By Denis Mitchell and Patrick Paultre Seismic Design

Transcription

1 11 By Denis Mitchell and Patrick Paultre Seismic Design 11.1 Introduction Seismic Design Considerations Loading Cases Design of a Six-Storey Ductile Moment-Resisting Frame Building Description of Building and Loads Determination of Design Forces Gravity Loading Seismic Loading Deflections, Drift Ratios and Torsional Sensitivity Design of Ductile Beam Determination of Design Moments Moment Redistribution and Moment Envelopes Design of Flexural Reinforcement at Critical Sections Design of Transverse Reinforcement in Beams Checking Extent of Plastic Hinging Bar Cut-offs Splice Details Design of Interior Ductile Column Column End-Actions from Analysis Factored Axial Loads and Moments Preliminary Selection of Column Reinforcement "Strong Column - Weak Beam" Requirement Design of Transverse Reinforcement in Column Splice Details Design of Interior Beam-Column Joint Determination of Factored Forces in Joint Check Factored Shear Resistance of Joint Transverse Reinforcement Required in Joint Bond of Beam Bars Analysis of a Ductile Core-Wall Structure Description of Building and Loads Analysis Assumptions Seismic Loading Minimum Lateral Earthquake Force

2 11 2 Seismic Design Accidental Torsion Degree of Coupling Check on Structural Irregularity Dynamic Analysis Deflections and Drift Ratios Design Forces Design of Coupling Beams Design Forces for Coupling Beams Design and Detailing of Coupling Beams Ductility of Coupling Beams Design of Ductile Walls Design Forces in E-W Direction Design Forces in N-S Direction Design of Base of Wall for Flexure and Axial Load Ductility of Walls Checking Wall Thickness for Stability (Clause ) Buckling Prevention Ties for Concentrated Reinforcement (Clause ) Design for Shear at Base of Walls (Clause ) Checking Sliding Shear Resistance at Construction Joints (Clause ) Determination of Plastic Hinge Region (Clause ) Changes in Horizontal Distributed Reinforcement Over the Height of the Walls (Clause ) Changes in Vertical Distributed Reinforcement Over the Height of the Walls (Clause ) Changes in Concentrated Vertical Reinforcement Over the Height of the Walls (Clause ) Frame Members Not Considered Part of the SFRS Slab-Column Connections (Clause ) Check on Design And Detailing of Columns (Clause 21.12) Comparisons With the Design Using the 1994 CSA Standard References

3 CAC Concrete Design Handbook Introduction The 2005 National Building Code of Canada (NBCC) gives the minimum lateral earthquake force for the equivalent static force procedure as: S Ta V R ( ) M I W S( 2.0) d v E R o MvIEW R R d o and for a Seismic Force Resisting System (SFRS) with an R d equal to or greater than 1.5, 2 S( 0. 2) IEW V need not be taken greater than 3 RdRo where: S ( T a ) design spectral response acceleration, expressed as a ratio to gravitational acceleration for a period of T a M v factor to account for higher mode effect on base shear I E earthquake importance factor for the structure T a fundamental period of vibration of the building in seconds in the direction under consideration W dead load, plus 25% of the design snow load, plus 60% of storage load and the full contents of any tanks. Minimum partition load need not exceed 0.5 kpa R d ductility-related force modification factor that reflects the capability of a structure to dissipate energy through inelastic behaviour R o overstrength-related force modification factor that accounts for the dependable portion of reserve strength in a structure. The designer chooses the type of SFRS, with the corresponding force modification factors, Rd and R o. The values of Rd and R o are a function of the type of lateral load resisting system and the manner in which the structural members are designed and detailed. Table 11.1 provides a guide for the required design and detailing provisions of CSA Standard A23.3 associated with the corresponding factors, R and R. d o

4 11 4 Seismic Design Table 11.1 Design and Detailing Provisions Required for Different Reinforced Concrete Structural Systems and Corresponding R and R Factors Type of SFRS Ductile moment resisting frames Moderately ductile moment resisting frames Ductile coupled walls Ductile partially coupled walls Ductile shearwalls Moderately ductile shearwalls Conventional construction: Moment resisting frames Conventional construction: Shearwalls Other SFRS(s) R d R o Summary of design and detailing requirements in CSA A Beams capable of flexural hinging with shear failure and bar buckling avoided. Beams and columns must satisfy ductile detailing requirements. Columns properly confined and stronger than beams. Joints properly confined and stronger than beams Beams and columns must satisfy detailing requirements for moderate ductility. Beams and columns to have minimum shear strengths. Joints must satisfy moderate ductility detailing requirements and must be capable of transmitting shears from beam hinging At least 66% of base overturning moment resisted by wall system must be carried by axial tension and compression in coupled walls. Coupling beams to have ductile detailing and be capable of flexural hinging or resist loads with diagonal reinforcement (shear failure and bar buckling avoided). Walls must have minimum resistance to permit attainment of nominal strength in coupling beams and minimum ductility level Coupling beams to have ductile detailing and be capable of flexural hinging or resist loads with diagonal reinforcement (shear failure and bar buckling avoided). Walls must have minimum resistance to permit attainment of nominal strength in coupling beams and minimum ductility level Walls must be capable of flexural yielding without local instability, shear failure or bar buckling. Walls must satisfy ductile detailing and ductility requirements Walls must satisfy detailing and ductility requirements for moderate ductility. Walls must have minimum shear strength Beams and columns must have factored resistances greater than or equal to factored loads. Columns and beams must satisfy minimum detailing requirements for conventional construction. Closely spaced hoops required in columns unless factored resistance of columns greater than factored resistance of beams or if R d R o Walls must have factored resistances greater than or equal to factored loads. Factored shear resistance must exceed shear corresponding to factored flexural resistance or shear corresponding to R d R o Walls must satisfy minimum detailing requirements for conventional construction d o

