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Coupling of Concrete Walls Using Post-Tensioned Precast Concrete Beams Brad D. Weldon 1 and Yahya C. Kurama 2 1. Graduate Research Assistant, Structural Systems Laboratory, Dept. of Civil Eng. and Geological Sciences, Univ. of Notre Dame, 156 Fitzpatrick Hall, Notre Dame, Indiana 46556; PH (574) 631-4307; FAX (574) 631-9236; email : [email protected] 2. Associate Professor, Dept. of Civil Eng. and Geological Sciences, Univ. of Notre Dame, 156 Fitzpatrick Hall, Notre Dame, Indiana 46556; PH (574) 631-5380; FAX (574) 631-9236; email: [email protected] Abstract This paper describes an analytical investigation on the nonlinear behavior of post-tensioned precast concrete beams to couple reinforced concrete walls. Different from conventional systems that use monolithic cast-in-place reinforced concrete coupling beams or embedded steel coupling beams, the nonlinear behavior of post- tensioned precast concrete coupling beams is governed by the opening of gaps at the beam ends. Steel top and seat angles are used at the beam-to-wall connections to yield and dissipate energy in the event of a large earthquake. Two types of analytical models are developed using the DRAIN-2DX and ABAQUS programs for verification of the results and more detailed evaluation of the expected behavior. Nonlinear reversed cyclic lateral load analyses of four coupling beam subassemblies are conducted. The main parameters investigated are the beam depth, the area of the post-tensioning steel, and the size of the top and seat angles. The results indicate that post-tensioned precast concrete coupling beams possess stable behavior through large rotations, significant self-centering capability due to the post-tensioning force, and significant energy dissipation provided by the yielding of the top and seat angles. Introduction Historically, concrete structural walls have performed extremely well as primary lateral load resisting systems under earthquake loading. When coupled together, structural walls demonstrate an increase in lateral strength and stiffness, allowing a smaller number of walls to be used to achieve the required design strength of a building. Coupling beams are used at the floor and roof levels to transfer shear forces between the wall piers and to dissipate energy over the height of the structure. Previous research on coupled wall systems has focused on cast-in-place reinforced concrete coupling beams monolithic with the wall piers and hybrid systems with steel

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Coupling of Concrete Walls Using Post-Tensioned Precast Concrete Beams

Brad D. Weldon1 and Yahya C. Kurama2

1. Graduate Research Assistant, Structural Systems Laboratory, Dept. of Civil Eng.and Geological Sciences, Univ. of Notre Dame, 156 Fitzpatrick Hall, Notre Dame, Indiana 46556; PH (574) 631-4307; FAX (574) 631-9236; email: [email protected]. Associate Professor, Dept. of Civil Eng. and Geological Sciences, Univ. of Notre Dame, 156 Fitzpatrick Hall, Notre Dame, Indiana 46556; PH (574) 631-5380; FAX (574) 631-9236; email: [email protected]

Abstract

This paper describes an analytical investigation on the nonlinear behavior of post-tensioned precast concrete beams to couple reinforced concrete walls. Different from conventional systems that use monolithic cast-in-place reinforced concrete coupling beams or embedded steel coupling beams, the nonlinear behavior of post-tensioned precast concrete coupling beams is governed by the opening of gaps at the beam ends. Steel top and seat angles are used at the beam-to-wall connections to yield and dissipate energy in the event of a large earthquake. Two types of analytical models are developed using the DRAIN-2DX and ABAQUS programs for verification of the results and more detailed evaluation of the expected behavior. Nonlinear reversed cyclic lateral load analyses of four coupling beam subassemblies are conducted. The main parameters investigated are the beam depth, the area of the post-tensioning steel, and the size of the top and seat angles. The results indicate that post-tensioned precast concrete coupling beams possess stable behavior through large rotations, significant self-centering capability due to the post-tensioning force, and significant energy dissipation provided by the yielding of the top and seat angles.

Introduction

Historically, concrete structural walls have performed extremely well as primary lateral load resisting systems under earthquake loading. When coupled together, structural walls demonstrate an increase in lateral strength and stiffness, allowing a smaller number of walls to be used to achieve the required design strength of a building. Coupling beams are used at the floor and roof levels to transfer shear forces between the wall piers and to dissipate energy over the height of the structure.

