ASIAN JOURNA OF CII ENGINEERING (BUIDING AND HOUSING) O. 8, NO. 3 (7) PAGES 3-34 A PROCEDURE FOR THE EAUATION OF COUPING BEAM CHARACTERISTICS OF COUPED SHEAR WAS D. Bhunia,. Prakah and A.D. Pandey Department of Earthquake Engineering, Indian Intitute of Technology Roorkee, Roorkee 47 667, India ABSTRACT The behavior of coupled hear wall i governed by coupling beam. Thi paper preent a imple technique for the purpoe of deign to determine an appropriate level of yield moment capacity for the coupling beam. Thi technique i checked againt nonlinear tatic puhover analyi performed uing DRAIN-3DX for the uual cae of ymmetric coupled hear wall with different type of coupling beam. The aumption of pinned bae in the hear wall with teel coupling beam yield reult which agree cloely with thoe of DRAIN-3DX. For the cae of fixed bae hear wall, the deign technique i expected to be conervative. Keyword: coupled hear wall, coupling beam, rotation, capacity. INTRODUCTION Coupled hear wall conit of two hear wall interconnected by beam along their height. The behavior of coupled hear wall i mainly governed by the coupling beam. The coupling beam are deigned for ductile inelatic behavior in order to diipate energy to provide damping during an earthquake. The bae of the hear wall may be deigned a pinned or may be deigned for ductile inelatic behavior. The amount of energy diipation depend on the yield moment capacity and platic rotation of the coupling beam. If the yield moment capacity i too high, then the coupling beam will undergo only limited rotation and diipate little energy. On the other hand, if the yield moment capacity i too low, then the coupling beam may undergo rotation much larger than their platic rotation capacitie. Therefore, the coupling beam hould be provided with an optimum level of yield moment capacitie depending on the platic rotation capacity available. The platic rotation capacity in coupling beam depend upon the type of coupling beam teel beam with hear-dominant coupling beam, teel beam with flexure-dominant coupling beam, R.C.C. beam with conventional flexural and hear reinforcement, R.C.C. beam with diagonal reinforcement, and R.C.C. beam with rhombic reinforcement. Coupling beam characteritic Email-addre of the correponding author: bhunideq@iitr.ernet.in
3 D. Bhunia,. Prakah and A.D. Pandey in a coupled hear wall are controlled by variou factor, i.e. type of material, ize, type of detailing, yield moment capacity and platic rotation capacity. Table ummarize the above mentioned factor which characterize the coupling beam a determined experimentally and analytically a per different ource [-8] and [4]. Table. Factor governing the coupling beam characteritic Platic Rotation Capacity (Radian) Type of material Size Type of detailing Shear capacity Moment Capacity ( M p ) Shear twlw fc IO S CP Reinforced concrete coupling beam α No limit <.5 a '.8 f cbhb p = λ '.8 f cbhb p = b λ Diagonal Reinforcement '.8 f cbhb p = λ p b p b 3.6.5. 6.5..6 3.6.8. 6.4.6.7 p b -.6.8.3.5 to 4. Rhombic Reinforcement '.8 f cbhb p = λ p b - NA NA NA Steel coupling beam.6m p e Shear dominant p =.6Fytw ( d t f ) p Z x F y.5 - b. b.5 b Steel coupling beam.6m e p p Flexure dominant M p p = Z x F y e b f 5 t f Fy h 48 and tw Fy b f 65 t f Fy h 64 and tw Fy θ y 6 θ y 8 θ y.5θ y θ y 3 θ y α=shear pan to depth ratio, a = Conventional longitudinal reinforcement with conforming tranvere reinforcement, b= Conventional longitudinal reinforcement with non-conforming tranvere reinforcement, IO = Immediate occupancy level, S = ife afety level, CP = Collape prevention level, Size = Ratio between clear pan and depth of coupling beam, ZxFyb θ y = =Yield rotation, E = 6EIb Modulu of elaticity of teel and I b = Moment of inertia of beam The capacity of a tructure depend on the trength and deformation capacitie of the individual component of the tructure. Nonlinear puhover analyi i required to obtain the capacity curve beyond elatic limit []. There are variou program like DRAIN-3DX [3] which can be ued to perform nonlinear analyi to determine the capacity curve.
