Critical orientation of earthquake loading for building performance assessment using lateral force analysis

Abstract

The orientation of the seismic action is known to have a non-negligible effect on the seismic behaviour of buildings. However, modern earthquake-related standards only partially cover the effect of this factor and results of practical utility on this topic are still limited. To address this issue, a methodology enabling the determination of the critical angle of seismic incidence in the context of Lateral Force Analysis is proposed. The analytical methodology determines the critical angle of seismic incidence for the storey displacements and the interstorey drifts of buildings that conform with standard-based provisions for linear static analysis. The applicability of the methodology is illustrated for three buildings comprising different typologies in plan and in elevation. The accuracy of the analytical results obtained is validated by performing a parametric lateral force analysis of the buildings for different orientations of the bidirectional seismic action.

Appendix: Derivation of the piecewise expression for the final demand

The terms F(α) and F(α + 90°) included in the expression of the final demand, which is expressed by Eq. (15) (for buildings with a real elastic axis), Eq. (28) (for buildings without a real elastic axis) and Eq. (30) (for the ISD), depend on the corner periods T_B and T_C of the response spectrum and on the principal periods of the structure T_I and T_II. As a result, the referred equations may have different forms according to the following scenarios:

• T_B ≤ T_II and T_I ≤ T_C—In this case all T_unc(α) fall on the horizontal branch of the spectrum and the spectral acceleration S_e is constant for all angles α. Therefore, F(α) and F(α + 90°) obtained from Eq. (11) are also constant and equal to F_const for all angles α. Equations (15), (28) and (30) have one branch in which the base shear can be omitted in the maximization procedure since it is not a function of the angle α. For example, Eq. (15) has the following form:

  $${\text{u}}^{\text{A}} ( {{\upalpha )}} = {\text{F}}^{ 2}_{\text{const}} \cdot \sqrt {\left( {{\text{u}}_{\text{I},\text{``}1\text{''}}^{\text{A}} ( {{\upalpha )}} \pm \beta \cdot {\text{u}}_{\text{I},\text{``}1\text{''}}^{\text{A}} ( {{\upalpha \,+\, 90}}^{\text{o}} )} \right)^{ 2} { + }\left( {{\text{u}}_{\text{II},\text{``}1\text{''}}^{\text{A}} ( {{\upalpha )}} \pm \beta \cdot {\text{u}}_{\text{II},\text{``}1\text{''}}^{\text{A}} ( {{\upalpha + 90}}^{\text{o}} )} \right)^{ 2} }$$

  (38)

• 0 < (T_II, T_I) ≤ T_B or T_C ≤ (T_II, T_I) ≤ T_D—In this case all T_unc(α) fall on the first or the third branch of the spectrum and S_e is a function of the angle α. Since both S_e (α) and S_e (α + 90°) belong to the same branch of the spectrum, F(α) and F(α + 90°) are expressed by the same equation. Hence, Eqs. (15), (28) and (30) have one branch in which F(α) and F(α + 90°) are expressed as functions of the angle α. For example, the terms F(α) and F(α + 90°) of Eq. (13) which lead to the final displacement in Eq. (15) are given by Eq. (39) assuming ω = 0.

  $${\text{F}}(\upalpha ) = \frac{{{\text{T}}_{\text{C}} \cdot {\text{a}}_{\text{g}} \cdot {\text{S}} \cdot {{\upeta }} \cdot 2. 5}}{{\frac{{{\text{T}}_{\text{I}} \cdot {\text{T}}_{\text{II}} }}{{\sqrt {{ \cos }\left( {{\upalpha }} \right)^{ 2} \cdot {\text{T}}_{\text{II}}^{ 2} + { \sin }\left( {{\upalpha }} \right)^{ 2} \cdot {\text{T}}_{\text{I}}^{ 2} } }}}} ,\quad {\text{F}}(\upalpha + 9 0^{\text{o}} ) { = }\frac{{{\text{T}}_{\text{C}} \cdot {\text{a}}_{\text{g}} \cdot {\text{S}} \cdot {{\upeta }} \cdot 2. 5}}{{\frac{{{\text{T}}_{\text{I}} \cdot {\text{T}}_{\text{II}} }}{{\sqrt {{ \cos }\left( {{{\upalpha \,+\, 90}}^{\text{o}} } \right)^{ 2} \cdot {\text{T}}_{\text{II}}^{ 2} + { \sin }\left( {{{\upalpha \,+\, 90}}^{\text{o}} } \right)^{ 2} \cdot {\text{T}}_{\text{I}}^{ 2} } }}}}$$