Previous research on coupled wall systems has focused on cast-in-place reinforced concrete coupling beams monolithic with the wall piers and hybrid systems with steel

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coupling beams embedded into the wall piers (e.g., Harries et al. 2000; Harries 2001). More recent research at the University of Notre Dame (Shen and Kurama 2000, 2002a,b; Kurama and Shen 2004; Kurama et al. 2004, 2005; Shen et al. 2005a,b) has introduced a new type of hybrid coupled wall system using steel beams that are not embedded into the walls. In this new system, coupling is achieved by post-tensioning the beams and the wall piers together at the floor and roof levels.

Post-tensioned coupling beams offer important advantages over conventional systems with monolithic concrete and embedded steel beams, such as simpler detailing for the beams and the wall piers, reduced damage to the structure, and an ability to self-center, thus reducing the residual lateral displacements of the structure after a large earthquake. This paper further extends the use of post-tensioning in coupled wallsystems by presenting analytical investigations on the nonlinear behavior of post-tensioned precast concrete coupling beams. The use of precast beams offers additional advantages over steel beams such as single trade construction, simpler beam-to-wall joint regions, better fire and environmental protection for the post-tensioning (PT) tendons, and simpler construction.

lw lw lb

2nd floor

3rd floor

4th floor

5th floor

6th floor

7th floor

8th floor

roof

hw

(a) (d)

A

coupling

PT tendon

wall

connection region

anchorage

angle

(b)

(c)

contact

gap opening

lw

confinement beam

region

region

reference line

beam-to-wall

confinement

A

lw lb

hs

C

C V

V

z db

lb

coupling beam

section at A-A

dda

Ta

Ta

Ca

b

b

Ca

V =lb

C z + (T +C ) daa

b abp

b a

b

b

b

bb

Figure 1: Coupled wall system (adapted from Shen and Kurama (2002b)) –(a) multi-story wall; (b) subassembly; (c) deformed shape; (d) coupling forces

As an example, Figure 1(a) shows a multi-story coupled wall system and Figure 1(b) shows a post-tensioned coupling beam subassembly consisting of a precast concrete coupling beam and the adjacent concrete wall regions at a floor level. The wall regions extend half-story above and half-story below the floor level. High-strength multi-strand tendons run through the wall piers and the beam providing the post-tensioning force to the system. The post-tensioning tendons are unbonded over their entire length (by placing the tendons inside ungrouted ducts) and are anchored to the structure only at two locations at the outer ends of the wall piers. The beam-to-wall connection regions include two top and two seat steel angles.

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If the coupled wall structure is displaced with lateral forces acting from left to right, the expected deformed configuration of the subassembly is shown in Figure 1(c). The non-linear deformations of the beam occur primarily due to the opening of gaps at the beam-to-wall interfaces. The application of the post-tensioning force develops a friction force at the beam-to-wall interfaces, supporting the beam. Under large displacements, a properly designed subassembly is expected to experience yielding of the steel top and seat angles, without significant damage to the wall piers or to the coupling beam. The angles provide redundancy in support of the beam as well as energy dissipation during an earthquake. The angles also provide a part of the moment resistance at the beam ends and prevent sliding of the beam against the wallpiers (together with friction induced by post-tensioning). The angles are designed to be sacrificial and can be replaced after the earthquake. Bonded longitudinal mild steel reinforcement is used at the beam ends to transfer the angle forces into the beam. Thebonded mild steel reinforcement is not continuous across the beam-to-wall interfaces, and thus, does not contribute to the beam end moment resistance.

Note that, as shown in Figure 1(c), kinking of the post-tensioning tendons may occur at the beam-to-wall interfaces as the beam rotates with respect to the wall piers. Experiments conducted by Priestley and MacRae (1996), Kurama et al. (2004, 2005), Morgen and Kurama (2004), and Shen et al. (2005a) have shown that kinking of multi-strand unbonded post-tensioning tendons does not affect the behavior of the system or the performance of the tendons.

