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Finite element analysis of reinforced concrete deep beam with large opening
BeniSuef University Journal of Basic and Applied Sciences volume 10, Article number: 25 (2021)
Abstract
Background
A series of nonlinear finite element (FE) analyses was performed to evaluate the different design approaches available in the literature for design of reinforced concrete deep beam with large opening. Three finite element models were developed and analyzed using the computer software ATENA. The three FE models of the deep beams were made for details based on three different design approaches: (Kong, F.K. and Sharp, G.R., Magazine of Concrete Res_30:8995, 1978), (Mansur, M. A., Design of reinforced concrete beams with web openings, 2006), and Strut and Tie method (STM) as per ACI 31814 (ACI318 Committee, Building Code Requirements for Structural Concrete (ACI31814), 2014). Results from the FE analyses were compared with the three approaches to evaluate the effect of different reinforcement details on the structural behavior of transfer deep beam with large opening.
Results
The service load deflection is the same for the three models. The stiffnesses of the designs of (Mansur, M. A., Design of reinforced concrete beams with web openings, 2006) and STM reduce at a load higher than the ultimate design load while the (Kong, F.K. and Sharp, G.R., Magazine of Concrete Res_30:8995, 1978) reduces stiffness at a load close to the ultimate design load. The deep beam designed according to (Mansur, M. A., Design of reinforced concrete beams with web openings, 2006) model starts cracking at load higher than the beam designed according to (Kong, F.K. and Sharp, G.R., Magazine of Concrete Res_30:8995, 1978) method. The deep beam detailed according to (Kong, F.K. and Sharp, G.R., Magazine of Concrete Res_30:8995, 1978) and (Mansur, M. A., Design of reinforced concrete beams with web openings, 2006) failed due to extensive shear cracks. The specimen detailed according to STM restores its capacity after initial failure. The three models satisfy the deflection limit.
Conclusion
It is found that the three design approaches give sufficient ultimate load capacity. The amount of reinforcement given by both (Mansur, M. A., Design of reinforced concrete beams with web openings, 2006) and (Kong, F.K. and Sharp, G.R., Magazine of Concrete Res_30:8995, 1978) is the same. The reinforcement used by the STM method is higher than the other two methods. Additional reinforcement is needed to limit the crack widths. (Mansur, M. A., Design of reinforced concrete beams with web openings, (2006)) method gives lesser steel reinforcement requirement and higher failure load compared to the other two methods.
1 Background
Designs of many of the new tower buildings include transfer beam elements with depths equal to a full floor depth to achieve the architectural requirements for both parking and upper floors. In many situations, the full beam depth cannot be utilized for the full span due to the presence of openings and space restrictions. In such cases, the structural engineer utilizes the available solid zone developing nonprismatic beams with different shapes and depths. These particular shapes create a challenge to understand the load transfer mechanisms and to estimate ultimate strengths and reinforcement details.
Elkareim et al [1] studied the strength and behavior of deep beams with openings. They reported experimental tests on flanged deep beams with different shear spans and openings. They considered the effect of flange on the behavior of these beams in terms of cracking, reinforcement strains, and deformations. They made a finite element model to predict the experimental behavior. Elkareim et al. [1] calculated the deep beams’ capacities using design equations. The finite element model was able to predict the experimental behavior. Hassan et al. [2] studied selfcompacted highstrength concrete deep beams with opening. They conducted experimental tests to study the effect of location, size, and shape of opening. Hassan et al. [2] commented on crack patterns, absorbed energy, and deflection. Yang et al. [3] analytically and experimentally evaluated the influence of web openings in reinforced concrete deep beams. They studied the effects of shear spantodepth ratio, depth and width of opening, and concrete strength. Their tests indicated that the effect of concrete strength on the shear capacity decreased for deep beams with openings compared to solid deep beams. Campione and Minafo [4] analytically and experimentally studied the effect of openings of circular shape in deep beams of low shear spantodepth ratio. The tested beams had different opening positions and reinforcement arrangements. Their study revealed that the benefit of reinforcement depends on its arrangements. They predicted the shear strength and corresponding deflection. Mohamed et al. [5] performed finite element analyses for deep beams with opening and without opening to study the effect of reinforcement pattern. They recommended avoiding web openings crossing the expected compression struts and limiting the opening depth to 20% of the deep beam depth. ElKassas et al. [6] studied deep beams with longitudinal opening for mechanical and electrical services. In their experimental work, they changed the opening size, shape, and location. They recorded crack patterns, load capacity, and deflection. The experimental work revealed that increasing the opening size decreases the beam capacity while changing its shape has a slight effect. Placing the opening in the compression zone gives more reduction in the loadcarrying capacity. Tseng et al. [7] developed an analytical method to evaluate the shear strength of deep beams with openings and obtain the failure mode. The proposed method is based on the strutandtie analysis. This method is used to study deep beams with different reinforcement ratios, concrete strengths, sizes of web openings, and spandepth ratios. The results proved that the proposed method is simple and accurate.
