ACI STRUCTURAL JOURNAL
TECHNICAL PAPER
Title no. 105-S43
Design and Analysis of Heavily Loaded Reinforced Reinforced Concrete Link Beams for Burj Dubai by Ho Jung Lee, Daniel A. Kuchma, William Baker, and Lawrence C. Novak This paper presents a study on the shear capacity and behavior of reinforced concrete link beams designed for the Burj Dubai Tower, which is the tallest building in the world and will be ready for occupancy in 2009. Several thousand reinforced concrete link beams were used in this structure to interconnect structural walls; in some cases, the factored shear forces in these link beams were up to three times the traditional nominal ACI shear force strength limit. This study presents an examination of the factors that control the design and behavior of heavily loaded reinforced concrete link beams. Nonlinear finite element analysis methods were used to validate and inform the design of the reinforced concrete link beams for Burj Dubai and to examine if the ACI nominal sectional shear force limit is appropriate for this type of member. The results illustrate the undue conservatism of the ACI design provisions and the role of nonlinear analyses in design.
The center hex reinforced concrete core walls provide the torsional resistance of the structure similar to a closed pipe or axle, as shown in Fig. 3. The center hex walls are
Keywords: beam; design; nonlinear finite element analysis; reinforced concrete; shear; strut-and-tie model.
INTRODUCTION The Burj Dubai Tower, when completed, will be the world’s tallest structure. Whereas the final height of this multi-use skyscraper is a “well-guarded secret,” it will comfortably exceed the current record holder of 509 meter (1671 ft) tall Taipei 101. The 280,000 m 2 (3,000,000 ft2) reinforced concrete tower will be used for retail, an Armani hotel, residences, and offices. The goal of the Burj Dubai Tower is not simply to be the world’s highest building—it is to embody the world’s highest aspirations. Designers purposefully shaped the structural concrete for the Burj Dubai to be Y-shaped in plan to reduce the wind forces as well as to keep the structure simple and foster constructibility. The structural system can be described as a buttressed core, as shown in Fig. 1 and 2. Each wing, with its own high-performance concrete core and perimeter columns, buttresses the others via a six-sided central core, or hexagonal hub. The result is a tower that is extremely stiff torsionally. The design team purposely aligned all the common central core and column elements to form a building with no structural transfers. Each tier of the building steps back in a spiral pattern that causes the tower’s width to change at each setback. The advantage of this stepping and shaping is to “confuse the wind.” The wind vortexes never become organized because at each new tier the wind encounters a different building shape that reduces the overall wind loads on the structure. Due to the tapering of the tower, the primary demand on the link beams is from gravity load redistribution, flow from the taller core to the perimeter of the structure. The 280,000 m 2 (3,000,000 ft2) tower and 185,000 m 2 (2,000,000 ft2) podium structures are currently under construction, as shown in Fig. 2. The project is scheduled for completion in 2009.
ACI Structural Journal/July-August 2008
Fig. 1—Tower rendering.
Fig. 2—Construction photo of tower. ACI Structural Journal , V. 105, No. 4, July-August 2008. MS No. S-2007-030 received January 19, 2007, and reviewed under Institute publication policies. Copyright © 2008, American Concrete Institute. All rights reserved, including the making of copies unless permission is obtained from the copyright proprietors. Pertinent discussion including author’s closure, if any, will be published in the MayJune 2009 ACI Structural Journal if the discussion is received by January 1, 2009.
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Ho Jung Lee is an Engineer with SC Solutions, Inc., Sunnyvale, CA. He received his BS and MS from the Seoul National University, Seoul, South Korea, and his PhD from the University of Illinois at Urbana-Champaign, Urbana, IL. Daniel A. Kuchma , FACI, is an Assistant Professor in the Department of Civil and Environmental Engineering at the University of Illinois at Urbana-Champaign. He received his PhD from the University of Toronto, Toronto, ON, Canada. He is a member of ACI Subcommittee 318-E, Shear and Torsion. William Baker is a Partner of Skidmore, Owings & Merrill LLP, Chicago, IL. LLP. Lawrence C. Novak is an Associate Partner of Skidmore, Owings & Merrill LLP.
