrico
Publication Date
2004-04-07
DOI
10.1002/nme.994)
Peer reviewed
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University of California
1
Finite Element Response Sensitivity Analysis Using Force-Based Frame Models
J. P. Conte
1
, M. Barbato
2
, and E. Spacone
3
.
SUMMARY
This paper presents a method to compute consistent response sensitivities of force-based finite element models of
structural frame systems to both material constitutive and discrete loading parameters. It has been shown that force-
based frame elements are superior to classical displacement-based elements in the sense that they enable, at no signif-
icant additional costs, a drastic reduction in the number of elements required for a given level of accuracy in the com-
puted response of the finite element model. This advantage of force-based elements is of even more interest in
structural reliability analysis, which requires accurate and efficient computation of structural response and structural
response sensitivities. This paper focuses on material nonlinearities in the context of both static and dynamic
response analysis. The formulation presented herein assumes the use of a general-purpose nonlinear finite element
analysis program based on the direct stiffness method. It is based on the general so-called Direct Differentiation
Method (DDM) for computing response sensitivities. The complete analytical formulation is presented at the element
level and details are provided about its implementation in a general-purpose finite element analysis program. The
new formulation and its implementation are validated through some application examples, in which analytical
response sensitivities are compared with their counterparts obtained using Forward Finite Difference (FFD) analysis.
The force-based finite element methodology augmented with the developed procedure for analytical response sensi-
tivity computation offers a powerful general tool for structural response sensitivity analysis.
KEY WORDS: Plasticity-based finite element model; material constitutive parameter; finite element response sen-
sitivity; force-based frame element; reliability analysis.
1. INTRODUCTION
Recent years have seen great advances in the nonlinear analysis of frame structures. Advances were led by
the development and implementation of force-based elements, which are superior to classical displace-
ment-based elements in tracing material nonlinearities such as those encountered in reinforced concrete
beams and columns (Spacone et al. [1-3], Neuenhofer and Filippou [4]). The state-of-the-art in computa-
tional simulation of the static and dynamic response of frame structures lies in the nonlinear domain to
account for material and geometric nonlinearities governing the complex behavior of structural systems,
especially near their failure range (i.e., collapse analysis).
1. Associate Professor, Dept. of Structural Engineering, 9500 Gilman Drive, University of California at San Diego, La
Jolla, California 92093-0085; E-mail: jpconte@ucsd.edu
2. Graduate Student, Dept. of Structural Engineering, 9500 Gilman Drive, University of California at San Diego, La Jolla,
California 92093-0085; E-mail: mbarbato@ucsd.edu
3. Associate Professor, CEAE Dept., University of Colorado, Boulder, CO 80309-0428; E-mail: spacone@colorado.edu
To appear in International Journal for Numerical Methods in Engineering
2
Maybe even more important than the simulated nonlinear response of a frame structure is its sensitivity to
loading parameters and to various geometric, mechanical, and material properties defining the structure.
Finite element response sensitivities represent an essential ingredient for gradient-based optimization
methods needed in structural reliability analysis, structural optimization, structural identification, and
finite element model updating (Ditlevsen and Madsen [5], Kleiber et al. [6]). Many researchers dedicated
their attention to the general problem of design sensitivity analysis, among others, Choi and Santos [7],
Arora and Cardoso [8], Tsay and Arora [9], Tsai et al. [10]. Consistent finite element response sensitivity
analysis methods have already been formulated for displacement-based finite elements (Zhang and Der
Kiureghian [11], Kleiber et al. [6], Conte et al. [12,13]). In the present paper, these methods are extended to
force-based finite elements, also called flexibility-based finite elements in the literature. The objective of
this work is to extend the benefits of force-based frame elements for nonlinear structural analysis to finite
element response sensitivity analysis.
The formulation presented here is based on the general so-called Direct Differentiation Method (DDM),
which consists of differentiating consistently the space (finite element) and the time (finite difference) dis-
crete equations of the structural response (Conte et al. [13]). It also assumes the use of a general-purpose
nonlinear finite element analysis program based on the direct stiffness method. This paper focuses on
materially-nonlinear-only static and dynamic structural response sensitivity analysis.
2. NONLINEAR STATIC AND DYNAMIC RESPONSE ANALYSIS OF STRUCTURES USING
FORCE-BASED FRAME ELEMENTS
After spatial discretization using the finite element method, the equations of motion of a materially-nonlin-
ear-only model of a structural system take the form of the following nonlinear matrix differential equation:
(1)
where t = time, θ = scalar sensitivity parameter (material or loading variable), u(t) = vector of nodal dis-
placements, M = mass matrix, C = damping matrix, R(u, t) = history dependent internal (inelastic) resist-
ing force vector, F(t) = applied dynamic load vector, and a superposed dot denotes one differentiation with
respect to time. In the case of “rigid-soil” earthquake ground excitation, the dynamic load vector takes the
form in which L is an influence coefficient vector and denotes the input ground
acceleration history. Without loss of generality, a single component ground excitation is considered herein.
The potential dependence of each term of the equation of motion on the sensitivity parameter θ is shown
explicitly in Equation (1). The numerical integration scheme used to integrate the static and dynamic equi-
librium equations (1) is summarized in Appendix A. It serves as starting point in deriving the analytical
sensitivities of the finite element structural response predictions.
