Nonlinear Analysis of RC Beam for Different Shear ... 11 I 01/1110301-2727 IJCEE-IJENS.pdfNonlinear...

International Journal of Civil & Environmental Engineering IJCEE-IJENS Vol: 11 No: 01 63

1110301-2727 IJCEE-IJENS © February 2011 IJENS I J E N S

Nonlinear Analysis of RC Beam for Different

Shear Reinforcement Patterns by Finite

Element Analysis I. Saifullah

1*, M.A. Hossain

2, S.M.K.Uddin

3, M.R.A. Khan

4 and M.A. Amin

5

1,2,3 Department of Civil Engineering, Khulna University of Engineering & Technolog y (KUET), Khulna-9203,

Bangladesh, email: [email protected]* 4,5

Undergraduate student, Department of Civil Engineering, Khulna University of Engineering & Technology

(KUET), Khulna-9203, Bangladesh.

Abstract-- Several methods have been utilized to study the

response of concrete structural components. Experimental

based testing has been widely used as a means to analyze individual elements and the effects of concrete strength under

loading. The use of finite element analysis to study these

components has also been used. This paper focuses on the

behavior of reinforced concrete beam for different pattern of

shear reinforcement to evaluate the effective shear reinforcement pattern and also compare the variation in

behavior of reinforced concrete beam for with and without

shear reinforcement with a simulation. To carry out the

analysis, six 3D beams without and with different patterns of shear reinforcement is built using comprehensive computer

software ANSYS 10 © 2005 SAS IP, Inc package. The static

non linear analysis is done to find out ultimate capacity,

formation of first crack and its distance from support,

initiation of diagonal crack and its distance from support. Load deflection response was also closely observed and

compared with the result from theoretical calculation. From

close observation of analyses results it was found that all types

of web reinforcements were almost same effective for static

loading condition.

Index Term-- ANSYS, shear reinforcement, finite element

analysis, diagonal crack.

I. INTRODUCTION

Concrete structural components exist in buildings and bridges

in different forms. Understanding the response of these

components during loading is crucial to the development of an

overall efficient and safe structure. Different methods have

been utilized to study the response of structural components.

Experimental based testing has been widely used as a means

to analyze individual elements and the effects of concrete

strength under loading. While this is a method that produces

real life response, it is extremely time consuming, and the use

of materials can be quite costly. The use of finite element

analysis to study these components has also been used.

Unfortunately, early attempts to accomplish this were also

very time consuming and in feasible using existing software

and hardware.

When a simple beam is loaded, bending moments and shear

forces develop along the beam. To carry the loads safely, the

beam must be designed for both type of forces. Flexural

design is considered first to establish the dimensions of the

beam section and the main reinforcement needed. The beam

is then designed for shear. If shear reinforcement is not

provided, shear failure may occur. Shear failure is

characterized by small deflections and lack of ductility,

giving little or no warning before failure [1]. On the other

hand, flexural failure is characterized by a gradual increase

in deflection and cracking, thus giving warning before total

failure. This is due to ACI Code limitation on flexure

reinforcement. The Design for shear must ensure that shear

failure does not occur before flexural failure [1]. The use of

FEA has been the preferred method to study the behavior of

concrete (for economic reasons). With the advent of

sophisticated numerical tools for analysis like the finite

element method (FEM), it has become possible to model the

complex behavior of reinforced concrete beams [2].

In recent years, however, the use of finite element analysis has

increased due to progressing knowledge and capabilities of

computer software and hardware. It has now become the

choice method to analyze concrete structural components.

The use of computer software to, model these elements are

much faster, and extremely cost-effective. To fully understand

the capabilities of finite element computer software, one must

look back to experimental data and simple analysis. Data

obtained from a finite element analysis package is not useful

unless the necessary steps are taken to understand what is

happening within the model that is created using the software.

Also, executing the necessary checks along the way is key to

make sure that what is being output by the computer software

is valid. By understanding the use of finite element packages,

more efficient and better

analyses can be made to fully understand the response of

individual structural components and their contribution to a

structure as a whole. This paper focuses on the behavior of

reinforced concrete beam for different pattern of shear

reinforcement to evaluate the effective shear reinforcement

pattern and also compare the variation in behavior of

reinforced concrete beam for with and without shear

reinforcement with a simulation.