As gaps open at the beam-to-wall interfaces, large compressive stresses due to post-tensioning are pushed toward the corners of the beam forming a diagonal compression strut. As shown in Figure 1(d), it is through the formation of this compression strut that the coupling shear force Vb is developed. The amount of coupling between the wall piers and the energy dissipation of the subassembly can be controlled by varying the total post-tensioning force Pb (which controls the compression force in the beam Cb), the tension and compression angle forces Ta and Ca, the beam depth db, and the beam length lb. To resist the large compression stresses that develop in the wall piers and the coupling beam due to post-tensioning and gap opening, concrete confinement is provided in the contact regions at the beam-to-wall interfaces. Note that as a result of the development of a diagonal compression strut along the length of the beam, the amount of shear reinforcement needed in a post-tensioned precast concrete coupling beam is expected to be significantly smaller than the amount of shear reinforcement needed in a monolithic cast-in-place reinforced concrete beam.

Before the initiation of gap opening, the post-tensioning force creates an initial lateral stiffness in the precast concrete coupling beam similar to the initial stiffness of a comparable monolithic cast-in-place reinforced concrete beam. Gap opening at the ends of the precast beam results in a reduction in the lateral stiffness, allowing the system to soften and undergo large nonlinear rotations without significant damage (except in the angles and minimal amounts of cover concrete spalling in the beam-to-wall contact regions). The post-tensioning force controls the size of the gaps and the contact depth between the wall piers and the coupling beam. As the wall is displaced

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laterally, the tensile forces in the post-tensioning tendons increase, thus, resisting gap opening. Upon removal of the lateral loads, the post-tensioning tendons provide a restoring force that tends to close the gaps, thus returning the beam and the wall piers to their undeformed configuration, leaving little or no residual displacements (i.e. self-centering capability). Unbonding of the post-tensioning tendons has two important advantages: (1) it results in a uniform strain distribution in the tendons, thus, delaying the nonlinear straining (i.e. yielding) of the steel; and (2) it significantly reduces the amount of tensile stresses transferred to the concrete, thus reducing or preventing cracking in the wall piers and the coupling beam.

Analytical Modeling

Analytical models of unbonded post-tensioned precast concrete coupling beam subassemblies were developed using the DRAIN-2DX program (Prakash et al. 1993) and the ABAQUS program (Hibbitt, Karlsson, and Sorensen Inc. 2002). These models are based on previous models developed for unbonded post-tensioned steel coupling beams as described in Shen and Kurama (2000, 2002a,b) and Shen et al. (2005a,b). The following sections focus on the model modifications needed for the investigation of precast concrete coupling beams instead of steel beams.

DRAIN-2DX Model. The DRAIN-2DX model (Figure 2) uses fiber beam-column elements to represent the wall piers and the coupling beam, zero-length spring elements to represent the top and seat angles, and truss elements to represent the unbonded post-tensioning tendons. Similar to the previous steel beam model described in Shen and Kurama (2000, 2002a,b) and Shen et al. (2005a,b), each wall pier region is modeled using two sets of fiber beam-column elements. The first set consists of elements that are in the vertical direction to model the axial-flexural and shear behavior of the wall region along its height. These elements, referred to as the “wall-height elements”, are used to model the cross-section of each wall pier in the horizontal X-Z plane.

The second set of fiber elements is used to model the local behavior of the wall contact regions to the left and right of the coupling beam. These elements, referred to as the “wall-contact elements”, are in the horizontal direction. Large concentrated compressive stresses develop in the contact regions between the beam and the wallpiers upon gap opening. The wall-contact elements model the deformation of the concrete in the wall contact regions under these large stresses. The fiber cross-section properties of the wall-contact elements are determined from the properties of an “effective” wall cross-section in the vertical Y-Z plane. The thickness of the effective

wall-beam elements

LEFT WALL REGION RIGHT WALL REGION

kinematicconstraint

heightelements contact

elements

lw

PT element

horizontal angle elem.vertical angle elem.

wall-

lb lw

X

Y

Z

A B DC

nodePT anchor

Figure 2: DRAIN-2DX fiber element model

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wall cross-section is assumed to be equal to the wall thickness, tw. The depth of the effective wall cross-section is equal to the beam depth, db at the beam-to-wall interface and is assumed to increase away from the interface with a slope of 1:3. The compressive stresses in a wall contact region decrease away from the interface as a result of an increase in the depth of the compression region inside the wall. The increase in the depth of the effective wall cross-section represents this increase in the depth of the compression region away from the interface.

