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MSC Software Confidential MSC Software Confidential
Analysis of Reinforced Concrete Beams
Using Nonlinear Finite Element Techniques
2013 Regional User Conference
Presented By: David R. Dearth
May 14, 2013
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During initial design of reinforced concrete beams, structural
engineers typically estimate the general sizing of the beam using
conventional hand equations. [Reference ACI 318]
Conventional hand analysis approaches involve using linear
elastic equations to compute equivalent, or transformed, cross
sectional properties.
Elastic equations are limited to estimating the onset of RC beam
cracking of the concrete and to some extent also approximating
ultimate failure of the RC beam after initial cracking.
To analyze the regions between initial cracking and ultimate
failure, nonlinear FE analysis techniques are required.
Introduction
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Before considering taking on the task of analyzing the nonlinear
response of RC beams, engineers should have at least a working
knowledge of how to perform a conventional linear analysis using
pencil, paper and a calculator per ACI code requirements.
When tasked with performing the nonlinear analysis one most likely will
look at a sample tutorial problem and simply follow the same steps with
their particular problem of interest substituting instructions from the
sample tutorial.
The real questions are : How can one relate the physical observations
witnessed in the environmental test lab to virtual testing developed
using nonlinear FEA techniques? Or how one can simulate actual
physical testing of RC beams using computer analyses software?
Baseline or Background Analysis
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The best approach would be to locate some actual test data. When it
comes to verifying the analytical results from analysis of RC beams
there is very little documented information showing results from actual
physical testing under tightly controlled laboratory conditions. Test data
on RC beams is very scarce.
A search through the available engineering literature found
comprehensive, documented data of actual physical testing under
tightly controlled laboratory conditions of several RC beams performed
by Foley and Buckhouse1. Wolanski2 provides analytical correlation to
the laboratory testing with detailed finite element analysis of the 1997
Buckhouse1 RC beam tests. The testing performed by Foley and
Buckhouse1 are cited in several other technical papers addressing FEA
of RC beams.
Analysis Results vs. Physical Tests
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RC Beam Definition RC Beam from Buckhouse Testing (1997) Marquette University1,2
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RC Beam from Buckhouse Testing (1997)
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Simple Supported Beam Tested at Marquette University
Reinforcement Layout
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There are generally three (3) methods for addressing stress and deflection in RC
beams using conventional hand equations per ACI 3183. The most common
methods are:
Linear Elastic Uncracked Approach: The linear elastic uncracked method assumes tension stress in the concrete remains below the cracking limit. Tension stresses are
assumed liner elastic and fully effective in an uncracked concrete section. This method is
used to calculate the state of stress and deflections when the RC beam structure is subjected
to normal anticipated service load conditions.
Elastic Cracked Approach: The elastic cracked method assumes concrete tension stress has exceeded cracking limits and neglects any concrete tension stress. Linear elastic
compressive stresses are balanced by tension stresses in the reinforcement.
Ultimate Cracked Approach: The ultimate cracked method assumes a simplified yielding stress criterion. For ultimate load carrying strength capability, tension stress in the
concrete is assumed nonexistent and maximum compressive strain is assumed to equal c = 0.003. The balancing tensile loading is assumed fully carried by the steel reinforcement with
the steel at yield, Fty.
Review Fundamental Principals ACI 318
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Stage 1: Linear Elastic Conditions
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Allowable compressive stress for concrete listed a value of fc = 4,800 psi
Per ACI 318 9.5.2.3 tension rupture stress
= 7.5 = 7.5 4,800 = 520
Calculated cracking moment, Mcr, and corresponding equivalent loading Pcr = 4,680 lbs.
The equivalent linear elastic deflections = 0.050; gross section properties per ACI.
Cracking moment, Mcr_tr, and corresponding equivalent loading Pcr_tr = 5,080 lbs. The equivalent
linear elastic deflections for this applied loading =
0.052; composite transformed section properties.
Initial Cracking per Linear ACI 318
RC Beam Linear Elastic Composite
Section Properties
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Stage 2: Elastic Cracked Section
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When the maximum tensile stress in the concrete exceeds
modulus of rupture, fr, the
cross section is assumed to be
"cracked" and all the tensile
stress is assumed to be
carried by the steel
reinforcement.
For the cross section shown properties for the composite
assembly is Icrack = 1,116 in4.
This effective inertia is used for
computing deflections after
crack initiation.
Balanced State of Stress Concrete & Rebar
RC Beam Elastic Cracked Section Properties
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Stage 3: Ultimate - Cracked Moment Mu
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To compute ultimate failure, cracked bending moment
tension stress in the concrete is
assumed nonexistent and
maximum compressive strain is
assumed to equal c = 0.003. Tensile loading the steel
reinforcement at yield stress.
The calculated ultimate moment capacity Mu = 826,740 in-lbs. Equivalent ultimate loading Pu, =
13,780 lbs. The equivalent
deflections at this applied
ultimate loading applied loading
= 0.548.
RC Beam Ultimate Cracked Section Properties
Cracked Moment Mu: Whitney Rectangular Stress Block
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Results: Deflections ACI 318 Hand Analyses
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The effective inertia, Ieff, is calculated after
crack initiation according to ACI 318 9.5.2.3. A
comparison of measured deflections at the
center line of the control beam to the
computed deflections using ACI 318 hand
equations is shown at right.