II. SCOPE

This study is focuses on the numerical simulation technique

of 3D approach of beams of without and with shear

reinforcement of different patterns and also a simulation

and compared with another group experimental and

analytical data. This 3D approach is extensible with making

variation on loading and support condition and is a basis for

the evaluation of the topics of interest for future study

mailto:[email protected]



includes providing the principles and guidelines to aid in the

optimization in a easier manner. The paper may also

provide low laborious procedure for modeling of versatile

RCC like structure.

III. EXPERIMENTAL STUDY

A. CRACKS IN CONCRETE MODEL

Concrete crack plots were created at different load levels to

examine the different types of cracking that occurred within

the concrete as shown in Fig. 1. The different types of

concrete failure that can occur are flexural cracks,

compression failure (crushing), and diagonal tension cracks.

Flexural cracks (Fig. 1a) form vertically up the beam.

Compression failures (Fig. 1b) are shown as circles.

Diagonal tension cracks (Fig. 1c) form diagonally up the

beam towards the loading that is applied. Crack develops in

concrete element when the concrete element stress exceeds

modulus of rupture of concrete (tensile strength of concrete).

Crash develops in concrete element when the concrete element

stress exceeds compressive crashing strength of concrete. This

study indicates that the use of a finite element program to

model experimental data is viable and the results that are

obtained can indeed model reinforced concrete beam behavior

reasonably well.

Fig. 1. Typical Cracking Signs in Finite Element Models: a) Flexural Cracks,

b) Compressive Cracks, c) Diagonal Tensile Cracks (Kachlakev, et al. 2001)

B. FAILURE CRITERIA FOR CONCRETE

The model is capable of predicting failure for concrete

materials. Both cracking and crushing failure modes are

accounted for. The two input strength parameters i.e., ultimate

uniaxial tensile and compressive strengths are needed to define

a failure surface for the concrete. Consequently, a criterion for

failure of the concrete due to a multiaxial stress state can be

calculated (William and Warnke 1975). A three-dimensional

failure surface for concrete is shown in Fig. 2.

Fig. 2. 3-D failure surface for concrete (William and Warnke 1975)

C. Finite Element Modeling of Steel Reinforcement

Tavarez (2001) discusses three techniques that exist to

model steel reinforcement in finite element models for

reinforced concrete is shown in fig. 3: the discrete model,

the embedded model, and the smeared model.

Fig. 3. Models for Reinforcement in Reinforced Concrete (Tavarez 2001):

(a) discrete; (b) embedded; and (c) smeared

D. ANSYS FINITE ELEMENT MODEL

T ABLE I

ELEMENT T YPES FOR WORKING MODEL

Material Type ANSYS Element

Concrete Solid65

Steel Plates and

Supports Solid45

Steel Reinforcement Link8



E. REAL CONSTANTS

The real constants for this model are shown in Table II. Note

that individual elements contain different real constants. No

real constant set exists for the Solid65 element.

T ABLE II

REAL CONSTANT FOR MODEL

F. MATERIAL PROPERTIES

Parameters needed to define the material models can be

found in Table III.

T ABLE III MATERIAL MODELS FOR THE CALIBRATION MODEL

Mate

rial

Mo

del

Nu

mb

er

Ele

men

t

Ty

pe

Material Properties

1

So

lid

65

Linear Isotropic

EX 3604974.865

PRXY 0.25

Multilinear Isotropic

Strain

(in/in)

Stress

(psi)

