# Nonlinear Dynamic Analysis of Seismically Base-Isolated Structures by a Novel OpenSees Hysteretic Material Model

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Seismically Base-Isolated Structures

#### 2.1. Elastomeric Bearings

#### 2.2. Sliding Bearings

## 3. Modeling of Elastomeric Bearings

#### 3.1. Algebraic Model Formulation

#### 3.2. OpenSees Uniaxial Model

`uniaxialMaterial HystereticPoly $matTag $ka $kb $a $b1 $b2 $tol`

`$matTag`is the progressive tag of the uniaxial material object,

`$ka,$kb,$a,$b1,$b2`represent, respectively, the five model parameters ${k}_{a}$, ${k}_{b}$, $\alpha $, ${\beta}_{1}$, ${\beta}_{2}$, whereas

`$tol`is the model numerical parameter ${\delta}_{k}$ which can be set equal to ${10}^{-20}$ [22].

- (a)
- $\$\mathtt{b}\mathtt{1}=\$\mathtt{b}\mathtt{2}=0$;
- (b)
- $\$\mathtt{b}\mathtt{1}/\$\mathtt{b}\mathtt{2}>0$ with $\$\mathtt{b}\mathtt{1}>0$ and $\$\mathtt{b}\mathtt{2}>0$;
- (c)
- $\$\mathtt{b}\mathtt{1}/\$\mathtt{b}\mathtt{2}>0$ with $\$\mathtt{b}\mathtt{1}<0$ and $\$\mathtt{b}\mathtt{2}<0$;
- (d)
- $\$\mathtt{b}\mathtt{1}/\$\mathtt{b}\mathtt{2}<0$ with $\$\mathtt{b}\mathtt{1}<\$\mathtt{b}\mathtt{2}$.

## 4. Nonlinear Time History Analyses in OpenSees

#### 4.1. Base-Isolated Structure Properties

#### 4.2. Applied Bidirectional Earthquake Excitation

#### 4.3. Hysteretic Material Models Parameters

`uniaxialMaterial BoucWen $matTag $alpha $ko $n $gamma $beta $Ao $deltaA $deltaNu $deltaEta`

`$matTag`is the progressive tag of the uniaxial material object, whereas

`$alpha`,

`$ko`,

`$n`,

`$gamma`,

`$beta`,

`$Ao`,

`$deltaA`,

`$deltaNu`,

`$deltaEta`represent the model parameters [35].

`$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.

#### 4.4. Numerical Results

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Typical SREB (

**a**) and FREB (

**b**): sectional view in deformed configuration (

**left**) and hysteresis loop shape (

**right**).

**Figure 3.**Typical FSSB (

**a**) and CSSB (

**b**): sectional view in deformed configuration (

**left**) and hysteresis loop shape (

**right**).

**Figure 4.**Types of hysteresis loop shapes that can be simulated with the OpenSees uniaxial HystereticPoly material model.

**Figure 5.**3D structural model (

**a**) of the analyzed base-isolated structure and related extruded view (

**b**).

**Figure 6.**SN (

**a**) and SP (

**b**) components of the ground acceleration recorded at the Jensen Filter Plant station during the Northridge earthquake of 17 January 1994.

**Figure 7.**Restoring force-displacement hysteresis loop (

**a**) obtained by applying a harmonic transverse displacement (

**b**) to each lead rubber bearing.

**Figure 10.**Restoring force-displacement hysteresis loops of the $RI$ along the X (

**a**) and Y (

**b**) directions.

BWM | $\$\mathtt{alpha}$ | $\$\mathtt{ko}\left[\mathrm{kN}{m}^{-1}\right]$ | $\$\mathtt{n}$ | $\$\mathtt{gamma}$ | $\$\mathtt{beta}$ | $\$\mathtt{Ao}$ | $\$\mathtt{deltaA}$, $\$\mathtt{deltaNu}$, $\$\mathtt{deltaEta}$ |

0.10 | 857.47 | 1 | 0 | 23 | 1 | 0, 0, 0 | |

AM | $\$\mathtt{ka}\left[\mathrm{kN}{\mathrm{m}}^{-1}\right]$ | $\$\mathtt{kb}\left[\mathrm{kN}{\mathrm{m}}^{-1}\right]$ | $\$\mathtt{a}$ | $\$\mathtt{b}\mathtt{1}\left[\mathrm{kN}{\mathrm{m}}^{-3}\right]$ | $\$\mathtt{b}\mathtt{2}\left[\mathrm{kN}{\mathrm{m}}^{-5}\right]$ | ||

1714.95 | 85.74 | 25 | 0 | 0 |

${\mathit{u}}_{\mathit{x}}^{\left({\mathit{RN}}_{\mathit{b}}\right)}\left[\mathbf{m}\right]$ | ${\mathit{u}}_{\mathit{y}}^{\left({\mathit{RN}}_{\mathit{b}}\right)}\left[\mathbf{m}\right]$ | ${\ddot{\mathit{u}}}_{\mathit{x}}^{\left({\mathit{RN}}_{3}\right)}\left[{\mathbf{ms}}^{-2}\right]$ | ${\ddot{\mathit{u}}}_{\mathit{y}}^{\left({\mathit{RN}}_{3}\right)}\left[{\mathbf{ms}}^{-2}\right]$ | |||||
---|---|---|---|---|---|---|---|---|

Max | Min | Max | Min | Max | Min | Max | Min | |

BWM | 0.1637 | $-0.2695$ | 0.1577 | $-0.1792$ | 2.1143 | −1.8308 | 2.8804 | $-2.4071$ |

AM | 0.1596 | $-0.2684$ | 0.1585 | $-0.1773$ | 2.1101 | −1.8415 | 2.8953 | −2.4074 |

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**MDPI and ACS Style**

Vaiana, N.; Capuano, R.; Sessa, S.; Marmo, F.; Rosati, L.
Nonlinear Dynamic Analysis of Seismically Base-Isolated Structures by a Novel OpenSees Hysteretic Material Model. *Appl. Sci.* **2021**, *11*, 900.
https://doi.org/10.3390/app11030900

**AMA Style**

Vaiana N, Capuano R, Sessa S, Marmo F, Rosati L.
Nonlinear Dynamic Analysis of Seismically Base-Isolated Structures by a Novel OpenSees Hysteretic Material Model. *Applied Sciences*. 2021; 11(3):900.
https://doi.org/10.3390/app11030900

**Chicago/Turabian Style**

Vaiana, Nicoló, Raffaele Capuano, Salvatore Sessa, Francesco Marmo, and Luciano Rosati.
2021. "Nonlinear Dynamic Analysis of Seismically Base-Isolated Structures by a Novel OpenSees Hysteretic Material Model" *Applied Sciences* 11, no. 3: 900.
https://doi.org/10.3390/app11030900