  (39)

• All other combinations of T_II and T_I—To illustrate this scenario, the case in which T_C ≤ T_I ≤ T_D and T_B ≤ T_II ≤ T_C (see Fig. 10a) is presented herein and the same rationale can be followed for any other combination.

  Fig. 10

  

  Position of T_I and T_II on the response spectrum (a) and relative position of the two orthogonal T_unc (b)

In this case, T_unc(α) falls on different branches of the spectrum depending on the angle α. As a result, S_e is expressed by different equations as a function of α and F(α) has multiple branches, where at least one of these branches is a function of α. The base shear of the perpendicular direction F(α + 90°) is expressed following the same rules. The relative position of T_unc(α) and T_unc(α + 90°) for a given angle α, shown in Fig. 10b, affects the development of Eqs. (15), (28) and (30). It can be seen in Fig. 10b that the angle θ_Tc corresponding to the corner period T_C determines the angle range of the branches of the final expression. To illustrate the formulation of Eqs. (15), (28) and (30), the relative position of T_unc(α) and T_unc(α + 90°) is presented in Fig. 11 for all ASIs and for all three cases with respect to the angle θ_Tc. The solid hatched areas represent the position of T_unc(α), while the dashed hatched areas represent the respective position of T_unc(α + 90°). As a result, Eqs. (15), (28) and (30) comprise eight branches with specific angle range limits. The terms F(α) and F(α + 90°) of Eq. (15) that are used to form the piecewise equation are presented in Eq. (40) for the first case of Fig. 11, i.e. θ_Tc < 45° and ω = 0.