buttressed by the wing walls and hammer head walls, which behave as the webs and flanges of a beam to resist the wind shears and moments. Outriggers at the mechanical floors allow the columns to participate in the lateral load resistance of the structure; hence, all of the vertical concrete is used to support both gravity and lateral loads. The walls had
concrete strengths ranging from C80 to C60 cube strength and contained portland cement and fly ash. Local aggregates were used for the concrete mixture design. The wall and column sizes were optimized using virtual work/LaGrange multiplier methods. 1 This results in a very efficient structure. The structure was analyzed for gravity (including P-Δ analysis and creep and shrinkage), wind, and seismic loads by a three-dimensional analysis model that consisted of the reinforced rein forced concrete walls, link beams, slabs, mats, mats, piles, and the spire structural steel system. The model consisted of over 73,500 shells and 75,000 nodes.
RESEARCH SIGNIFICANCE This paper examines methods for the design and analysis for reinforced concrete link beams that are cast integral with wall piers. In particular, the appropriateness of the strut-and-tie method and shear design stress limits for this class of member are reviewed. The results of this investigation indicate that much higher shear stress levels should be permitted in ACI 318, as this would greatly extend the utility of this class of member.
OVERVIEW OF LINK BEAM DESIGN
Fig. 3—Three-dimensional view of single floor.
The demands on the link beams vary greatly and are dependent on the location of the link beam relative to a setback with the largest shear forces being generated in the beams closest to a setback. The typical link beams used in the Burj Dubai are quite stocky with a shear-span ratio ( l /2 /2h) of 0.85, a width of 650 mm (2.13 ft), and a height of 825 mm (2.7 ft). For the design of reinforced concrete link beams, the conventional deep beam design method in the ACI 318-99 2 and the strut-and-tie method in ACI 318-02 3 were used, with Appendix A enabling the design of link beams somewhat beyond the conventionally designed maximum deep beam stre stress ss limi limitt of 10 f c′ in psi psi (0.8 (0.83 3 f c′ in MPa) MPa),, whic which h is based on ACI 318-99,2 Section 11.8.4, as will be discussed in the following. In the case of members subjected to very large shear forces, embedded built-up structural steel sections were provided within the core of the concrete link beams to carry the entire shear and flexure demand. This study was principally conducted during the structural design of the Burj Dubai and used to check and inform the design by the different methods and to examine the condition and stiffness of the reinforced concrete link beams under service and factored loads. In addition, a study was made of the appropriateness of nominal shear stress limits for link beams. A series of nonlinear analyses were conducted that can account for the influence of many factors on response including the amount of flexural reinforcement, the distribution of vertical and horizontal web reinforcement, the span -depth ratio, and the confinement confinement provided by walls at the ends of the link beams. The nonlinear finite element analysis tools used in this investigation were ABAQUS, 4 ADINA,5 and VecTor2.6
DESIGN DETAILS OF LINK BEAMS
Fig. 4—Design details for analyzed link beams. (Note: 1 mm = 0.0394 in.)