Point 1 0.00049931 1800

Point 2 0.00065 2158.06

Point 3 0.00080 2552.24

Point 4 0.001 2996.43

Point 5 0.0012 3347.11

Point 6 0.0014 3609.99

Point 7 0.0016 3794.94

Point 8 0.0018 3913.71

Point 9 0.002 3978.22

Point 10 0.0022 3999.57

Point 11 0.002219 4000

Point 12 0.003 4000

Concrete

ShrCf-Op 0.3

ShrCf-Cl 1

UnTensSt 474.34

UnCompSt -1

BiCompSt 0

HydroPs 0

BiCompSt 0

UnTensSt 0

TenCrFac 0

2

So

lid

45

Linear Isotropic

EX

29,000,000

psi

PRXY 0.3

3

Lin

k8

Linear Isotropic

EX

29,000,000

psi

PRXY 0.3

Bilinear Isotropic

Yield Stress 60,000 psi

Tangent

Modulus 2,900 psi

The Solid65 element requires linear isotropic and multi-

linear isotropic material properties to properly model

concrete. The multi-linear isotropic material uses the von

Real

Co

nst

ant

set

Ele

men

t

Ty

pe

Constants

1

So

lid

65

Real

Co

nst

an

ts

for

Reb

ar

1

Real

Co

nst

an

ts

for

Reb

ar

2

Real

Co

nst

an

ts

for

Reb

ar

3

Mate

rial

Nu

mb

er

0 0 0

Vo

lum

e

Rati

o

0 0 0

Ori

en

tati

on

An

gle

0 0 0

Ori

en

tati

on

An

gle

0 0 0

2

So

lid

45

Cro

ss-

secti

on

al

Are

a,

(in

2)

1.0

Init

ial

Str

ain

(in

./in

.)

0

3

Lin

k8

Cro

ss-

secti

on

al

Are

a (

in2 )

0.11

Init

ial

Str

ain

(in

./in

.)

0



Mises failure criterion along with the Willam and Warnke

(1975) model to define the failure of the concrete. EX is the

modulus of elasticity of the concrete (Ec), and PRXY is the

Poisson‟s ratio (µ). The modulus of elasticity was based on

the equation,

Ec = 57000√f’c (1)

with a value of f’c equal to 4,000 psi. Poisson‟s ratio was

assumed to be 0.25. The compressive uniaxial stress -strain

relationship for the concrete model was obtained using the

following equations to compute the multi-linear isotropic

stress-strain curve for the concrete (MacGregor 1992)

(2)

(3)

(4)

Where;

f = stress at any strain ε, psi

ε = strain at stress f

= strain at the ultimate compressive strength, f’c

The multi-linear isotropic stress-strain implemented requires

the first point if the curve to be defined by the user. It might

satisfy Hook‟s Law;

(5)

The multi-linear curve is used to help with convergence of

the nonlinear solution algorithm.

Fig. 4. Uniaxial Stress-Strain Curve

Fig. 4 shows the stress-strain relationship used for this study

and is based on work done by Kachlakev,et al. (2001).

MacGregor Nonlinear model curve Point 1, defined as ' 0.45

fc’ is calculated in the linear range (Equation 4). Other

points are calculated from Equation 2 with ε0 obtained from

Equation 3.Last point is defined at f‟c and ε0=0.003 in./in.

indicating traditional crushing strain for unconfined

concrete.

Fig. 5. Idealized Stress-Strain Curve of Reinforcing Steel

G. MODELING

Fig. 6. Typical Beam Dimensions

Fig. 7. Quarter Beam for Model



Fig. 8. Different types of shear reinforcements

Fig. 9. Reinforcement Detailing for Beam Model

Fig. 10. Mesh of the Concrete, Steel Plate and Steel Support

Link8 elements were used to create the flexural and shear

reinforcement. Only half of the stirrup is modeled because of the

symmetry of the beam. Fig. 10 illustrates that the rebar shares the

same nodes at the points that it intersects the shear stirrups. The

element type number, material number, and real constant set

number for the calibration model were set for each mesh as shown

in Table IV.

Fig. 11. Reinforcement Configuration and Meshing for Type 1

T ABLE IV MESH ATTRIBUTES FOR THE MODEL

Mo

del

Part

s

Ele

men

t

Ty

pe

Mate

rial

Nu

mb

er

Real

Co

nst

ant

Set

Concrete Beam 1 1 1

Steel Plate 2 2 N/A

Steel Support 2 2 N/A

Longitudinal

Reinforcement 3 3 2

Shear

Reinforcement 3 3 3

Fig. 12. Reinforcement Configuration and Meshing for without shear

reinforcement






Fig. 16. Different Patterns of Shear Reinforcement in ANSYS

III. ANALYTICAL STUDY

For this purpose it is eventual to compare the develop model

with an existing one. And here this simulation was made by

using the data given by „Anthony J. Wolanoski B.S.‟ in his

thesis paper [2]. Where, he used following specification-

1. Beam size – The width and height of beam

were 10 in. and 18 in respectively

2. Clear span length – 15 ft

3. Area of steel – 0.93 in.2

4. Yield Stress of Steel, fy = 60,000 psi

5. 28-days Compressive Strength of Concrete,

f’c = 4800 psi

The detail of Wolanoski‟s beam and also the beam for

simulation is given below:

Fig. 17. Reinforcement Detailing of Wolanoski‟s Beam

Fig. 18. Load-Deflection curve comparison of ANSYS and Backouse

(1997) [2]

The graph of present analysis of Wolanoski‟s thesis is given

bellow:

Fig. 19. Load-Deflection Curve after simulation

The comparison of Wolanoski‟s analysis and present

analysis are given in table.