Fig. 11

Determination of the branches of the piecewise equation

$$\left\{ {\begin{array}{*{20}l} {\begin{array}{*{20}c} { 2 {{\uppi }} -\uptheta_{{{\text{T}}_{\text{c}} }} \le {{\upalpha }} \le\uptheta_{{{\text{T}}_{\text{c}} }} } \\ {\text{and}} \\ {\uptheta_{{{\text{T}}_{\text{c}} }} \le {{\upalpha \,+\, 90}}^{ \circ } \le {{\uppi }} -\uptheta_{{{\text{T}}_{\text{c}} }} } \\ \end{array} } \hfill & \Rightarrow \hfill & {{\text{F}}(\upalpha ) = \frac{{{\text{T}}_{\text{C}} \cdot {\text{a}}_{\text{g}} \cdot {\text{S}} \cdot {{\upeta }} \cdot 2. 5}}{{\frac{{{\text{T}}_{\text{I}} \cdot {\text{T}}_{\text{II}} }}{{\sqrt {{ \cos }\left( {{\upalpha }} \right)^{ 2} \cdot {\text{T}}_{\text{II}}^{ 2} {\text{ + sin}}\left( {{\upalpha }} \right)^{ 2} \cdot {\text{T}}_{\text{I}}^{ 2} } }}}},} \hfill & {{\text{F}}(\upalpha + 9 0^{ \circ } )= {\text{a}}_{\text{g}} \cdot {\text{S}} \cdot {{\upeta }} \cdot 2. 5} \hfill \\ {\begin{array}{*{20}c} {\uptheta_{{{\text{T}}_{\text{c}} }} \, \le \,\,{{\upalpha }}\, \le \,{{\uppi }}\,{ - }\,\uptheta_{{{\text{T}}_{\text{c}} }} } \\ {\text{and}} \\ {\uptheta_{{{\text{T}}_{\text{c}} }} \le {{\upalpha \,+\, 90}}^{ \circ } \le {{\uppi }} -\uptheta_{{{\text{T}}_{\text{c}} }} } \\ \end{array} } \hfill & \Rightarrow \hfill & {{\text{F}}(\upalpha ) = {\text{a}}_{\text{g}} \cdot {\text{S}} \cdot {{\upeta }} \cdot 2. 5,} \hfill & {{\text{F}}(\upalpha + 9 0^{ \circ } )= {\text{a}}_{\text{g}} \cdot {\text{S}} \cdot {{\upeta }} \cdot 2. 5} \hfill \\ {\begin{array}{*{20}c} {\uptheta_{{{\text{T}}_{\text{c}} }} \, \le \,\,{{\upalpha }}\, \le \,{{\uppi }}\,{ - }\,\uptheta_{{{\text{T}}_{\text{c}} }} } \\ {\text{and}} \\ {{{\uppi }}\,{ - }\,\uptheta_{{{\text{T}}_{\text{c}} }} \, \le \,{{\upalpha }}\,{ + }\, 9 0^{ \circ } \, \le \,{{\uppi }}\,{ + }\,\uptheta_{{{\text{T}}_{\text{c}} }} } \\ \end{array} } \hfill & \Rightarrow \hfill & {{\text{F}}(\upalpha ) = {\text{a}}_{\text{g}} \cdot {\text{S}} \cdot {{\upeta }} \cdot 2. 5,} \hfill & {{\text{F}}(\upalpha \,+\, 90^{ \circ } )= \frac{{{\text{T}}_{\text{C}} \cdot {\text{a}}_{\text{g}} \cdot {\text{S}} \cdot {{\upeta }} \cdot 2. 5}}{{\frac{{{\text{T}}_{\text{I}} \cdot {\text{T}}_{\text{II}} }}{{\sqrt {{ \cos }\left( {{{\upalpha \,+\, 90}}^{ \circ } } \right)^{ 2} \cdot {\text{T}}_{\text{II}}^{ 2} + { \sin }\left( {{{\upalpha \,+\, 90}}^{ \circ } } \right)^{ 2} \cdot {\text{T}}_{\text{I}}^{ 2} } }}}}} \hfill \\ {\begin{array}{*{20}c} {\theta_{{T_{c} }} \, \le \,\alpha \, \le \,\pi \, - \,\theta_{{T_{c} }} } \\ {\text{and}} \\ {\pi \, + \,\theta_{{T_{c} }} \, \le \,\,\alpha \, + \,90^{ \circ } \, \le \,2\,\pi \, - \,\theta_{{T_{c} }} } \\ \end{array} } \hfill & \Rightarrow \hfill & {{\text{F}}(\upalpha ) = {\text{a}}_{\text{g}} \cdot {\text{S}} \cdot {{\upeta }} \cdot 2. 5,} \hfill & {{\text{F}}(\upalpha \,+\, 90^{ \circ } )= {\text{a}}_{\text{g}} \cdot {\text{S}} \cdot {{\upeta }} \cdot 2. 