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Analysis of RC Beams using Nonlinear Finite Element Techniques 10 October 2012

Applied Analysis & Technology Page i

Analysis of Reinforced Concrete (RC) Beams

Using Nonlinear Finite Element Techniques

MSC/Marc

10 October 2012

Prepared By:

David R. Dearth, P.E.

Applied Analysis & Technology, Inc 16731 Sea Witch Lane

Huntington Beach, CA 92649-3054

Telephone (714) 846-4235

E-Mail [email protected]

Web Site www.AppliedAnalysisAndTech.com

mailto:[email protected]

http://www.appliedanalysisandtech.com/


Applied Analysis & Technology, Inc. Page 1 of 20

"Analysis of Reinforced Concrete (RC) Beams using Nonlinear Finite Element Techniques"

The purpose of this article is to develop an understanding of the arithmetic involved in the design solution to

analyzing reinforced concrete (RC) beams. This article provides a single source review to outline the steps

necessary to perform analysis of RC beams beginning at the linear elastic region into the nonlinear analyses

and to ultimate failure.

In this analysis no prior knowledge of the cracked RC beam is assumed. The analytical results are compared

to actual test data from a RC beam tested under closely monitored laboratory conditions. A finite element

model is also constructed to produce a simplified nonlinear analysis of the RC beam using MSC Marc.

Introduction

During the process of designing reinforced concrete beams structural engineers typically estimate the

general sizing of the beam using conventional hand equations. These conventional hand analysis approaches

involve using linear elastic equations to compute equivalent, or transformed, cross sectional properties. The

linear elastic approaches have been utilized for many years and have for the most part been very successful.

These 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.

When it is desired to calculate the regions between initial cracking and ultimate failure, nonlinear analysis

techniques are required. When nonlinear approaches are desired, finite element analysis (FEA) techniques

are employed. Before considering taking on the task of calculating 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. 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. In essence this simply becomes a case of parroting the

steps outlined in the sample tutorial problem without fully understanding what is going on.

So how might one develop confidence in performing these types of nonlinear analysis problems? The best

way is to locate a sample real life problem with a known, documented solution that one can be work through

using hand calculations and also develop a FEA model.

So the question is.... How can one relate the physical observations witnessed in the environmental test lab

to virtual testing calculated using nonlinear FEA techniques? Or how one can simulate actual physical

testing of RC beams using computer analyses software?

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.



Sample Problem: RC Beam from Buckhouse Testing (1997) Marquette University1,2

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. Wolanski

2 provides analytical correlation to the laboratory testing with detailed finite element

analysis of the 1997 Buckhouse1 RC beam tests. The investigations performed by Foley and Buckhouse

1 are

cited in several other technical papers addressing FEA of RC beams.

Figure 1a shows a sketch of the1997 Buckhouse RC beam geometry, loading and boundary conditions.

Figure 1b shows the layout for the internal reinforcement, rebar & stirrups.

Figure 1a RC Beam Tested at Marquette University

Figure 1b RC Beam Reinforcement Layout



Thumbnail Course Review of Fundamental Principals ACI 318 Analysis of RC Beams

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 linear 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 ε c = 0.003. The balancing tensile

loading is assumed fully carried by the steel reinforcement with the steel at yield.

Stage 1: Linear Elastic Conditions – Normal Service Life & Initial Crack Stress

During normal service life conditions no cracking of the concrete is assumed. Stresses and deflections of RC

beams can be performed using a conventional linear elastic approach. For normal service life conditions it

is assumed that stress in the RC remain in the linear elastic range and the only difficulty is computing the

sections properties, EcI, for the RC beam. In computing the section properties, Young’s modulus, Ec, is

taken as the 28 day strength value. The inertia, I, for the composite section of concrete and steel

reinforcement is computed using the conventional “transformed section method”.

Analysis for the uncracked RC beam can be performed by treating the RC beam as a composite assembly of

concrete and steel reinforcement. A conservative approach is to neglect the stiffness contribution from the

reinforcement and consider the gross section properties of the concrete only. When the steel reinforcement

is included, it is assumed the reinforcement steel maintains intimate contact with the surrounding concrete.

With intimate contact or bonding maintained, the steel reinforcement and concrete will maintain the same

strain compatibility during loading. So long as the maximum tensile stress in the concrete remains below the

maximum the tensile capacity stress of the concrete, fr, (fr is also referred to as the modulus of rupture)

then the RC beam will act as a conventional composite assembly.

