# On the Distribution in Height of Base Shear Forces in Linear Static Analysis of Base-Isolated Structures

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{i}at the i-th story can be determined as a function of both the total design base shear (V

_{b}) and the story masses m

_{i}[17,34]:

_{b}that can be described as follows:

_{i}in Equation (2) [44,46] or combining Equations (1) and (2) [41,42,48].

## 2. Research Methodology: Parametric Analysis

#### 2.1. Superstructure

- linear-elastic behavior of the superstructure, which is generally accepted in BI-structures;
- rigid diaphragm constraints are set at all floor;
- soil-structure interaction is neglected;
- torsional effects are also neglected, as the superstructure is regular both in plan and height.

^{2}and is divided into three equal bays along the X-direction and three equal bays along the Z-direction. The story height is 3.10 m (Figure 2).

_{1}and G

_{2}) at the intermediate floor is equal to 7.17 kN/m

^{2}, at the base levels 8 kN/m

^{2}and at the roof levels 4.30 kN/m

^{2}. Live load (Q

_{k}) is the same for each level and it is equal to 2 kN/m

^{2}. Moreover, the infill walls are not considered as structural elements and are modelled with a uniform load of 7.25 kN/m. Therefore, the total masses of the three-, five- and seven-floor superstructures are 1142 t, 1817 t and 2555 t, respectively.

_{s}/M

_{b}and the superstructure fundamental period T

_{bf}.

#### 2.2. Seismic Isolation System

_{1}(before yielding/sliding), the post-yield stiffness K

_{2}and the intersection Q between the post-elastic branch and the vertical axis (Figure 3). The other parameters are the yield force and displacement, F

_{y}and X

_{y}, and the NLTH design displacement demand X

_{d}. The ratios α = K

_{2}/K

_{1}and μ = X

_{d}/X

_{y}are the post-yield hardening and ductility ratio of the IS, respectively. Specifically, for FPB, the mechanical behavior is described by an elastic-hardening curve with a high value of the initial stiffness, K

_{2}is given by the W/R ratio and the characteristic force Q is equal to υW, where the W is the weight supported by the bearings, R is the effective curvature radius of the concave surface, and υ is the friction coefficient.

_{eff}and effective damping ratio ξ. The effective stiffness is defined as the secant stiffness to the design displacement X

_{d}and is related to the effective period T

_{eq}of the BI-building. The hysteretic damping ratio is given by the energy dissipation principle, based on the equivalence between the energy dissipated by one cycle of the bilinear model, EH, and the damping energy of the linear damped system, ED, related to the maximum displacement value [57].

_{eff}and the effective damping ratio ξ, at the displacement X

_{d}are calculated by the following equations:

_{eq}, and various values of initial stiffness K

_{1}of the isolators. For the sake of clarity, the ranges of α and ξ parameters assumed in this study are wider than those typically associated to the IS in order to get a better knowledge of the BI structural dynamics and evaluated the applicability of the formulae over a larger selection. In fact, the α values vary between 0 and 1 and the ξ values, with respect to the limit values of the previous parameter, can increase from 0 to 0.63. On the one hand, the lower effective damping value, related to α equal to 1, represents a linear behavior, on the other the upper limit, related to α equal to 0, describes an elastic perfectly plastic curve. It is worth noting that the typical values of ξ parameter range from 0.05 to 0.35.

_{eq}), ductility ratio (μ), initial stiffness and period (K

_{1}, T

_{1}), the stiffness ratio (α) and, obviously, the hysteretic damping ratio (ξ).

#### 2.3. Ground Motions

_{g}, F

_{0}, and T

_{C}* are provided to generate the horizontal elastic acceleration and displacement spectra. More details (the shape of an acceleration spectrum and the meanings of the different parameters) can be found in Chapter 3 of NTC 2018 [34].

_{d}, defined according to the simplified approach suggested by the Italian Building Code, is obtained from the elastic design spectrum for a reference return period T

_{r}= 975 years. Only the limit state of collapse prevention (SLC) with a low probability of occurrence of the 5% in the design working life of 50 years is considered as seismic input. The seismic parameters for the design spectra are listed in Table 4. As for geotechnical parameters, soil type B and topology type T

_{1}are considered.

_{eq}: the average spectrum is never lower than 10% and higher than 30% of the target response spectrum.

#### 2.4. Methods

- Step 1:
- NLTH analyses of base isolated building configurations by considering a set of seven spectrum-compatible accelerograms;
- Step 2:
- Average of the maximum story shear values at the different levels derived from the selected seven accelerograms;
- Step 3:
- Conversion of the median peak story shears V
_{i}at the i-th level to median lateral force F_{i}, according to the following equations:$${\mathrm{V}}_{\mathrm{i}}={\displaystyle \sum}_{\mathrm{j}=1}^{\mathrm{NColumn}}{\mathrm{F}}_{\mathrm{j}}{}^{\left(\mathrm{i}\right)}$$$${\mathrm{F}}_{\mathrm{i}}{=\mathrm{V}}_{\mathrm{i}}{-\mathrm{V}}_{\mathrm{i}+1}$$_{j}are the shear forces recorded at the column of each i-th level, V_{i}and V_{i+1}are the average of the maximum story shears at the i-th and (i + 1)-th levels and the F_{i}are the lateral force applied at the i-th level; - Step 4:
- Normalization of the story shear and lateral force distributions by base shear (assuming V
_{b}= 1), in order to focus on relative distribution of force rather than their values.

