Three-Dimensional Nonlinear Finite Element Analysis of Reinforced Concrete Horizontally Curved Deep Beams

Three-Dimensional Nonlinear Finite Element Analysis of Reinforced Concrete Horizontally Curved Deep Beams Abstract This research deals with the analysis of reinforced concrete horizont

Three-Dimensional Nonlinear Finite Element Analysis of Reinforced Concrete Horizontally

Curved Deep Beams

Abstract

This research deals with the analysis of reinforced concrete horizontally curved deep beams, loaded transversely to its plane, using a three-dimensional nonlinear finite element model in the pre and post cracking levels and up to the ultimate load The 20-node isoparametric brick element with sixty degrees

of freedom is employed to model the concrete, while the reinforcing bars are modeled as axial

members embedded within the concrete brick element Perfect bond between the concrete and the reinforcing bars is assumed The behavior of concrete in compression is simulated by an elasto-plastic work hardening model followed by a perfect plastic response, which is terminated at the onset of crushing In tension, a fixed smeared crack model has been used with a tension-stiffening model to represent the retained post-cracking tensile stress Also, a shear retention model that modifies the shear modulus after cracking is used.

Numerical study is carried out to investigate some of effects on the behavior of reinforced concrete horizontally curved beams such as the shear length to effective depth ratio (a/d) on the ultimate load resisted by curved beams and the effect of the central subtended angle, boundary conditions, amount of transverse reinforcement, and using additional longitudinal bars (horizontal shear reinforcement) on the behavior of reinforced concrete horizontally curved beams with different shear length to effective depth ratios (a/d).

Keywords :Finite Element; Nonlinear Analysis; Reinforced Concrete; Curved Beam; Deep

Beam

ةصلاخلا

لوانتت هذه ساردلا ة ليلحت تابتعلا ناسرخلا ي ة ةحلسملا ةقيمعلا

ةسوقملا ايقفأ

تحت ريثأت لامممحأ

ةطلسم ةروصب ةيدومع ىلع ىوتسم اهسوقت مادختساب اجذومن

لا

ًايطخ يثلاث داعبلأا مرصانعلل ةددممحملا

يف

لحارم ليمحتلا لبق

دعبو ققشتلا و

ىلإ دح لمحلا ىصقلأا

مت مادختسا رصنعلا

يقوباطلا يذ

نيرممشعلا

ةدقع عم نوتممس ةجرد ةيرح ليثمتل ةناممسرخلا امأ

ديدح حيلممستلا دقف

لثم رصانعب ةيروحم ةرومممطم لخاد

مرصانعلا ةيقوباطلا عم

ضارتفا دوجو طبارت مات نيب ةناسرخلا ديدحو

.حيلستلا ربتعا فرممصت ةناممسرخلا يف

طاغضنلاا

ًافرصت

ًاندل-ًانرم هعبتي

ًافرصت مًاندل

ًامات يهتني دنع مشهت ةناسرخلا امأ

ليثمتل كولممس ةناممسرخلا

تحت تاريثأت تاداهجا دشلا

دقف مت ينبت جذومن ققشتلا رشتنملا تباثلا

( Fixed Smeared Crack Model

)

لمعتساو جذومنلا

بلصت دشلا ( Tension Stiffening Model )

باسحل تاداممهجا

دشلا يقبتملا دعب ثودممح

ققشتلا متو

ينبت جذومنأ ساممبتحا صقلا

( Shear Retention Model مم)

يذمملاو موممقي ضيفختب ةميق

لماعم

صقلا يقبتملا عم رارمتسا ليمحتلا

يف ةلحرم ام عب د .ققشتلا

مت ءارجإ ةسارد ةيليلحت ىلع تابتعلا ةيناسرخلا ةحلسملا

ةسوقملا ايقفأ

ةساردل ريثأت ريغت ةبسن

ءاضف صقلا ىلإ قمعلا لاعفلا ( a/d ىلع ) فرصتلا لمحلاو

ىصقلأا كلتل

و تابتعلا كلذك

مت ةسارد ريثأممت

ةيواز ةقيرط ,سوممقتلا ةيمك ,دانممسلإا

ديدح حيلممستلا ةفاممضلإاب ,ضرعتممسملا

ىلإ لامعتممسا نابممضق

ديدح

حيلست ةيلوط ةيفاضإ فرصتتل حيلستك

صق يقفأ ىلع متابتعلا ةيناسرخلا ةحلممسملا

ةسوقملا ايقفأ

بسنل

( a/d ةفلتخم )