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The geometry, factored loads, and design methods of four Burj Dubai link beams, LB1 to LB4, are shown in Table 1. These link beams, which have the same external dimensions, capture the range of typical shear design force levels for which different design solutions were used. Table 2 and Fig. 4 present details on Link Beams LB1 and LB2, as well as Link Beams LB2A and LBRCMAX that are more heavily reinforced and hypothetical link beams whose behavior is evaluated in the
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nonlinear analyses. Link Beam LB1 was designed by the deep design method specified in ACI 318-99. 2 Link Beam LB2 was designed by the strut-and-tie model in Appendix A of ACI 318-02.3 Vertical shear or tie reinforcement in Link Beams LB1 and LB2 was determined using the selected design approaches to support their factored design loads. Horizontal web reinforcement close to the minimum amount suggested for deep beams in ACI 318-02 3 was used in Link Beams LB1 and LB2. The design of Link Beam LB2 using the strut-and-tie method in the ACI 318-02 3 provisions, unchanged in ACI 318-05,7 is described in the next section. As previously mentioned, a pure reinforced design concrete solution was not possible for all members by ACI 318-02, 3 or would be by ACI 318-05,7 such that composite members with steel-embedded sections were used to support the shear design in members subjected to very large shear and flexural forces, including Link Beams LB3 and LB4. An objective of this study was to investigate whether or not it was possible to develop a pure reinforced concrete solution to support the very large shear forces in Link Beams LB3 and LB4. To this end, the behavior of more heavily reinforced members, Link Beams LB2A and LBRCMAX, as described in Table 2 and Fig. 4, will also be examined. The concrete cylinder compressive strength used in the design of these link beams was f c′ = 64 MPa (9280 psi); a concrete cube strength of 80 MPa (11600 psi) was specified and actual cube and cylinder breaks indicate considerably stronger concrete. The design yield strength of flexural reinforcement used in the link beams was 460 MPa (67 ksi) and of the vertical stirrups and horizontal web reinforcement was 420 MPa (61 ksi). Reinforcement with a yield strength of 460 MPa (67 ksi) was actually provided, but in accordance with Section 11.5.2 of ACI 318-02, 3 the effective strength was taken as 420 MPa (61 ksi). Nonlinear finite element analyses of these reinforced concrete link beams (LB1, LB2, LB2A, and LBRCMAX) were performed as will be presented.
The strut-and-tie model used for the design of Link Beam LB2 is shown in Fig. 5. Th e design yields the required amount of horizontal and vertical tie reinforcement and the required strength of diagonal concrete struts and nodal zones. Minimum horizontal web reinforcement was provided in the link beams even though it was not specifically required when the strut-and tie design procedure is used. In this link beam design, the amount of horizontal and vertical ties can simply be increased to support larger shear loads until the strength is limited by the strength of the diagonal struts and nodal zones. The use of Appendix A in ACI 318-02 3 and ACI 318-057 provides for a direct design of the diagonal strength of struts so as to avoid a diagonal compressive failure. This permits a member to be designed for a higher shear stress than the limit for deep beams in ACI 318-99.2 This deep beam stress limit was also set to guard against diagonal compression failures but is not considered to be necessary for deep beams as will be discussed in the following. In the design of Link Beam LB2 by Appendix A of ACI 318-02, the conservative assumption of narrow bottle-shaped diagonal struts were made even though the results of the analyses indicate that there is a uniform field of diagonal compression throughout these members. Thus, it was considered that the nominal capacity calculated for this strut-and-tie model by ACI 318-02 would lead to a conservative design.
MAXIMUM SHEAR STRENGTH SPECIFIED IN CODE PROVISIONS There is a large variation in the nominal shear design stress limit that is specified in codes of practice even though the reason for this limit is the same as guarding against a diagonal
Table 1—Geometry, loading, and design methods for link beams Geometry
STRUT-AND-TIE MODEL USED IN LINK BEAM The strut-and-tie method has recently developed as a rational method in the design of discontinuity (D)-regions in structural concrete such as deep beam, squat walls, pile caps, and other elements in which plane sections do not remain plane. The strut-and-tie method provides a conceptually simple design methodology based on the lower-bound theorem of limit analysis. 8 Provisions for using the strut-and-tie method were included as Appendix A in ACI 318-02.3
Factored loads
Beam Width, Depth, Span, Shear, Moment ID mm mm mm kN kN-m
Design method used
LB1
650
825
1400 1705
1194
Conventional (ACI 318-99, Section 11.8)
LB2
650
825
1400 2805
1164
Strut-and-tie (ACI 318-02, Appendixes A and C)
LB3
650
825
1400 3750
2625
Steel plate
LB4
650
825
1400 5250
3675
Built-up steel I-beam
Notes: Factored loads are equal for both ends of link beams. Ratio of ultimate load to sustained day-to-day (gravity only) service loads is approximately 2.5. Walls adjacent to link beams are 650 mm thick and are typically reinforced with a minimum of T20mm at 350 mm on each vertical and horizontal face. 1 in. = 25.4 mm; 1 kip = 4.448 kN; and 1 ft-kip = 0.356 kN-m.
