A] "

Alexandria Engineering Journal (2014) xxx, xxx-xxx

FACULTY OF ENGINEERING ALEXANDRIA UNIVERSITY

Alexandria University Alexandria Engineering Journal

www.elsevier.com/locate/aej www.sciencedirect.com

ORIGINAL ARTICLE

Prediction of the behavior of reinforced concrete deep beams with web openings using the finite element method

Ashraf Ragab Mohamed *, Mohie S. Shoukry, Janet M. Saeed

Structural Engineering Department, Faculty of Engineering, Alexandria University, Alexandria 21544, Egypt Received 8 December 2013; revised 11 February 2014; accepted 2 March 2014

KEYWORDS

Deep beam; R.C.; Opening; Finite element; Reinforcement

Abstract The exact analysis of reinforced concrete deep beams is a complex problem and the presence of web openings aggravates the situation. However, no code provision exists for the analysis of deep beams with web opening. The code implemented strut and tie models are debatable and no unique solution using these models is available. In this study, the finite element method is utilized to study the behavior of reinforced concrete deep beams with and without web openings. Furthermore, the effect of the reinforcement distribution on the beam overall capacity has been studied and compared to the Egyptian code guidelines. The damaged plasticity model has been used for the analysis. Models of simply supported deep beams under 3 and 4-point bending and continuous deep beams with and without web openings have been analyzed. Model verification has shown good agreement to literature experimental work. Results of the parametric analysis have shown that web openings crossing the expected compression struts should be avoided, and the depth of the opening should not exceed 20% of the beam overall depth. The reinforcement distribution should be in the range of 0.1-0.2 beam depth for simply supported deep beams.

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University.

1. Introduction

Structural bending members can be broadly divided into two regions. The first region is the Bernoulli regions (B-Region), where the strain distribution across the section is linear. The second re-

* Corresponding author. Tel.: +20 1005424456; fax: +20 34240456. E-mail address: ashrafrm04@yahoo.com (A.R. Mohamed). Peer review under responsibility of Faculty of Engineering, Alexandria University.

gion is the D- or Disturbed regions, where the strain distribution is nonlinear as the case of deep beams. Reinforced concrete deep beams have many useful applications in building structures such as transfer girders, wall footings, foundation pile caps, floor diaphragms, and shear walls. The use of deep beams at the lower levels in tall buildings for both residential and commercial purposes has increased rapidly because of their convenience and economical efficiency. It is recognized that the distribution of the strain across the section of deep beams is nonlinear and hence, these structural elements belong to the D-Regions, Nagarajan and Madhavan [1]. Traditionally, the D-Regions have been designed using empirical formulae or past experience. Recently, the Strut-and-Tie Model (STM) has been recognized as an effective tool for the design of both B- and D-Regions and it has found place in many design codes.

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The strut and tie model (STM) provides design engineers with a more flexible and intuitive option for designing structural elements. The complex stress flows in a cracked concrete structure are approximated with simple truss elements that can be analyzed and designed using basic structural mechanics. Though the STM is effective for the design of D-Regions, the method has not yet been widely implemented due to many reasons such as: (1) the difficulty in fixing an optimum truss configuration for a given structural member with given loading, (2) the complexity and approximation of the solution and the inability of the STM to predict the failure modes of deep beams, Tan et al. [2] and Yang et al. [3].

It has been recognized that the finite element method can provide realistic and satisfactory solutions for nonlinear behavior of reinforced concrete structures. Therefore, the finite element software, ABAQUS [4], has been used to study the behavior of reinforced concrete deep beams with and without web opening under monotonic loading actions. First, the modeling technique has been verified by comparing the model prediction to experimental work in the literature. A parametric study has been conducted to predict the behavior of simply supported and continuous reinforced concrete deep beams under 3-points and 4-points bending configurations. Also, it examines the effect of the location of web openings in both simple and continuous deep beams. Finally, the effect of the reinforcement distribution on the overall capacity of the beam has been conducted. The results of this study have been compared with the ACI 318-08-Appendix A [5], and the Egyptian Code (EC 203-2006) [6] recommendations.

