## 4 Nonlinear Time History Analyses in OpenSees
### Hysteretic Material Models Parameters  
The adopted lead rubber bearings exhibit the restoring force-displacement hysteresis loop of Figure 7a when subjected to a sinusoidal transverse displacement having amplitude \(D_{d}=0.40\) m and unit frequency, as the one illustrated in Figure 7b. Such a hysteresis loop is characterized by an effective (or secant) stiffness and an equivalent viscous damping ratio that allow the analyzed base-isolated structure to have a \(T_{eff}=3.00\) s and a \(\zeta_{eff}=0.30\) at \(D_{d}=0.40\) m.  
To perform the NLTHAs in OpenSees, each elastomeric bearing is modeled by inserting, under each superstructure column, two truss elements parallel to the \(X\) and \(Y\) axes, respectively. In particular, the axial hysteretic behavior of such elements is simulated by using both the HystereticPoly material model, presented in Section 3, and the widely used OpenSees BoucWen material model whose command line is:  
uniaxialMaterial BoucWen $matTag $alpha $ko $n $gamma $beta $Ao $deltaA  $deltaNu $deltaA  
where $matTag is the progressive tag of the uniaxial material object, whereas $alpha, $ko, $n, $gamma, $beta, $Ao, $deltaA, $deltaNu, $deltaAe represent the model parameters [35].  
Specifically, the parameters adopted in the Algebraic Model (AM) and in the BoucWen Model (BWM), selected to reproduce the hysteresis loop illustrated in Figure 7a, are listed in Table 1.  
Figure 7: Restoring force-displacement hysteresis loop (a) obtained by applying a harmonic transverse displacement (b) to each lead rubber bearing.  
We note that the AM parameters have a clear mechanical significance which may allow such a model to be easily employed in actual applications. Indeed, when \(\$b1=\$b2=0,\$ka(\$kb)\) represents the value of the tangent stiffness at the beginning (end) of the generic loading or unloading curve, whereas \(\$a\) defines the rate of variation of the tangent stiffness from \(\$ka\) to \(\$kb\). Conversely, the interpretation of the BWM parameters is not straightforward.  
The current OpenSees library offers other four material objects suitable for the simulation of the uniaxial transverse response of elastomeric bearings: (1) Hysteretic material, (2) MultiLinear material, (3) KikuchiAikenHDR material, and (4) KikuchiAikenLRB material.  
The first two models are characterized by a lower level of accuracy compared to the Bouc-Wen one since, differently from the latter, they are not smooth models and, for such a reason, their typical hysteresis loops are made of a series of straight lines instead of continuous curves. In addition, since the first derivative of their output variable is a noncontinuous function, the convergence of the nonlinear dynamic analyses may be drastically compromised [35].  
The last two models are as accurate as the Bouc-Wen one but require more parameters which can be calibrated only by performing complicated and expensive experimental tests, as explained in [36].  
For all those reasons, the numerical accuracy of the proposed OpenSees material object is demonstrated by comparing the results with those obtained by employing the Bouc-Wen material.