M θ()u
··
t θ,()C θ()u
·
t θ,()Rut θ,()θ,()++ F t θ,()=
F t() MLu
··
g
t()–=
u
··
g
t()
3
2.1 Force-Based Frame Element
The last few years have seen the rapid development of force-based elements for the nonlinear analysis of
frame structures. In a classical displacement-based element, the cubic and linear Hermitian polynomials
used to interpolate the transverse and axial frame element displacements, respectively, are only approxima-
tions of the actual displacement fields in the presence of non-uniform beam cross-section and/or nonlinear
material behavior. On the other hand, force-based frame element formulations stem from equilibrium
between section and nodal forces, which can be enforced exactly in the case of a frame element. The exact
flexibility matrix can be computed for an arbitrary variation of the cross-section and for any section consti-
tutive law. The main issue with force-based frame elements is their implementation in a general-purpose
nonlinear finite element program, typically based on the stiffness method. Spacone et al. [1,2] presented a
consistent solution to this problem. They propose a state determination based on an iterative procedure that
is basically a Newton-Raphson scheme under constant nodal displacements. During the iterations, the
deformation fields inside the element (mainly curvature and axial strains) are adjusted until they become
compatible (in an integral sense) with the imposed nodal deformations. Neuenhofer and Filippou [4]
showed that the iterations are not necessary at the element level at each global (structure level) iteration
step, since the element eventually converges as the structure iteration scheme converges. The first (full
iteration) procedure is more robust near limit points and computationally more demanding at the element
level, but may save iterations at the global level. The second procedure is generally faster.
The force-based element formulation proposed by Spacone et al. [1,2] is totally independent of the section
constitutive law. The section state determination is identical to that required for a displacement-based ele-
ment. The section module must return the section stiffness and the section resisting forces corresponding to
the current section deformations. Different section models have been implemented, notably layer and fiber
sections and section with nonlinear resultant force-deformation laws. Appendix B presents the features of
the force-based frame element formulation, which are needed in deriving the analytical sensitivities of
force-based finite element models of structural frame systems.
Geometric nonlinearities are not included in this paper, whose focus is on material nonlinearities. Two
frameworks for including geometric nonlinearities in a force-based beam formulation have been proposed,
one by de Souza [14] with earlier work by Neuenhofer and Filippou [15], who uses a corotational formula-
tion to include large displacements, the other by Sivaselvan and Reinhorn [16], who modify the shape of
the force interpolation functions to include the geometric effects.
4
3. RESPONSE SENSITIVITY ANALYSIS AT THE STRUCTURE LEVEL
The computation of finite element response sensitivities to material and loading parameters requires exten-
sion of the finite element algorithms for response computation only. Let r(t) denote a generic scalar
response quantity such as displacement, acceleration, local or resultant stress, local or resultant strain, or
local/global cumulative plastic deformation. By definition, the sensitivity of r(t) with respect to the mate-
rial or loading parameter θ is mathematically expressed as the partial derivative of r(t) with respect to the
variable θ, i.e., where denotes the nominal value taken by the sensitivity parameter θ
for the finite element response analysis.
Assume that the response of a frame type structure modeled using force-based frame elements is computed
according to the element state determination algorithm described in Appendix B, Section B.2, imple-
mented within a general-purpose nonlinear finite element analysis program based on the direct stiffness
method, employing suitable numerical integration techniques such as Newton-Raphson or modified New-
ton-Raphson at the structure level and Gauss or Gauss-Lobatto at the element level. At each time step, after
convergence of the response computation, the consistent response sensitivities are computed. Following
the Direct Differentiation Method (DDM) (Conte [13]), this requires to differentiate exactly the finite ele-
ment numerical scheme for the response calculation (including the numerical integration scheme for the
material constitutive law) with respect to the sensitivity parameter θ in order to obtain the “exact” sensitiv-
ities of the computationally simulated system response, which itself is an approximation of the exact but
unknown system response
1
. As shown elsewhere for the displacement-based finite element methodology
(Zhang and Der Kiureghian [11]; Kleiber et al. [6]; Conte et al. [13]) and shown below for the force-based
finite element methodology, this procedure consists in computing first the conditional derivatives of the
element and material history/state variables, forming the right-hand-side (RHS) of the response sensitivity
equation at the structure level, solving it for the nodal displacement response sensitivities and updating the
unconditional derivatives of all the history/state variables. The response sensitivity calculation algorithm
propagates across the various hierarchical layers of finite element response calculation: (1) the structure
level, at which the general framework of response sensitivity computation is organized and the response
sensitivity equation is solved, (2) the element level, at which the element formulation (e.g., displacement-
based, force-based) is defined, (3) the section level (or integration/Gauss point level), at which the sec-
tional constitutive relations are defined, and (4) the material level characterized by the material constitutive
law (in differential form), its numerical integration, and the consistent/exact differentiation of the constitu-
1. The exact system response would require the exact solution of the (time continuous - space continuous) governing par-
tial differential equations for the physical model of the structure under consideration.
rt()∂θ
θθ
0
=
∂⁄
θ
0