5 CAC Concrete Design Handbook Seismic Design Considerations Seismic design is concerned not only with providing the required strength but also with providing minimum levels of ductility and choosing appropriate structural systems. These goals may be achieved by: (i) choosing structural systems which are as symmetrical as possible in plan and as uniform as possible in elevation (minimizing structural irregularities); (ii) designing the primary lateral load resisting structural components so that desirable energy dissipating systems will form (e.g., "weak-beam strong-column"); (iii) detailing the energy dissipating regions of the primary lateral load resisting components to ensure that substantial inelastic deformations can be achieved without significant loss of strength, and (iv) ensuring that secondary members which are not part of the lateral load resisting system can maintain their gravity load carrying capacity as they undergo the required lateral deformations. In the design of ductile members it is necessary to determine the hierarchy of strengths of different members. To ensure that certain hierarchy of strengths are achieved the CSA Standard defines "probable", "nominal" and "factored" resistances. Table 11.2 summarizes the various types of flexural resistances used in the CSA Standard and suggests approximate relationships between these resistances. Table 11.2 Factored, Nominal and Probable Moment Resistances Type of flexural resistance M r factored resistance M n nominal resistance M p probable resistance Calculated using Where used φ c 65 All members must φ s 85 satisfy Mr Mf φ c 1.0 To ensure columns φ s 1.0 stronger than beams φ c 1.0 s 1.0 f 1. 25f φ s y Note: the relationship between level of axial load Mn and Approximate relationships for flexure M 1. 2 n M r M p M r M r for the case of flexure and axial load depends on the 11.3 Loading Cases For loading combinations including earthquake, the factored load combinations shall include: Principal loads: 1. 0D E and either of the following combinations of principal and companion loads: 1) For storage occupancies, equipment areas and service rooms: 1. 0D E L S 2) For other occupancies: 1. 0D E L S

6 11 6 Seismic Design 11.4 Design of a Six-Storey Ductile Moment-Resisting Frame Building Description of Building and Loads The six-storey reinforced concrete frame building shown in Fig is located in Vancouver and is to be designed as a ductile moment resisting frame structure. The six-storey reinforced concrete office building has 7-6 m bays in the N-S direction and 3 bays in the E-W direction which consist of 2-9 m office bays and a central 6 m corridor bay. The interior columns are all 500 x 500 mm while the exterior columns are 450 x 450 mm. The one-way slab floor system consists of a slab 110 mm thick spanning in the E-W direction supported by beams in the N-S direction. The secondary beams supporting the slab are 300 mm wide x 350 mm deep (from top of slab to bottom of beam). The beams of both the N-S and E-W frames are 400 mm wide x 600 mm deep for the first three storeys and 400 x 550 mm for the top three storeys. Material Properties Concrete: normal density concrete with f c 30 MPa Reinforcement: f 400 MPa Live loads Floor live loads: 2.4 kn/m 2 on typical office floors 4.8 kn/m 2 on 6 m wide corridor bay y Roof load 2.2 kn/m 2 snow load, accounting for parapets and equipment projections 1.6 kn/m 2 mechanical services loading in 6 m wide strip over corridor bay Dead loads self-weight of reinforced concrete members calculated as 24 kn/m kn/m 2 partition loading on all floors 0.5 kn/m 2 mechanical services loading on all floors 0.5 kn/m 2 roofing Wind loading 1.84 kn/m 2 net lateral pressure for top 4 storeys 1.75 kn/m 2 net lateral pressure for bottom 2 storeys The fire-resistance rating of the building is assumed to be 1 hour.

7 CAC Concrete Design Handbook 11 7 Fig Six-storey structure located in Vancouver Determination of Design Forces Gravity Loading To determine the member forces, the structure was analyzed using ETABS. To make allowances for cracking, member stiffnesses were assumed to be 0.4 of the gross I for all beams as required by CSA A23.3. To account for the influence of the axial load level on the column stiffnesses, average estimated cracked moments of inertia of 0.6 and 0.7 of I g were used for the columns in the top three storeys and bottom three storeys, respectively (Clause ). The analysis models and the gravity loading are illustrated in Fig

8 11 8 Seismic Design To illustrate the requirements for the design of a ductile moment-resisting frame, components of a typical interior E-W frame will be designed in the following examples. Fig Unfactored loading cases considered in design of typical interior frame Seismic Loading Minimum Lateral Earthquake Force The structure is located in Vancouver and is founded on very dense soil and soft rock. Therefore the site classification is C and the acceleration-based and velocity-based site coefficients are F a 1. 0 and F v 1. 0, respectively. The seismic response factor, S, is dependent on the fundamental period, T, of the structure S a 2.0 are 0.94, The 5% damped spectral response accelerations, S a ( ), S a ( ), S a ( ) and ( ) 0.64, 0.33 and 0.17, respectively. The design spectral response accelerations are given by the product of the site coefficients and S as shown in Fig a Figure 11.3 Design spectral response acceleration The empirical fundamental lateral period, T a, for concrete moment frames is given by: T 3 / 4 3 / 4 a n 075h s

9 CAC Concrete Design Handbook 11 9 The calculated period for this structure, using the computer program ETABS is 1.35 s. The value of the fundamental lateral period cannot be taken greater than , and hence use T a s. The corresponding value of S ( T a ) is (see Fig. 11.3). For this office building, the earthquake importance factor, I E The values of Mv and J depend on the ratio of Sa ( 0. 2) / Sa ( 2. 0) 94 / and the value of T a. For this case M v 1.0 and J For this ductile moment resisting frame structure R d 4. 0 and R o Hence the seismic base shear, V, is: ( ) S Ta MvIEW W V 0453W R R d o ( 2. 0) S MvIEW W Vmin 025W R R d o ( 0. 2) 2 S IEW W Vmax 092W 3 R R d o For this structure, W 44, 765 kn. Hence, V W , kn. The portion of V concentrated at the top of the building is Ft 0. 07Ta V kn, but need not be taken greater than 0. 25V kn. The calculations of the seismic lateral forces at each floor level are summarized in Table Table 11.3 Lateral Load Calculations for Each Floor Level Floor h x, m W x, kn h x W x, kn m F x, kn V x, kn T x, kn m 6 roof , , , , , , , , , , ,026 Total - 44, ,068 2, Accidental Torsion The 3-D model shown in Fig was used to calculate accidental torsional effects by applying the lateral forces F x (see Table 11.3) at an accidental eccentricity of ± 0.1D nx, where D nx is the plan dimension of the building at level x, perpendicular to the direction of seismic loading. This gives an accidental torsional eccentricity of m, from the centre of mass (same as centre of rigidity) for loading in the E-W direction. The resulting floor torques, T x, are given in Table 11.3.