Three wall-contact elements are used between the center of the left wall region (Node A) and the beam-to-wall interface (Node B) as shown in Figure 2. The Y-translational degree-of-freedom (DOF) of Node B is kinematically constrained to Node A assuming no slip occurs at the interface. The rotational and X-translational DOFs of Node B are not constrained. The length of the first wall-contact element adjacent to Node B is equal to 0.5db and the length of the second element is equal to db. The total length of the wall-contact elements between Nodes A and B is equal to one half of the wall length, lw. The modeling of the right wall region is similar to the left wall region.

It is assumed that the coupling beam is designed and detailed with an adequate amount of bonded longitudinal mild steel reinforcement to transfer the angle forces into the beam and an adequate amount of transverse reinforcement to prevent diagonal tension (i.e., shear) failure. The beam-to-wall connections are assumed to beproperly designed to prevent shear slip at the beam ends. The angle-to-wall and angle-to-beam connections are also assumed to be adequate to resist and transfer the forces that develop as the subassembly is displaced.

Note that the bonded longitudinal mild steel reinforcement provided inside the coupling beam does not contribute to the flexural resistance at the beam ends since it does not cross the beam-to-wall interfaces. Thus, the fiber elements modeling the beam include concrete fibers only, without any fibers representing the mild steel reinforcement. To model the gap opening behavior at the beam-to-wall interfaces, the concrete at the beam ends is assumed to have no strength in tension (see Shen and Kurama (2002b) and Shen et al. (2005a) for a more detailed discussion on the modeling of gap opening). Away from the beam ends, little, if any, cracking occurs in the beam since most of the tensile deformations of the beam occur through gap opening. Thus, the concrete away from the beam ends is assumed to be linear elastic in tension. The nonlinear behavior of the unconfined and confined concrete in compression is represented using a multi-linear idealization of a smooth uniaxial stress-strain model developed by Mander et al. (1988).

Modeling of the top and seat angles and the post-tensioning tendons is the same as the previous model for steel beam subassemblies and is described in detail in Shen and Kurama (2002b) and Shen et al. (2005a,b). Two zero-length translational spring elements are used to represent the behavior of each angle as shown in Figure 2. Kinking of the post-tensioning tendons at the beam-to-wall interfaces is included in the model. Analytical models of multi-story wall systems can be constructed by combining the subassembly models for the different floor and roof levels as described in Kurama and Shen (2004) and Shen et al. (2005a).

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ABAQUS Model. Finite element models of unbonded post-tensioned precast concrete coupling beam subassemblies were developed using the ABAQUS program(Hibbitt, Karlsson, and Sorensen Inc. 2002), similar to previous models for steel coupling beam subassemblies as described in Shen and Kurama (2002b) and Shen et al. (2005a). These models are used for the following two purposes: (1) verification of the DRAIN-2DX fiber element model described above; and (2) more detailedassessment of the stress distributions inside the coupling beam and the beam-to-wall contact regions.

As shown in Figure 3, the ABAQUS finite element model uses two dimensional nonlinear rectangular plane stress elements to represent the wall regions and the coupling beam, truss elements to represent the unbonded post-tensioning tendons, and gap/contact surfaces to represent the gap behavior at the beam-to-wall interfaces. Since the subassembly deformations are concentrated in the beam-to-wall contact regions, a finer finite element mesh is used in these regions of the model. The same assumptions used in the DRAIN-2DX model above are used in the ABAQUS model. An adequate amount of friction is used at the beam-to-wall interfaces to prevent sliding of the beam with respect to the walls piers. The concrete is assumed to be linear elastic in tension.

Note that the top and seat angles are not included in the ABAQUS model because of the difficulties involved in accurately representing the nonlinear behavior of the angles, in particular the boundary conditions adjacent to the angle legs, prying, friction, slip, and interaction between the angles, bolts, and nuts (Sims 2000).

