Seismic Performance of RC Building Using Spectrum Response and Pushover Analyses
Abstract
Classical seismic design codes consider behavior factor to assess seismic response of structures proposing simplified equivalent static method and dynamic spectral modal method using response spectrum. Both use a global behavior factor to take into account nonlinear effect of elements response. Nowadays Performance based Seismic Design considers nonlinear static analyses using Pushover analysis with different performance levels. They permit to determine the bearing capacity of structures and distinguish whether they will be able to withstand major earthquakes. The first mode is in general considered in such kind of analysis.
This study will compare the shear base forces, drift and absolute displacements using dynamic response spectrum analysis and pushover analysis according to Eurocode 8. The results are carried in accordance with the Algerian seismic design Code in force RPA99/version 2003 and ETABS 2013 program.
1 Introduction
The Algerian seismic code propose simplified methods to assess the response of structures under earthquakes actions, such as equivalent static method and spectrum response method which permit to evaluate first lateral forces, and then to check lateral displacements. The nonlinear behavior of dissipative elements is taken into account through the behavior factor R, which consider dissipative take. The 2003 Boumerdes major earthquake, magnitude 6.9 showed some shortcomings of these simplified methods. In the recent years, new methods of seismic assessment and design have been developed, particularly with respect to real nonlinear behavior of structural elements using nonlinear static analysis (pushover) with several performance levels. The basic theory consider capacity design approach. The literature propose many procedures like ATC40, FEMA 354, FEMA 440, N2 method (Eurocode 8).
1.1 Linear Response Spectrum Method
Response spectrum method is one of the useful tools of earthquake engineering for analyzing the performance of structures especially in earthquakes, since many systems behave as single degree of freedom systems. Thus, once the natural frequency of the structure evaluated, then the peak response of the building can be estimated by reading the value from the ground response spectrum for the appropriate frequency. In most building codes in seismic regions, this value forms the basis for calculating the forces that a structure must be designed to resist seismic analysis. Response spectrum method is a linear dynamic statistical analysis method which measures the contribution from each natural mode of vibration to indicate the likely maximum seismic response of an essentially elastic structure. Response spectrum analysis provides insight into dynamic behavior by measuring pseudospectral acceleration, velocity, or displacement as a function of structural period for a given time history and level of damping. It is practical to envelope response spectra such that a smooth curve represents the peak response for each realization of structural period (RPA 2003).
 S_{a}:

inelastic response spectrum in terms of acceleration.
 g:

gravity.
 T:

vibration period of a linear single degree of freedom system.
 T_{1}:

lowest characteristic period of site.
 η:

damping correction factor.
 Q:

quality factor.
 R:

behavior factor.
 T_{2}:

highest characteristic period of site.

square root of the sum of the squares (SRSS).

complete quadratic combination (CQC). A method that is an improvement on SRSS for closely spaced modes.
1.2 Nonlinear Static Analysis
Linear static analysis assumes that the relationship between loads and the induced response is linear. For example, if you double the magnitude of loads, the response (displacements, strains, stresses, reaction forces, etc.), will also double. All real structures behave nonlinearly at some level of loading. In some cases, linear analysis may be adequate. In many other cases, the linear solution can produce erroneous results because the conservatism assumptions. Nonlinear analysis methods are best applied when either geometric or material nonlinearity is considered (Iqbal 1999).
Geometric Nonlinearity:
This is a type of nonlinearity where the structure is still elastic, but the effects of large deflections cause the geometry of the structure to change, so that linear elastic theory breaks down. Typical problems that lie in this category are the elastic instability of structures, such as in the Euler bulking of struts and the large deflection analysis of a beamcolumn member. In general, it can be said that for geometrical nonlinearity, an axially applied compressive force in a member decreases its bending stiffness, but an axially applied tensile force increases its bending stiffness. In addition, PDelta effect is also included in this concept.
Material Nonlinearity:
In this type of nonlinearity, material undergoes plastic deformation. Material nonlinearity can be modeled as discrete hinges at a number of locations along the length of a frame (beam or column) element and a discrete hinge for a brace element as discrete material fibers distributed over the crosssection of the element, or as a series of material points throughout the element.
Static Pushover Analysis:

Fully Operational: No significant damages has occurred to structural and nonstructural components. Building is suitable for normal occupancy and use.

Operational: no significant damage has occurred to structure, which retains nearly all of its preearthquake strength and stiffness. Non structural components are secure and most would function.

Life Safety: significant damage to structural elements, with substantial reduction in stiffness, however, margin remains against collapse. Nonstructural elements are secured but may not function. Occupancy may be prevented until repair can be instituted.

