Analysis of Reinforced Concrete (RC) Beams Using Nonlinear Finite Element

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

Applied Analysis & Technology

MSC Software Confidential

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

2



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

3



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

4



RC Beam Definition RC Beam from Buckhouse Testing (1997) Marquette University1,2

5



RC Beam from Buckhouse Testing (1997)

6

Simple Supported Beam Tested at Marquette University

Reinforcement Layout



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

7



Stage 1: Linear Elastic Conditions

8

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



Stage 2: Elastic Cracked Section

9

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



Stage 3: Ultimate - Cracked Moment Mu

10

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



Results: Deflections ACI 318 Hand Analyses

11

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



FEA Model Definition Nonlinear Cracking to Ultimate using MSC/Marc

12



FEM Definition: Nonlinear Cracking

13

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



Concrete: Basic Isotropic Properties

14

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



Concrete: Nonlinear Cracking Properties

15

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



Steel Reinforcement

16

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



Applied Loading

17

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



Solution Parameters

18

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



FEA Model Results Nonlinear Cracking to Ultimate using MSC/Marc

19



Results: Nonlinear Finite Element Analysis

20

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



Results: Notes

21

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.



Results: Crack Initiation Comparison

22

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



Crack Progression Crack, Strain Vector Plots

23

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



References

24

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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Analysis of Reinforced Concrete (RC) Beams Using Nonlinear Finite Element