2. Background

Deep beams are defined as members loaded on one face and supported on the opposite face so that compression struts can develop between the loads and the supports. Their clear spans are either equal to or less than four times the overall member depth; or regions with concentrated loads within twice the member depth from the face of the support, ACI 318-08 [5]. The EC 203-2006 [6] adopts the same definition as ACI 318-08, whereas the Euro Code [7] defines a deep beam as a member whose span is less to or equal to 3 times the overall section depth. These structural elements belong to D (Disturbed) regions, which have traditionally been designed using empirical formulae or using past experience.

STM is a recent development in the analysis and design of reinforced concrete structural elements. In STM, the reinforced concrete member is replaced by an equivalent truss, where the compression and tension zones are converted into equivalent struts and ties connected at the nodes to form a truss resisting the applied loads.

Design codes provide an extensive explanation and illustration of the struts, ties and nodes' shapes, classification and detailing. In addition to the permissible stresses in struts and nodes and the corresponding cross sectional areas of struts and nodes [5-7]. Fig. 1 illustrates a schematic representation of the STM developed for deep beams under 3- and 4-points bending configurations respectively.

The STM has been subjected to ongoing debates due to the difficulty in constructing the optimum truss configuration for a given loading. Traditionally, STM has been developed using load path method or with the aid of stress trajectories.

However, this STM is not unique and varies with the designer's intuition and past experience. In order to overcome the limitations associated with the development of the STM, the Finite Element Method (FEM), is applied in the present study to predict the behavior of reinforced concrete deep beams. Results are compared to the corresponding code provisions for the design of deep beams using the STM.

FEM has proven to be a versatile tool for studying the nonlinear behavior of reinforced concrete structures. Current advances in computational capabilities have motivated the development of large number of commercial finite element codes. These codes have shown the adequate reliability and accuracy to study the behavior of reinforced concrete structures. In the present study, the damaged plasticity model, as implemented in the general purpose finite element software ABAQUS [4], is used to study the behavior of reinforced concrete deep beams. This constitutive modeling has proved to be the most stable regime for modeling concrete nonlinear behavior. It shows the ability to capture the whole concrete behavior up to failure with reliable accuracy when compared to the experimental results, Saeed [8].

The concrete damaged plasticity model in ABAQUS [4] is based on the models proposed by Lubliner et al. [9] and Lee and Fenves [10]. The model uses the concepts of isotropic damaged elasticity in combination with isotropic tensile and com-pressive plasticity to represent the inelastic behavior of concrete. The model consists of the combination of non-associated multi-hardening plasticity and scalar (isotropic) damaged elasticity to describe the irreversible damage that occurs during the fracturing process. The elastic behavior of the material is isotropic and linear. The model is a continuum, plasticity-based, damage model for concrete. It assumes that the main two failure mechanisms are tensile cracking and com-pressive crushing of the concrete material. The evolution of the yield (or failure) surface is controlled by two hardening variables linked to failure mechanisms under tension and compression loading, respectively. Fig. 2 shows the uniaxial tensile and compressive behavior of concrete, respectively, used in the concrete damaged plasticity model. As depicted from the figure, if the concrete is unloaded at any point on the softening branch, the elastic stiffness is reduced. The effect of the damage is different in tension and compression, and the degraded response of concrete is taken into account by introducing two independent scalar damage variables for tension and compression respectively.

3. Research program

The research program includes two parts; the first part is the validation of the proposed model using experimental data from literature. The second part is concerned with the parametric study.