To maintain strain compatibility between the steel reinforcement and surrounding concrete, the steel

reinforcement having a greater modulus of elasticity, Esteel > Econcrete, experiences a greater magnitude in

stress distribution across this composite section than does the concrete. The ratio in stresses between the

steel and concrete is called the modular ratio, n. Where n = Esteel / Econcrete. The transformed section

properties for the composite assembly of concrete and steel rebar is computed by replacing the steel area by

an equivalent concrete area.

Figure 2 shows a cross section of the concrete and rebar for the 1977 Buckhouse RC beam tested. For the

cross section shown in Figure 2, the transformed cross sectional properties for the composite assembly is Itr

= 5,138 in4. Appendix “A” lists detailed arithmetic to compute the properties listed in Figure 2.



Figure 2 RC Beam Linear Elastic Composite Section Properties

As a general rule of thumb concrete tension stress is approximately 1/10 the compressive values. For the

allowable compressive stress for concrete used, Wolanski2 listed a value of f’c = 4,800 psi. ACI 318 9.5.2.3

3

computes tension stress or modulus of rupture stress, fr, computed based on the maximum compressive

stress using the following:

√ √

The onset of initial cracking of the concrete is computed per ACI 9.5.2.3 using the gross section properties.

The threshold cracking stress is computed from ACI 318 equations for the rupture stress, fr. For the gross

section the calculated cracking moment, Mcr, and corresponding equivalent loading Pcr = 4,680 lbs(1.)

. The

equivalent linear elastic deflections for this applied loading = 0.050”.

For Figure 2 transformed section properties the cracking moment, Mcr_tr, and corresponding equivalent

loading Pcr_tr = 5,080 lbs. The equivalent linear elastic deflections for this applied loading = 0.052”.

(1.) The calculated value of 4,680 lb is within 4% of the average physical observed loading to crack initiation equal to 4,500 lb.

quoted by Foley and Buckhouse1.



Stage 2: Elastic Cracked Section - Balanced State of Stress; Concrete & Rebar

Cracks begin to form when the tensile stress in the concrete exceed the maximum capacity of the concrete to

react tension stress at the modulus of rupture, fr. 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.

When a flexural crack occurs, it begins at the tension face when the tensile capacity is exceeded and the

crack propagates upward until the concrete is in compression. The section properties change as the crack

propagates, causing the increased tensile stress forcing the crack upwards toward the compression region.

Equilibrium is achieved once the crack stops propagating. Compressive stresses are still assumed to remain

in the elastic region.

For the cross section shown in Figure 3, the cross sectional properties for the composite assembly is Icracked

= 1,116 in4. Appendix A lists detailed arithmetic to compute the properties listed in Figure 3. The value of

Icrack it used to compute the instantaneous effective inertia, Ieff per ACI 318 9.5.2.3. This effective inertia is

used for computing deflections after crack initiation.

Figure 3 RC Beam Elastic Cracked Section Properties



Stage 3: Ultimate - Cracked Moment ϕMu: Whitney Rectangular Stress Block

To compute ultimate failure, cracked bending moment, a simplified yielding stress criterion is assumed. For

ultimate load carrying strength capability, tension stress in the concrete is assumed nonexistent and

maximum compressive strain is assumed to equal εc = 0.003. This magnitude of compressive strain is

representative of concrete with compressive strength from 2,000 < f‘c < 6,000 psi. The balancing tensile

loading is assumed carried by the steel reinforcement with the steel material at yield stress.

Figure 4 shows the cracked Whitney equivalent stress block cross section. The calculated ultimate moment

capacity ϕMu = 826,740 in-lbs. Using this calculated moment value, ϕMu, an equivalent ultimate loading Pu,

= 13,780 lbs(2.)

can be calculated. Appendix A lists detailed arithmetic to compute the ultimate moment ϕMu

values for Figure 4. The equivalent deflections at this applied ultimate loading applied loading = 0.548”.

Figure 4 RC Beam Ultimate Cracked Section

(2.) In the Foley and Buckhouse1 testing a theoretical ultimate load capacity value equal to 14,600 lb. is quoted. No

documentation could be found on how this 14,600 lb. value was computed; 13,780 lb. is within 6% of 14,600 lb. Foley and

Buckhouse1 noted the following: “The moment due to the dead weight of the beam was subtracted from Mult to give the

moment capacity of the beam due to superimposed live load”



Results: Deflections from ACI 318 Hand Analyses

The effective inertia, Ieff, is calculated after crack initiation according to ACI 318 9.5.2.3(3.)