T ABLE V COMPARISON BETWEEN ANTHONY J.WOLANOSKI ANALYSIS AND

PRESENT STUDY BY ANSYS

A.

CRA

CK

DEV

ELO

PED

IN

THE

CON

CRE

TE

BEA

MS

At

first

the

crack

is

forme

d in

the concrete beams because of flexural stress. For the

increasing of loads the diagonal tension crack is initiated

after the formation of 1st

crack. The crack increase with the

increase of loads and the steel stress reach to its yielding

stress. The failure of concrete beams also observes by the

formation of crack which is shown in fig.s 20, 21, 22, 23

and 24.

(a) 1

st Crack of the Concrete Model (load 9686 lb)

(b) Initiation of Diagonal Tension crack (load 20423 lb)

(c) Yielding of Reinforcement (load 57533 lb)

(d) Failure of the Concrete beam (load 61615 lb)

Fig. 20 (a),(b),(c)&(d). Represents Cracks Formation in Beam of present study for Without Shear Reinforcement in different stages during the

application of load

(a

) 1st

Crack of the Concrete Model (load 9658 lb)


Mo

del

Ex

trem

e

Ten

sio

n F

iber

Str

ess

(p

si)

Rein

forc

em

en

t

Ste

el

Str

ess

(p

si)

Cen

terl

ine

Defl

ecti

on

(in

)

Lo

ad

at

Fir

st

cra

ck

ing

(lb

)

Fro

m t

hesi

s p

ap

er

of

An

tho

ny

J. W

ola

no

ski

B.S

. [2

] Man

ual

calc

ula

tio

n

53

0

30

24

0.0

52

9

51

18

AN

SY

S

53

6

28

40

0.0

53

4

52

16

Pre

sen

t

stu

dy

Sim

ula

ted

AN

SY

S

52

5

28

43

.8

0.0

53

4

52

12





Fig. 21. (a),(b),(c)&(d) represents Cracks Formation in Beam of present study for Shear Reinforcement Type 1 in different stages during the

application of load

(a) 1





Fig. 22. (a),(b),(c)&(d) represents Cracks Formation in Beam of present

study for Shear Reinforcement Type 2 in different stages during the application of load

(a) 1







Fig. 23. (a),(b)(c)&(d) represents Cracks Formation in Beam of present

study for Shear Reinforcement Type 3 in different stages during the

application of load

(a) 1



(c)

Yielding of Reinforcement (load 57451 lb)

(d) Failure of the Concrete beam (load 61964 lbs) Fig. 24. (a),(b),(c)&(d) represents Cracks Formation in Beam of present

study for Shear Reinforcement Type 4 in different stages during the

application of load

B. LOAD-DEFLECTION CURVE

Fig. 25. Combined Load-Deflection Curve for Different patterns of shear

Reinforcement

Load-Deflection Curve is linear with a sharp slope up to

9,000-10,000 lb. Within this load first cracking occur. The

graph changes its nature after first cracking i.e. its slope is

changed continuously. This is due to change in crack depth

with the load increment. The location of initiation of the

diagonal tension cracking of concrete in curves is in

between the 1st

cracking loads and steel yielding loads. This

crack is observed from concrete cracks and crushing plots

which is within 17400 lb to 23050 lb. The cracks & curves

were observed and the data from cracks & curves were

listed as tabular form in results.



IV. RESULTS AND DISCUSSIONS T ABLE VI

CRACK FORMATION AND DISTANCE OF CRACK FROM SUPPORT AND

DEFLECTION AT FAILURE LOADS ON THE BASIS OF ANALYSIS

Type

1st

crack

Init

iati

on

of

dia

go

nal

ten

sio

n c

rack

Lo

ad

at

Fail

ure

(lb

)

Defl

ecti

on

at

Fail

ure

(in

.)

Lo

ad

(l

b)

Dis

tan

ce f

rom

sup

po

rt (

in.)

Lo

ad

s (

lb)

Dis

tan

ce fr

om

sup

po

rt (

in.)