Studies on deep beams with web openings are generally limited and international codes do not include provisions for the design of these elements. Other previous studies such as the works of Maxwell and Breen [8], Kong and Sharp [9], Kong and Sharp [10], Ray. D. P [11], Ashour and Rishi [12], Ray [13], and Tan et. al. [14] have developed formulas to calculate the ultimate strength of deep beams using the modified MohrCoulomb’s criterion and the shear friction theory. Kong. F. K [15] provided recommendations for applying his method with limitations on the opening size and location. A simplified procedure was also suggested by Mansur [16] which proposed an equivalent stiffness of the beam opening segment based on the analysis of a continuous beam with large web openings. Another alternative for designing deep beams with web openings is the use of strutandtie (STM) models. The complex stress flow in a cracked concrete structure is approximated with simple truss model that can be analyzed and designed by structural mechanics (Tan et al. [14]). The ACI 31814 [17] code substituted the use of empirical equations for predicting the shear strength of deep beams by the use of the Strut and Tie (STM) method.
2 Methods
The objective of this paper is to compare the mechanical behavior such as cracking, yielding of reinforcement, and deflection of transfer deep beam with large opening designed according to three different approaches. This is to decide on the merits of each design approach. To the writers’ knowledge, this type of comparison and evaluation of merits of the three design models is not done in the literature. The finite element software ATENA Program Documentation [18] is used in the analysis. This software can simulate the actual behavior of concrete elements including cracking and plasticity phases. The software is validated previously using laboratory research carried by El Maaddawy and Sherif [19] for deep beams with openings and Markou and AlHamaydeh [20] for deep beams without shear reinforcement.
3 Details of beam geometry and reinforcement
A typical transfer beam in a highrise building consisting of two levels for basement and ground and fourteen typical floors was analyzed for this study. The beam was introduced from ground to firstfloor level to take the columns’ location which was restricted in the basement by the driveway and in the first floor by the setback, as shown in the crosssection in Fig. 1. Reinforced details were provided using the three selected approaches by Kong. F.K [15], Mansur [16], and Strut and Tie method (STM) as per ACI 31814 [17].The dimensions of the beam were based on the maximum allowed shear value and the possible concrete dimensions that meet architectural conditions. The beam was designed for a concentrated factored load of 10 MN. The cubic concrete strength (f_{cu}) is assumed to be 50 MPa, and both flexural and shear reinforcement bars were assumed to have a yield strength f_{y} of 460 MPa. The provided reinforcements in the support zones were similar in all approaches to prevent failure in these zones.
The beam was considered as a simply supported deep beam with overall dimensions of 10,000 × 5900 × 500 mm (length × depth × width) with a large opening of 4100 × 3100 mm. The transfer load was applied at shear span to overall depth ratio of 0.25.
3.1 Beam reinforcement using the Kong. F.K [15] approach
3.1.1 Flexural
The beam was analyzed as a simple beam in which the internal forces were obtained. The ultimate flexural strength for the beam was calculated using the formula proposed in the CIRIA (Construction Industry and Research Information Association), CIRIA Guide 2 [21] for a beam with a solid web as explained in the section below.
where A_{st} is the flexural reinforcement, M is the design moment at ultimate state, and z is the lever arm for a single span. For preliminary design, it is recommended to use z = 0.6D, where D is the overall depth and f_{ys} is the yield strength of longitudinal steel reinforcement. Therefore, the reinforcement will be distributed over a height of 0.2D, which was taken as 800 mm due to the presence of an opening. The equation above is applicable for beams with openings; however, a capacity reduction factor of 0.65 is recommended (Ray, [13]). The principal reinforcement is calculated and rounded to the nearest significant figure.