10 11 10 Seismic Design Dynamic Analysis This symmetrical structure has no structural irregularities in the vertical or horizontal directions and in addition is not sensitive to torsion (see Section ). Therefore, In accordance with NBCC a dynamic analysis is not required. However, a dynamic analysis was carried to determine the lateral period of vibration (see above). This dynamic analysis was also used to determine the design forces for the members and to estimate the lateral displacements. The purpose of carrying out a dynamic analysis in this example is to illustrate the approach required and to obtain a more realistic design force distribution. The first step is to determine V e from a linear dynamic analysis. The design base shear Vd is obtained from: Ve V d IE Rd Ro However for this regular structure, V d shall not be taken less than 0.8V. All forces and deflections obtained from the linear dynamic analysis are scaled by the factor V d / V e to obtain the design values. However, in order to obtain realistic values of anticipated deflections and drifts, the design values need to be multiplied by R d Ro / IE. Fig shows the 3-D model for the dynamic analysis, using ETABS. In the analysis, rigid end offsets were used to simulate the dimensions of the joints and rigid diaphragms were assumed. The total mass for each floor was concentrated at the centre of mass (coincident with the centre of rigidity for this structure). To account for sway effects (P-Delta) the ETABS program option, accounting for second order effects by the addition of the so-called geometric stiffness, which is a function of the compression forces in the columns from gravity loads, was used. These compressive forces were obtained from the consistent loading case of1. 0D L S, with live load reduction factors. The first three lateral modes in the E-W direction are shown in Fig. 11.5, together with the associated periods of vibration and the modal participating mass ratios. Note that the sum of these ratios is 96.9% of the total mass and hence exceeds the minimum required ratio of 90% of the total mass (NBCC). Spectral modal superposition, using SRSS for the first three modes in the E-W direction was used to determine all forces and deformations.

11 CAC Concrete Design Handbook Figure D Model used for dynamic analysis Mode 1 T MPMR 82 Mode 2 T 453 MPMR 11 Mode 3 T 250 MPMR 04 Figure 11.5 Mode shapes, corresponding lateral periods of vibration and modal participating mass ratios The base shear determined by dynamic analysis is V kn. Therefore: V d kn However for this regular building Vd shall not be taken less than 0. 8V kn. Hence, all forces and deflections obtained from the dynamic analysis shall be multiplied by V / V / d e e

12 11 12 Seismic Design Deflections, Drift Ratios and Torsional Sensitivity The deflections obtained from the dynamic analysis need to be multiplied by the factor to account for the total anticipated displacements, including the inelastic effects. It is necessary to multiply these deflections by the factor Rd Ro / IE to obtain the design values. Note that the deflections obtained from dynamic analysis include P-Delta effects. The deflections arising from accidental torsional eccentricity shall be added to the deflections from the dynamic analysis. To determine if the structure is sensitive to torsion, the value of B x is determined from the maximum and average displacements of the structure at level x in the E-W and N-S directions. The maximum value, B, of the B x values is at the first floor level for loading in the E-W direction (an average displacement of 5.1 mm and a maximum displacement of 6.8 mm), giving: δ max 6. 8 B δ 5. 1 ave Because B is less than 1.7, the structure is not sensitive to torsion. The maximum interstorey drift ratio occurs in Frames 1 and 8 in the second storey for the E- W direction of loading. From the dynamic analysis the maximum interstorey drift ratio is and the interstorey drift ratio from accidental torsion at this level is , for a maximum interstorey drift ratio of Therefore the anticipated interstorey drift ratio, including inelastic effects is / This anticipated maximum interstorey drift ratio is less than the NBCC limit of Design of Ductile Beam To illustrate the procedures involved in designing a beam in a ductile moment-resisting frame, a typical first storey interior beam will be designed below. For illustration purposes frame 2 will be designed. This frame, although it has a smaller torsional shear than frame 1, will require more reinforcement than frame 1 because it carries larger dead and live loads. The details of the beam and column framing are given in Fig Fig Typical beam and column framing

13 CAC Concrete Design Handbook Determination of Design Moments The moments in the beams resulting from dead load, D, live load, L, and earthquake loading, E, as determined from frame analyses, are in Fig Note that the moments are given at the face of the columns. Since most of the gravity loading in beams AB, BC and CD is introduced at the locations of the secondary beams, the small uniformly distributed loading has been approximated by additional concentrated loads at the secondary beam locations. Fig Loading cases on typical interior beam at second floor level Table 11.4 gives the unfactored moments at critical locations and also gives the factored moment combinations which need to be considered Moment Redistribution and Moment Envelopes Instead of designing each of the critical sections for the maximum factored moments given in Table 11.4, moment redistribution will be used to reduce some of the maximum design moments. Since the beams in a ductile moment-resisting frame structure are designed and detailed to exhibit considerable ductility the maximum redistribution of 20%, permitted by Clause 9.2.4, will be used. While it is possible to redistribute the earthquake moments, care must be exercised to ensure that the total column shears in any one storey remain unchanged after redistribution. A simpler approach is to redistribute only the dead load and live load moments. In order to reduce the magnitude of the negative moments at location BA, the dead and live load support moments are reduced at this location by the maximum permitted amount (20%). The positive moments in span AB are increased by the appropriate amounts. The resulting moments are summarized in Table 11.5.