Table 2—Reinforcing details of link beams analyzed in study Reinforcement
Fig. 5—Strut-and-tie model used in design of Link Beam LB2. (Note: (Note: 1 in. = 25.4 mm; 1 kip = 4.448 kN; 1 psi = 6.89 × 10 –3 MPa; and 1 ft-kip = 1.356 kN-m.)
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Beam ID
Top Top bars
Bottom bars
LB1
5-T32
5-T32
Stirrups
Side bars Spacing, each each face face Size Size mm 5-T12
T16
150
Type Two hoops
LB2
12-T32
12-T32
4-T12
T16
125
Two hoops
LB2A
18-T32
18-T32
4-T12
T16
80
Three hoops
LBRCMAX 27-T32
27-T32
4-T12
T16
75
Five ho hoops
Notes: T32, T20, T16, and T12 are deformed reinforcing bars with respective diameters of 32, 20, 16, and 12 mm. In LB2, top and bottom bars were used in two layers each. LB2A and LRCMAX were not used in Burj Dubai project. They are included for purpose of examining appropriateness of current ACI 318-05 limit on maximum shear stress. LB2A has significantly more longitudinal tension reinforcement and transverse reinforcement than LB2. LBRCMAX is analyzed to figure out maximum shear capacity of reinforced concrete link beam. In LB2A and LBRCMAX, top and bottom bars were placed in two layers and three layers each, respectively. 1 in. = 25.4 mm.
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compression failure. For the limits shown in Table 3, the range in maximum permitted shear stress is more than a factor of two for the 64 MPa (9280 psi) concrete used in the design of the Burj Dubai. The ACI conventionally designed beam shear limit of 10 f c′ (psi) (0.83 f c′ [MPa]) was derived from simply supported beam tests in which there is a funneling and thus magnification of diagonal compressive stresses as they flow from the top of the beam to its support and in which anchorage failures are common. The AASHTO LRFD9 and CSA (Canadian) code 10 shear stress limit of 0.25 f c′ for reinforced concrete members was derived from the modified compression field theory. 11) In this approach, the influence of diagonal tensile strains on the capacity of a uniform diagonal compression field is directly considered. The flow of diagonal forces in link beams, which are supported over their ends, are expected to be uniform as illustrated later and thus the limit in the LRFD and CSA provisions is more appropriate. For the reinforced concrete link beams used in the Burj Dubai, there is the further structural advantage of the confinement effect from the pier walls. The walls on both sides help to prevent vertical expansion of the link beams at their ends, which limits diagonal cracking. The use of a higher shear design stress limit in these link
Fig. 6—Geometry and detailing of link beam tested by Galano et al.14 (Note: 1 in. = 25.4 mm; and 1 psi = 6.89 × 10 –3 MPa.)
beams is investigated in the nonlinear finite element analyses that are presented in the next section.
