Kachlakev_RC Control Beam MSC-Marc_wAppendix

Applied Analysis & Technology © 2013
16 September 2013 : D2
Rev “x”
Slide 1 of 19
Analysis of Reinforced Concrete (RC) “Control Beam” Using Nonlinear
Finite Element Techniques MSC/Marc
Prepared By:
David R. Dearth, P.E.
Applied Analysis & Technology, Inc.
16731 Sea Witch Lane
Huntington Beach, CA 92649
Telephone (714) 846-4235
E-Mail AppliedAT@aol.com
Web Site www.AppliedAnalysisAndTech.com

Applied Analysis & Technology © 2013 Slide 2 of 19
Introduction
 Kachlakev et. al. 2001 (1.) tested a reinforced concrete (RC) “Control Beam” used for
baseline calibration of analysis to compute effects of adding fiber reinforced polymer
(FRP) composites to strengthen full-size reinforced concrete beams. The purpose of the
original work was to predict improvements to adding FRP composite reinforcement
similar to the transverse beams use at the Horsetail Creek Bridge. The beams were
fabricated and tested at Oregon State University (Kachlakev and McCurry 2000).
 The purpose of this summary is to present results of revising the RC “Control Beam”
and computing the load deflection curve using MSC/Marc for comparison to the
experimental test data. Results using Ansys and Abaqus are also compared.
 As additional information, the loading to produce (a.) initial cracking and (b.) ultimate
capacity is computed using ACI 318.
 For comparison purposes the finite element idealization mesh density used in the
Kachlakev paper is reproduced as a closely as possible.
Rev “x”

“Control Beam” Geometry with Rebar Definition
from Reference 1 No Scale
Rev “x”
Figure 2.13: Typical steel reinforcement locations (not to scale) (McCurry and
Kachlakev 2000)
P/2 P/2

Rev “x”
Slide 4 of 19
Quarter Symmetric RC Beam with Boundary Conditions & Loading
X-Y Symmetric
Plane, BC = Tz
Symmetric Loading,
Ptot/4 for Qtr Sym
Idealization
Vertical Reaction,
BC=Ty
Y-Z Symmetric
Plane, BC = Tx

Rev “x”
Slide 5 of 19
Concrete : Isotropic Properties
The concrete is idealized using 3D solid elements. Young’s modulus of elasticity for the concrete is given as:
Concrete Material Properties
Es= 2.806x 106 psi ν =0.2
Critical Cracking Stress (Rupture Stress) fr = 329 psi
Crushing Strain, εc = 0.0017 in/in
Note: Plasticity definition data for MSC/Marc is defined as post-yield, or plastic, portion of the stress strain curve; e.g. yield
stress  zero net plasticity. Typical engineering data for stress-strain curves are defined as total nominal strain.

Rev “x”
Slide 6 of 19
Concrete : Isotropic Properties
The concrete is idealized using 3D solid elements. Young’s modulus of elasticity for the concrete is given as:
Concrete Material Properties
Elastic : Es= 2.806x 106 psi ν =0.2
Cracking : Critical Cracking Stress (Rupture Stress) fr = 369 psi
Crushing Strain, εc = 0.0017 in/in
Plasticity : Elastic-Plastic, Isotropic Hardening, Buyukozturk Concrete
Concrete Isotropic Material Input Dialog

Rev “x”
Slide 7 of 19
Quarter Symmetric RC Beam Rebar Idealization
#7 Rebar
Area = 0.60 in2
Y-Z Symmetric
Plane, BC = Tx
#7 Rebar at Plane of
Symmetry
Area/2 = 0.30 in2
#6 Rebar
Area = 0.44 in2
Symmetry
Area/2 = 0.155 in2
Symmetry
Area/2 = 0.155 in2
2- #5 Rebar at Plane of
Symmetry (Merged)
2* (Area/2) = 0.31 in2
Rebar Material Properties
Es= 29x 106 psi ν =0.3
Yield Stress Fty = 60,000 psi
Bi-Linear-Plastic Modulus = Perfectly Plastic

Comparison ACI 318 Hand Calculations to Kachlakev Control Beam 16 September 2013 : D2
Rev “x”

Rev “x”Kachlakev Control Beam Test Deflections vs Marc FEA

Rev “x”Concrete Crack Progression FEA to Kachlakev Control Beam
22K Last Load Step Prior to Cracks
Crack Progression vs. Total Beam Loading

Rev “x”Comparison FEA to Kachlakev Control Beam Test: Ansys, Abaqus & Marc

References
1) Kachlakev, D., Miller, T. , Yim, S., Chansawat, K. , Potisuk, T. “Finite Element
Modeling of Reinforced Concrete Structures Strengthened with FRP Laminates”;
California Polytechnic State University Oregon State University, for Oregon
Department of Transportation, May 2001
2) Sinaei, H., Shariati, M., Abna, A.H., Aghaei, M. and Shariati, A., “Evaluation of
reinforced concrete beam behavior using finite element analysis by ABAQUS”,
Islamic Azad University, Sirjan, Iran. 10 January, 2012
Rev “x”

Appendix A
Summary ACI 318 Hand Calculations & Analysis Notes
Rev “x”

Analysis of RC Beams using Nonlinear Finite Element Techniques 9 September 2013
RC Beam Tested by Kachlakev 2001
Stage 1: Linear Elastic Moment of Inertia Calculations for Composite Section
The concrete compressive strength at 28 days is given as:
fc 2423 psi
Using ACI 318 8.5.1 the modulus of elasticity of the concrete is calculated as follows:
Ec 57000 fc 2.806 10
6
 psi 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.
fr 7.5 fc 369 psi Concrete Cracking Stress for normal weight concrete
Beam Section Gross Dimensions
bc 12 inches, Base
hc 30.25 inches, Height
Beam Section Gross Moment of Inertia
Ic_gross
bc hc
3
 