Near Collapse: substantial structural and nonstructural damage. Structural strength and stiffness substantially degraded. Little margin against collapse. Some falling debris hazards may have occurred.
The capacity curve can then be combined with a demand curve, typically in the form of an Acceleration Displacement Response Spectrum (ADRS). This combination essentially reduces the problem to an equivalent single degree of freedom system. Static pushover analysis is most suitable for systems in which the fundamental mode dominates behavior. Results provide insight into the ductile capacity of the structural system, and indicate the mechanism, load level, and deflection at which failure occurs.
1.3 Pushover Analysis According to EC8
Pushover analysis is performed under two lateral load patterns. A load distribution corresponding to the fundamental mode shape and a uniform distribution proportional to masses (CEN EC8 2003).
 1.
Cantilever model of the structure with concentrated masses with elastic behavior (uncracked cross sections).
 2.
Determination of the fundamental period of vibration.
 3.
Determination of the fundamental mode shape (Eigen vectors) normalized in such a way that ϕ_{n} = 1.
 4.
Determination of the modal mass coefficient for the first natural mode.
 5.
Determination of the lateral displacements at each level for the first natural mode.
 6.
Determination of seismic forces at each level for the first natural mode.
 7.Transformation of the multi degree of freedom system to an equivalent single degree of freedom with an equivalent mass m* determined as:$$ {\text{m}}^{*} = \sum {{\text{m}}_{\text{j}}\upphi_{\text{j,1}} = \sum {{\bar{\text{F}}}_{\text{j}} } } $$(1)
 8.Determination of the modal participation factor of the first natural mode:The force F* and displacement d* of the equivalent single degree of freedom are computed as:$$ \Gamma = \frac{{{\text{m}}^{ *} }}{{\sum {{\text{m}}_{\text{j}}\upphi_{\text{j,1}}^{ 2} } }} = \frac{{\sum {{\bar{\text{F}}}_{\text{j}} } }}{{\sum {\left( {\frac{{{\bar{\text{F}}}_{\text{j}}^{2} }}{{{\text{m}}_{\text{j}} }}} \right)} }} $$(2)where F_{b} and d_{n} are, respectively, the base shear force and the control node displacement of the multi degree of freedom system.$$ \text{F}^{ * } = \frac{{\text{F}_{\text{b}} }}{\Gamma }\;\;\text{et}\;\;\text{d}^{ * } = \frac{{{\text{d}}_{\text{n}} }}{\Gamma } $$(3)
 9.Determination of the idealized elastic perfectly plastic force displacement relationship. The yield force \( {\text{F}}_{\text{y}}^{ *} \), which represents also the ultimate strength of the equivalent single degree of freedom system, is equal to the base shear force at the formation of the plastic mechanism. The initial stiffness of the equivalent single degree of freedom system is determined in such a way that the areas under the actual and the equivalent single degree of freedom system force displacement curves are equal. Based on this assumption, the yield displacement of the equivalent single degree of freedom system \( d_{y}^{ * } \) is given by:where \( E_{\text{m}}^{ *} \) is the actual deformation energy up to the formation of the plastic mechanism. Figure 1 shows the principle of energies idealization.$$ {\text{d}}_{\text{y}}^{ * } = 2\left( {{\text{d}}_{\text{m}}^{ * }  \frac{{{\text{E}}_{\text{m}}^{ * } }}{{{\text{F}}_{\text{y}}^{ * } }}} \right) $$(4)
 10.Determination of the period of the equivalent single degree of freedom system T* as:$$ {\text{T}}^{ * } = 2\pi \sqrt {\frac{{{\text{m}}^{ * } {\text{d}}_{\text{y}}^{ * } }}{{{\text{F}}_{\text{y}}^{ * } }}} $$(5)
 11.Determination of the target displacement for the equivalent single degree of freedom system as:where S_{e}(T*) is the elastic acceleration response spectrum at the period T*. For the determination of the target displacement \( d_{t}^{ * } \) for structures in the short period range and for structures in the medium and long period ranges, different expressions should be used.$$ {\text{d}}_{\text{et}}^{ * } = {\text{S}}_{\text{e}} ( {\text{T}}^{ * } )\left( {\frac{{{\text{T}}^{ * } }}{{ 2\uppi}}} \right)^{ 2} $$(6)
 (a)T^{*} < T_{C} (short period range)
 If \( {\text{F}}_{\text{y}}^{ * } / {\text{m}}^{ * } \ge {\text{S}}_{\text{e}} ( {\text{T}}^{ * } ) \), the response is elastic and thus:$$ {\text{d}}_{\text{t}}^{ *} {\text{ = d}}_{\text{et}}^{ *} $$(7)
 If \( {\text{F}}_{\text{y}}^{ *} / {\text{m}}^{ *} \;{ < }\;{\text{S}}_{\text{e}} ( {\text{T}}^{ *} ) \), the response is nonlinear and:where q_{u} is the ratio between the acceleration in the structure with unlimited elastic behavior S_{e}(T*) and in the structure with limited strength \( {\rm F}_{y}^{*} /{\rm m}^{*} \).$$ \begin{aligned} {\text{d}}_{\text{t}}^{ *} = & \frac{{{\text{d}}_{\text{et}}^{ *} }}{{{\text{q}}_{\text{u}} }}\left( { 1+ \left( {{\text{q}}_{\text{u}}  1} \right)\frac{{{\text{T}}_{\text{C}} }}{{{\text{T}}^{ *} }}} \right) \ge {\text{d}}_{\text{et}}^{ *} \\ & {\text{with}}\quad {\text{q}}_{\text{u}} = \frac{{{\text{S}}_{\text{e}} ( {\text{T}}^{ *} ) {\text{m}}^{ *} }}{{{\text{F}}_{\text{y}}^{ *} }} \\ \end{aligned} $$(8)