NONLINEAR FINITE ELEMENT ANALYSIS The structural behavior of reinforced concrete link beams is influenced by many factors including the amount of flexural reinforcement, vertical and horizontal shear reinforcement, the span-depth ratio, and the confinement effect of the adjacent pier walls. Nonlinear finite element analyses can be used to evaluate the effect of these parameters. In this section, the nonlinear finite analyses of the Burj Dubai link beams (LB1 and LB2) and the more heavily reinforced link beams (LB2B and LBRCMAX), as described in Table 2, are presented. Three different programs, ABAQUS, ADINA, and VecTor2, were used for the finite element analysis of these reinforced concrete link beams. In ABAQUS, the concrete damaged plasticity model was selected. This model is based on the work of Lubliner et al. 12 and Lee and Fenves. 13 The concrete damaged plasticity model is intended to provide a general capability for the analysis of concrete structures subjected to static and dynamic loading under low confining pressures. It is based on the combination of nonassociated multi-hardening plasticity and scalar damaged elasticity models. The program ADINA provided the option to adjust various concrete strength and ductility parameters as well as failure envelopes to better represent the concrete behavior for a specific application. The program VecTor2 implements the rotating-angle smeared-crack model of the modified compression field theory and quadrilateral elements are used in the analyses. Before conducting the finite element analyses of these link beams, the finite element models were calibrated and validated using experimental results that accounted for the effects of complex geometries, loadings, and edge effects. Because this study focuses on the strength of the link beams, it was investigated to see if the selected parameters for the concrete material model in ABAQUS, ADINA, and VecTor2 predict well the strength of link beams tested to failure in laboratories. One of the comparisons used the link beam tested under monotonic loading by Galano et al. 14 that had a shear span ratio of 0.75. This ratio was the closest to that of the Burj Dubai link beams (shear ratio = 0.85) for the identified test data. The geometry and reinforcement details of the tested
Table 3—Maximum nominal shear design stress permitted by code provisions Design code
Maximum shear strength, V n, max
ACI 318-02 and ACI 318-05
V c + 8 f c b w d
AASHTO STD1
V c + 8 f c b w d
AASHTO LRFD3
0.25 f c b v d v + V p
AASHTO segmental bridges2
12 f c b w d
CSA (1994) 10
0.25 f c b v d v + V p
CSA (2004) 11
0.25 f c b v d v + V p
JSCE14
15 f c b w d
′
′
′
′
′
′
′
Notes: V c equals nominal shear resistance provided by concrete, f c equals concrete compressive strength, bw equals width, d equals distance from compression face to centroid of longitudinal tension reinforcement, bv equals web width including adjustment for presence of ducts, d v equals effective shear depth, and V p equals component in direction of applied shear of effective prestressing force. ′
Fig. 7—Predicted load-deformation response of link beam tested by Galano et al. 14 (Note: 1 in. = 25.4 mm; and 1 kip = 4.448 kN.)
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link beam are shown in Fig. 6. Four-node plane stress isoparametric elements were used in the analyses. Reinforcing steel was modeled as two-node truss elements. The top section of the left wall is restrained and the shear force is applied to the bottom section of the right wall by displacement control; the same prescribed vertical displacement is used along the section. The analysis results are presented in Fig. 7 and the strengths predicted by ABAQUS, ADINA, and VecTor2 are shown to predict the experimentally measured capacity reasonably well. The results of other validation studies were similar. Therefore, the selected parameters for the concrete model in ABAQUS, ADINA, and VecTor2 were taken to be sufficiently accurate for capacity evaluation of the four link beams in this study. The load and boundary conditions of the finite element model for the Burj Dubai link beams are shown in Fig. 8. The wall boundary is extended vertically up to the depth of the link beam and horizontally up to the length of the link beam. This extension with the constraints put in the finite element model provides a realistic boundary for the investigation of link beam capacity that was expected to lead to conservative (lower bound) estimations of capacity. The load and boundary conditions are the same as that of the link beam tested by Galano et al. 14 except the stiff perimeter steel was used to account for the constraining effect of the wall as the boundary of the model. The additional benefit of axial compression in the walls, as investigated by other analyses not presented in this paper, was found to only have a very minimal beneficial effect on the capacity of the link beams; it is the link beam end restraint rather than axial compression in the wall piers that influences the behavior of the link beams. Thus, the axial load acting on the walls was not considered in the reported analyses. The strain variation of reinforcing steel identified in Fig. 8 was investigated as well as the crack patterns and capacity of the link beams. Additional information on the finite element models is now presented and this is followed in the next section by the predicted capacity and behavior of the link beams. In creating the finite