= (
)3 + 1
3
Deflections ACI 318 Hand Analyses
Compare ACI 318 Calculations to Test Data
Reproduced Test Data
ACI Computed Values
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FEA Model Definition Nonlinear Cracking to Ultimate using MSC/Marc
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FEM Definition: Nonlinear Cracking
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For comparison purposes it was decided to duplicate as closely as possibly the RC beam test article and FEA model definition described by Foley & Buckhouse1 and Wolanski2.
Due to the symmetry of loading and geometry, the full RC beam can be idealized using quarter symmetric idealization; symmetric boundary conditions (constraints) are denoted.
Full RC Beam Geometry Quarter Symmetric RC Beam Geometry
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Concrete: Basic Isotropic Properties
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The concrete is idealized using 3D solid elements. Youngs modulus of elasticity is computed using ACI 318 8.5.1.
= 57,000 = 57,000 4,800
= 3.949 106
The stress-strain curve data for the concrete is shown at right. To maintain
consistency with Wolanski2 analysis, a
Poissons ratio for concrete = 0.3 is assumed. It is recognized, however, that a
Poissons value of = 0.18 to 0.2 may be more representative for concrete.
Concrete Compressive Stress-Strain Data
Concrete Properties
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Concrete: Nonlinear Cracking Properties
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The non-linear concrete cracking formulation used by MSC/Marc is based on Buyukozturk model. The typical strain-softening relationship of concrete and idealized forms are shown below. The area
under the tension-softening region represents fracture energy Gf. When tension-softening, Es, is not
included, material loses all load-carrying capacity; stress goes to zero upon cracking. Assuming the
characteristic length for the RC concrete beam equals the depth of the beam, hc = 18 inches. Then
fracture energy Gf can be calculated from the following: = 1
2
2= 0.62 /
Typical stress-strain Uniaxial Stress-Strain Diagram
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Steel Reinforcement
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The steel reinforcement (rebar & stirrups) is idealized using Rod/Truss elements with discrete
idealization of rebar with the concrete; i.e. rebar & concrete elements sharing common nodes.
Linear Youngs Modulus, Es = 29,000,000 psi Poissons Ratio, = 0.3 Yield Stress, Fty, = 60,000 psi
Bi-Linear Elastic-Plastic Modulus, E1 = 2,900 psi (nearly perfectly plastic)
Quarter Symmetric RC Beam Rebar & Stirrups
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Applied Loading
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Buckhouse1 lists the ultimate recorded loading at failure equal to 16,300 lbs. To ensure uniform
deflections at the load points, individual concentrated loading is distributed as shown below.
Concentrated Nodal Loading Distribution
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Solution Parameters
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The MSC/Marc nonlinear solution Load Increment Parameters were set to Adaptive increment
type as shown below. The Iteration Tolerance Parameters for convergence were set to
Residual Force = 10%.
Adaptive Load Increment Dialog Inputs
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FEA Model Results Nonlinear Cracking to Ultimate using MSC/Marc
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Results: Nonlinear Finite Element Analysis
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The figure shows a comparison of
measured deflections at the center line of
the Buckhouse1 control beam to the
computed deflections from the FEA model
developed using MSC/Marc4.
Deflections from FEA Analyses
Compare FEA Results to Test Data
Reproduced Test Data
FEA Computed Values
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Results: Notes
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The nonlinear FEA MSC/Marc solution contains only 17 output steps using adaptive load stepping
In the analysis performed by Wolanski2 the iteration parameters were adjusted during selected load steps to ensure the analytical results better fit the experimental data. Having
prior knowledge of the solution to the nonlinear response is not what is generally available
to analysts attempting to predict the response of beams before they are built.
For the analysis outlined herein, no prior knowledge of the solution is assumed and it was decided to perform the analysis by applying the full ultimate loading and letting the program
solution determine what happens in between zero load and full ultimate loading.
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Results: Crack Initiation Comparison
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Maximum principal stress contour plot of the
concrete at the onset of crack initiation is
shown. The load step increment to the onset of
cracking is Incr =11, Time =0.32150 of total loading. The corresponding applied loading is
16,300 x 0.32150 = 5,240 lbs. This value is
within +3% of the hand calculations using the
composite properties for the transformed
section (concrete & rebar) Pcr_tr = 5,080 lbs. The
corresponding computed stress value of 490 psi
is within 6% of the maximum allowable tension
stress, or rupture stress fr = 520 psi defined per
ACI 318 9.5.2.3. This figure illustrates the
concrete stress distribution at the last linear-
elastic load step before cracks begin to develop.
Compare FEA Results to Test Data
Stress at Crack Initiation Comparison to Hand Calculations
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Crack Progression Crack, Strain Vector Plots
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Crack Propagation Resultant Crack Strain
The progressive pictures shown to the right
illustrate typical propagation of the concrete
cracks by displaying Vector plots of Resultant
Crack Strain.
Crack Progression
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References
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1. Christopher M. Foley and Evan R. Buckhouse, Strengthening Existing Reinforced Concrete Beams for Flexure Using Bolted External Structural
Steel Channels, Structural Engineering Report MUST-98-1, January 1998.
2. Anthony J. Wolanski, B.S., Flexural Behavior of Reinforced and Prestressed Concrete Beams Using Finite Element Analysis, Masters Thesis, Marquette University, Milwaukee, Wisconsin May, 2004.
3. ACI 318-08, Building Code Requirements for Structural Concrete and Commentary ACI Manual of Concrete Practice, Part 3, American Concrete Institute, Detroit, MI, 1992.
4. MSC/Marc Reference Manuals & Finite Element Analysis System: Volumes
A, B, C, D" MSC Software Corporation, 2 MacArthur Place, Santa Ana, California
92707
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