A PROCEDURE FOR THE EAUATION OF COUPING BEAM 33. PROPOSED FORMUATION. Aumption. Coupled hear wall exhibit flexural behavior.. Point of contra flexure occur at mid point of clear pan of the beam. 3. Axial deformation of the beam can be neglected. 4. Coupling beam carrie axial force, hear force and bending moment. 5. The lateral loading ha a triangular variation. 6. The horizontal diplacement in each point of wall i equal to the horizontal diplacement in each point of wall due to the preence of coupling beam. 7. All coupling beam have identical moment capacitie. They are platified or carry equal amount of hear force imultaneouly before collape mechanim i formed, i.e. all beam reach the rotational level at collape prevention imultaneouly a well a all coupling beam reach the rotational level at yield point at the ame intant. 8. The curvature of the two wall are ame at any level. In Figure (a), the coupled hear wall are ubjected to a triangular variation of point loading in each torey with amplitude of F at the roof level. The value of F could be determined o that all coupling beam reach their rotational limit for collape prevention level a well a yield level imultaneouly, then ubequently the bae hear, roof diplacement and hear force developed in coupling beam could be determined. The procedure including tep a well a mathematical calculation ha been illutrated a follow with initial value of F a :. Step ) Select type of coupling beam and determine it hear capacity. ) Determine the fraction of total lateral loading ubjected on wall and wall repectively. 3) Determine hear force developed in coupling beam for different bae condition. 4) Determine wall rotation in each torey. 5) Check for occurrence of platic hinge at the bae of the wall. For wall hinged at the bae thi check i not required. 6) Calculate coupling beam rotation in each torey. 7) Check if coupling beam rotation lie at yield level or collape prevention level. 8) Modify the value of F for next iteration tarting from tep () if tep (7) i not atified a per the aumption (7). Otherwie go to tep (9). 9) Calculate bae hear and roof diplacement..3 Mathematical Calculation Step : For variou type of reinforced (conventional, diagonal or rhombic) coupling beam, limiting value of hear capacity i given by table, p.8f = λ c bh b ()
34 D. Bhunia,. Prakah and A.D. Pandey Where, breadth of coupling beam i b; depth of coupling beam i h b ; f c i pecified compreive trength of concrete and young modulu of concrete E c depend on f c and λ i a factor of value.5. For Steel hear dominant type of coupling beam, limiting value of hear capacity i given by table, p y w ( d t ) =.6F t () f For Steel flexural dominant type of coupling beam, limiting value of hear capacity i given by table, M p p = e (3) Where, M p = Z xfy, Z x i platic ection modulu, Fy i yield tre of teel and young modulu of teel E depend on F y, t w i web thickne, d i the depth of the ection and t f i flange thickne. Step : In Figure (b), free body diagram of coupled hear wall ha been hown; α and β are fraction of total lateral loading incident on wall and wall repectively, uch that, α+ β =. (4) Baed on the aumption (8), following equation can be written a M E c ( x) M ( x) I = (5) E ci or, M I ( x) M ( x) = (6) I If = torey number counted from the roof of the ytem (ranging from to n), n i total number of torie, b + w =, and b + w = then the moment about the center line of wall at a ditance x from roof in fig. (b) for x>, i F ( ) = { ( )( ) } α M x H (i )h x (i )h i i= H i= (7) Similarly, the moment about the center line of wall, i
A PROCEDURE FOR THE EAUATION OF COUPING BEAM 35 F ( ) = { ( )( ) } β M x H (i )h x (i )h i i= H i= (8) Subtituting (7) & (8) into (6) lead to the following equation I i= I F i= n { ( H (i )h )( x (i )h ) } = i= α H I F H { β ( H (i )h )( x (i )h ) } I i= i i (9) For implifying the above and conidering equation (4) in conjunction, α = I I + I + F n ( I + I ) ( H (i )h )( x (i )h ) H i= i= i ( I I ) () Step 3: The definition of degree of coupling could be written a [9], T l DC = () M ot Where, T i the axial force at the bae of the wall and M ot i total overturning moment. For fixed bae condition the degree of coupling varie from to wherea for the cae of pinned bae condition the degree of coupling i. So baed upon the above criteria and conidering equation (), hear force developed in the coupling beam could be determined a follow. Here = ince there i no coupling beam beyond the roof a per figure (b). Fixed bae condition: b M ot ( H) *[. ] n ( w ) + w = () = l Where, M ot i total overturning moment at the bae due to the lateral loading. Therefore, hear force in th coupling beam at a ditance x from roof i, ( x) M = ot ( x) b *[. ] (w + w ) i (3) l i=