element models, it was necessary to select and specify material models and select convergence limits. In all analyses, the response of the reinforcement was linear elastic and perfectly plastic. The default value of the convergence limit in VecTor2 was used, a convergence value of up to 0.01 was used in ADINA, whereas the stability command was used in ABAQUS, but its use was found to have little effect on the predicted behavior. The philosophy of VecTor2 is that the user is not required to select nonbasic material parameter values and thus only the compressive cylinder strength was specified and all other default values were accepted. ADINA and ABAQUS require the user to select many other parameter values. The selections made in these programs are given in the following. In ADINA, the required concrete material input parameters were tangent modulus at zero strain (44,000 MPa [6380 ksi]), Poisson’s ratio (0.15), uniaxial cut-off tensile stress (4.83 MPa [700 psi]), post-cracking uniaxial cut-off tensile stress (4.83 MPa [700 psi]), uniaxial maximum compressive stress (–64 MPa [–9280 psi] for unconfined concrete; –102.4 MPa [–14848 psi] for confined concrete) and corresponding compressive strain str ainss (–0.002 for unconf unconfined ined concre concrete; te; –0.0032 for confined co nfined concrete), ultimate uniaxial compressive stress (–54.4 MPa [–7888 psi] for unconfined concrete; –87 MPa [–12615 psi] for confined concrete), and ultimate uniaxial compressive strain (–0.003 for unconfined concrete; –0.0048 for confined
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concrete). These input parameters define the stress-strain curve of concrete. In ABAQUS, the compressive and tensile response was defined similarly as in ADINA, but the prediction by ABAQUS was based on the concrete damaged plasticity model. In this model, the value of the flow potential eccentricity parameter has a significant effect on the concrete response. This value was set to be 20 based on the calibration with experimental data, whereas whereas the dilation dilation angle angle was similarly selected selec ted to be 36. Default values were used for the other input parameters in the concrete damaged plasticity model.
DISCUSSION OF PREDICTED LINK BEAM BEHAVIOR The finite element analysis results of Link Beam 1 (LB1) are shown in Fig. 9 through 11. ABAQUS, ADINA, and VecTor2 all predicted similar responses for Link Beam LB1. According to Fig. 9, the predicted capacity is much larger than the factored design load. Therefore, the use of the deep beam design method as specified in the ACI 318-992 Code is considered to lead to a very conservative design for this link beam. Crack patterns at the loading steps similar to the factored design load and at the ultimate load are shown in Fig. 10. At the factored design load, flexural cracks occur at the boundary regions of the link beam under tension,
Fig. 8—Load and boundary conditions used in modeling of Burj Dubai link beams.
Fig. 9—Predicted load-deformation response of Link Beam LB1. (Note: 1 in. = 25.4 mm; and 1 kip = 4.448 kN.)
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whereas no shear cracking was observed in the central portion of the link beam. The development of straining in the longitudinal and shear reinforcement is presented in Fig. 11. The longitudinal tensile strain increased gradually with increasing shear force until the capacity of the member was reached at the point of longitudinal yielding. As shown in Fig. 11(b), the transverse reinforcement strain rapidly increased with the formation of diagonal cracks, but the transverse strain at the capacity of the member was less than the yield strain of 0.0021. At the ultimate load, severe flexural cracks occurred and extensive shear cracking was predicted. In summary, the finite element analysis results predict that the failure mode of Link Beam LB1 was yielding of longitudinal tension reinforcement and that shear failures did not occur. It is also observed that Link Beam LB1, as designed by the deep beam design method in ACI 318-99, 2 was more conservatively designed for shear than for flexure. The predicted load-deformation response of Link Beam 2 (LB2) by the nonlinear finite element analyses is shown in Fig. 12 in which the predicted capacity is considerably larger than the factored design load. The strut-and-tie method specified in ACI 318-057 was therefore observed to lead to a conservative conservativ e design for this link beam. ABAQUS, ADINA, and VecTor2 provide similar predictions of behavior of this link beam. Two different concrete models were used in the ADINA analyses. One is an unconfined concrete stress-strain model and the other is a confined concrete model. ADINA and ABAQUS do not directly consider the confinement effect by the pier walls at the end of the link beams. Thus, if the unconfined model is used in ABAQUS and ADINA for the
analysis of Link Beam LB2, the capacity of Link Beam LB2 is slightly decreased due to concrete crushing at the ends of the link beam. This is not expected in practice because the pier walls provide significant restraint to the vertical expansion of the link beam and this increases the concrete compressive strength at the ends of these link beams. VecTor2 directly considers concrete confinement and softening without additional input from users. When the confined concrete
Fig. 11—Predicted development of reinforcement strains in Link Beam LB1. (Note: 1 kip = 4. 448 kN.)