3.1. Model validation

In order to validate the ability of the selected concrete model to study the tensile and compressive behavior of reinforced concrete deep beams, a benchmark test has been carried out using one of the deep beams (Beam SS-1), studied by Hong et al. [11] for the evaluation of shear strength of deep beams. This test serves as a source for comparison with the existing

Design Equation (ACT 318-08)

wa=wT cos e + LB sine

WT = 2C

c = effective cover to steel reinforcement Permissible compressive stresses in Struts & Nodes fcs = 0.85fckps fm = 0.85fckpn

Design Equations (BS EN 1992-1-1: 2004)

Design strength of struts without transverse tension

CTRd,max =

Design strength of struts with transverse tension CTRd,max = 0-6 V fcd

Desien compressive stresses within nodes

Kvfcd v= l-fck/250

fck= characteristic compressive cylinder strength

K = force reduction factor for different types of node

** Strut and Tie widths are evaluated based on

node equilibrium

Design Equations (EC 203-2006)

Wt = cps + 2c + (n-1). s

cps = diameter of the bars in the tie,

c = the cover to the surface of the bars,

n = the number of rebar layers, and

s = the spacing of bars

Permissible compressive stresses in Struts & Nodes fd = 0.67 MWYc) fed = 0.67 pn (fcu / yc)

Figure 1 Schematic representation of STM.

experimental results. In the study conducted by Hong et al. [11], simply supported beams were instrumented to measure the mid span deflections and loads. Fig. 3 illustrates the cross section and loading configuration of the tested beam. An 8-node solid element with one point integration was utilized to create the concrete beam mesh. An embedded truss reinforcement a 2-node linear 3D truss element was used to model steel rebars. The mesh used in this validation is shown in Fig. 4.

Fig. 5 illustrates the load-deflection response of the studied beam in comparison with the experimental results obtained by Hong et al. [11]. The modeled response verifies the ability of the selected model to capture the whole beam's behavior up to failure and shows a good agreement to the experimental results. The results of the model can be used in validating and guiding experimental work, in addition to exploring concrete

response under complicated loading conditions such as the behavior of reinforced concrete deep beams with and without web opening introduced in the current study.

3.2. Parametric study

The research program for the parametric study conducted in this paper consists of (9) deep beams. All the studied beams had a clear span (lc) of 6000 mm, a depth (d) of 2000 mm deep and a width (b) of 500 mm. Fig. 6 shows the studied beams' geometry and dimensions, while Fig. 7 shows typical example for the meshing of deep beam DS3-1W.

The studied beams were categorized in three main groups; the first group is for simply supported deep beams with and without web openings under 3-points bending (Beams DS3-0,

2 D= 10 mm

Bottom Rebars (4 D=19mm)

№ «

-p Stirrups

10mm Diam. h = 600 mm

Figure 3 Cross section and loading configuration of beam SS-1 [11].

Figure 4 The applied mesh for beam SS-1.

2 3 4 5 6

Mid-Span Deflection (mm)

Figure 5 Comparison of load-deflection response for the applied material model vs. experimental results.

DS3-1W, DS3-2W). The second group includes simply supported deep beams with and without web openings under 4-points bending (Beams DS4-0, DS4-1W. and, DS4-2W). On the other hand, the third group contains continuous deep beams with and without web openings (Beams DC-0, DC-1W and DC-2W). The mechanical properties of the concrete and reinforcing bars for all studied beams are summarized in Table 1. The material properties of concrete are taken as proposed by Hong et al. [11].

The beams were designed in accordance with the recommendation of the EC 203-2006 [6]. Table 2 summarizes the reinforcement detailing for the beams. The deep beams have been modeled using the damaged plasticity model in ABAQUS/Stan-dard. Concrete tensile behavior has been modeled as a quasi-brittle material and the beam reinforcement has been modeled as embedded truss elements. The considered variables in this parametric study are the loading scheme, the location of web openings and the reinforcement distribution. The obtained results are compared to both the ACI 318-08 [5] and the EC 203-2006 [6] provisions, in terms of the strut and tie widths and the effective reinforcement distribution depth.