1 Introduction

Reinforced concrete horizontally curved beams are extensively used in many fields, such as in the construction of modern highway intersections, elevated

freeways, the rounded corners of buildings, circular balconies,….etc In some of these cases, large depths are needed for curved beams in order to resist high loads or to fulfill some aesthetic purposes The analytical analysis of such members is very complex due to the fact that those members are subjected to combined action of bending, shear and torsion Furthermore, non-homogeneous nature of the materials involved contributes to the complexity of the problem Therefore, it becomes

necessary to employ numerical analysis procedures, such as the finite element

method, to satisfy the safety and the economy requirements

Haider A A Al-Tameemi

University of Kufa

Ammar Y Ali

University of Babylon

Ali N Attiyah

University of Kufa

A horizontally curved beam, loaded transversely to its plane, is subjected to torsion in addition to bending and shear Furthermore, in deep beam the plane section does not remain plane after bending because of high stresses and warping occurs Therefore, special features of analysis and design for horizontally curved deep beams

is necessary to include the effect of above mentioned factors Several methods of collapse analysis (Khalifa 1972, Jordaan et al 1974, Badawy et al 1977, Hsu et al

1978, and Abul Mansur and Rangan 1981 ) were proposed for analysis of specific cases of reinforced concrete curved beams However, till yet studies concerning reinforced concrete horizontally curved deep beams are rare

At present, with the application of digital computers beside the development

of numerical methods, the mathematical difficulties associated with curved deep beam have been largely overcome

One of the most effective numerical methods utilized for analyzing reinforced concrete members is the finite element method Using this method, many aspects of the phenomenological behavior of reinforced concrete structures can be modeled rationally These aspects include the tension-stiffening, non-linear multiaxial material properties, modeling of cracking and crushing, and many other properties related to the behavior of reinforced concrete members under stresses An important utilization

of the finite element method is the modeling of the degradation of concrete

compressive strength in the presence of transverse tensile straining as happens in members subjected dominantly to torsion or shear stresses Therefore, the present study adopted a three dimensional non-linear finite element model to investigate the behavior and the load carrying capacity of reinforced concrete horizontally curved deep beams

2.Finite Element Model

The 20-node isoparametric brick element shown in Fig.1 is used in the current study

to model the concrete Each node of this element has three degrees of freedom (u, v, and w) in the (x, y, and z) directions, respectively The isoparametric definition of the brick element is(Al- Shaarbaf, 1990):





20

1

, , ,

,

u N

u       ,     





20

1

, , ,

,

v N

v      ,





20

1

, , ,

,

i

i

N

w      (1)

where Ni (ξ, η, >) is the shape function at the i-th node and ui, vi, wi are the

corresponding nodal displacements The shape functions for the 20 node brick

element which are adopted to map the element are given in Table 1.

The Gauss-Legender quadrature numerical integration scheme has been found

to be accurate and a convenient technique to carry out the finite element analysis The integration rule, which has been used in this study , is the 15-point rule

The weights and abscissa of the sampling points are listed in Table 2 The relative distribution of the Gaussion points over the element is given in Fig 2.

Figure(2)Distribution of sampling points

(Al- Shaarbaf, 1990)

Figure (1) 20-node brick element

4 Modeling Of Material Properties

The material model used in the present work is suitable for the three-dimensional

nonlinear analysis of reinforced concrete structures under monotonically increasing

load The behavior of concrete in compression is presented by an elastic-plastic work

hardening model followed by a perfectly plastic response, which is terminated at the

initiation of crushing The growth of subsequent loading surfaces is described by an

isotropic hardening rule A parabolic equivalent uniaxial stress-strain curve shown in

Fig.3 has been used to represent work hardening stage of behavior and the plastic

straining is controlled by an associated flow rule A yield criterion suitable for

analyzing reinforced concrete members has been used This criterion was used

successfully can be expressed as(Al- Shaarbaf, 1990):

  cI1 cI123 J 22  

f (2)

Where c and β are material parameters to be determined by fitting biaxial test

results.Using the uniaxial compression test and the biaxial test under equal

compressive stresses I 1 and J 2 are the first stress and second deviatoric stress

invariants and σ0 is the equivalent

effective stress taken from uniaxial tests

In tension, linear elastic behavior is assumed to occur prior to cracking Crack

initiation is controlled by a maximum tensile stress criterion A smeared crack model

Table(1)Shape functions of the quadratic 20-node brick

element.(Cook,1974, Carlos, 2004)

Location ξ η > Ni (ξ, η, >) Corner

nodes ±1 ±1 ±1 (1+ ξ ξi)(1+ η ηi)(1+ > > i) (ξ ξi+ η ηi + > >i-2 )/8 Mid-side

nodes 0 ±1 ±1 (1- ξ2)(1+ η ηi )(1+ > >i ) /4 Mid-side

nodes ±1 0 ±1 (1- η2 )(1+ ξ ξi)(1+ > >i )/4 Mid-side

nodes ±1 ±1 0 (1- >2 )(1+ ξ ξi)(1+ η ηi )/4

Table (2) Weights and abscissa of sampling points

(Al- Shaarbaf, 1990).