36 Pinned bae condition: D. Bhunia,. Prakah and A.D. Pandey n = ( H) Mot n = (4) l Therefore, hear force in coupling beam at a ditance x from roof i, ( x) M ot ( x) = i (5) l i= F F *(Hh )/H F *(H- h )/H F *(H- 3h )/H Wall I, A I, A F α C/ of Wall W Mid-point of C/ of F β W F *(H-(n-)h )/H Wal W W M A l M B Figure (a). Coupled hear wall Figure (b). Free body diagram of coupled hear wall Step 4: After getting α, β and at each torey for the particular value of F, bending moment value in each torey could be determined for each wall. After that, curvature diagram for each wall i generated from which wall rotation in each torey for the wall could be determined. Step 5: i. Tenile force in wall a well a compreive force in wall are calculated due to lateral loading in each level. ii. Compreive load in wall and wall are calculated in each torey due to gravity loading. iii. Net axial force in wall and wall in each torey are calculated. iv. Then, according to thee net axial force for the particular value of f ck, b, d and p, the limiting moment value in each torey in wall and wall could be determined from P- M interaction curve [-]. Where f ck, b, d and p are yield trength of concrete, breadth of a ection, depth of that ection and percentage of minimum reinforcement in that particular ection repectively. All thee limiting value are baically for linear
A PROCEDURE FOR THE EAUATION OF COUPING BEAM 37 behavior of that particular ection. v. So if calculated bending moment value at the bae i greater than limiting moment value, then platic hinge in that particular torey would be formed otherwie no platic hinge would be formed. vi. If bae moment of wall i zero then real hinge would be formed at the bae of wall. Step 6: The rotation of coupling beam in each torey i determined a follow: Rotation of coupling beam at any level x for ymmetrical wall [], θ bx = θ wx + w (6) b Where, θwx i rotation of wall at any level x, w = w = w = depth of wall, b = length of coupling beam. Equation (6) can be written a follow, or ( ) = ( θ ) w ( θ ) + ( θ ) wx θ bx wx + (7) b wx w ( bx ) = ( θ bx ) θ (8) Equation (7) and (8) are for unymmetrical wall. For pot-yield rotation at the bae of the wall ( θ wp ), Equation (6) and (7) could be written a, θ = ( θ + θ ) (9) bx wb wx wp ( ) = ( θ ) = {( θ ) + ( )} θ () bx bx wx θ wp For real hinge rotation at the bae of wall ( θ w ), Equation (6) and (7) could be written a, θ = ( θ + θ ) () bx wb wx w ( ) = ( θ ) = {( θ ) + ( )} θ () bx bx wx θ w
38 D. Bhunia,. Prakah and A.D. Pandey w + + b Where, = wb + w and b = Step 7: The rotational limit for collape prevention level & immediate occupancy level (auming yield of different type of RC coupling beam and teel beam are given in table. Here auming rotational limit for rhombic reinforced type of coupling beam i equal a rotational limit for diagonal reinforced coupling beam. Check whether the rotation of all beam lie at yield level or collape prevention level, otherwie go to tep 8 where magnitude of F i being modified for different type of bae condition of wall. Step 8: The modified F will be a follow: w (F ) yieldlevel n j j= = n (3) i i= F = Where, the above equation repreent modified value of F for yield level and none or few or more beam carry equal amount of hear capacity in beam at the yield level. (F ) collapeprevention n = (4) n i i= F = Where, the above equation repreent modified value of F for collape prevention level; i hear capacity in coupling beam. Step 9: The roof diplacement can be calculated a per following equation: ( Δ ) = [ θ + θ +... + ] roof h wx wx θ wxn (5) ( ) = [ θ ) + ( θ ) +... + ( ) ] Δ (6) roof h ( wall wx wx θ wxn ( roof ) ( ) wall = Δ roof wall Δ (7) The equation (5) i for ymmetrical coupled hear wall; (6) and (7) are for unymmetrical coupled hear wall. The Bae hear can be calculated a follow:
A PROCEDURE FOR THE EAUATION OF COUPING BEAM 39 B ( H h ) H + F ( H h )/ H + + F ( H ( n ) h )/ H = F + F / K K (8) The methodology dicued above i referred a Technique hereafter. 3. CASE STUDY The reult of capacity curve a well a hear force ditribution in coupling beam at collape prevention level (cp and at yield level are compared by Technique and DRAIN-3DX for ymmetrical coupled hear wall. Thee wall are ubjected to triangular variation of lateral loading. The dimenion are depth of wall D w = 4. m, length of beam b =.8 m, depth of beam H b = 6 mm, total wall height h w = 6 m (n=), and wall thickne t w = 3 mm = b b breadth of coupling beam. Note that E c = 7 GPa; dead load, D = 6.7 kn/m and live load, =.4 kn/m [9], f c = 9.6 MPa; auming young modulu of teel E = GPa. The figure have given a Figure (a) and Figure (b). Figure (a). Plan view of building Figure (b). Coupled hear wall 3. Modeling in DRAIN-3DX Wide column frame analogy ha been ued for modeling in DRAIN-3DX a per following Figure. In thi analogy, hear wall element are repreented a two line element (centre line of hear wall) and beam are repreented a line element (centre line of beam) and connected with each other with rigid link. Beam column elatic element (Type-7) and inelatic element (Type-5) are ued for modeling. F y = 45 MPa i ued for the cae of reinforced concrete ection and F y = 5 MPa i ued for the cae of teel ection.
3 D. Bhunia,. Prakah and A.D. Pandey Figure 3. Modeling in drain-3dx 4. RESUTS AND DISCUSSIONS a) Conventional type of reinforced coupling beam: Auming longitudinal reinforcement with conforming tranvere reinforcement. It ha been een from the Figure 4(a) that for the cae of RC beam with fixed bae condition, the reult of the initial part of the capacity curve are nearly matched but there are mall difference of the end part of the capacity curve obtained both from technique and DRAIN-3DX repectively. The reult of the hear force ditribution of RC coupling beam are not matched obtained both from technique and DRAIN-3DX repectively a per the Figure 5(a). It ha been alo een from the Figure 4(b) that for the cae of RC beam with pinned bae condition, the reult of the capacity curve i in lower ide in the cae of technique againt the reult obtained from DRAIN-3DX. The reult of the hear force ditribution of coupling beam are nearly imilar pattern obtained both from DRAIN-3DX and technique repectively a per the Figure 5(b). 4 9 Bae Shear(kN) 8 6 4..4.6 Roof Diplacement(m) Technique DRAIN-3DX Bae Shear(kN) 8 7 6 5 4 3..4 Roof Diplacement(m) Technique DRAIN-3DX Figure 4(a). Capacity curve for fixed bae condition Figure 4(b). Capacity curve for pinned bae condition
A PROCEDURE FOR THE EAUATION OF COUPING BEAM 3 Storey evel 5 5 5 5 Shear Force(kN) in Coupling Beam Technique(Yield Technique(CP DRAIN-3DX(Yield DRAIN-3DX(CP Storey evel 5 5 5 5 Shear Force(kN) in Coupling Beam Technique(Yield Technique(CP DRAIN-3DX(Yield DRAIN-3DX(CP Figure 5(a). Shear force ditribution in coupling beam for fixed bae condition Figure 5(b). Shear force ditribution in coupling beam for pinned bae condition b) Diagonal/Rhombic type of coupling beam: Auming rotational level for rhombic type i ame a rotational level for diagonal type. Bae Shear(kN) 4 8 6 4..4.6 Technique DRAIN-3DX Bae Shear(kN) 9 8 7 6 5 4 3..4.6 Technique DRAIN-3DX Roof Diplacement(m) Roof Diplacement(m) Figure 6(a). Capacity curve for fixed bae condition Figure 6(b). Capacity curve for pinned bae condition It ha been een from the above figure that for the cae of RC beam with fixed bae condition, the reult of the initial part of the capacity curve are nearly matched but there are mall difference of the end part of the capacity curve obtained both from technique and DRAIN-3DX repectively. Wherea for the cae of RC beam with pinned bae condition, the reult of the capacity curve i in lower ide in the cae of technique againt the reult obtained from DRAIN-3DX.