4. Results and discussions

4.1. Load-deflection response

Figs. 8-10 show the load deflection response of the simply supported deep beams under 3-points and 4-points bending and continuous deep beam respectively. It can be depicted from these figures that the presence of web openings crossing the compression struts developed between the load and the supports (case of beams DS3-2W, DS4-2W, DC-1W) resulted in a substantial reduction in the failure load and the beam's overall capacity. This reduction ranges from 33% in case of the simply supported deep beam under 3-points bending to 36% in case of the simply supported deep beam under 4-points bending. However, when the web openings were located outside the expected line of action of compression struts, the reduction in the failure load ranged from 3% in case of beam DS4-1W to 15% in case of beam DS3-1W

In order to study the effect of the size of the web opening on the beam's overall capacity, four different web openings were considered in Beam DS3-1W with length ranging from 0.1 lc to 0.2 lc and a total depth from 0.075d to 0.3d. Fig. 11 illustrates the load-deflection response of Beam DS3-1W for different web opening sizes and the corresponding failure load. As depicted from this figure, for deep beams having the same web opening depth, the reduction in the failure load due to the increase in the web opening width from B1 = 600 mm to B2 = 2B1 = 1200 mm is negligible. On the other hand, for deep beams having a constant web opening width and a variable depth from H1 to H2 = 2H1, the reduction in the failure load and the beam capacity was about 7%. Therefore, the most important observation that can be deduced from Fig. 11 is that, as long as the opening does not interfere with the load path or stress trajectories, i.e. compression struts, the depth is the most important parameter influencing the beam's overall capacity. Hence, for an overall reduction in the beam's capacity not exceeding 10%, the depth of the central opening should not exceed 20% of the beam overall depth (0.2d).

(a) Beam DS3-0

(b) Beam DS3-1W

, 1200 , LI =2000. | .1200,

600£ ^^^ ^^^ J600

(c) Beam DS3-2W

1500 Î 3000

(d) Beam DS4-0

1500 Î 3000

(e) Beam DS4-1W

y. 1:>UU Jç 3UUU -J- 1SUU . |

1200. i.LI = 2000.1.1200.

600^ [

] JôOO

(f) Beam DS4-2W

1500 3000

| 3000

(g) Beam DC-0

1500 3000

(h) Beam DC-1W

1500 3000

600 Ll= 2400 600

------ ---

□ 1«

3000 3000

i) Beam DC-2W

Figure 6 Beams geometry and dimensions.

Figure 7 The applied mesh for beam DS3-1W. 4.2. Crack pattern

The general failure modes of reinforced concrete members are represented by the occurrence and development of cracks and crushing of concrete. Fig. 12 illustrates the crack pattern at failure for the studied beams. It should be noted that the concrete damaged plasticity model does not have the notion of cracks developing at the material integration point. However, it is possible to introduce the concept of an effective crack direction with the purpose of obtaining a graphical visualization of the cracking patterns in the concrete structure. The criteria adopted in this model are the assumption that cracking initiates at points where the tensile equivalent plastic strain is

greater than zero, and the maximum principal plastic strain is positive. Based on these criteria, the direction of the vector normal to the crack plane is assumed to be parallel to the direction of the maximum principal plastic strain. This direction can be viewed in the visualization module of the post-processor ABAQUS/CAE.

4.3. The developed struts' width

Compression struts have developed between the load and supports. These struts' widths obtained from the model are compared with the corresponding values evaluated using the ACI 318-08 [5] and the EC (203-2006) [6] equations. Tables 3-5 summarize the comparison between the calculated and the obtained strut's widths from the model, for the studied deep beams without web openings.

These results show that the code provisions for the design of reinforced concrete deep beams substantially underestimate the strength of the reinforced concrete members. Therefore the actual capacity of the structure is greater than that of the idealized truss, and, hence, designs based on STM are always on the safer side. Moreover, as the current study has shown that the evaluated strut width is wider than the expected strut width, as obtained by the model, this will allow more openings' width to be considered in deep beams.