Subassembly Details

This section describes the details of the post-tensioned precast concrete coupling beam subassemblies investigated in the paper. The wall pier properties in the subassemblies are assumed to be the same, with a length of lw=3048 mm (120 in.) and thickness of tw=381 mm (15 in.). A total of four subassemblies are considered based on a prototype subassembly (referred to as Subassembly 1) with a coupling beam width of bb=381 mm (15 in.), depth of db = 712 mm (28 in.), and length of lb = 2286 mm (90 in.). These beam dimensions were chosen to meet typical beam length to depth aspect ratios in coupled wall building structures in the U.S. (ranging between 2 and 4 as determined from personal communications with practicing engineers from the western U.S.). The prototype subassembly was designed to have a flexural strength similar to the unbonded post-tensioned steel coupling beam subassemblies

Figure 3: ABAQUS finite element model

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investigated in previous research by Shen and Kurama (2000, 2002a,b), Kurama and Shen (2004), and Shen et al. (2005a).

Twelve 15.2 mm (0.6 in.) diameter high strength strands with a total area of Abp = 1680 mm2 (2.604 in2) are used to apply the post-tensioning force. The strands are prestressed to an initial stress of fbpi = 0.50fbpu where fbpu = 1862 MPa (270 ksi) is the design ultimate strength of the strands. A total of four 8x8x3/4 top and seat angles are used at the beam-to-wall interfaces. The gage length for the angle-to-wall connections (i.e., the length measured from the heel of the angle to the center of the innermost angle-to-wall connectors) is equal to lgv = 127 mm (5 in.). The yield strength for the angle steel is taken as fay = 328 MPa (47.5 ksi).

The unconfined compressive strength of the beam and wall pier concrete is assumed to be f’c = 41.4 MPa (6 ksi). The concrete confinement in the beam-to-wall contact regions consists of 12.7 mm (0.5 in.) diameter steel reinforcing hoops and transverse ties placed at a pitch of 76 mm (3 in.) extending over a length of approximately 0.5db

into the beam and wall piers. The yield strength of the confinement reinforcement is assumed to be fhsy = 414 MPa (60 ksi). The compressive stress-strain relationship of the confined concrete was estimated using a uniaxial stress-strain model by Mander et al. (1988), resulting in a peak strength of f’cc = 116 MPa (16.8 ksi) and ultimate (crushing) strain of εccu=0.047.

Three parametric variations of Subassembly 1 are considered as shown in Figure 4 and Table 1. The following parameters are varied: (1) total area of the post-tensioning steel, Abp; (2) thicknessof the top and seat angles; and (3) beam depth, db. The increase in the area of the post-tensioning steel is achieved by using a total of sixteen strands in the subassembly, while keeping the initial stress in the strands constant. The increase in the angle thickness is used to investigate the contribution of the angles to the strength and energy dissipation of the system. Finally, the increase in the beam depth is used to investigate the effect of the coupling beam aspect ratio on the behavior of the structure. The confined concrete compressive stress-strain relationship is assumed to remain the same in the four subassemblies.

711 mm(28 in.)

381 mm (15 in.)

Subassembly 2L8x8x3/4Abp = 2240 mm2

(3.472 in.2)confinedconcrete

711 mm(28 in.)

381 mm (15 in.)

Subassembly 3L8x8x1Abp = 2240 mm2

(3.472 in.2)confinedconcrete

914 mm(36 in.)

381 mm (15 in.)

Subassembly 4L8x8x3/4Abp = 2240 mm2

(3.472 in.2)confinedconcrete

711 mm(28 in.)

381 mm (15 in.)

