# Failure Analysis of a Concrete Anchor under Severe Seismic Action
## 2 Response Modification and Overstrength Factors
This overstrength indicates the extent to which the yield base shear force exceeds the design base shear force. When including overstrength in seismic design, the overstrength coefficient (\(\Omega_{o}=F_{y}/F_{d}\)) is used; this is the ratio of the design load to the yield load. Thus, the seismic design criteria determine the value of the inelastic energy absorption coefficient (\(R_{\mu}\)) multiplied by the overstrength  
Figure 1: Dynamic factors in different structural periods, \(S_{\text{D1}}\): the design spectral response acceleration parameter at a period of 1.0 s.  
coefficient (\(\Omega_{\rm o}\)); this is the response modification factor \(R\)([PERSON] et al., 1999, ATC-19, 1995, ASCE 7-16) of Equation (4). \[R=R_{\mu}\times\Omega_{\rm o} \tag{4}\]  
Elastic analysis using the response modification coefficient under a reduced seismic load shows that the displacement does not accurately reflect the inelastic behavior. Therefore, seismic design uses an approximation of inelastic displacement; the deflection amplification factor (\(C_{d}\)) is multiplied by the design deflection. During real-world structural design, \(F_{d}=F_{E}/R\) is exceeded given the cascading increases in cross-section size or back-up mass; this is termed \"design overstrength\". As the structural material strength exceeds the nominal strength, the structure behaves elastically under higher lateral forces than expected (\"material overstrength\"). As the lateral force gradually increases, the structure or its member moves out of the elastic range and can resist the formation of plastic hinges either locally or globally. This is attributable to structural redundancy, and is the \"system overstrength\" illustrated in Figure 2.  
Equations (5) and (6) are relevant for the design of anchors in compression/tension zones. Care must be taken when using Equations (5) and (6) in the presence of negative vertical acceleration (7) to (10) below due to potential uplift. ASCE 7 defines the load combination when the earthquake load (\(E\)) is primary. \(D\), \(L\), and \(S\) are the dead, live, and snow loads, respectively. The basic earthquake load E is equivalent to \(Q_{E}\) (\(=F_{E}\)). During elastic design, this horizontal earthquake load can be expressed as \(E_{h}=\rho Q_{E}\), where \(\rho\) is a redundancy factor. On the other hand, during ductile design, the horizontal earthquake load can be expressed as \(E_{mh}=\Omega_{\rm o}Q_{E}\), where \(\Omega_{\rm o}\) is an overstrength factor and \(\rho\) is set to 1. This reduces the response modification factor \(R\) to \(R_{\mu}\) (the decrease is expressed as \(\Omega_{\rm o}\)). This increases the horizontal earthquake load, \(Q_{E}\), because of the overstrength imparted by the design, material, and structural systems in a ductile design. Figure 3 shows graphical versions of the load combinations for elastic and ductile design. \[U=1.2D+1.0E+L+0.2S \tag{5}\]  
\[U=0.9D+1.0E \tag{6}\]  
where, \(E=E_{h}\pm E_{\rm v}\), \(E_{h}=\rho Q_{E}\), \(E_{\rm v}=0.2S_{DS}D=0.5A\cdot D\) and the resulting load combinations are:  
\[U=(1.2+0.2S_{DS})D+\rho Q_{E}+L+0.2S \tag{7}\]  
Figure 1.0  
Figure 1.1  
Figure 1.2 Response modification, overstrength and ductility.