. Figure 5 shows

a comparison of measured deflections at the center line of the control beam C11,2

to the computed

deflections using ACI 3183 hand equations.

[ (

)

]

Figure 5 Comparison of ACI 318 Hand Calculations to 1977 Buckhouse Laboratory Test Data

(3.) When crack initiation calculations are performed using the transformed section properties, Itr, ACI 318 9.5.2.3 instead of

gross section properties, the effective inertia results in slightly lower deflections.



FEA Model Definition for Nonlinear Cracking to Ultimate using MSC/Marc

For comparison purposes it was decided to duplicate as closely as possibly the RC beam test article and

FEA model definition described by Foley and Buckhouse1 and Wolanski

2. MSC/Marc

4 is one FEA program

that can be utilized to address the nonlinear characteristics confronted when analyzing RC beams. Figures

6a & 6b show isometric views of the RC beam geometry per Figures 1a & 1b. Due to the symmetry of

loading and geometry, the full RC beam can be idealized using quarter symmetric idealization as shown in

Figure 6b; symmetric boundary conditions (constraints) are denoted. For convenience the initial FEA

model was constructed using Femap v10.15 and imported into Patran 2010

4 using Marc Preferences

4.

Figure 6a

Full RC Beam Geometry

Figure 6b Quarter Symmetric RC Beam Geometry



Concrete: Basic Isotropic Properties

The concrete is idealized using 3D solid elements. Young’s modulus of elasticity is computed using ACI

318 8.5.1. Modulus of elasticity of the concrete is …

√ √

The stress-strain curve data for the concrete is shown in Figure 7. To maintain consistency with Wolanski2

analysis, a Poisson’s ratio for concrete ν= 0.3 is assumed. It is recognized, however, that a Poisson’s value

of ν= 0.18 to 0.2 may be more representative for concrete.

Figure 7 Concrete Compressive Stress-Strain Data from Wolanski

2



Concrete: Nonlinear Cracking Properties for MSC/Marc

The non-linear concrete cracking formulation used by MSC/Marc is called “Buyukozturk” model. The

typical strain-softening relationship of concrete shown in Figure 8.1 is idealized as shown in Figure 8.2. In

Figure 8.2 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:

(

) (

)

Figure 8.1 Figure 8.2 Typical stress-strain relationship for concrete Uniaxial Stress-Strain Diagram

MSC/Marc Crack Data Input Dialog

Critical cracking stress, fr = 520 psi

Tension softening modulus = Non Specified Brittle Cracking

Crushing strain = Non Specified No Plasticity

Shear retention factor = Neglected No Shear Retention after Cracks Develop (Conservative4)

Fracture energy, Gf = 0.62 lb/in (Reference)

(4.) Shear reinforcement is reacted by vertical stirrups.



Steel Reinforcement

The steel reinforcement (rebar & stirrups) is idealized using Rod/Truss elements with the node points

defined each rebar element sharing common nodes with the concrete solids. This approach is called discrete

idealization of rebar with the concrete. The steel material is defined using the nonlinear stress-strain curve

data listed below. The nonlinear material properties are entered using von Mises yield criteria.

Linear Young’s Modulus, Es = 29,000,000 psi

Poisson’s Ratio, ν = 0.3

Yield Stress, Fty, = 60,000 psi

Bi-Linear Elastic-Plastic Modulus, E1 = 2,900 psi (nearly perfectly plastic)

Figure 9 shows the layout for rebar and stirrups (5.)

per Figure 1b.

Figure 9 Quarter Symmetric RC Beam Rebar & Stirrups

(5.) In the analysis performed by Wolanski2 an additional stirrup is shown at the mid-span location of the beam. The nonlinear

MSC/Marc FEA model developed herein was processed with and without the additional mid-span stirrup. The net results

were a negligible difference in the solutions.



Applied Loading & Solution Parameters

Foley and 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 in Figure 10.

Figure 10 Concentrated Nodal Loading Distribution

The MSC/Marc nonlinear solution Load Increment Parameters were set to Adaptive increment type as

shown in Figure 11. The Iteration Tolerance Parameters for convergence were set to Residual Force = 10%.