Subassembly 1L8x8x3/4Abp = 1680 mm2

(2.604 in.2)confinedconcrete

PT duct and strands confined

concreteconfinedconcrete

confinedconcrete

confinedconcrete

Figure 4: Subassembly beam cross sections

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Table 1: Subassembly Properties

Subassem. Beam Depth, db

mm (in.)Angle Size

PT Area, Abp

mm2 (in.2)fbpi/fbpu

lgv

mm (in

1 711 (28) L8x8x3/4 1680 (2.604) 0.50 127 (5

2 711 (28) L8x8x3/4 2240 (3.472) 0.50 127 (5

3 711 (28) L8x8x1 2240 (3.472) 0.50 127 (5

4 914 (36) L8x8x3/4 2240 (3.472) 0.50 127 (5

Analysis Results

This section presents the results from the nonlinearthe four precast concrete coupling beam subassemblies abovand Kurama (2002b), the subassembly analyses are conductleft wall pier region is fixed and by moving the right wall pwithout any rotation (i.e., the right wall region is allowed to horizontal directions, but it is not allowed to rotate). With thethe subassembly displacements are similar to the idealizerespect to the reference line in Figure 1(c). An exaggeratedfinite element model at a beam chord rotation of θb=6 % is sh

SuSubassembly 3

Su

(c)

beam c

beam chord rotation, θ (%) b beam c

kN (

kip

s)co

up

ling

sh

ear

forc

e, V

b

kN (

kip

s)co

up

ling

sh

ear

forc

e, V

b

kN (

kip

s)co

up

ling

sh

ear

forc

e, V

b

Subassembly 1

(a)beam chord rotation, θ (%) b

kN (

kip

s)co

up

ling

sh

ear

forc

e, V

b

0-7 7

2758(400)

-2758(-400)

-7

2758(400)

-2758(-400)

0-7 7

2758(400)

-2758(-400)

-7

2758(400)

-2758(-400)

Figure 5: Predicted Vb-θb behavior – (a) Subassembly 1;(c) Subassembly 3; (d) Subassembly 4

7”

.)

Aspect Ratio

Varied Parameter

) 3.21 prototype beam

) 3.21 PT area

) 3.21 angle thickness

) 2.50 beam depth

lateral load analyses of e. As discussed in Shen ed by assuming that the ier region up and down move in the vertical and se boundary conditions, d displaced shape with displaced shape of the own in Figure 3.

bassembly 4

bassembly 2

(b)

(d)

hord rotation, θ (%) b

hord rotation, θ (%) b

0 7

0 7

(b) Subassembly 2;

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Figure 5 shows the predicted coupling beam shear force versus chord rotation (Vb-θb) behaviors from the DRAIN-2DX models of Subassemblies 1-4. The coupling beam chord rotation is calculated by dividing the vertical displacement of the right end of the beam with the beam length. The analysis results indicate that the subassemblies possess stable behavior through large rotations, significant self-centering capability due to the post-tensioning force, and significant energy dissipation due to the yielding of the top and seat angles. Failure of the subassemblies occurs as a result of the crushing of the confined concrete at the beam ends.

Similarly, Figure 6 shows the DRAIN-2DX analytical results for the total forces in the post-tensioning tendons of Subassemblies 1-4. The total post-tensioning force is normalized with respect to the design ultimate strength of the tendons, Abpfbpu. The analytical models capture the increase in the tendon forces as gap opening occurs at the beam-to-wall interfaces. Note that little, if any, yielding occurs in the post-tensioning steel since the tendons are not bonded to the concrete. As a result, the initial post-tensioning force levels in the four subassemblies are preserved even after large nonlinear beam rotations.

Figure 7 shows the principal compression stresses in the coupling beam as Subassembly 4 is monotonically displaced to a beam chord rotation of θb=6 % in the ABAQUS model. The compression stresses are concentrated in the contact regions

Subassembly 1

(a)beam chord rotation, θ (%) b

Pb

p/A

bpf b

pu

0-7 7

1

0

Subassembly 2

(b)beam chord rotation, θ (%) b

1

0

Subassembly 3

(c)beam chord rotation, θ (%) b

1

0

Subassembly 4

(d)beam chord rotation, θ (%) b

1

0

0-7 7

0-7 7 0-7 7

Pb

p/A

bpf b

pu

Pb

p/A

bpf b

pu

Pb

p/A

bpf b

pu

Figure 6: Total force in PT tendons – (a) Subassembly 1; (b) Subassembly 2;(c) Subassembly 3; (d) Subassembly 4

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(i.e., the corners of the beam in contact with the wall piers) and spread out into the beam creating a compression strut and developing the coupling force. The compression stresses away from the beam-to-wall contact regions are relatively small, and thus concrete confinement is only needed in the contact regions.