3 D. Bhunia,. Prakah and A.D. Pandey Storey evel 5 5 5 5 Shear Force(kN) in Coupling Beam Technique(Yield Technique(CP DRAIN-3DX(Yield DRAIN-3DX(CP Storey evel 5 5 5 5 Shear Force(kN) in Coupling Beam Technique(Yield Technique(CP DRAIN-3DX(Yield DRAIN-3DX(CP Figure 7(a). Shear force ditribution in coupling beam for fixed bae condition Figure 7(b). Shear force ditribution in coupling beam for pinned bae condition The reult of the hear force ditribution of RC coupling beam are not matched which are obtained both from technique and DRAIN-3DX repectively a per the Figure 7(a). The reult of the hear force ditribution of coupling beam are nearly imilar pattern obtained both from DRAIN-3DX and technique repectively a per the Figure 7(b). c) Steel hear dominant type of coupling beam: Bae Shear(kN) 4 8 6 4 3 Technique DRAIN-3DX Bae Shear(kN) 9 8 7 6 5 4 3 3 Technique DRAIN-3DX Roof Diplacement(m) Roof Diplacement(m ) Figure 8(a). Capacity curve for fixed bae condition Figure 8(b). Capacity curve for pinned bae condition
A PROCEDURE FOR THE EAUATION OF COUPING BEAM 33 Storey evel 5 5 5 4 Shear Force(kN) in Coupling Beam Technique(Yield Technique(CP DRAIN-3DX(Yield DRAIN-3DX(CP Storey evel 5 5 5 4 Shear Force(kN) in Coupling Beam Technique(Yield Technique(CP DRAIN-3DX(Yield DRAIN-3DX(CP Figure 9(a). Shear force ditribution in coupling beam for fixed bae condition Figure 9(b). Shear force ditribution in coupling beam for pinned bae condition The above figure how that for the cae of teel coupling beam the reult by propoed technique and DRAIN-3DX are nearly ame. It i oberved that the ue of teel coupling beam, in contrat with conventional RC, lead to increaed to roof diplacement while the bae hear i only marginally affected. It i therefore imperative that the type of coupling beam to be adopted be judiciouly elected. 5. CONCUSIONS. The aumption of pinned bae in the hear wall with teel coupling beam yield reult which agree cloely with thoe of DRAIN-3DX.. For the cae of fixed bae hear wall, the deign technique i conervative. 3. The type of coupling beam i judiciouly choen to make the deign of the coupled hear wall optimal for a particular zone. 4. The reult are encouraging and the imple technique propoed may be effectively employed in deign office practice. REFERENCES. Englekirk, R.E., Seimic of Reinforced and Precat Concrete Building, Reearch Studie Pre (John Wiley), NY, 3.. Galano,. and ignoli, A., Seimic behavior of hort coupling beam with different reinforcement layout, ACI Structural Journal, 97() 876-885. 3. Park, R. and Paulay, T., Reinforced Concrete Structure, Reearch Studie Pre (John Wiley), NY, 975. 4. Harrie, K.A., Mitchell, D., Cook, W.D. and Redwood, R.G., Seimic repone of teel beam coupling concrete wall, Journal of Structural Engineering, ASCE, 9(993) 36-369.
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