Wit

ho

ut

shear

rein

forc

em

en

t

96

86

78

20

42

3

51

61

61

6

2.3

1

Ty

pe 1

96

58

75

23

04

8

33

62

02

0

4.0

34

0

Ty

pe 2

96

46

82

.5

19

94

9

42

61

85

2

3.3

88

5

Ty

pe 3

96

57

75

.75

17

45

3

37

.5

61

88

0

3.6

87

9

Ty

pe 4

96

58

75

20

31

3

36

61

96

4

3.4

60

8

T ABLE VII COMPARISON BETWEEN THEORETICAL CALCULATION AND ANSYS

T ABLE VIII

FORMATION OF 1ST CRACK AND RESPECTIVE DEFLECTION & STEEL

STRESS IN FINITE ELEMENT ANALYSIS

Mo

del

Rein

forc

ing

(main

bar)

Ste

el

Str

ess

(psi

)

Cen

terl

ine

Defl

ecti

on

(in

.)

Lo

ad

at

Fir

st

Cra

ck

(fl

ex

ure

cra

ck

)

(lb

)

*M

an

ual

Calc

ula

tio

n

AN

SY

S

* M

an

ual

Calc

ula

tio

n

AN

SY

S

*H

Man

ual

Calc

ula

tio

n

AN

SY

S

Wit

ho

ut

Sh

ear

Rein

forc

em

en

t

27

72

.34

27

88

.0

0.0

53

94

0.0

54

78

6

94

45

.5

96

86

Ty

pe 1

27

88

.4

0.0

54

7

96

58

Ty

pe 2

27

88

.5

0.0

54

75

1

96

46

Ty

pe 3

27

88

.6

0.0

54

75

0

96

57

Ty

pe 4

27

88

.5

0.0

54

74

8

96

58

Model

Lo

ad

at

Fir

st

Cra

ck

(lb

)

Cen

terl

ine

Defl

ecti

on

(in

.)

Rein

forc

ing

Ste

el

Str

ess

(p

si)

Without Shear

Reinforcement 9686 0.056181 2859.0

Type 1 9658 0.064117 3870.0

Type 2 9646 0.057770 3077.0

Type 3 9657 0.058253 3138.3

Type 4 9658 0.062322 3651.1



T ABLE IX FLEXURAL STEEL STRESS ON THE BASIS OF ANALYSIS

Mo

del

Ste

el

Str

ess

at

yie

ldin

g o

f st

eel

(psi

)

Defl

ecti

on

at

yie

ldin

g o

f st

eel

(in

.)

Lo

ad

s o

n b

eam

at

yie

ldin

g o

f

steel

(lb

)

Ste

el

Str

ess

at

fail

ure

(p

si)

Lo

ad

s o

n b

eam

at

fail

ure

(lb

)

*T

heo

reti

cal

calc

ula

tio

n

60

00

0

-

57

70

3

- -

Wit

ho

ut

shear

rein

forc

em

en

t

60

00

9

0.8

58

35

0

57

53

3

60

10

8

62

02

0

Ty

pe 1

60

01

0

0.9

17

14

57

89

8

60

14

3

62

02

0

Ty

pe 2

60

00

4

0.8

29

57

8

57

45

0

60

16

3

61

85

2

Ty

pe 3

60

01

0

0.8

84

73

6

57

57

6

60

11

4

61

88

0

Ty

pe 4

60

00

4

0.8

55

752

57

45

1

60

12

8

61

96

4

A. COMPARISON

From another thesis group [7] performing on “Experimental

and Analytical Investigation of Flexural Behavior of

Reinforced Concrete Beam” got the results as follows:

T ABLE X

1ST CRACK FORMATION DISTANCE FROM SUPPORT (WITHOUT SHEAR

REINFORCEMENT) [7]

From present analysis: T ABLE XI

1ST CRACK FORMATION DISTANCE FROM SUPPORT FOR THIS ANALYSIS

B. COMMENTS ON RESULTS

Initiation of diagonal tension crack occurs in Type

1 at larger loads in compare to others.

For the beam without shear reinforcement diagonal

tension crack initiates at larger distance from

support with compared to others.

The ultimate load carrying capacity is larger for

Type 1 with respect to other types and also

showing large deflection for its better ductile

property.

Theoretical calculation and ANSYS analysis give

almost same results for steel stressing at 1st

crack.