It is noteworthy that no code provisions have been provided in the ACI 318-08, EC 203-2006 or BS EN 1992-1-1: 2004. Hence, for the design of deep beams with web openings, no comparison could be made for the stress distribution or the developed struts widths for the studied reinforced concrete deep beams with web opening. Fig. 13 illustrates the developed compression struts between the load and supports for all the studied deep beams.

Table 1 Material properties for the studied deep beams [11].

Concrete properties:

1. Elastic properties

Young's modulus 20 GPa

Poisson's ratio 0.2

2. Uniaxial compression values

Characteristic compressive strength (f c) 23.5 MPa

3. Uniaxial tension values

Cracking failure stress 2 MPa

4. Details of softening behaviora Tensile stress (rt) Displacement across crack (uo)

2.00 0

1.5753 5.3633E—005

1.0559 9.9559E-005

0.70781 0.00014054

0.47446 0.00017843

0.31804 0.00021451

0.21319 0.00024966

0.1429 0.00028447

0.095791 0.00031939

0.064211 0.00035472

0.043042 0.00039072

5. Details of tension damage (dt)a Damage parameter (dt) Displacement across crack (uo)

0.381217 5.3633E—005

0.617107 9.9559E—005

0.763072 0.00014054

0.853393 0.00017843

0.909282 0.00021451

0.943865 0.00024966

0.965265 0.00028447

0.978506 0.00031939

0.9867 0.00035472

0.99177 0.00039072

Steel (rebar) properties:

Young's modulus 230 GPa

Poisson's ratio 0.3

Yield stress 392 MPa

a Not listed by the reference.

4.4. The stress and strain distributions

The stress and strain distributions for all the studied deep beams at failure, have been extracted and evaluated. Typical example of these distributions is shown in Fig. 14. As expected, the strain distribution is nonlinear and the stress distribution in tension is limited to the assumed tensile strength of concrete, while the compressive stress simulates that assumed by the code.

4.5. Effect of reinforcement distribution

Since 2003, the EC 203-2006 Committee has issued a handbook for the detailing of structural members. This handbook introduces some guidelines for reinforcement detailing of different structural members. For simply supported deep beams under uniformly distributed loads, it is recommended that the main tension reinforcement should be distributed along a depth ranging from 0.15 to 0.3 of the overall beam depth. This

range varies according to the type of the applied loads and the boundary conditions. In the present study, this guideline was reviewed for simply supported deep beams by distributing the main tension reinforcement over varying depths ranging from 0.05 to 0.4 of the total beam depth (H). The effect of reinforcement distribution on the beam capacity has been studied and the stress distribution in reinforcing steel has been monitored. Fig. 15 illustrates the different reinforcement distribution applied in the present study.

The simply supported reinforced concrete deep beam under 3-points bending, (Beam DS3-0) illustrated in Fig. 6a, was studied for the different reinforcement distribution. Fig. 16 illustrates the load-deflection response of the beam and the observed reduction in the beam capacity with increasing reinforcement depth.

From the figure, it can be depicted that no significant reduction in the beam's capacity was observed when the reinforcement depth increased from 0.05H to 0.1H. However, a reduction of 6.0% in the beam capacity was detected when the upper limit specified in the handbook for reinforcement

Table 2 Reinforcement of tested deep beams.

Beam ref. Tension longitudinal Compression longitudinal Horizontal Vertical

reinforcement reinforcement reinforcement reinforcement

DS3-0 12 /16 6 /16 /16@275 mm /12@275 mm

DS3-1W 12 /16 6 /16 /16@275 mm /16@275 mm

DS3-2W 12 /16 6 /16 /16@275 mm /16@275 mm

DS4-0 12 /16 6 /16 /16@275 mm /12@275 mm

DS4-1W 12 /16 6 /16 /16@275 mm /16@275 mm

DS4-2W 12 /16 6 /16 /16@275 mm /16@275 mm

DC-0 12 /16 12 /16 /16@275 mm /12@275 mm

DC-1W 12 /16 12 /16 /16@275 mm /16@275 mm

DC-2W 12 /16 12 /16 /16@275 mm /16@275 mm

^ 3000 —■■

Beam DS3 ^^ Bear -0 DS3-1W

Beam DS3-2W

-Beam DS3-0 -Beam DS3-1W - Beam DS3-2W

0 2 4 6 8 10

Mid-Span Deflection (mm)

Figure 8 Load-deflection response of simply supported deep beams under 3-points bending configuration.