14 11 14 Seismic Design Table 11.4 Moments at Critical Locations (kn m) Before Redistribution AB a b BA BC c D L Accidental Torsion ± 30 ± 11 ± 9 ± 28 ± 41 ± 15 E without accidental ± 98 ± 36 ± 30 ± 92 ± 131 ± 48 torsion E with accidental ± 128 ± 47 ± 39 ± 120 ± 172 ± 63 torsion 1.25D+1.5L D+1.0E D-1.0E D+0.5L+1.0E D+0.5L-1.0E Note: controlling load combinations shown in bold Table 11.5 Moments at Critical Locations (kn m) after Redistribution AB a b BA BC c D L Accidental Torsion ± 30 ± 11 ± 9 ± 28 ± 41 ± 15 E without accidental ± 98 ± 36 ± 30 ± 92 ± 131 ± 48 torsion E with accidental ± 128 ± 47 ± 39 ± 120 ± 172 ± 63 torsion 1.25D+1.5L D+1.0E D-1.0E D+0.5L+1.0E D+0.5L-1.0E Note: controlling load combinations shown in bold It is noted that, after redistribution, earthquake loading governs at all negative moment sections at the second floor level Design of Flexural Reinforcement at Critical Sections Top bars at column faces In deciding on the appropriate top reinforcement, note that Clause limits the diameter, d b, passing through the joint to l j / 24 for this normal density concrete structure and uncoated bars. Thus for this case the maximum diameter of beam bars passing through the interior columns is 500 / mm. Hence the maximum beam bar size is 20M. At column A, a factored moment resistance of at least 328 kn m is required. Assuming a flexural lever arm of 0. 75h m, the required area of top bars would be / ( ) 2144 mm 2. If it is assumed that slab reinforcement within a distance of 3 hf from the sides of the beam is effective, then 4 10M bars in the slab are effective.

15 CAC Concrete Design Handbook It is assumed that these 10M bars in the flange are effective under reversed cyclic loading even though there is no anti-buckling reinforcement for these bars. Note that larger bars may not be effective. The additional reinforcement required is then 1744 mm 2. Note that it is unwise to be too conservative when designing the top reinforcement since beam shears, joint shears, column moments and column shears are all increased if the flexural capacity at the end of the beam is increased. Let us try an arrangement of 6 20M bars as shown in Fig Keeping in mind that the positive moment resistance of the beam needs to be at least one-half of the negative moment resistance, try using 4 20M bars on the bottom of the beam. Fig Beam cross section near column face Accounting for the presence of the compression reinforcement, the depth of compression, c, is found to be 92 mm and the factored negative moment resistance is 359 kn m. Hence the moment capacity is satisfactory. The required minimum top and bottom reinforcement, A s, min, from Clause is: As,min 1. 4bw d / fy / mm mm 2 O.K. The maximum reinforcement permitted is: bw d mm mm 2 O.K. Note that in choosing the arrangement of the beam bars at column faces the following factors must be considered: (a) the need to restrain the longitudinal bars from buckling by providing lateral restraint in the form of hoops and ties. (b) the need to pass the beam bars through the column cage, and (c) the need to provide adequate space between top bars to permit placement and vibration of concrete. Since the magnitudes of the negative moment resistances required at column faces AB, BA and BC are all about the same, we will use the same reinforcing arrangement at these three locations.

16 11 16 Seismic Design Bottom bars for positive moment regions Span BC: For span BC, the effective compressive flange width is 1600 mm (see Clause ). For 4 20M bars ( A s 1200 mm 2 ), M r at the column face, accounting for the large amount of top reinforcement, is 229 kn m which is larger than one half of 359 kn m (i.e., M r at column face where A s 2200 mm 2 ). As 4 20M bottom bars are provided at column faces AB, BA and BC, use 4 20M bars in span BC. The positive moment resistance M r is 229 kn m in the midspan regions of span BC. As 229 kn m exceeds 112 kn m (Table 11.5), 4-20M bars will be satisfactory. Span AB: For span AB, the effective compressive flange width is 2200 mm (see Clause ). For M r 252 kn m (Table 11.5) try 6-20M bottom bars. Neglecting the top reinforcement, the depth of the equivalent rectangular stress block is 18 mm and M r 315 kn m (see Fig. 11.9). Accounting for the large amount of top reinforcement, the positive moment M r at the column face is 246 kn m. Fig Positive and negative moment capacities of beam

17 CAC Concrete Design Handbook Design of Transverse Reinforcement in Beams Shear requirements Determine the shears corresponding to the development of flexural hinging at both ends of the beam. For the chosen reinforcement at the beam ends the probable moment resistances are: (i) Probable negative moment resistance, M pr Using a strain compatibility approach to calculate M pr results in a depth of compression, c 95 mm, a stress block depth of 85 mm and M pr 528 kn m (see Fig. 11.9). Note that the probable moment resistance of the beams can be estimated by multiplying M r by the ratio / In this case M - pr would be kn m. This simple approach is sufficiently accurate for design purposes. + M pr (ii) Probable positive moment resistance, The factored moment resistance at the ends in span BC is 229 kn m and hence we can + estimate the probable moment resistance as M pr kn m. Similarly, the probable moment resistance at the ends of span AB is kn m. (iii) Determine factored shears The shear diagrams shown in Fig are drawn for lateral forces acting in the west direction. For lateral forces acting in the east direction the shear diagrams will be "mirror images" of those shown (e.g., the shear at B would be 234 kn). Fig Determinaton of shears corresponding to flexural hinging

18 11 18 Seismic Design Using θ 45 and β 0 gives (Clause ): V r φ s A f v y d v / s At the column faces the transverse reinforcement consists of 4 10M legs, hence A 400 mm 2. v At the ends A and B the required spacing for shear is: s 276 mm At the ends B and C the required spacing for shear is: 234 s mm. 262 If 2-legged 10M stirrups are used as transverse reinforcement ( A 200 mm 2 ) the required spacings in the middle regions of beams AB and BC are 344 mm and 228 mm, respectively. Other shear design requirements: (i) Maximum shear (Clause ) Vr,max 0. 25φ cfc bw dv kn (ii) Minimum amount of stirrups (Clause ): for 4 stirrup legs: Av f y 400 x 400 s 1217 mm 0.06 f b x 400 c w v for 2 stirrup legs: s 608 mm (iii) Spacing limits (Clause ): Since V < φ f b d x X 30 x 400 x0. 9 f c c w v kn Then s 600 mm or 0. 7dv mm. max Note that near the ends of the beams the stirrup spacing required for shear cannot exceed 276 and 247 mm for spans AB and BC, respectively. "Anti-buckling requirements (Clause ) Hoops to prevent buckling of longitudinal bars are required over a length of 2d from the face of the columns. The spacing of the hoops shall not exceed: (i) d / / mm (ii) 8 d bl mm (iii) 24 d bh mm (iv) 300 mm Note that the 4-legged arrangement of transverse reinforcement satisfies Clause Hence use a spacing of 130 mm for 4-legged hoops over a length of at least 2 d mm.