Figure 11 Adaptive Load Increment Dialog Inputs



Results: Nonlinear Finite Element Analysis MSC/Marc

Figure 12 shows a comparison of measured deflections at the center line of the control beam C1 of Foley

and Buckhouse1 to the computed deflections from the FEA model developed using MSC/Marc

4.

Figure 12 Comparison of MSC/Marc Results to 1977 Buckhouse Laboratory Test Data

Notes: (6.) The nonlinear FEA MSC/Marc solution contains only 17 output steps using adaptive load stepping

(7.) 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: Stress at Crack Initiation – Comparison to Hand Calculations

Figure 13 shows maximum principal stress contour plot of the concrete at the onset of crack initiation. As

indicated on the contour plot legend, the load step increment to the onset of cracking is “Incr =11, Time

=0.32150” of total loading. The applied loading corresponding to Figure 13 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.

Figure 13 illustrates the concrete stress distribution at the last linear-elastic load step before cracks begin to

develop.

Figure 13 Maximum Principal Stress Contour Plot

Last Load Step Prior to Crack Propagation



Results: Crack Progression – Crack, Strain Vector Plots

The progressive pictures shown in Figure 14 illustrate typical propagation of the concrete cracks by

displaying Vector plots of Resultant Crack Strain.

Figure 14 Crack Propagation – Resultant Crack Strain



Conclusions & Recommendations:

1.) This report demonstrates the use of linear and nonlinear finite element analysis techniques to

analyze RC beams. It is concluded that FEM code MSC/Marc can be utilized with a high degree of

confidence in performing these types of analysis investigations.

2.) In this report it was desired to simulate the element mesh density and procedures outlined in the

Wolanski2 analysis for purposes of direct comparison. It is recommended extended analysis be

performed using fine grid mesh density to improve the solution accuracy.

3.) In this report, the effects of creep, shrinkage, moisture intrusion, thermal cycling, fatigue, etc… on

concrete have not been included. It is recommended to extend analysis to include these effects.



References

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”, Master’s 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

5.) Femap – Finite Element Modeling and Post Processing: Siemens PLM Software Incorporated, 5800

Granite Parkway Suite 600, Plano, TX 75024



LIST OF SYMBOLS

As_eff = equivalent effective area of concrete due to presence of rebar

As_rebar = cross sectional area of each rebar

Aconc = total cross sectional area of concrete section

f’c = 28 day compressive strength of concrete

fr = tensile capacity of concrete per ACI 318 9.5.2.3 also modulus of rupture

bc = base distance of concrete section

hc = vertical distance of concrete section

ccrack = c = distance for elastic compression of cracked section

Ec = modulus of elasticity for concrete per ACI 318 8.5.1

Es = modulus of elasticity for steel rebar

Gf = fracture energy

n = modular ratio of steel vs concrete

nrebar = number of rebar

drebar = reference distance to center of area for rebar

Ic_gross = cross sectional moment of inertia for uncracked concrete only

Icrack = cross sectional moment of inertia for cracked composite section

Itr = cross sectional area moment of inertia for uncracked composite section, concrete & rebar

Ieff = effective cross sectional area moment of inertia for composite section per ACI 318 9.5.2.3

Mcr_gross = bending moment to crack initiation based on gross section properties

Mcr_tr = bending moment to crack invitation using transformed cross sectional properties

φMu = bending moment to crack invitation using transformed cross sectional properties

Pcr_gross = loading to crack initiation gross cross sectional properties concrete only

Pcr_tr = loading to crack initiation using transformed cross sectional properties

Pu = maximum loading to ultimate using Whitney stress block cross sectional properties

yc_ref = reference distance to center of area for concrete

ybar = location of center of area for composite section, concrete & rebar

yu = beam deflection due to ultimate loading, Pu and Ieff.

ycr_gross = beam deflection due to crack initiation loading, Pcr_gross and Ic_gross.

ycr_tr = beam deflection due to crack initiation loading, Pcr_tr and Itr.

σcon_elastic = maximum elastic tension stress of concrete

φ1 = moment reduction factor

β1 = coefficient for depth of equivalent Whitney stress block

au = depth of equivalent Whitney stress block

cu = distance to location of equivalent Whitney stress compression



Applied Analysis & Technology, Inc

Applied Analysis & Technology was founded in 1982 and has extensive design and analysis experience in

mechanical engineering, aerospace, computer peripherals, medical components, piping networks, other

high-tech related fields, custom software, test equipment and prototype development. Our customer base

extends throughout the U.S.A. and as far away as Europe and South America. The primary focus of our

consulting firm is in finite element, FEA, analyses and computer aided design, CAD. Applied Analysis &

Technology utilizes specialized consultants in traditional design, metallurgy, failure analysis, conventional

mechanical analyses and environmental testing and other disciplines as required to achieve the desired

technical evaluation or design.