Fig. 10—Predicted crack patterns by ADINA for Link Beam LB1. (Note: 1 kip = 4.448 kN.)
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Fig. 12—Predicted load-deformation response of Link Beam LB2. (Note: 1 psi = 6.89 × 10 –3 MPa, 1 in. = 25.4 mm; and 1 kip = 4.448 kN.)
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model is used in ABAQUS and ADINA for the analysis of the link beams, it i t gives similar results res ults to those of the the VecTor2 VecTor2 default model. The predicted development of reinforcement strains for Link Beam LB2 is shown in Fig. 13. For Link Beam LB2, the strain in the transverse reinforcement reaches yield before yielding of the longitudinal reinforcement. The capacity and mode of failure of Link Beam 2 is predicted to occur at the point of yielding of both the longitudinal and transverse reinforcement. The finite element analysis results of hypothetical Link Beams LB2A and LBRCMAX, which contained higher levels of longitudinal and shear reinforcement, are shown in Fig. 14 through 17. ABAQUS, ADINA, and VecTor2 provide similar predictions of the behavior of these link beams. Without consideration of the confinement effect by the walls, the analysis results predict a maximum shear stress capacity limit that is 70% larger than the conventionally designed beam shear stress limit of 10 f c′ (psi) (0.83 .83 f c′ [MPa [MPa]) ]) that that is specified in ACI 318-05. 7 If the confinement effect by the pier walls is considered in the analysis through the use of the confined concrete stress-strain model, the shear stress capacity of the reinforced concrete link beam can be 2.5 times the ACI shear stress limit. This is illustrated in the predicted behavior of Link Beam LBRCMAX in which very significant amounts of longitudinal and transverse reinforcement were provided. For both Link Beams LB2A and LBRCMAX, the predicted transverse reinforcement
strains were considerably smaller than yield strain at the point of yielding of the longitudinal reinforcement. Therefore, it is predicted that the capacity of these link beams is limited by their flexural capacities. Sectional analyses of Link Beams LB2A and LBRCMAX were also performed using the sectional analysis program Response 2000.15 Response 2000 15 is also based on the modified compression field theory and employs employs the engineering engineer ing beam theory assumption that plane sections remain plane. Unlike with the nonlinear finite element analyses, Response 200015 cannot account for the beneficial effects of confinement and strut action. Under combined shear and bending loads, the sectional capacities predicted by Response 200015 for Link Beams LB2A and LBRCMAX were 6026 and 7600 kN (1355 and 1709 kips), respectively. The state of cracking in Link Beam LBRCMAX was also predicted by program Response 2000 15 as shown in Fig. 18 at its ultimate capacity. The member analysis by Response 200015 for Link Beam LBRCMAX supports the conclusion from the nonlinear finite element analyses that a shear force close to the LRFD and CSA limit can be supported by a reinforced concrete link beam of the dimensions used in the Burj Dubai. The strength results of this investigation are summarized in Fig. 19, which presents a comparison of the factored shear
Fig. 14—Predicted load-deformation response of hypothetical Link Beam LB2A. (Note: 1 in. = 25.4 mm; and 1 kip = 4.448 kN.)
Fig. 13—Predicted development of reinforcement strains in Link Beam LB2. (Note: 1 kip = 4.448 kN.)
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Fig. 15—Predicted load-deformation response of hypothetical Link Beam LBRCMAX. (Note: 1 in. = 25.4 mm; and 1 kip = 4.448 kN.)