19 CAC Concrete Design Handbook Checking Extent of Plastic Hinging The moment diagrams corresponding to plastic hinging at both ends of beams AB and BC are shown in Fig Hinging can spread over a distance of about 3.43 m from the face of column B in beam AB. Since the earthquake loading can reverse, provide 4-legged hoop reinforcement spaced at 130 mm as shown in Fig The bottom 4-20M bars can only be spliced near midspan of the beam (Clause ). Therefore, to satisfy the maximum spacing requirements of Clause for regions of lap splices, provide 2-legged hoops spaced at 100 mm in the middle region of beam AB (see Fig ). For span BC, provide 4-legged hoops at a spacing of 130 mm over a distance of 2d (1054 mm) from the column faces. Outside of these regions, the provision of 2-legged hoops at a spacing of 100 mm satisfies both the shear requirements and the confinement requirements for lap splices (see Fig ) Bar Cut-offs The locations of bar cut-offs are determined from the moment diagrams corresponding to the formation of plastic hinges at the ends of the beams. The theoretical cut-off location is located at a distance of 1.51 m from the face of the column (see Fig ). From Clause it is required to provide an embedment length beyond the theoretical cut-off point of at least d or 12d b. Hence the minimum length required is d mm. Continue the 6 20M top bars a distance of 3 m from the column face such that the bars are terminated in a region of lower shear. For span BC, extend the 6-20M top bars a distance of 2.0 m from the face of the column Splice Details Flexural reinforcement cannot be spliced within a distance of 2d from the column face nor within a distance d from of a potential plastic hinge location (Clause ). In evaluating cutoff locations, d was taken as 527 mm. In determining locations of bar cut-offs and splices we will consider the moment diagram corresponding to the formation of hinges at the ends of the beams (see Fig ). The splices for the top bars will be located in a region of the beam where the bars are predicted to remain in compression. However, as it is required to have a minimum negative and positive moment resistance at the face of the joint (Clause ) the splice length will be calculated as for a tension splice. (a) Splicing of the 2 20M "continuous" top bars The required minimum moment capacity along the length of the beam (Clause ) is kn m. M r for 2-20M top bars is 175 kn m. Hence for the classification of tension lap splices in accordance with Clause (A s provided)/ A s required) is 175 / Hence Class B splices are required. The development length l d for these top bars from Table 12-1 is: fy 400 l d 45k1 k 2k 3k 4 d b mm f 30 c Thus the splice length is mm. (b) Splicing of the 4 20M "continuous" bars The development length for these bottom bars is: fy 400 l d 45k1 k 2k 3k 4 d b mm f 30 c

20 11 20 Seismic Design Thus the splice length is mm. The details of the reinforcement in the beam are illustrated in Fig Fig Reinforcement details in beams

21 CAC Concrete Design Handbook Design of Interior Ductile Column To illustrate the procedures involved in designing a column in a ductile moment resisting frame a typical first storey interior column in Frame 2 or 7 of the building described in Section 11.4 will be designed. The column that will be designed is shown in Fig Fig Beam-column framing Column End-Actions from Analysis The column end actions obtained from analysis are summarized in Table The earthquake forces given are those for lateral seismic forces acting in the E-W direction. For the live load values, pattern loading and live load reduction values were considered. Table 11.6 Column End Actions 2 nd floor bottom of column 1 st floor top of column 1 st floor bottom of column P D P L P E M D M L M E kn kn kn kn m kn m kn m kn kn kn ± ± ± ± ± ± ± ± ±94 V D V L V E

22 11 22 Seismic Design Factored Axial Loads and Moments The column must be designed to resist the appropriate combinations of axial load and moment. From Table 11.6, it is evident that the factored moments at the base of the column will be larger than that at the top. For the base of the column at the ground floor level the factored axial load and moment combinations are given in Table Table 11.7 Factored Axial Load and Moments at Column Bases Case D +1.5L Case 2 1.0D +1.0E Case 3 1.0D -1.0E Case 4 1.0D+0.5L +1.0E Case 5 1.0D+0.5L -1.0E P f kn M f kn m Note that for this member A g f c / 10 ( ) 30 / kn. As P f exceeds this value, the requirements of Clause 21.4 apply (Clause ) Preliminary Selection of Column Reinforcement In selecting the column bars, recall that the diameter of these bars must satisfy the requirements that d b l j / / mm (Clause ) for this normal density concrete and for uncoated bars. Hence the maximum bar size is 25M. Try using 8-25M bars as shown in Fig Fig Column reinforcement details For this arrangement of reinforcement A mm 2. From Clause st the minimum area of longitudinal steel is mm 2 and the maximum area of longitudinal steel outside of lap splice regions (assuming lap splicing with an equal area of steel) is mm 2. Hence this steel arrangement satisfies these requirements.

23 Checking Column Capacity CAC Concrete Design Handbook The axial load-moment interaction diagram for the chosen column section is shown in Fig It can be seen that the column has adequate capacity to resist the various combinations of P and M which occur at the base of the column. f f Fig P r - M r interaction diagram for column section Although there is a considerable excess of moment capacity in the column, this additional capacity is needed at the top of the column in order to ensure that the columns are stronger than the beams (see below) "Strong Column - Weak Beam" Requirement The flexural capacity of the columns must exceed the flexural capacity of the beams so that M nc M pb Hence it is necessary to first determine the probable resistances of the beams framing into the column.