12
27680.6 inches4
Beam Cross Section A-A Through Constant Moment Region
Per ACI 9.5.2.3 the crack initiation moment Mcr_gross=
Mcr_gross
fr Ic_gross 
hc
2






675645 in lbs
Pcr_gross
Mcr_gross
72
9384 lbs Loading to crack initiation neglecting DW of concrete & rebar
The Total maximum Beam loading =PTot_Crack 2 Pcr_gross 18768 lbs Total
Applied Analysis & Technology, Inc. Page 14 of 19

To calculate stress in the rebar then Transformed section properties are needed.
Compute modular ratio, n, to be used for transformed inertia
Ec 2.806 10
6
 psi Concrete Es 29 10
6
 psi Steel
n
Es
Ec
10.336 Where "n" is modular ratio of Esteel/Econcete
Transform area of Steel to equivalent or effective area of concrete, As_eff_Lwr & As_eff_Upr
Lower Steel #6-2 #7-3 As_6 0.44 in2 for each #6 rebar nrebar_6 2 number of #6 rebar
As_7 0.60 in2 for each #7 rebar nrebar_7 3 number of #7 rebar
As_eff_Low n nrebar_6  As_6 nrebar_7  As_7  27.7 in2 drebar_L 2.5 in from Bottom
Upper Steel #5-2 As_5 0.31 in2 for each #5 rebar nrebar_5 2 number of #5 rebar
As_eff_Upr n nrebar_5 As_5  6.408 in2 drebar_U 20 in from Bottom
Concrete Area Aconc hc bc 363 in2 yc_ref
hc
2






15.125 in
The location of the centroid of area for the effective composite section, concrete & steel rebar
ybar
Aconc yc_ref As_eff_Low drebar_L As_eff_Upr drebar_U 
Aconc As_eff_Low As_eff_Upr 
14.323 inches
measured from the
Lower surfaceThe transformed composite area moment of inertia is computed
using parallel axis theorem
Itr Ic_gross Aconc yc_ref ybar 2




As_eff_Low drebar_L ybar 2





As_eff_Upr drebar_U ybar 2





31993 inches4

Compute the equivalent loading, P lbs, to just exceed the maximum allowable concrete tension
stress to initiate first cracking.
Recall: σcon_elastic fr 369 psi
Using bending equation σcr_tr = (Mcr_tr *yc)/Itr, where Mcr_tr = Pcr_tr*72 in-lbs
Mcr_tr
fr Itr 
hc ybar 
741575 in lbs
Pcr_tr
Mcr_tr
72
10300 lbs Loading to crack initiation using transformed section properties and
Neglecting DW of concrete and rebar
The Total maximum Beam loading = PTot_cr_tr 2 Pcr_tr 20599 lbs Total
The corresponding stress in the steel rebar at this loading is σrebar = n(Mcr_tr*yrbar)/Itr
σs_elastic
n Mcr_tr  drebar_U ybar  
Itr
1360 psi

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 Lower 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".
ccrack
As_eff_Low As_eff_Low 2
4
bc
2






As_eff_Low  hc drebar_L 






2
bc
2







9.243 in
The moment of inertia of this transformed area w.r.t. the neutral axis for "cracked" section is
calculated using the following for Lower reinforcement only in the RC section; i.e Neglecting the 2
#5 Upper Compression Rebar:
Icrack
bc ccrack 3




3
As_eff_Low hc drebar_L ccrack 2
 12646 inches4
Neglecting Upper
Compression Rebar

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 neglected, set equal to 1.0 to compute Ultimate moment
Uniform distribution rectangular stress block, stress intensity factor β1.
ϕu 1
β1 1.05 .05
fc
1000






 0.929 fs_ty 60000 psi, rebar steel yield stress
au
nrebar_6 As_6 nrebar_7 As_7   fs_ty 
.85 fc bc 
6.506 inches
cu
au
β1
7.005 inches to N-A
ϕMu ϕu nrebar_6 As_6 nrebar_7 As_7  fs_ty hc drebar_L 
au
2







 3939095 in lbs
The maximum loading at Each Load Pad, Pu
ϕMu
72
54710 lbs
The Total maximum Beam loading =PTot_Ult 2 Pu 109419 lbs Total

Calculate Deflections from Elastic Moment of Inertia Calculations
for Cracked Section
At the estimated 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 27681 in
4
Recall Mcr_gross 675645 in lbs
Ieff
Mcr_gross
ϕMu






3
Ic_gross 1
Mcr_gross
ϕMu






3







Icrack 12722 in
4
ab 72 in Lb 3 ab 216 in
Recall ultimate loading on Each Load PadPu 54710 lbs
yu
Pu ab  4 ab
2
 3 Lb
2
 
24Ec Ieff 
0.548 inches
Note: When the transformed section properties (Itr & Mcr_tr) are used in place of gross (Ic_gross
& Mcr_gross) properties deflections at ultimate loading equal -0.546".
Linear Elastic Deflection at Mid-Span using gross section properties =
ycr_gross
Pcr_gross ab  4 ab
2
 3 Lb
2
 
24Ec Ic_gross 
0.043 inches
Linear Elastic Deflection at Mid-Span using transposed section properties =
ycr_tr
Pcr_tr ab  4 ab
2
 3 Lb
2
 
24Ec Itr 
0.041 inches

Kachlakev_RC Control Beam MSC-Marc_wAppendix

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