The area of reinforcement required to be concentrated around the opening is based on the reduced tensile capacity of the opening, a ∗ b ∗ f_{t} where, a is the half dimension of the opening perpendicular to reinforcement direction considered, b is the beam thickness, f_{t} is the cylindersplitting tensile strength of the concrete, \( {f}_t=0.52\ \sqrt{f_{\mathrm{cu}}} \) , and f_{cu} characteristic strength of concrete.
3.1.2 Shear strength
The ultimate shear equation proposed by Kong. F.K [15] is based on the shear resistance of a solid deep beam and is a function of the concrete resistance and the tension and web reinforcements. The effect of web opening on the concrete shear strength is accounted by the constants λ_{1}, λ_{2}, and λ_{3,} as follows:
where λ_{1} is a failure mode parameter that mainly depends on the shear span as follows:
λ_{1}= \( \left[1\frac{1}{3}\left(\frac{K_1{X}_n}{K_2D}\right)\right]\kern1em \)for \( \frac{{\mathrm{K}}_1{x}_n}{K_2D}\le 1 \), and λ_{1}=2/3 for \( \frac{K_1{X}_N}{K_2D}\ge 1 \)
in which K_{1}x_{n} and K_{2}D are the dimensions from the edge of the practical region to the opening center as shown in Fig. 2.
λ_{2} is a constant that accounts for the interception of the opening into the natural path load (critical diagonal crack) that can be expressed as:
in which m is the ratio of the path length intercepted to the total path length along the natural path.
λ_{3} is a constant that accounts for the combined effect of the size of the opening with respect to its location as follows:
e_{x} and e_{y} are the eccentricity of the center of the opening in relation to the center point of the critical diagonal crack, X_{net} and Y_{net} are the dimensions of solid shear zone in the X and Y directions after subtracting the dimensions of the web opening in the respective directions, and a_{1}x and a_{2}D are the opening dimention in X and Y directions, respectively, measured as indicated in Fig. 2 a.
The sign convention adopted as negative when the opening is in the quarter where the load is applied and positive when is located in the quarter with no load. As shown in Fig. 2a the loaded quadrants are the regions located closer to the load and support bearing blocks.
ψ_{s} and ψ_{w}are empirical coefficients assumed as 0.65 and 0.5, respectively.
ρ_{s} (ρ_{s} = \( \frac{A_{st}}{\left( bD\ast 100\right)} \)) is the main steel ratio in the concrete deep beam, and ρ_{w} (ρ_{w} = \( \sum \frac{A_w}{\left( bD\ast 100\right)} \)) is the web steel ratio in the concrete deep beam. A_{w} is the area of individual web reinforcement either vertical , horizontal or inclined intersecting with the critical diagonal crack.
K_{w} = 0.85 for horizontal web reinforcement, Cotβ for vertical web reinforcement, and 1.15 for inclined web reinforcement.
β is the critical diagonal crack inclination angle with the horizontal axis.
Figure 2 presents the structural idealization and reinforcement details based on the empirical formula proposed by Kong. F.K [15].
The above equations represent the general relationship used for plain beams and for beams with and without web reinforcements. Ray [13] considered that this equation favors the design engineering practice.
A system of orthogonal web reinforcement is required with bars in each face where the minimum area of vertical and horizontal reinforcement are as follows: \( \frac{A_v}{\left(b{S}_v\right)}\ge 0.20,\mathrm{and}\ \frac{A_h}{\left(b{S}_h\right)}\kern0.5em \) ≥ 0.20
where S_{v} and AS_{h} are the vertical and horizontal reinforcement spacing, respectively.
3.2 Beam reinforcement using the Mansur [16] method
In this method, the beam is analyzed as a simple determinate structure in which the nonprismatic shape does not affect the straining action values. The design procedure for the opening segment is based on Vierendeel behavior of chord members at the opening. Contra flexure points are assumed at the mid span of the top and bottom chord members for which the axial load is obtained by dividing the moment at the center of the opening by the distance between the plastic centroids of the chord members, as follows:
where Y_{ct} is the distance between the plastic centroids of the bottom chord and top chord and M is the ultimate bending moment at the middle of the opening.
The shear force at the center of the opening, V, is distributed between the top chord V_{t} and the bottom chord V_{b} according to their relative flexural stiffness. The moments at the ends of the top chord, M_{t}, and the bottom chords, M_{b}, are calculated using statics as follows:
in which I_{b} and I_{t} are the gross moments of inertia at the bottom and top chords, respectively, and L is the opening length.
The critical sections were designed for bending, axial, and shear in the usual manner using the interaction diagrams. Figure 3 shows the free body diagram of the opening segment and the required reinforcement details.
3.3 Beam reinforcement using the STM method as per ACI 31814
The deep beam was designed using the strut and tie method (STM) described in Chapter 23 of ACI 31814. There can be many possible truss models for the given loading and geometry of the deep beams. However, the assumed theoretical STM for the applied loads is shown in Fig. 4. Selecting the truss model of the STM to use depends on the experience of the designer. The location of struts and ties represent the elastic flow of forces within the structural components. The tie is placed at the position of the bars centroid; the struts such as the horizontal struts along the top of the member are placed based upon the depth, a, of the rectangular compression stress block as determine from the typical flexural analysis as follows:
where
A_{s} = area of tension reinforcement in mm^{2},
\( {A}_s^{\prime } \) = area of compression reinforcement in mm^{2},
f_{s} = stress in tension reinforcement in MPa,
\( {f}_s^{\prime } \) = stress in compression reinforcement in MPa,
\( {f}_c^{\prime } \) = specified cylinder compressive strength of concrete in MPa, and
b = width of member’s web in mm.
3.3.1 Tie strength
where
F_{u} = Factored ultimate force in the tie,
f_{ys} = yield strength of the steel, and
Φ = resistance factor of 0.75.
3.3.2 Strength of the nodal zones
The nominal compression strength at the face of a nodal zone or at any section through the nodal zone shall be
where A_{n} is taken as the area of the face of the nodal zone that the strut force F_{u} acts on, f_{ce}is the effective compressive strength of the concrete in the nodal zone and is calculated as f_{ce} = 0.85 β_{n} f′_{c}.
The coefficients β_{n} can be obtained from Table 23.9.2 in ACI 31814 [17].
3.3.3 Strength of the strut
The nominal compressive strength of a strut without longitudinal reinforcement shall be taken as
where A_{cs} is the crosssectional area at the end of the strut that the strut force F_{u} acts on, f_{ce}is the effective compressive strength of the concrete in the nodal zone and is calculated as f_{ce} = 0.85 β_{s} f′_{c}.
The coefficients β_{s} can be obtained from Table 23.4.3 in ACI 31814 [17].
The verification of the strut and nodal zone strengths can be obtained by comparison of the available strut or nodal area against the required values. Generally, the thickness of the beam is assumed constant and verification can be provided by comparing the strut or nodal width, w_{prov}, with the required value w_{req}. The critical nodes and struts have been manually checked for this deep beam however the whole truss is analyzed and checked using cast software.
4 Finite element model
The previously verified finite elements program ATENA [18] is used in this research. Threedimensional (3D) finite element (FE) models were developed for the analysis of the transfer deep beam. All the models had the same material properties with different reinforcement arrangements and quantities according to the designs of the three methods. The concrete was modeled using 3D solid brick elements with CC3DNonLinCementitious2 Model of the ATENA program. The steel bars were modeled as discrete reinforcement of bilinear stressstrain relationship using truss elements. A sensitivity study was performed to analyze the effect of mesh size. Large size was chosen for the mesh and a finite element run was performed. The mesh size was then reduced in successive runs until the stresses obtained from the successive runs showed no significant change. Finally, it is selected to use four elements for the beam width as recommended by ATENA manual to capture the flexural behavior and to use an element length of 25 cm. The finite element model is shown in Fig. 5. The steel loading plates used in the FE model covered the entire width of the beam. The model is supported on two plates, one at each end. One end support plate is restrained from movement in the vertical direction (Z) and in the out of plane direction (Y) with a point support placed at the bottom center of the plate. The other support is restrained in all directions (Z, X, and Y) to maintain the model stability and prevent out of plane distortion. A displacementcontrolled incremental loading method is employed in the analysis. The iterative solution procedure is based on the NewtonRaphson method.
4.1 Constitutive model for structural concrete
Two stages describe the nonlinear behavior of concrete in the case of static loading. The initial stage is prior to the crack initiation where the material is modeled using a linearelastic relationship. After cracking, several constitutive relationships can describe the 3D nonlinear behavior of structural concrete such as elasticplastic, fractureplastic, and smearedcrack failure.
The nonlinear behavior is to be modeled using the fractureplastic model (CC3DNonLinCementitious2) which can describe all phases of concrete including cracking, crushing, and plastic behavior. This model combines the constitutive relations for tensile (fracture) and compressive (plastic) responses as shown in Fig. 6.
The input parameters of the finite element (FE) model have been used as the default calculated according to EuroCode2 (EC2) and CEPFIP MC90 [22] model code expressions as indicated in Table 1.
4.2 Reinforcement constitutive model
In this work, discrete bar elements were used to model the reinforcement assuming a bilinear stressstrain relationship that considers strain hardening which allows the stresses to increase after yielding. The input data for the bilinear constitutive model are shown in Fig. 7 where the yield stress is 460 MPa, the ultimate stress is 560 MPa, and the ultimate strain is 0.025.
5 Results of crack patterns and failure modes