Integration rule Samplingpoint

number

Natural coordinates

Weight

15a-point rule 2,31 1.00.0 0.00.0 0.00.0 1.564440.35556

8-15 0.6714 0.6714 0.6714 0.53778

with fixed orthogonal cracks has been adopted to represent the behavior of cracked

sampling points The retained post-cracking tensile stress and the reduced shear

modulus are calculated according to Fig.4 and Fig.5 respectively Details of the

plasticity based model in compression and the smeared crack model in tension can be

found elsewhere(Al-Tameemi, 2005)

Figure(3) Uniaxial stress-strain curve for concrete(Al- Shaarbaf, 1990).

Figure (4) Post-cracking model for concrete

(Al- Shaarbaf, 1990).

4 Analysis Of Reinforced Concrete Horizontally Curved Beams

In this section, reinforced concrete horizontally curved beams subjected to single

load have been analyzed using the finite element technique and the models discussed

in the pervious sections The computer program 3DNFEA (3-Dimensional Nonlinear

Finite Element Analysis) has been used in the present study This program has been

originally developed by Al-Shaarbaf (Al- Shaarbaf,1990).The analytical results are

compared with the available experimental results on load-deflection curves In the

following sections a description of the concrete horizontally curved beams and the

validity of the finite element analysis are presented

Fig.(5) Shear retention model for concrete

( Al- Shaarbaf, 1990 )

.



1 

cr

n

3 1 1 3 2











   







cr n

1.0

2

3

4.1 Jordaan et al (1974) Reinforced Concrete Horizontally Curved Beam

In this study, a reinforced concrete horizontally curved beam subjected to single

point load tested by (Jordaan et al., 1974) was selected for the analysis using the

present computer program The geometry and loading conditions for this beam are

shown in Fig 6 The beam was fully fixed at the two supports Fig 10 shows the

cross section details The total length of the beam was considered in the finite element analysis The beam was modeled using 20-quadratic brick elements mesh The finite

element meshes used, boundary conditions and loading arrangement are shown in Fig.

8 The external force was modeled as line loads distributed across the width of the

beam The material properties adopted in the analysis are given in Table 3.

Table (3) Material properties used in the analysis of Jordaan et al (1974)

curved beam

Young's modulus, Ec (MPa) * 29725 Young's modulus, Es (MPa) 200000 Compressive strength ( f c')

Poisson's ratio () 0.2 Diameter of longitudinal bars (mm) 22

Diameter of stirrups bars (mm) 6.35

Figure(6) Jordaan et al.(1974) reinforced concrete horizontally curved beam,

dimensions and loading.

all dimensions in mm

229 mm

mm251

mm503

22 Ø2

2

22 Ø

mm 001@ 6 Ø

Figure(7) Cross section details of Jordaan et al.(1974) reinforced concrete

horizontally curved beam

Figure(8) Finite element idealization of Jordaan et al.(1974) curved beam.

4.1.1 Results of The Analysis

The experimental and numerical load-deflection curves obtained for curved beam

tested by (Jordaan et al.,1974) is shown in Fig 9 This figure generally reveal that the

finite element solution is in good agreement with the experimental results throughout the entire range of behavior It can be noted that, the behavior was relatively more brittle compared with the experimental results On the other hand, it was observed that the numerical ultimate load was lower than the experimental ultimate load by (1%) However, the computed failure load is very close to the corresponding experimental ultimate load

all dimensions in mm

0 20 40 60 80 100 120 140 160 180

Deflection under load (mm)

Experimental Present study

Figure( 9) Experimental and numerical load-deflection curves of Jordaan et al.

(1974) curved beam.

4.2 Badawy et al.(1977) Reinforced Concrete Horizontally Curved Beam

In this study, a reinforced concrete horizontally curved beam subjected to single point load tested by (Badawy et al., 1977)was selected for the analysis using the present computer

program The geometry and loading conditions for this beam are shown in Fig 10 The beam

was completely fixed at one end, while the flexural and torsional fixity were removed at the

other end (simple support) Figure 11 shows the cross section details The total length of the

beam was considered in the finite element analysis The beam was modeled using 20-quadratic brick elements mesh The finite element meshes used, boundary conditions and

loading arrangement are shown in Fig 12 The external force was modeled as line loads

distributed across the width of the beam The material properties adopted in the analysis are

given in Table 4

Figure(10) Badawy et al.(1977) reinforced concrete horizontally curved beam,

dimensions, boundary conditions and loading.

all dimensions in mm

229 mm

mm251

m503 m

91 Ø2

2

91 Ø

mm 301@ 1.7 Ø

Figure(11) Cross section details of Badawy et al.(1977) reinforced concrete

horizontally curved beam

Figure(12) Finite element idealization of Badawy et al (1977)curved beam Table (4) Material properties used in the analysis of Badawy et al.