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design loads for link beams (LB1 to LB4) in the Burj Dubai and the predicted shear capacities by the nonlinear finite element analyses for Link Beams LB1, LB2, LB2A, and LBRCMAX. The shear force corresponding to the ACI 318-057 conventionally designed beam shear stress limit of 10 f c′ (psi) (0.38 f c′ [MPa]) is also shown. The results of the analyses predict that the designs are very conservative
for Link Beam LB1 that was designed by the ACI 318-99 2 deep beam provisions and for Link Beam LB2 that was designed by the strut-and-tie model of Appendix A in ACI 318-02.3 In the Burj Dubai, composite link beams with large embedded sections were used to support the shear and flexure demand when this demand was in excess of what could be designed for by ACI 318 sectional or strut-and-tie design procedures. The results of the analyses for hypothetical Link Beams LB2A and LBRCMAX suggest that it would have been possible to support the large shear force demands in Link Beams LB3 and LB4 with reinforced concrete link beams; however, reinforcing bar congestion and constructibility concerns would need to be evaluated for Link Beams LB2A and LBRCMAX.
CONCLUSIONS In this paper, the design and capacity of heavily loaded reinforced concrete link beams were investigated by nonlinear finite element analyses using the programs ABAQUS,4 ADINA,5 and VecTor2.6 The following is a summary of results from this study: 1. The effect of the pier walls at the ends of the link beams was predicted by the analyses to reduce the vertical transverse expansion at the ends of the members and thereby reduce the demands on transverse reinforcement. The pier walls also provided confinement that enabled the link beams to support larger compressive stresses at their ends. Furthermore, they enabled a more uniform field of diagonal compression and vertical distribution of shear over the depth of the member throughout the entire length of the link beam;
Fig. 16—Predicted development of reinforcement strains in hypothetical Link Beam LB2A. (Note: 1 kip = 4.448 kN.)
Fig. 17—Predicted development of reinforcement strains in hypothetical Link Beam LBRCMAX (Location A in Fig. 8). (Note: 1 kip = 4.448 kN.)
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Fig. 18—Predicted cracking in hypothetical Link Beam LBRCMAX by Response 2000 at ultimate load (V = 1709 kips [7600 kN]). (Note: 1 in. = 25.4 mm; and 1 kip = 4.448 kN.)
Fig. 19—Comparison of shear demands and capacities in link beams.
ACI Structural Journal/July-August 2008
2. The use of the deep beam design method in the ACI 318-992 provisions leads to very conservative shear designs for link beams as suggested by the results of multiple nonlinear finite element analyses. The ACI 318-99 2 shear stress limitation for deep beams, which is a function of the span-depth ratio and imposed to guard against a diagonal compression failure, unnecessarily limits the load-carrying capacity of stocky link beams supported over their heights such as those used in the Burj Dubai project; 3. The strut-and-tie model in Appendix A of ACI 318-02 3 and ACI 318-057 permits reinforced concrete link beams to be designed for substantially higher loads than would be possible by the use of ACI sectional design methods. In the use of the strut-and-tie model, a direct check is made to ensure that the diagonal compressive struts have adequate capacity. Therefore, a design completed by the strut-and-tie model negates the need to satisfy any sectional shear stress limitation; 4. The results of nonlinear finite element analyses indicate that reinforced concrete link beams have significantly larger shear capacities than the nominal strengths calculated by the strut-and-tie model of ACI 318-05. 7 The capacity of the members was predicted to be controlled by yielding of the longitudinal reinforcement; reinforcement; the compressive and nodal stress limits in ACI 318-05 7 were found to be conservative; and 5. There is a wide variation in the maximum shear stress limits in codes of practice. The difference is more than a factor of two between the sectional design models in ACI 318-057 (10 f c′ [psi] [0.83 f c′ (MPa)]) and th the Canadian and AASHTO LRFD Design Codes (0.25 f c′ ) for link beams cast with 64 MPa (9280 psi) concrete, as used in the Burj Dubai. The higher shear stress limit in the Canadian and AASHTO LRFD code was found to be appropriate for short link beams that are supported over their height at their ends by continuous wall piers. piers.
ACKNOWLEDGMENTS The analyses presented in this paper were performed for Skidmore, Owings & Merrill LLP (SOM) LLP, Chicago, IL, under the direction of W. Baker, L. Novak, A. Ozkan, and S. Korista.
ACI Structural Journal/July-August 2008
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