# Nonlinear Analysis of Cross Sections under Axial Load and Biaxial Bending

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```					 ACI STRUCTURAL JOURNAL                                                                                TECHNICAL PAPER
Title no. 107-S38

Nonlinear Analysis of Cross Sections under Axial Load and
Biaxial Bending
by Marco Di Ludovico, Gian Piero Lignola, Andrea Prota, and Edoardo Cosenza

The paper presents a method and an integration procedure                    The proposed computation method specifically focuses on
specifically targeting the analysis of cross sections under biaxial      the analysis of regular or irregular shaped cross sections (for
bending and axial load, whatever the shape—including internal            example, made of concrete) internally or externally reinforced
voids—and made of different materials (for example, concrete)            by mild steel (reinforced concrete [RC]) and prestressing
internally or externally reinforced by mild steel, prestressing          tendons (prestressed concrete [PC]) as well as by fiber-reinforced
tendons, and fiber-reinforced polymers. Surface integration of
meshed cross sections allows moment-curvature relationships to
polymers (FRP). The developed algorithm provides the
be computed using nonlinear constitutive relationships (also with        moment-curvature diagrams for cross sections subjected to
softening) and three-dimensional (3D) interaction domains P-Mx-My        axial load and biaxial bending using linear and nonlinear
to be drawn. Iteration procedures and convergence criteria are           stress-strain relationships, also with softening. Moreover, it
herein proposed to rapidly solve the highly nonlinear problem.           allows cross sections to be designed and/or checked by
Numerical tests showed fast convergence of the algorithm and             determining the ultimate flexural moments Mx-My given a
good agreement with experimental results elsewhere; comparisons          constant axial load P and orientation angle β of the resultant
between the numerical outcomes and the simplified expressions            moment; the Mx-My domain for a given axial load and,
available in the literature have been performed to point out the         consequently, the 3D failure domain can be plotted.
current limits of the available codes’ formulas.                            A review of the literature’s methods and a description of
the proposed numerical method, along with model validation
Keywords: biaxial bending; cross section; nonlinear; moment-curvature;   by comparison with results found elsewhere, are outlined in
three-dimensional interaction domain.                                    the following sections.

INTRODUCTION                                                     RESEARCH SIGNIFICANCE
The study of axially loaded members under biaxial                         This study deals with reliable and rapid nonlinear analyses
bending is of particular interest in the case of structures              in terms of strength and deformability of cross sections of
subjected to earthquake motion. In such cases, the presence              regular or irregular shape and made of different materials. The
of biaxial bending moments reduces the cross section                     provided algorithm is intended as a tool, adopting robust
capacity both in terms of strength and ductility, usually                procedures, well suited to study RC cross sections under axial
evaluated only under uniaxial loads. The current seismic                 load and biaxial bending (for example, seismic actions). In
codes, however, allow cross sections to be designed under                particular, the procedure may be used to assess the main
biaxial bending by analyzing such sections as subjected to               characteristics of the nonlinear behavior of cross sections,
two uniaxial bendings and axial loads acting separately.                 namely, cracking, yielding point, flexural strength, ultimate
In particular, FEMA 356 1 and Italian seismic guideline                  curvature, moment-curvature relationship also with softening,
D.M. 14/01/20082 provide simplified equations to                         curvature ductility, and 3D interaction domains P-Mx-My).
approximate the effective three-dimensional (3D) failure                    The results provided by the developed procedure have
surface, P-Mx-My; Eurocode 83 suggests treating biaxial                  been compared to experimental results on symmetric and
bending by carrying out the checks separately in each direction,         irregular cross sections available in the literature. Comparisons
with the uniaxial flexural strength reduced by 30%.                      between the numerical results and the simplified expressions
These are some of the several approximate methods that                 available in the literature indicate the current limits of the
have been proposed in recent decades. The use of simplified              available codes’ formulas.
approaches is unsafe in some cases and cannot give any
information about fundamental properties (that is, cracking                      REVIEW OF LITERATURES’ METHODS
and yielding point and curvature ductility) of the cross                   The design and/or check procedures to analyze cross
section. They are generally justified by the difficulty in               sections under biaxial load are iterative and require the integration
manually computing the cross section capacity under biaxial              of the stress field over the cross section at each step. To avoid
bending and axial load. Such computation requires the                    such a time-consuming procedure, several simplified methods
integration of stresses associated with strain-based failure             were previously developed to provide an approximate
criteria that can be numerically performed by discretizing the           analytical expression for the failure surface. One of these,
cross section and iteratively locating the neutral axis depth.           the load contour (LC) method, was provided by Bresler,4
Thus, an appropriate numerical integration procedure that
can be implemented in a computer analysis program,                         ACI Structural Journal, V. 107, No. 4, July-August 2010.
MS No. S-2008-222.R5 received September 25, 2009, and reviewed under Institute
cross sections under axial load and biaxial bending, has been            including the making of copies unless permission is obtained from the copyright proprietors.
Pertinent discussion including author’s closure, if any, will be published in the May-June
proposed in this study.                                                  2011 ACI Structural Journal if the discussion is received by January 1, 2011.

390                                                                                             ACI Structural Journal/July-August 2010
Marco Di Ludovico is an Assistant Professor of structural engineering at the University       ⎛ -------- ⎞ + ⎛ -------- ⎞ ⎛ 1 – β⎞ – 1 = 0 if ⎛ -------- ⎞ > ⎛ -------- ⎞
Mx
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Description: The paper presents a method and an integration procedure specifically targeting the analysis of cross sections under biaxial bending and axial load, whatever the shape-including internal voids-and made of different materials (for example, concrete) internally or externally reinforced by mild steel, prestressing tendons, and fiber-reinforced polymers. Surface integration of meshed cross sections allows moment-curvature relationships to be computed using nonlinear constitutive relationships (also with softening) and three-dimensional (3D) interaction domains P-M^sub x^-M^sub y^ to be drawn. Iteration procedures and convergence criteria are herein proposed to rapidly solve the highly nonlinear problem. Numerical tests showed fast convergence of the algorithm and good agreement with experimental results elsewhere; comparisons between the numerical outcomes and the simplified expressions available in the literature have been performed to point out the current limits of the available codes' formulas. [PUBLICATION ABSTRACT]
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