^ 8000 z

o 6000 75

2 4 6 8 10

Mid-Span Deflection (mm)

Figure 10 Load-deflection response of continuous deep beams.

4000 -

Mid-Span Deflection (mm) Mid-Span Deflection (mm)

Figure 9 Load-deflection response of simply supported deep Figure 11 Load-deflection response of beam DS3-1W for beams under 4-points bending configuration. different web openings.

(a) Beam DS3-0

(b) Beam DS3-1W

(c) Beam DS3-2W

(g) Beam DC-0 (h) Beam DC-1W (i) Beam DC-2W

Figure 12 Visualization of crack pattern.

(g) Beam DC-0 (h) Beam DC-1W (i) Beam DC-2W

Figure 13 The developed compression struts.

0 -2 -4

Stress (MPa)

(a) Stress Distribution of Beam DS3-0 at failure

0.003 0.0025 0.002 0.0015 0.001 0.0005 0 -0.0005 Strain

(b) Strain Distribution of Beam DS3-0 at Failure

Figure 14 Stress and strain distribution of simply supported deep beam without web opening under 3-points bending (Beam DS3-0) at beam mid span at failure.

• •••

B = 50mm

Figure 15 Reinforcement distribution.

distribution; 0.3H, was applied. Table 6 summarizes the applied reinforcement distribution depths and the corresponding reduction in the beam's overall capacity. From this comparison, it is suggested that the reinforcement distribution should be in the range of 0.1-0.2H for simply supported deep beams.

Central Deflection (mm)

Figure 16 Load-deflection response of simply supported deep beam with various tension reinforcement distributions.

Table 3 The strut and tie width for beam DS3-0. z!

Current study ACI 318-08 EC (203-2006)

Strut width (Ws) mm 471.73 670.3 761.8

Tie width (WT) mm 278.1 444.6 677

Table 4 The strut and tie width for beam DS4-0.

Current study ACI 318-08 EC (203-2006)

Strut width (Ws) mm 518.3 659.76 726.3

Tie width (WT) mm 381.05 320.8 488.6

Table 5 The strut and tie width for beam DC-0.

Current ACI EC

study 318-08 (203-2006)

Strut width (Ws1) mm 549.35 316.76 482.2

Strut width (Ws2) mm 607.4 950 1446.7

Tie width (WT1) -top mm 357.5 412.5 628.25

Tie width (W-n) - bottom mm 356.8 275 419

Table 6 Reduction in beam capacity due to reinforcement distribution.

Tension reinforcement depth % Reduction in beam capacity

0.1H (200 mm) 0.34

0.2H (400 mm) 2.1

0.3H (600 mm) 5.87

0.4H (800 mm) 12.16

re a. S

"ÜT 200 «

—•—R1 @ 75mm —■—R2 @ 100mm

0 1000 2000 3000 4000 5000 6000 7000 Length (mm)

Figure 17 Stress distribution in rebars (distribution depth = 0.05H).

"!) 250 re CL S

IT 200 «

® 150

0 1000 2000 3000 4000 5000 6000 7000 Length (mm)

Figure 18 Stress distribution in rebars (distribution depth = 0.1H).

TO Q. S

m 200 tfl eu

®> 150

0 1000 2000 3000 4000 5000 6000 7000 Length (mm)

Figure 19 Stress distribution in rebars (distribution depth = 0.2H).

IT 200 «

25 150

0 1000 2000 3000 4000 5000 6000 7000 Length (mm)

Figure 20 Stress distribution in rebars (distribution depth = 0.3H).