In this part, the output from the finite element models is used to evaluate the three design approaches and determine the most efficient design method for a transfer deep beam with large opening.
Failure modes and cracking patterns varied among the specimens. The location and amount of the main flexural and shear reinforcement change from one approach to another which affect the cracks’ initiation, crack pattern at failure, and failure mode. Generally, the cracks initiate at the chord which does not contain the main flexural reinforcement. Figure 8 and Table 2 show the crack propagation with the load and crack width at service load for the three design approaches: Kong. F.K [15], Mansur [16], and STM, respectively. Figure 9 and Table 3 show principal concrete compression strain at failure, and steel tensile stress at failure resulted from the numerical analysis for the three approaches.
The empirical approach suggested by Kong. F.K [15] produces main reinforcement at the bottom chord as shown in Fig. 2. At an approximate load of 2 MN, a vertical crack starts to develop at the top left corner of the opening below the point of load application. At an approximate load of 3.4 MN, more cracks were developed at the beam top right surface and near the bottom right corner of the opening. With the load increase, diagonal cracks start to form and spread over the top chord. The failure of this specimen mainly occurred due to extensive shear cracks present at the top chord on the right side. At a load of 9.8 MN, the reinforcement starts yielding at the top left corner of the opening and the top right surface of the beam. Most of the shear reinforcement of the top chord reach the yield point at the same load. The bottom crack was narrow due to the presence of the main reinforcement. At the service load of approximately 6 MN (assumed to be 60% of the ultimate design load), the crack width at the top chord was about 0.8 mm while the crack width at the bottom chord was 0.2 mm. Fig. 8b1 and Table 2 show the crack width at service load.
In the specimen designed and detailed using the Mansur [16] method, the vertical cracks start at the top left and bottom right corners of the opening at an approximate higher load (compared to Kong. F.K [15] model) of 2.3 MN. At a load of 4.0 MN, diagonal cracks start to form at the top left corner of the opening in addition to the development of vertical cracks at the top right and bottom left surfaces of the beam. At the service load of 6.0 MN, the maximum crack width at the bottom chord was 0.3 mm while the one at the top was 0.6 mm as shown in Table 2. The yield point of the reinforcement starts at approximately 11.8 MN at the left upper corner of the opening. Failure occurs due to the presence of extensive shear crack at the left side of the upper chord.
The specimen detailed using the STM method starts to develop cracks at the lower chord at an approximate load of 2.5 MN. The cracks at the upper chord initiate at a higher load of 3.8 MN. At load of 14.81 MN, the concrete in the model reaches its tensile strength cracking at the right section of the upper chord and some sudden deflection occurs resulting in a redistribution of the forces. Finally, more reinforcements undergo yield. Parts a3 and b3 of Fig. 9 and Table 3 show the steel stress and concrete strain at the ultimate load. The crack width at the service load for the bottom chord is 0.3 mm while the one at the top chord is 0.6 mm. The yield point of the reinforcement at the lower chord starts at an approximate load of 9.0 MN while at the upper chord, the reinforcement starts yielding at a higher load of 14 MN. The location of the yielded reinforcement is at the left part of the tie and stirrups on the right. It can be observed that the model details using the STM method are capable to develop and spread the yield of reinforcement over a large area in both shear and bending.
6 Deflection results
The loaddeflection curves resulted from the numerical analyses for the three design approaches are presented in Figure 10. The failure loads, deflection at service and ultimate loads, and crack widths are summarized in Table 4. From the loaddeflection curves it can be observed that the deflection is approximately the same for all the three models at the service loads. At loads close to the ultimate design value of approximately 9 MN, the stiffness of the beam reduces and exhibits faster deflection for the specimen designed and detailed using the Kong. F.K [15] model with the failure occurring close to the ultimate design load. The other two specimens behave similarly up to the failure load. The stiffness is reduced at a load of 12 MN, which is higher than the ultimate design load.
7 Deflection discussion
After the failure, the load decreases suddenly for all specimens except for the specimen designed and detailed using the STM method. The specimen designed according to the STM method restores its capacity and develops more deflection after the initial drop and this indicates that the load can be redistributed and find other alternative load paths after failure in any of its parts. The deflection value can be compared at the serviceability limit. The criterion for the deflection limit is the amount of displacement compared to the span over 500. All these designs satisfy the deflection serviceability limit.
8 Conclusions

