Kachlakev_RC Control Beam MSC-Marc_wAppendix

Applied Analysis & Technology © 2013

16 September 2013 : D2

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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 [email protected]

Web Site www.AppliedAnalysisAndTech.com

mailto:[email protected]

http://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.


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“Control Beam” Geometry with Rebar Definition from Reference 1 No Scale


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Figure 2.13: Typical steel reinforcement locations (not to scale) (McCurry and

Kachlakev 2000)

P/2 P/2



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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



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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.



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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



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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

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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


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Appendix A

Summary ACI 318 Hand Calculations & Analysis Notes


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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 2.806 106 psi Concrete Modulus

The tensile capacity stress of the concrete, fr, is defined using ACI 318 9.5.2.3. This value fris 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_grossbc 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_grossfr Ic_gross

hc

2

675645 in lbs

Pcr_grossMcr_gross

729384 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. 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 2.806 106 psi Concrete Es 29 10

6 psi Steel

nEs

Ec10.336 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_refhc

2

15.125 in

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

ybarAconc yc_ref As_eff_Low drebar_L As_eff_Upr drebar_U

Aconc As_eff_Low As_eff_Upr 14.323 inches

measured from theLower 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 tensionstress 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_trfr Itr

hc ybar 741575 in lbs

Pcr_trMcr_tr

7210300 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_elasticn Mcr_tr drebar_U ybar

Itr1360 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 steelreinforcement. 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 24

bc

2

As_eff_Low hc drebar_L

2bc

2

9.243 in

The moment of inertia of this transformed area w.r.t. the neutral axis for "cracked" section iscalculated using the following for Lower reinforcement only in the RC section; i.e Neglecting the 2#5 Upper Compression Rebar:

Icrackbc ccrack 3

3As_eff_Low hc drebar_L ccrack 2 12646 inches4

Neglecting UpperCompression Rebar



Stage 3: Ultimate Strength Calculations for Cracked Section

For ultimate load carrying strength capability tension stress in the concrete is assumed nonexistentand 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 materialat 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 .05fc

1000

0.929 fs_ty 60000 psi, rebar steel yield stress

aunrebar_6 As_6 nrebar_7 As_7 fs_ty

.85 fc bc 6.506 inches

cuau

β17.005 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, PuϕMu

7254710 lbs

The Total maximum Beam loading = PTot_Ult 2 Pu 109419 lbs Total



Calculate Deflections from Elastic Moment of Inertia Calculationsfor 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 in4 Recall Mcr_gross 675645 in lbs

IeffMcr_gross

ϕMu

3

Ic_gross 1Mcr_gross

ϕMu

3

Icrack 12722 in4

ab 72 in Lb 3 ab 216 in

Recall ultimate loading on Each Load PadPu 54710 lbs

yuPu ab 4 ab

2 3 Lb2

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_grossPcr_gross ab 4 ab

2 3 Lb2

24Ec Ic_gross 0.043 inches

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

ycr_trPcr_tr ab 4 ab

2 3 Lb2

24Ec Itr 0.041 inches


Kachlakev_RC Control Beam MSC-Marc_wAppendix

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