At steel yielding the steel stress is almost same to

the theoretical value. These data was collected

from ANSYS output after analysis.

Steel stress at failure is maximizing for Type 2

shear reinforcement. These data was collected from

ANSYS output after analysis.

Compare with another group, the behavior of 1st

crack formation, is found satisfactory level.

From combined load deflection curve, the 1st

cracking point and the steel yielding point for with

and without different patterns of shear

reinforcement are almost same.

V. CONCLUSION

The project emanated with an aim to find out the ultimate

load carrying capacity of beams of without and with

different patterns of shear reinforcements and also find out

the different behaviors of beams for different stages of

loading. The project is expected to generate reasonable

solutions of focused problem defined under some parametric

condition. Initially some parameters are chosen for these

beams by analysis with finite element method. The ultimate

load carrying capacity is then determined by without

considering and considering different patterns of shear

reinforcement with a constant flexural reinforcement. After

completing the analysis curves are drawn for without and

with different patterns of shear reinforcement, to find out

various parameters (1st

crack formation in beams, initiation

1st

Crack Formation Distance from

Support

Without shear

reinforcement 0.413L

Type 1 0.396L

Type 2 0.437L

Type 3 0.401L

Type 4 0.396L

1st

Crack Formation Distance from Support

Lab Test 0.421L

ANSYS 0.414L



of diagonal cracks, failure load etc.) for finding effective

shear reinforcement pattern for beam to this loading

condition. Also a simulation and comparison to another

group is done to the satisfactory use of finite element

modeling in structural components. The following

conclusion can be stated based on the evaluation of the

analyses:

ANSYS 3D concrete element is very good

concerning the flexural and shear crack

development but poor concerning the crushing

state. However this deficiency could be eas ily

removed by employing a certain multi-linear

plasticity options available in ANSYS.

From close observation of analyses results it can be

concluded that all types of web reinforcements are

almost same effective for static loading condition.

VI. REFERENCES [1] Nilson, Arthur H.; Darwin, David; Dolan Charles W., 2006

“Design Of Concrete Structures”, McGraw-Hill, 13th

Edition. [2] Wolanski, Anthony J., B.S., 2004, “Flexure Behavior of

Reinforced and Prestressed Concrete Beams Using Finite

Element Analysis”, Faculty of Graduate School, Marquette University, Milwaukee, Wisconsin, May.

[3] SAS (2005) ANSYS 10 Finite Element Analysis System, SAS IP, Inc.

[4] Hossain, M. Nadim, 1998, “ Structural Concrete; Theory & Design”, Addison-Wesley Publishing Company.

[5] Nakasone, Y., Yoshimoto, S., Stolarski, T . A., 2006, “ENGINEERING ANALYSIS WITH ANSYS SOFTWARE”,

ELSEVIER, 1st Published.

[6] Kachlakev, D.; Miller, T.; Yim, S., May, 2001, “Finite Element Modeling of Reinforced Concrete Structures Strengthened With FRP Laminates”, California Polytechnic State University, San

Lius Obispo, CA and Oregon State University, Corvallis, OR for Oregon Department of Transportation, May.

[7] Nasir-Uz-Zaman, M, Sohel Rana, M, 2009 “Experimental And

Analytical Investigation of Flexural Behavior of Reinforced Concrete Beam”, Undergraduate Thesis Report, Department of Civil Engineering, Khulna University of Engineering and Technology, Khulna, April.

[8] Willam,K. J. and Warnke, E. P. (1975), “Constitutive models for the triaxial behavior of concrete”, Proceedings of the International Assoc. for Bridge and Structural Engineering , vol 19, pp. 1- 30.

[9] Murdock, L. J., Brook, K. M. and Dewar, J. D., “Concrete: Materials and Practice”, 6th Edition, Edward Arnold, London, 1991

[10] American Concrete Institute, “Material and General Properties

of Concrete”, ACI Manual of Concrete Practice, part 1, 1996 [11] Tavarez, F.A., (2001), “Simulation of Behavior of Composite

Grid Reinforced Concrete Beams Using Explicit Finite Element

Methods,” Master‟s Thesis, University of Wisconsin-Madison, Madison, Wisconsin.

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Nonlinear Analysis of RC Beam for Different Shear ... 11 I 01/1110301-2727 IJCEE-IJENS.pdfNonlinear...

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