1
The results of the finite element nonlinear analyses of transfer deep beams with large openings show that the designs obtained with the three different approaches (Kong. F.K [15], Mansur [16], and the STM method (ACI 31814 [17]) provide sufficient ultimate load capacity values.

2
At service load for all the three design models, the maximum width of cracks was observed near the midspan of the top chord. The maximum crack width for beams designed according to Mansur [16] and STM (ACI 31814 [17]) was approximately 0.6 mm and for beams designed according to Kong. F.K [15] was 0.8 mm. Additional reinforcement may be required to limit the crack width to the design value of 0.30 mm at the service stage.

3
The approach proposed by Mansur [16] may be considered efficient for the design of deep beams with large openings as it considers that the concrete members around the opening will follow a Vierendeel truss action and deals with chord members above and below the opening as an eccentrically loaded members. Designs using this method show considerable higher failure loads compared with the design ultimate load with costbeneficial reinforcement ratios.

4
Comparing the Kong. F.K [15] and Mansur [16] approaches, it is observed that the amount of steel used in both approaches is approximately the same, with the difference in the arrangement of the steel.

5
The results from the analyses of the designs according to the STM method (ACI 31814 [17]) and Mansur [16] method provide an adequate margin in the model capacity in which the numerical load/design load = 1.52 and 1.48, respectively. This indicates that placing the main reinforcement in the top chord in the present transfer beam was most effective in transferring the loads to the supports. Further investigation is recommended to study various stiffness ratios between the top and bottom chords.

6
The reinforcement amount used in the STM method was approximate 25% more than the reinforcement utilized by the other two methods which had the same amount of reinforcement.

7
The deep beam designed according to STM method (ACI31814) shows better ductility compared to the designs of the Kong. F.K [15] and Mansur [16] methods.

8
The three design models give deep beam which satisfy the deflection limit.

9
The current study considers transfer beams under static loads. Further investigations can be extended to consider the effect of cyclic loads.
Availability of data and materials
All data are presented.
Abbreviations
 FE:

Finite element
 STM:

Strut and tie method
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Ismail, E., Issa, M.S. & Elbadry, K. Finite element analysis of reinforced concrete deep beam with large opening. BeniSuef Univ J Basic Appl Sci 10, 25 (2021). https://doi.org/10.1186/s4308802100104z
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DOI: https://doi.org/10.1186/s4308802100104z