Extended Modal Pushover Analysis: EMPA

9. Interpolate with urn + u_G (being uG the control point displacement in the dominant direction due to gravity loads) from the database of pushover anal-ysis, to obtain the combined eects of lateral loads and gravity due to n-mode contribution (rn+G).

10. Obtain the contribution of n-mode to the seismic response exclusively, by extracting the eect due to the self-weight: rn = r_n+G− rG, where rG is the contribution considering the self-weight acting alone.

11. Repeat steps 4 to 10 for all modes with frequencies below or equal f_gov. 12. Obtain the total nonlinear seismic response by combining the contribution of

each mode with frequency below f_gov, using an appropriate combination rule, here CQC rule is selected for this purpose: rnl.

13. Compute the higher mode seismic response (rel) by means of spectrum analysis (MRSA) including modes with frequencies higher than fgov and lower or equal than fmax= 25Hz.

14. Combine the participation of rst modes with the higher mode eect (between f_gov and f_max), employing SRSS combination rule to obtain the dynamic re-sponse of the structure: rd≈q

r_nl² + r²_el. Frequencies higher than fmax= 25 Hz are ignored.

15. Calculate the total demand (r) by combining the self-weight eect (rG) with the dynamic contribution exclusively due to the earthquake (rd). Since the sign of seismic forces is lost in pushover procedures², two hypotheses are made, considering both positive and negative signs in the earthquake response to take into account that the seismic input, and the consequent structural behaviour, have alternating sign reversals.

r≈ max (rG± rd) (G.4)

G.2 Extended Modal Pushover Analysis: EMPA

EMPA proposed procedure is summarized next, repeating for completeness some steps introduced in MPA:

1. Apply rst the gravity loads considering geometric nonlinearities (P-∆ eects).

2. Perform modal analysis from the deformed conguration of the structure, com-puting frequencies fn, participation factors Γ^jn(j = X, Y, Z) and mode shapes φ_n up to the maximum limiting frequency; fmax = 25 Hz. Next, a study about the participation of each mode below fmax in the global response in

2Pushover is equivalent to spectrum analysis in the elastic range.

404 Appendix G. Step-by-step description of advanced pushover

terms of forces or displacements is conducted (see section6.4.4.1), identifying the governing longitudinal (fnX) and transverse modes (fnY), EMPA covers both dominant modes. The limit frequency which marks the end of the range where pushover is to be conducted is established as fgov= max(f_nX, f_nY). 3. For the nth mode, included in the range studied by pushover analysis (fn ≤

f_gov), develop the base shear-control point displacement curve, ¯V_bn− ¯urn3, by means of three-dimensional nonlinear static analysis of the structure when the load distribution s^∗n = mφn is incrementally applied (self-weight included), now fully considering its three-dimensional characteristics. Three pushover curves are obtained while the structure is being pushed beyond the linear range, each one associated with the longitudinal, transverse and vertical di-rections; (V_bn^X− u^Xrn), (V_bn^Y − u^Yrn),(V_bn^Z− u^Zrn)(see gure6.14). The base shear V_bn^j is the sum of the shear recorded in each connection of the model to the ground in j-direction during the pushover analysis of nth mode, whilst the control point displacement u^jrnis consequently the displacement of the control point in j-direction; this point, is selected as the node with maximum modal displacement regardless of the direction where it is recorded.

4. Transform each V_bn^j − u^jrn pushover curves to Fsn^j /M_n− qn^j coordinates using expressions (6.46) and (6.47), repeated below:

Fsn^j 5. Idealize the real Fsn^j /M_n− qn^j curve for the equivalent SDOF system associ-ated with n-mode in j-direction. Here, a bi-linear curve considering a modied

`Equal Area' rule presented in gure 6.11(a) is proposed because of its sim-plicity (more realistic curves could be considered). The kinematic properties of the nonlinear spring behaviour in the SDOF system need to be addressed (see appendixH), dening the loading and unloading branches appropriate for the structural system and material.

6. Obtain a modular ¯F_sn/M_n− ¯qn from the three-directional results using ex-pressions (6.48) and (6.49), repeated next:

F¯sn

3The bar symbol means `magnitude' of the three directional displacement or shear components.

G.2. Extended Modal Pushover Analysis: EMPA 405

7. Compute the peak generalized modular displacement ¯qn of the n-mode in-elastic SDOF system. Here, the integration of the SDOF dierential equation (6.45) of the corresponding mode in time domain (repeated below) is proposed, following the scheme presented in appendix H.

¨¯

q_n+ 2ξ_nω_nq˙¯_n+F¯_sn

M_n =−¨u^∗g,n(t) (G.9) Where ¨u^∗g,n(t) = Γ^X_nu¨^X_g (t) + Γ^Y_nu¨^Y_g (t) + Γ^Z_nu¨^Z_g (t), and ¨u^jg the accelerogram 3D components.

8. Obtain the peak modular control point displacement ¯urn with:

¯

u^max_rn = ¯φ_rnmax

t [¯q_n(t)] (G.10)

9. Interpolate with ¯urn+ ¯u_G(being ¯uGthe modular displacement of control point due to gravity loads; ¯uG =

q

(u^X_G)²+ (u^Y_G)²+ (u^Z_G)²) from the database of the three-dimensional pushover analysis, in order to obtain the combined eects of lateral loads and gravity due to n-mode contribution rn+G.

10. Obtain the contribution of n-mode to the seismic response exclusively, by extracting the eect due to the self-weight: rn = r_n+G− rG, where rG is the contribution of gravity loads acting alone.

11. Repeat steps 3 to 10 for all modes with frequencies below or equal f_gov. 12. Obtain the total nonlinear seismic response (rnl) by combining the

contribu-tion of each studied mode using an appropriate combinacontribu-tion rule, here CQC rule is selected for this purpose.

13. Compute the higher mode seismic response (rel) by means of spectrum analysis (MRSA), including modes with frequencies higher than f_govand lower or equal than fmax= 25Hz.

14. Combine the participation of rst modes with the higher mode eect by means of SRSS combination rule to obtain the dynamic response of the structure:

r_d≈q

r_nl² + r²_el.

15. Calculate the total demand by combining the eect due to self-weight (rG) with the dynamic contribution due to the earthquake exclusively (rd). Since the sign of earthquake forces is lost in pushover procedures, two hypotheses are made, considering both positive and negative signs in the earthquake response, which takes into account the alternating nature of the seismic response.

r≈ max (rG± rd) (G.11)

406 Appendix G. Step-by-step description of advanced pushover