CHAPTER III EVALUATION BASED ON LINEAR ANALYSIS
3.2 COMPUTATIONAL MODEL
3.3.2 Response spectrum analysis
The equations of motion associated with the response of a structure to ground motion are given by:
( )t + ( )t + ( )t = u tgx( )+ u tgy( )+ u tgz( )
x x x
Mu Cu Ku m m m (3.11)
Here, M is the diagonal mass matrix, C is the proportional damping matrix, K is the stiffness matrix, u, u and u are the relative (with respect to the ground) acceleration, velocity and displacement vectors, respectively, mx, my, and mz are
the unit acceleration loads and ugx, ugy and ugz are the components of uniform ground acceleration.
The objective of response spectrum analysis is to obtain the likely maximum response from these equations. The earthquake ground acceleration in each direction is given as a response spectrum curve*. According to IS 1893: 2002, high rise and irregular buildings must be analysed by the response spectrum method. However, this method of linear dynamic analysis is also recommended for regular buildings.
*
The response spectrum is a plot of the maximum response (maximum displacement, velocity, acceleration or any other quantity of interest) to a specified load function for all possible single degree-of-freedom systems. The abscissa of the spectrum is the natural period (or frequency) of the system and the ordinate is the maximum response. It is also a function of damping. Figure 3.3 shows the design response spectra given in IS 1893: 2002 for a 5% damped system.
Response spectrum analysis is performed using mode superposition, where free vibration modes are computed using eigenvalue analysis. The maximum modal response (λk) of a quantity (considering the mass participation factor) is obtained
for each mode of all the modes considered. Sufficient modes (r) to capture at least 90% of the participating mass of the building (in each of the orthogonal horizontal directions), have to be considered in the analysis. The modal responses of all the individual modes are then combined together using either the square root of the sum of the squares (SRSS) method or complete quadratic combination (CQC) method. The SRSS method is based on probability theory and is expressed as follows. 2 1 ( ) r k k= λ =
∑
λ (3.12) If the building has very closely spaced modes then the CQC method is preferable.The base shear is calculated for response spectrum analysis in the following manner. The Sa/g value corresponding to each period of all the considered modes
is first calculated from Figure 3.4. The base shear corresponding to a mode is then calculated as per Section 3.3.1.5. Each base shear is multiplied with the corresponding mass participation factor and then combined as per the selected mode combination method, to get the total base shear of the building.
If the base shear calculated from the response spectrum analysis (V is less than B) the design base shear (V calculated from Equation 3.7, then as per IS 1893: B) 2002, all the response quantities (member forces, displacements, storey shears and base reactions) have to be scaled up by the factor V V . B/ B
3.4 EVALUATION RESULTS
The demands (moments, shears and axial forces) obtained at the critical sections from the linear analyses are compared with the capacities of the individual
elements. The capacities of RC members are to be calculated as per IS 456: 2000, incorporating the appropriate “knowledge factors” (Table 3.2). The demand-to- capacity ratio (DCR) for each element should be less than 1.0 for code compliance. Muy Pu Mux A B C DCR = AB/AC
Figure 3.6: Demand to capacity ratio for column flexure
For a beam, positive and negative bending moment demands at the face of the supports and the positive moment demands at the span need to be compared with the corresponding capacities. For a column, the moment demand due to bi-axial bending under axial compression must be checked using the P-Mx-My surface (interaction surface), generated according to IS 456: 2000. The demand point is to be located in the P-Mx-My space and a straight line is drawn joining the demand point to the origin. This line (extended, if necessary) will intersect the interaction surface at the capacity point. The ratio of the distance of the demand point (from the origin) to the distance of the capacity point (from the origin) is termed as the DCR for the column (Figure 3.6).
EL
(c) Shear force demand in beam (sway to left) (b) Shear force demand in beam (sway to right)
(a) Loading on beam
0.5 wu ln 0.5 wu ln 0.5 wu ln 0.5 wu ln wu = 1.2 (wDL + wLL) ln 1.4(M –uR, left + M +uR, right)/ln 1.4(M+uR, left + M –uR, right)/ln 1.4 M +uR, right 1.4 M –uR, left plastic hinge 1.4 M –uR, right 1.4 M+uR, left plastic hinge EL
Figure 3.7: Calculation of shear force demand in beams
The shear demand should be calculated as per IS 13920: 1993 recommendations. For beam, the shear demand will be the larger of the shear force from analysis and the shear force corresponding to the beam reaching its flexural capacity (formation of moment hinges at both ends of the beam). This concept is called the capacity based design.
The shear demands (Vu) at the support faces (left or right) are obtained as follows
(Clause 6.3.3, IS 13920: 1993).
(
)
, ,0.5 1.4 ,
u left u n uR left uR right n
V = w l + M − +M + l
(3.13a)
(
)
, ,0.5 1.4 ,
u right u n uR left uR right n
V = w l + M + +M − l
(3.13b) Here, ln is the clear span, and wu is the factored load as shown in the Figure 3.7.
The factor 1.4 is intended to account for the higher flexural capacity than the calculated value. The flexural capacity is higher because the actual yield strength of the steel is higher than the characteristic strength and the steel undergoes strain hardening.
Similarly for the columns, the shear demand should be calculated as the larger of the shear force from analysis and the shear force in the column corresponding to the beams (framing into the column) reaching their flexural capacities. The shear demand (Vu) is given by the following expression (Clause 7.3.4, IS 13920: 1993).
(
, 1 , 2)
1.4
u uR b uR b st
V = M +M h (3.14) Here, MuR, b1 and MuR, b2 are the factored moments of resistance of beam ends ‘1’
and ‘2’ framing into the column from opposite faces, and hst is the storey height
(Figure 3.8).
The shear demands for beams and columns should be checked with the corresponding shear capacities. The shear capacities for beams and columns can be calculated using the procedure outlined in Appendix C.
The axial force demands for the ‘equivalent struts’ should be compared with their capacities. The capacity of the equivalent strut can be calculated according to Appendix B.
The storey drift for every storey due to the design lateral force, with partial load factor of 1.0, should satisfy the limitation of 0.4% of the storey height (Clause 7.11.1, IS 1893: 2002).
Vu
1.4MuR, b2
1.4MuR, b1
hst
Vu