David R. Dearth is President of Applied Analysis & Technology, Huntington Beach, CA and can be

contacted by e-mail at [email protected] or at www.AppliedAnalysisAndTech.com.

mailto:[email protected]



Appendix A

Summary Hand Calculations

& Analysis Notes


Applied Analysis & Technology, Inc. Page A1 of A6

Stage 1: Linear Elastic Moment of Inertia Calculations for Composite Section

The concrete compressive strength at 28 days is given as:

Using ACI 318 8.5.1 modulus of elasticity of the concrete is computed as:

√ Concrete Modulus

The tensile capacity stress of the concrete, fr, is defined using ACI 318 9.5.2.3. This value fr is also referred

to as the modulus of rupture.

√ Concrete Cracking Stress for normal weight concrete

Beam Section Gross Moment of Inertia

, Base

, Height

Beam Cross Section

Per ACI 9.5.2.3 the crack initiation moment based on concrete gross section properties

Loading to crack initiation



To calculate stress in the rebar, Transformed section properties are needed.

Compute modular ratio, n, to be used for transformed inertia

Concrete and Steel

Where "n" is modular ratio of Esteel/Econcete. Transform area of Steel to equivalent or effective area of

concrete, As_eff

for each #5 rebar number of rebar

The location of the centroid of area for the effective composite section, concrete & steel rebar is:

The transformed composite area moment of inertia is computed using parallel axis theorem

[ ( ) ] [

]



Compute the equivalent loading, P lbs, to just exceed the maximum allowable concrete tension stress to

initiate first cracking.

Recall:

Using bending equation σcr_tr = (Mcr_tr *yc)/Itr, where Mcr_tr = Pcr_tr*60 in-lbs

Loading to crack initiation using transformed section properties

The corresponding stress in the steel rebar at this loading is σrebar = n(Mcr_tr*yrbar)/Itr



Stage 2: Elastic Moment of Inertia Calculations for Cracked Section

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. The

compressive stress in the remaining concrete is assumed to remain elastic. Calculate the location of the

neutral axis for the cracked section from the top of the beam, "ccrack".

[ √( ) (

) ( ) ]

(

)

The moment of inertia of this transformed area w.r.t. the neutral axis for "cracked" section is calculated

using the following for single reinforcement RC section:

[

]



Stage 3: Ultimate Strength Calculations for Cracked Section

For ultimate load carrying strength capability tension stress in the concrete is assumed nonexistent and

maximum compressive strain is assumed to equal εc = 0.003. The magnitude of compressive strain is

representative of concrete with compressive strength from 2,000 < f'c < 6,000 psi. The balancing tensile

loading is assumed fully carried by the steel reinforcement with the steel material at yielding at fs_ty.

Calculate the location of the neutral axis for the cracked section from the top of the beam, "ccrack".

Equivalent Whitney Stress Block definitions

Moment Reduction factor ϕu set equal to 1.0 to compute Ultimate moment,

Uniform distribution rectangular stress block, stress intensity factor β1

(

) rebar steel yield stress

[ ) ]

The maximum loading,



Calculate Deflections from Elastic Moment of Inertia Calculations for Cracked Section

At the computed Ultimate Moment capacity, effective inertia is calculated using ACI 318 9.5.2.3. To be

conservative, the gross section properties, Ic_gross, and concrete modulus, Ec, are used.

Recall Ic_gross = 4,860 in4 and Mcr_gross = 280,592 in –lbs

[ (

) ]

Where: and

Recall ultimate loading Pu = 13,779 lbs

Note: When the transformed section properties are used in place of gross properties deflections at ultimate

loading equal -0.548".

Linear Elastic Deflection at Mid-Span using gross section properties =

( )

Linear Elastic Deflection at Mid-Span using transposed section properties =

( )

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Analysis of Reinforced Concrete (RC) Beams Using …pages.mscsoftware.com/rs/mscsoftware/images/Paper_Analysis of... · Analysis of RC Beams using Nonlinear Finite Element Techniques