(1977)curved beam

Young's modulus, Ec (MPa) * 25743 Young's modulus, Es (MPa) 200000 Compressive strength ( f c')

Tensile strength ( f )(MPa) t 2.7 Yield stress (MPa) 475 Poisson's ratio () 0.2 Diameter of longitudinal bars (mm) 19

Diameter of stirrups bars (mm) 7.1

4.2.1 Results of The Analysis

all dimensions in mm

The experimental and numerical load-deflection curves obtained for curved

beam tested by Badawy et al.(1977) is shown in Fig 13 This figure generally reveal

that the finite element solution gives good agreement with the experimental results throughout the entire range of behavior The computed failure load is close to the corresponding experimental ultimate load Furthermore, experimental results showed more ductile behavior of the tested beam than the numerical behavior A relatively stiffener response has been obtained in the initial cracking stage of behavior for curved beam On the other hand, it was observed that the numerical ultimate load was lower than the experimental ultimate load by (4%)

0 10 20 30 40 50 60 70 80 90

0 20 40 60 80 100

Deflection under load (mm)

Experimental Present study

Figure(13) Experimental and numerical load-deflection curves of Badawy et

al.(1977)curved beam.

5 Numerical Study Of Reinforced Concrete Horizontally Curved Deep Beams

This section illustrates a numerical study that was carried out on reinforced concrete horizontally curved beams with different depths to investigate the effect of some important parameters on the load-deflection response of curved beams and the

ultimate load resisted by those beams The parameters included in this study were the total depth of the beam, subtended angle, boundary conditions, amount of transverse steel reinforcement ,use additional longitudinal bars, besides change the location of load The reinforced concrete horizontally curved beam tested by (Jordaan et al., 1974), subjected to single point load was adopted in this numerical study

5.1 The Influence of the Depth of the Beam

The effect of increasing the total depth (h) on the load-deflection response and the ultimate load was investigated In this section the total depth (h) was increased from (305 mm) to (400 mm), (500 mm), (600 mm), (700 mm), and (750 mm) The result of this study leads to the conclusion that increasing the total depth has a significant rule

on load-deflection and ultimate load of curved beams This effect of increasing the total depth becomes more significant when the total depth exceeds 600 mm

Table 5 shows the results of the ultimate load for different total depths with

Figure(14) Shows the calculation of shear length (a)

0 100 200 300 400 500 600 700

(a/ d)

Figure(16) Influence of (a/d) ratio on the

ultimate load

Figure(15) Effect of depth (h) on load-deflection

behavior

of curved beams

their ratios of the shear length (length of curved segment of beam)to the effective

depth (a/d) Calculation of shear length for curved beams is shown in Fig 14.

Fig 15 shows the influence of total length (h) increasing on the

load-deflection response for the curved beams This figure reveals that both initial and post

cracking stiffness and the ultimate load are significantly increased as the total depth

increased This can be attributed to the fact that when the total depth is increased, the

internal lever arm between the compression force in the concrete and the tensile force

in tension reinforcement is significantly increased Also , the capacity of the curved

beam cross section in shear and torsion is increased as the area enclosed by the

centerline of stirrups legs increases

Fig 16 shows the influence of the ratio of (a/d) on the ultimate load of curved

beam It can be concluded according to this figure that the ultimate load resisted by

curved beams increases with decreases (a/d) For values of (a/d) lower than two (h

>600 mm), the ultimate load increasing at a sharp slope with decreasing (a/d) This

can be attributed to the effect of arch action on the behavior of the reinforced concrete

curved beams Consequently, curved beam with (a/d) ratio less than two can be

considered as a deep beam

5.2 Effect of the Central Subtended Angle

Table (5) Effect of increasing depth (h) on the ultimate load

Total depth (h) (mm) 305 400 500 600 700 750

Ultimate load (kN) 159 252 335 425 570 610

% of increase in

0

100

200

300

400

500

600

700

Deflection under load (mm )

Exp.( h=305 m m) Study(h=305 m m) Study(h=400 m m) Study(h=500 m m) Study(h=600 m m) Study(h=700 m m) Study(h=750 m m)