5} \hfill \\ {\begin{array}{*{20}c} {{{\uppi }}\,{ - }\,\uptheta_{{{\text{T}}_{\text{c}} }} \, \le \,\,{{\upalpha }}\, \le \,{{\uppi }}\,{ + }\,\uptheta_{{{\text{T}}_{\text{c}} }} } \\ {\text{and}} \\ {{{\uppi }}\,{ + }\,\uptheta_{{{\text{T}}_{\text{c}} }} \, \le \,\,{{\upalpha }}\,{ + }\, 9 0^{ \circ } \, \le \, 2 {{\uppi }}\,{ - }\,\uptheta_{{{\text{T}}_{\text{c}} }} } \\ \end{array} } \hfill & \Rightarrow \hfill & {{\text{F}}(\upalpha ) = \frac{{{\text{T}}_{\text{C}} \cdot {\text{a}}_{\text{g}} \cdot {\text{S}} \cdot {{\upeta }} \cdot 2. 5}}{{\frac{{{\text{T}}_{\text{I}} \cdot {\text{T}}_{\text{II}} }}{{\sqrt {{ \cos }\left( {{\upalpha }} \right)^{ 2} \cdot {\text{T}}_{\text{II}}^{ 2} {\text{ + sin}}\left( {{\upalpha }} \right)^{ 2} \cdot {\text{T}}_{\text{I}}^{ 2} } }}}},} \hfill & {{\text{F}}(\upalpha \,+\, 90^{ \circ } )= {\text{a}}_{\text{g}} \cdot {\text{S}} \cdot {{\upeta }} \cdot 2. 5} \hfill \\ {\begin{array}{*{20}c} {{{\uppi }}\,{ + }\,\uptheta_{{{\text{T}}_{\text{c}} }} \, \le \,\,{{\upalpha }}\, \le \, 2 {{\uppi }}\,{ - }\,\uptheta_{{{\text{T}}_{\text{c}} }} } \\ {\text{and}} \\ {{{\uppi }}\,{ + }\,\uptheta_{{{\text{T}}_{\text{c}} }} \, \le \,\,{{\upalpha }}\,{ + }\, 9 0^{ \circ } \, \le \, 2 {{\uppi }}\,{ - }\,\uptheta_{{{\text{T}}_{\text{c}} }} } \\ \end{array} } \hfill & \Rightarrow \hfill & {{\text{F}}(\upalpha ) = {\text{a}}_{\text{g}} \cdot {\text{S}} \cdot {{\upeta }} \cdot 2. 5,} \hfill & {{\text{F}}(\upalpha \,+\, 90^{ \circ } )= {\text{a}}_{\text{g}} \cdot {\text{S}} \cdot {{\upeta }} \cdot 2. 5} \hfill \\ {\begin{array}{*{20}c} {{{\uppi }}\,{ + }\,\uptheta_{{{\text{T}}_{\text{c}} }} \, \le \,\,{{\upalpha }}\, \le \, 2 {{\uppi }}\,{ - }\,\uptheta_{{{\text{T}}_{\text{c}} }} } \\ {\text{and}} \\ { 2 {{\uppi }}\,{ - }\,\uptheta_{{{\text{T}}_{\text{c}} }} \, \le \,\,{{\upalpha }}\,{ + }\, 9 0^{ \circ } \, \le \,\uptheta_{{{\text{T}}_{\text{c}} }} } \\ \end{array} } \hfill & \Rightarrow \hfill & {{\text{F}}(\upalpha ) = {\text{a}}_{\text{g}} \cdot {\text{S}} \cdot {{\upeta }} \cdot 2. 5,} \hfill & {{\text{F}}(\upalpha \,+\, 90^{ \circ } )= \frac{{{\text{T}}_{\text{C}} \cdot {\text{a}}_{\text{g}} \cdot {\text{S}} \cdot {{\upeta }} \cdot 2. 5}}{{\frac{{{\text{T}}_{\text{I}} \cdot {\text{T}}_{\text{II}} }}{{\sqrt {{ \cos }\left( {{{\upalpha \,+\, 90}}^{ \circ } } \right)^{ 2} \cdot {\text{T}}_{\text{II}}^{ 2} + { \sin }\left( {{{\upalpha \,+\, 90}}^{ \circ } } \right)^{ 2} \cdot {\text{T}}_{\text{I}}^{ 2} } }}}}} \hfill \\ {\begin{array}{*{20}c} {{{\uppi }}\,{ + }\,\uptheta_{{{\text{T}}_{\text{c}} }} \, \le \,\,{{\upalpha }}\, \le \, 2 {{\uppi }}\,{ - }\,\uptheta_{{{\text{T}}_{\text{c}} }} } \\ {\text{and}} \\ {\uptheta_{{{\text{T}}_{\text{c}} }} \, \le \,\,{{\upalpha }}\,{ + }\, 9 0^{ \circ } \, \le \,{{\uppi }}\,{ - }\,\uptheta_{{{\text{T}}_{\text{c}} }} } \\ \end{array} } \hfill & \Rightarrow \hfill & {{\text{F}}(\upalpha ) = {\text{a}}_{\text{g}} \cdot {\text{S}} \cdot {{\upeta }} \cdot 2. 5,} \hfill & {{\text{F}}(\upalpha \,+\, 90^{ \circ } )= {\text{a}}_{\text{g}} \cdot {\text{S}} \cdot {{\upeta }} \cdot 2. 5} \hfill \\ \end{array} } \right.$$

(40)