24 11 24 Seismic Design (a) Probable negative moment resistance, M pb Note that the probable resistance, M pb, can be approximated as 1. 47M r kn m. This simple approach is sufficiently accurate for design purposes as can be seen from Fig (b) Nominal positive moment resistance, + M pb kn m. + M pb (c) Determination of M nc To determine M nc for a particular loading case we need to calculate the nominal moment resistance of the column above and below the beam-column joint. The lowest flexural resistance will occur at either the highest or lowest axial load, that is, load cases 3 and 4 need to be investigated (load case 1 does not involve lateral load). The axial load corresponding to cases 3 and 4 are given in Fig along with the column nominal moment resistances corresponding to these axial loads (from the P-M interaction diagram). Fig Capacity design of columns and factored loads on column Thus the requirement that M nc M pb is satisfied. Note that for simplicity the above calculations have neglected the influence of the beam and column shears acting at the joint faces Design of Transverse Reinforcement in Column Shear requirements The column must have a factored shear resistance, V r, which exceeds the column shear corresponding to the probable moment resistance in the beams and which exceeds the shear

25 CAC Concrete Design Handbook forces due to factored loads (Clause ). From Table 11.6, load case 4 ( 1. 0D L E) gives the maximum factored shear of 132 kn. The moment at the top of the column corresponding to the development of the probable moment resistances of the beams may be estimated from: + Kc 1/ Mc ( Mpr + Mpr ) ( ) 445 kn m K 2 1/ c However since we are designing a ground storey column a different approach is required at the base of this column. It is assumed that the column frames into a substructure that is considerably stronger and stiffer than the column and hence the possibility of hinging at the column base must be accounted for. To ensure adequate column shear capacity, it is necessary to determine the maximum probable moment resistance corresponding to all axial loads. Because the axial loads for all the seismic load cases are close to the balanced axial load level, the moment at the base of the column will be taken as the probable moment resistance corresponding to the balanced loading conditions (i.e., the highest probable moment resistance possible). The calculations involved in determining this moment resistance are summarized in Fig Fig Determination of factored shear strength in ground storey column

26 11 26 Seismic Design The column actions which will correspond to the formation of hinges in the beams at the top of the column and the formation of a hinge at the base of the column are shown in Fig From Clause the shear carried by the concrete is determined with values of β and θ taken from Clause 11 but limited to a maximum of 0.10 and a minimum of 45, respectively. For this column containing greater than minimum amounts of transverse reinforcement and subjected to axial compression, Clause applies, but the limits for β and θ given above control. The shear resistance attributed to the concrete, assuming that d v 72h, is: V φ λβ f b d kn c c c w v The required V s is equal to kn. Using the transverse reinforcement arrangement shown in Fig with square and diamond shaped hoops, the effective area of o shear reinforcement is A v ( 2 + 2cos 45 ) mm 2. Hence, the required stirrup spacing can be found from Equation (11-7) as: φs A s v fy d V s v cot θ cot v o 142 mm Since V f of 359 kn is less than λφ cfcbw d 439 kn, then from Clause , the maximum spacing of the shear reinforcement is the smaller of mm or 600 mm. In order to satisfy the minimum shear reinforcement requirements of Clause , the maximum spacing of the 10M stirrups is: A s v fy f b c w mm Therefore for shear a spacing of 142 mm controls. (b) Confinement requirements Since the column under consideration is at the base of the structure confinement reinforcement must be provided over the full height of the column (Clause ). From Clause , the total cross-sectional area of rectangular hoop reinforcement depends on the following factors: nl 8 k n n P l 2 ( A A ) + f A ( ) o α 1 fc g st y st. kn k p Pf / Po 2119 / Hence, the total area of confinement reinforcement is: Ag fc Ash 2k nk p shc s 3. 34s A f ch yh but not less than: fc 30 Ash 0. 09shc s 2. 84s f 400 yh For A 341mm 2, s 341 / mm. sh

27 CAC Concrete Design Handbook From Clause the spacing of the hoops shall not exceed: (i) h / / mm (ii) 6 d mm b (iii) s ( 350 h )/ ( ) / x x mm Hence use 10M hoops at 100 mm centres as shown in Fig The chosen arrangement of hoops and longitudinal reinforcement also satisfies Clauses and The details of the first-storey column reinforcement are given in Fig Fig Details of reinforcement in first-storey column

28 11 28 Seismic Design Splice Details We will splice the column bars at mid-height of the column with tension lap splices in accordance with Clause The development length, l d, can be found from Table 12-1 as: fy 400 l d 45k1 k 2k 3k 4 d b mm f 30 c Provide a lap length of 1 3l mm (see Fig ).. d Design of Interior Beam-Column Joint To illustrate the procedures involved in designing a beam-column joint in a ductile momentresisting frame, an interior joint in the structure described in Section 11.4 will be designed. A description of the joint details is given in Fig Fig Geometry of interior beam-column joint Determination of Factored Forces in Joint In accordance with Clause , assume that the tensile force in the beam reinforcement is 1.25A s f. y

29 CAC Concrete Design Handbook To estimate the corresponding shear, V col, in the column above the joint, assume that flexural hinging occurs in the beams at the first and second storey levels. The calculations are summarized in Fig Fig Determination of factored shear resistance in joint

30 11 30 Seismic Design Check Factored Shear Resistance of Joint Since four equal depth beams frame into the joint and each covers more than 3/4 of each face of the joint, the joint is considered to be externally confined (Clause ). Hence the factored shear resistance of the joint is taken as: Vr 2. 2λφ c fc A j kn As the design shear in the joint of 1408 kn is less than 1958 kn, the shear resistance of the joint is adequate Transverse Reinforcement Required in Joint As the joint is framed by four equal depth beams which provide confinement, only one-half of the confinement steel required for the column is required through the joint (Clause ). The spacing required for confinement in the joint is therefore 200 mm. However the spacing limits of Clause control ( s max h / mm). Hence provide 3 sets of 10M hoops between the flexural bars in the beams as shown in Fig Fig Details of joint reinforcement