R1@75mm

R2@256mm

R3@437mm

R4@619mm

R5@800mm

tfl EH

0 1000 2000 3000 4000 5000 6000 7000 Length (mm)

Figure 21 Stress distribution in rebars (distribution depth = 0.4H).

Furthermore, the stress distribution in the tension reinforcement was monitored and recorded for different reinforcement distribution schemes. Figs. 17-21 illustrate the stress distribution in each rebar layer along the rebar length.

From this set of figures, it can be observed that, for smaller distribution depth (0.05H), the stresses in the rebar layers are almost identical. A reduction in the reinforcement stresses is observed when the rebar depth increases and this reduction is proportional to the rebar position.

5. Summary and conclusions

Based on this study, it is recognized that the exact analysis of reinforced concrete deep beams is a complex problem and the presence of web openings aggravates the situation. The application of the damaged plasticity model implemented in ABA-QUS/Standard for the analysis of simply supported deep beams under 3-points and 4-points bending and continuous deep beams with and without web openings provided useful information about the responses of reinforced concrete deep beams under monotonic loadings. The most important conclusions of the conducted parametric study can be summarized as follows:

1. The model validation with experimental work from literature has shown that the model is capable of capturing the entire response reasonably.

2. The web openings crossing the expected compression struts developed between the load and the supports cause approximately about 35% reduction in the beam's capacity in all the studied cases and hence it should be avoided.

3. When the introduced web opening does not interfere with the load path or stress trajectories, i.e. compression struts, the observed reduction the beam's capacity ranged from 6% to 8% depending on the opening dimensions.

4. The depth of the opening is the most important parameter influencing the beam's overall capacity. Therefore for an overall reduction in the beam's capacity not exceeding 10% of the capacity the beam without web opening, the depth of the opening should not exceed 20% of the beam overall depth (0.2d).

5. The effect of reinforcement distribution on the beam capacity was studied and the stress distribution in reinforcing steel has been monitored. A reduction, ranging from 0.3% to 12% in the beam capacity, was observed, when the tension reinforcement distribution depth was increased from 0.1H to 0.4H. Thus, it is suggested that the reinforcement distribution should be in the range of 0.1-0.2H for simply supported deep beams.

References

[1] P. Nagarajan, P.T.M. Madhavan, Development of strut and tie models for simply supported deep beams using topology optimization, Songklanakarin J. Sci. Technol. 30 (5) (2008) 641-647.

[2] K.H. Tan, K. Tong, C.Y. Tang, Consistent strut-and-tie modeling of deep beams with web openings, Mag. Concr. Res. 55 (1) (2003) 572-582.

[3] K. H Yang, H.C. Eun, H.S. Chung, The influence of web openings on the structural behavior of reinforced high-strength concrete deep beams, Eng. Struct. 28 (13) (2006) 1825-1834.

[4] ABAQUS User Manual, Version 6.7, ABAQUS Inc., Pawtucket, Rhode Island, 2007.

[5] ACI 318-08, Building Code Requirements for Structural Concrete and Commentary.

[6] EC, Egyptian Code for the Design and Construction of Concrete Structures (203-2006).

[7] BS EN1992-1-1: 2004, Eurocode2: Design of concrete structures - Part 1-1: General rules and rules for buildings.

[8] J.M. Saeed, Modeling of the Mechanical Behavior of Concrete, M.Sc. Thesis, Structural Eng. Dept., Faculty of Engineering, Alexandria University, 2012, p. 163.

[9] J. Lubliner, J. Oliver, S. Oller, E. Onate, A plastic-damage model for concrete, Int. J. Solids Struct. 25 (3) (1989) 299-326.

[10] J. Lee, G.L. Fenves, A plastic-damage model for cyclic loading of concrete structures, J. Eng. Mech., ASCE 124 (8) (1998) 892900.

[11] S. Hong, D. Kim, S. Kim, N. Hong, Shear strength of reinforced concrete deep beams with end anchorage failure, ACI Struct. J. 99 (1) (2002) 12-22.