 (b)T^{*} ≥ T_{C} (medium and long period range), the response is nonlinear and:$$ {\text{d}}_{\text{t}}^{*} = {\text{d}}_{\text{et}}^{*} \quad {\text{with}}\quad {\text{d}}_{\text{t}}^{*} \le 3\,{\text{d}}_{\text{et}}^{*} $$(9)
 (a)
 12.The target displacement of the multi degree of freedom system is then:$$ {\text{d}}_{\text{t}} =\Gamma {\text{d}}_{\text{t}}^{ *} $$(10)
2 Case Study

The grade of concrete is f_{c28} = 20 MPa.

The grade of steel is f_{e} = 235 MPa.

The roof dead load is GT = 6.53 KN/m^{2}.

The roof live load is QT = 1 KN/m^{2}.

The current level dead load is GE = 5.51 KN/m^{2}.

The current level live load is QE = 2.5 KN/m^{2}.
Dimensions of beams and columns
Story  Beam (cm^{2})  Rebar  Column (cm^{2})  Rebar 

1 to 4  30 × 40  6HA12  40 × 40  8HA14 
5 to 8  30 × 40  6HA12  30 × 30  84HA14 
3 Results
3.1 Linear Static Analysis
Natural periods and modal participating mass ratios
Direction  T (s)  α_{1} (%) 

Long (XX)  1.962  80.087 
Trans (YY)  2.125  79.978 
Mode shapes for the two first modes
Story  First mode shapes  

Long (XX)  Trans (YY)  
8  1.000  1.000 
7  0.994  0.985 
6  0.976  0.951 
5  0.634  0.886 
4  0.818  0.760 
3  0.660  0.604 
2  0.474  0.420 
1  0.269  0.225 
Absolute displacements with LRSM
Story  \( \delta_{x}^{k} \left( m \right) \)  \( \delta_{y}^{k} \left( m \right) \) 

8  0.248  0.273 
7  0.237  0.261 
6  0.214  0.236 
5  0.182  0.200 
4  0.141  0.157 
3  0.112  0.123 
2  0.079  0.085 
1  0.043  0.044 
3.2 Nonlinear Static Analysis
Modal participation factors
Direction  Long (XX)  Trans (YY) 

Γ_{1}  1.187  1.223 
Main parameters of equivalent SDOF system
Direction  Long (XX)  Trans (YY) 

\( {\text{d}}_{\text{m}}^{*} \left( {\text{m}} \right) \)  0.200  0.066 
\( {\text{d}}_{\text{y}}^{*} \left( {\text{m}} \right) \)  0.059  0.027 
\( {\text{T}}_{\text{C}} \left( {\text{s}} \right) \)  0.500  0.500 
\( {\text{T}}^{*} \left( {\text{s}} \right) \)  2.816  2.168 
\( {\text{S}}_{\text{e}} \left( {{\text{T}}^{*} } \right)\left( {\% {\text{g}}} \right) \)  0.262  0.312 
\( {\text{d}}_{\text{et}}^{*} \left( {\text{m}} \right) \)  0.516  0.364 
\( {\text{d}}_{\text{t}} \left( {\text{m}} \right) \)  0.613  0.446 
Absolute displacements with NLSA
Story  \( \delta_{x}^{k} \left( m \right) \)  \( \delta_{y}^{k} \left( m \right) \) 

8  0.613  0.446 
7  0.610  0.439 
6  0.599  0.424 
5  0.573  0.395 
4  0.502  0.339 
3  0.405  0.269 
2  0.290  0.187 
1  0.165  0.100 
4 Discussion

The linear analysis with linear spectrum method in terms of absolute displacements showed that the demand is less than the capacity for the longitudinal direction (XX).

The linear analysis with linear spectrum method in terms of absolute displacements showed that the demand is greater than the capacity for the transversal direction (YY).

The nonlinear analysis with pushover analysis in terms of absolute displacements showed that the demand is greater than the capacity for both longitudinal (XX) and transversal (YY) directions.

Even though all the modes were considered in the linear analysis, the results showed a great difference compared to the nonlinear analysis.

The nonlinear static analysis showed that the obtained values of absolute displacements are more than double than those obtained for the linear static analysis.

For adjacent buildings, the dimension of the seismic joint is a critical choice. In case of a small gap, the hammering effect could be dangerous. Otherwise, in case of a large gap, it’s not economic.
5 Conclusion
The analysis of absolute displacements of a building using linear response spectrum methods, taking into account indirectly the nonlinear behavior of structural elements by introducing the behavior factor, and the nonlinear static analysis using pushover procedure have been performed. The results showed a large difference between the two methods. The question remains, is the response spectrum response more conservative, or the nonlinear static analysis less conservative?
One can say that the type of loading in the nonlinear static analysis has a predominant role to determine the capacity curve.
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