31 CAC Concrete Design Handbook Bond of beam Bars As the beam bars pass through the joint their bond characteristics are checked by the requirement in Clause that the bar diameters be not greater than l j / / mm (normal density concrete and uncoated bars). Since this exceeds the actual bar diameter of 20 mm this requirement is met Analysis of a Ductile Core-Wall Structure Description of Building and Loads The twelve-storey reinforced concrete building shown in Fig is located in Montreal and is founded on stiff soil. Fig Plan and elevation of twelve-storey office building

32 11 32 Seismic Design The twelve-storey reinforced concrete office building has a centrally located elevator core. Each floor consists of a 200 mm thick flat plate with 6 m interior spans and 5.5 m end spans. The columns are all 550 x 550 mm and the thickness of the core wall components 400 mm. The 400 mm thick wall thickness was initially chosen such that it exceeds l u / / mm (Clause ). This value is checked in Section The core wall measures 6.4 m by 8.4 m, outside to outside of the walls. Two 400 mm wide x 900 mm deep coupling beams connect the two C-shaped walls at the ceiling level of each floor. The core walls extend one storey above the roof at the 12 th floor level forming an elevator penthouse at the 13 th floor level. The slab has a 100 mm overhang. Material Properties Concrete: normal density concrete with f 30 MPa Reinforcement: f 400 MPa Gravity and Wind Loadings Floor live load: Roof load: y c 2.4 kn/m 2 on typical office floors 4.8 kn/m 2 on 12 m by 12 m corridor area around core 2.2 kn/m 2 full snow load 1.6 kn/m 2 mechanical services loading in 6 m wide strip over corridor bay Dead loads: self-weight of members calculated at 24 kn/m kn/m 2 partition loading on all floors 0.5 kn/m 2 ceiling and mechanical services loading on all floors 0.5 kn/m 2 roofing Wind loading: varies from 1.1 to 1.37 kn/m 2 net lateral pressure over the height of the building The building is to be designed with a fire-resistance rating of 2 hours Analysis Assumptions To determine the forces in the walls and the coupling beams and the periods of vibration, the three-dimensional core wall system was analyzed using ETABS. To make allowances for cracking, member stiffnesses were based on effective properties equal to 0.25I g for the moment of inertia and 0.45Ag for the shear area for all diagonally reinforced coupling beams as required by Clause The walls were modeled with an effective flexural stiffness of 0.7EIg and an effective axial stiffness of 0.7EAg, determined as a function of the axial loading at the base of the walls (see Clause ) Seismic loading For the force modification factors, R d and R o, we will assume that the core-wall system will take 100% of the lateral loads as allowed by the NBCC. In the N-S direction we will design and detail the walls as ductile shear walls and hence R d 3. 5 and R o In the E-W direction we will design and detail the coupling beams and walls as a ductile coupled wall system and hence R d 4.0 and R o In order for the E-W direction to qualify as a ductile coupled wall system we must check the degree of coupling as determined by analysis of the structure.

33 CAC Concrete Design Handbook Minimum Lateral Earthquake Force The structure is located in Montreal and is founded on stiff soil. Therefore the site classification is D. The acceleration-based site coefficient F a and the velocity-based site coefficient F v The seismic response factor, S ( T a ), is dependent on the fundamental period, T a, of the structure. The 5% damped spectral response accelerations, S a ( T ), for Montreal are given in Table Table 11.8 also gives the design spectral response accelerations, S ( T ), obtained from the product of the site coefficients and Sa as shown in Fig Table 11.8 Spectral response accelerations and design spectral response accelerations T 0.2 T 5 T 1. 0 T 2. 0 T 4. 0 S a ( T ) S ( T ) Figure Design spectral response acceleration The empirical fundamental lateral period, T a, for this shear wall structure in the N-S and E-W directions, is given by: 3 / 4 3 / 4 T 05h s a n N-S Direction The calculated period for this structure in the N-S direction, using the computer program ETABS, is 1.83 s. Note that a 3-D model including the walls, the slabs and the columns was also analysed and resulted in a period of 1.75 s. Because this period is within 15% of the periods of the walls alone, then a period of 1.83 s was used. The value of the fundamental lateral period cannot be taken greater than and hence use T a s. From linear interpolation, S( T a ) 0981 (see Fig ). The values of Mv and J depend on the ratio of Sa ( 0. 2) / Sa ( 2. 0) 69 / and the value of T a. It is necessary to interpolate the value of S ( T a ) M v and the value of J between periods of 1.0 and 2.0 s and 0.5 and 2.0 s, respectively. This interpolation results in S( Ta ) Mv 170, M v and J 505.

34 11 34 Seismic Design This office building has an earthquake importance factor, I E For this ductile shear wall R d 3.5 and R o Hence the seismic base shear, V, is: ( ) S Ta MvIEW W V 0304W R R d o ( 2. 0) S MvIEW W Vmin 0291W R R d o ( 0. 2) 2 S IEW W Vmax 0923W 3 R R d o For this structure, W kn. Hence V W kn. The portion of V concentrated at the top of the building is Ft 0. 07TaV kn, but need not be taken greater than 0. 25V kn. The calculations of the seismic lateral forces at each floor level are summarized in Table E-W Direction The calculated period for this structure in the E-W direction, using the computer program ETABS is 1.72 s. It is noted that this period may be used because the period for the full 3-D structure (walls, slabs and columns) is within 15% of this value. The value of the fundamental lateral period cannot be taken greater than and hence use T a s. From linear interpolation, S( T a ) 101 (see Fig ). It is necessary to interpolate the value of S ( T a ) M v and the value of J between periods of 1.0 and 2.0 s and 0.5 and 2.0 s, respectively. This interpolation results in S( Ta ) Mv 110, M v and J 757. For the ductile coupled wall system in the E-W direction R d 4. 0 and R o Hence the seismic base shear, V, is: ( ) S Ta MvIEW W V 0162W R R d o ( 2. 0) S MvIEW W Vmin 0115W R R d o ( 0. 2) 2 S IEW W Vmax 0760W 3 R R d o For this structure W kn. Hence V W kn. The portion of V concentrated at the top of the building is Ft 0. 07TaV kn, but need not be taken greater than 0. 25V kn.