Similarly, Figure 8 shows the principal tension stresses developed in the coupling beam from Subassembly 4 at a beam chord rotation of θb=6 %. As a result of the reduced coupling beam aspect ratio, Subassembly 4 has the largest beam shear force versus end moment ratio and is the most critical beam for shear among the four subassemblies in Table 1. The red shaded areas of the beam in Figure 8 indicate regions where cracking is expected to occur. Due to the opening of gaps at the beam-to-wall interfaces and the development of a diagonal compression strut, the tensile stresses along the length of the beam remain small even under large nonlinear rotations. The most critical tensile regions of the beam are the beam ends where bonded transverse mild steel reinforcement is needed to resist the tension stresses and the top and bottom surfaces near the beam ends where longitudinal reinforcement is needed. Note that larger tension stresses are expected to develop at the top and bottom surfaces near the beam ends due to the transfer of the top and seat angleforces into the beam; however, the angles were not included in the ABAQUS modelsas described previously. The bonded longitudinal mild steel reinforcement at the beam ends should be designed to resist the forces from the angles.

Comparison of DRAIN-2DX and ABAQUS Models. Figure 9(a) compares the beam shear force versus chord rotation plots from the fiber element (i.e., DRAIN-2DX) and finite element (i.e., ABAQUS) models for Subassembly 4. For this comparison, the top and seat angles are excluded from the fiber element model since the finite element model does not include the angles. Similarly, Figure 9(b) comparesthe predicted contact depth at the beam-to-wall interfaces from the two models for Subassembly 4. The contact depth is normalized with respect to the beam depth, db. It is observed that the models provide similar results in terms of both global (i.e., shear force versus rotation) and local (i.e., contact depth) behaviors. As results from ongoing laboratory experiments at the University of Notre Dame become available, these analytical models will be further verified and refined as necessary.

Summary and Conclusions

Unbonded post-tensioned coupling beams offer many advantages including reduced damage to the overall structure, significant self-centering capability, and

Figure 7: Principal compression stresses Figure 8: Principal tension stresses

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simpler design and construction for the beams and the wall piers. Analytical models were developed and verified to investigate the nonlinear behavior of four parametric post-tensioned precast concrete coupling beam subassemblies under lateral loading. The results show that the coupling beams have stable behavior through large rotations, significant self-centering capability due to the post-tensioning force, and significant energy dissipation due to the yielding of the top and seat angles provided at the beam-to-wall connections.

Different from conventional systems with monolithic cast-in-place reinforced concrete coupling beams, the lateral resistance of unbonded post-tensioned precast concrete coupling beams develops through the formation of a diagonal compression strut along the span, upon the opening of gaps at the beam-to-wall interfaces. As a result of these gaps, the tensile stresses in the beam and the wall piers remain relatively small even under large nonlinear displacements, thus, significantly reducing the amount of bonded mild steel reinforcement needed inside the beam. Current work as part of this research project at the University of Notre Dame is focused on half-scale experiments of unbonded post-tensioned precast concrete coupling beam subassemblies similar to the subassemblies investigated in this paper, as well as further analytical studies, including nonlinear static and dynamic time-history analyses of multi-story coupled wall structures.

Acknowledgements

This research is funded by the National Science Foundation (NSF) under Grant No. NSF/CMS 04-09114 and is also part of a Precast/Prestressed Concrete Institute (PCI) Daniel P. Jenny Fellowship. The support of the NSF Program Director Dr. P.N. Balaguru and members of the PCI Research and Development Committee is gratefully acknowledged. The authors also acknowledge the technical assistance provided by Cary Kopczynski of Cary Kopczynski and Company, Inc. in the design of the prototype structures. The opinions, findings, and conclusions expressed in the paper are those of the authors and do not necessarily reflect the views of the organizations and individuals acknowledged above.

0 7 beam chord rotation, θb (%)

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Figure 9: Comparison of DRAIN-2DX and ABAQUS models –(a) beam shear force versus chord rotation; (b) beam-to-wall contact depth

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