35 CAC Concrete Design Handbook The calculations of the seismic lateral forces at each floor level using the equivalent static force procedure are summarized in Table The weight of the penthouse has been included at the roof level. Table 11.9 Lateral Load Calculations for Each Floor Level Floor h i, m W i, kn h i W i, kn m Total F x N-S T x F x E-W T x Accidental Torsion The 3-D model shown in Fig was used to calculate accidental torsional effects by applying the lateral forces F x (see Table 11.9) at an accidental eccentricity of ± 0.1D nx, where D nx is the plan dimension of the building at level x, perpendicular to the direction of seismic loading. This gives an accidental torsional eccentricity of m, from the centre of mass (same as centre of rigidity) for loading in the N-S and E-W directions. The values of T x are given in Table The structure was analysed with a 3-D model of the core-wall structure for both wind and seismic loading, with and without eccentricity. In these analyses the participation of the flat plate and columns was neglected Degree of Coupling In the calculations of the base shear it was assumed that there was sufficient coupling of the walls in the E-W direction to qualify this wall system as a ductile coupled wall system rather than a partially coupled wall system. To check the degree of coupling by the wall system in the E-W direction, the base overturning moment resisted by axial tension and compression forces in the walls (resulting from shear in the coupling beams), divided by the total base overturning moment is determined. Although the design forces were obtained from dynamic analysis, it is not

36 11 36 Seismic Design appropriate to use these values to determine the degree of coupling because the values obtained from modal combination (e.g., SRSS or CQC) does not satisfy static equilibrium. The degree of coupling was determined using static analysis with the F x forces from the equivalent static force procedure, giving: Tl M + M + Tl where T axial tension and compression acting at centroid of coupled walls l distance between centroids of coupled walls, equal to 6.5 m for this example The degree of coupling is 74%, which is exceeds the minimum limit for ductile coupled walls of 66%. Hence R d 4. 0 and R o 1. 7, as assumed above Check on Structural Irregularity To determine if the structure is sensitive to torsion, the values of B need to be determined at all levels from the maximum and average displacements of the structure at in the E-W and N-S directions. The maximum value, B (determined at the extreme points of the structure), in the N-S direction occurs in the first storey, with a displacement due to accidental torsion of 1.04 mm and a displacement due to F x of 1.30 mm. Hence: δ B δ ave max Because B is greater than 1.7, the structure is sensitive to torsion and hence is designated as irregular. The maximum value of B in the E-W direction occurs in the first storey and is Note that a 3-D analysis of the structure, including the columns and slabs as well as the actual mass distributions indicates that the first and fourth modes of vibration are torsional with periods of 1.89 and 0.54 s, respectively. This confirms that the structure is indeed torsionally sensitive. This design example illustrates the steps necessary to design this common type of structure, that is torsionally sensitive Dynamic Analysis The NBCC requires that the Dynamic Analysis Procedure be used except that the Equivalent Static Force Procedure may be used for structures that meet any one of the three conditions in parts (a), (b) and (c) of Clause For this building, the term IE FaSa ( 0.2) is greater than 0.35 and hence the condition in part (a) is not satisfied. The presence of the structural irregularity due to torsion sensitivity means that the Equivalent Static Force Procedure cannot be used (part (b) of ). Part (c) of is also not satisfied. Accordingly, the Equivalent Static Force Procedure is not permitted as an alternative to the Dynamic Analysis Procedure for this example building. The first step is to determine V e from a linear dynamic analysis. The design base shear Vd is obtained from: Ve V d R R d o I E

37 CAC Concrete Design Handbook Because this is an irregular structure, that requires dynamic analysis (NBCC ), Vd shall not be taken less than 1.0V rather than 0.8V, permitted for a regular structure. All forces and deflections obtained from the linear dynamic analysis are scaled by the factor V d / V e to obtain the design values. However, in order to obtain realistic values of anticipated deflections and drifts, the design values need to be multiplied byr d Ro / IE. Fig shows the 3-D ETABS model that considers only the core wall system (SFRS) and is used for the dynamic analysis. The second model used is the entire structure including the frame members not considered part of the SFRS (columns and the slabs) to check the ductility and strength of these members subjected to seismically induced deformations. The total mass for each floor was concentrated at the centre of mass (same as centre of rigidity for this example) and rigid diaphragms were assumed at each floor level. Sway effects (P-Delta) were included using the ETABS program option. For this analysis, compressive loads on the walls were obtained from the consistent loading case of 1. 0D L S, with live load reduction factors. The first three lateral modes in the N-S and E-W directions are shown in Fig , together with the associated periods of vibration and the modal participating mass ratios. Note that the sum of these ratios is 94.9% and 93.8% of the total mass in the N-S and E-W directions, respectively. These ratios exceed the minimum required ratio of 90% of the total mass (NBCC). Spectral modal superposition, using SRSS for the first three modes in the both directions was used to determine all forces and deformations. The base shear in the N-S direction determined by dynamic analysis is Ve kn. Therefore: 1. 0 V d kn However for this irregular building Vd shall not be taken less than V kn. Hence, all forces and deflections obtained from the dynamic analysis shall be multiplied byvd / Ve / in the N-S direction. The base shear in the E-W direction determined by dynamic analysis is Ve 11021kN. Therefore: Vd kN However for this irregular building Vd shall not be taken less than V kN. Hence, all forces and deflections obtained from the dynamic analysis shall be multiplied by V / V 1621 / in the E-W direction. d e

38 11 38 Seismic Design Figure D Model used for dynamic analysis

39 CAC Concrete Design Handbook Mode 1 T MPMR 67 Mode 2 T 339 MPMR 22 Mode 3 T 143 MPMR 07 (a) N-S direction Mode 1 T MPMR 72 Mode 2 T 435 MPMR 17 Mode 3 T 199 MPMR 05 (b) E-W direction Figure Mode shapes, corresponding lateral periods of vibration and modal participating mass ratios in the N-S and E-W directions