## 3. Results

#### 3.1. Base-Isolated (BI) Structure Response: Ground Motions

_{i}normalized with respect to the mean of the base shear V

_{b}obtained from NLTH analysis.

#### 3.2. Equivalent Distribution of Lateral Forces

_{i}and F

_{i}, analytically defined by Equations (5) and (6), respectively, can be determined for each NLTH analysis and the corresponding average value can be determined for the seven accelerograms considered in this study. The base shear V

_{b}= V

_{i|i=1}can also be determined.

_{i}/V

_{b}(left column) and F

_{i}/V

_{b}(right column) for the three buildings with 3, 5, and 7 stories. It is worth highlighting that the resulting F

_{i}/V

_{b}gives a clear picture of the resulting seismic lateral force distribution, which can be deducted from NLTH analyses. The dashed and the dash-dotted lines reported in the same diagrams correspond, respectively, to the uniform acceleration pattern of Equation (1) and the inverted triangular distribution of Equation (2).

#### 3.3. Effective Height

## 4. Discussion

#### 4.1. Comparison between Different Lateral Distribution

#### 4.2. Statistical Analysis and Considerations

_{NLTH}and F

_{LSA}represent, respectively, the peak of the story shear force obtained from NLTHA and the static force provided by the uniform distribution for the i-th level.

_{eff, u}is the effective height assessed for uniform distribution.

_{eff,t}is the effective height assessed for triangular distribution.

_{eff,u}and h

_{eff,t}reduces to 0.50 h

_{tot}and 0.67 h

_{tot}, respectively. Conversely, the expression of h in Equation (11) covers the case of more general distribution of lateral forces resulting from the variability of relevant structural parameters.

## 5. Conclusions

- the lateral force distribution currently recommended in EuroCode and NTC 2018, which neglects the contribution of higher modes, significantly underestimates the shear forces at the upper levels of the superstructures, even when the IS exhibits weakly non-linear response;
- the lateral force distribution proposed by ASCE 7–10 provides results that are too conservative compared with those of dynamic analyses for low and medium equivalent damping ratios;
- the degree of non-linearity of the isolation system strongly influences the seismic response of the base-isolated buildings. As the equivalent damping ratio increases, the shear envelope increasingly bulges because of more significant higher mode effects;
- some formulations available in the literature provide more accurate predictions of the peak seismic forces throughout the height of buildings and hence of the lateral force distributions because of their explicit dependence on IS parameters;
- an accurate vertical distribution can be achieved as a function of the relevant parameters of the superstructures and isolation systems;
- a simplified formula for the vertical distribution of the base shear, combining both uniform and linear distributions, is proposed as a function of the equivalent damping ratio (Equation (9));
- the formula provides slightly conservative seismic story forces, resulting in a more economic design compared to the procedure of the ASCE and in a safer method than that proposed in NTC 2018 for buildings that comply with the codes’ limitation.

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

F_{i} | equivalent lateral force at the i-th level |

m_{i} | story mass of the i-th floor |

V_{b} | total design base shear force |

N | number of stories |

h_{i}, h_{j} | height of the i-th and j-th stories -from the base level |

V_{i} | shear force at the i-th level/median peak story shear at the i-th level |

G_{1}, G_{2} | self-weight and permanent loads, respectively |

Q_{k} | live loads |

M_{s} | mass of the superstructure |

M_{b} | mass of the base level |

T_{bf} | fundamental period of vibration of the superstructure assumed fixed at the base |

T_{eq} | effective fundamental period of the base isolated building |

K_{1}, K_{2} | initial stiffness of the isolators, post yield stiffness |

α | stiffness ratio K_{2}/k_{1} - post yield hardening |

Q | intersection force of hysteresis cycle with vertical axis |

υ | friction coefficient of sliding bearings or friction pendulum bearings (FPB) |

W | weight supported by the bearings |

R | effective curvature radius of the concave surface of FPB |

X_{y}, F_{y} | yield displacement and force of the isolators |

X_{d} | design displacement demand of the IS effective stiffness center |

µ | ductility ratio X_{d}/X_{y} of the isolators |

K_{eff} | effective stiffness of the isolation system at a displacement X_{d} |

ξ | equivalent damping ratio |

T_{r} | reference return period of the design spectrum |

a_{g} | reference peak ground acceleration on type A ground |

g | acceleration of gravity |

F_{0} | amplification factor of the design spectrum |

T_{c}* | period at the end of the constant acceleration branch of the elastic spectrum |

S_{e}(T) | elastic horizontal ground acceleration response spectrum |

S_{De}(T) | elastic displacement response spectrum |

h_{eff} | effective height |

h_{eff,u} | effective height for the uniform lateral forces distribution |

h_{eff,t} | effective height for the inverted triangular lateral forces distribution |

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**Figure 4.**Ground motion records and horizontal elastic displacement (S

_{De}) and acceleration (S

_{e}) response spectra.

**Figure 5.**Distribution in height of the maximum inter-story drift ratio obtained from NLTH analyses for each base-isolated configuration: three-story (

**a**), five-story (

**b**) and seven-story building (

**c**).

**Figure 6.**Story shear at each level obtained from the NLTH analyses under the seven spectrum-compatible ground motions for different base-isolated configurations.

**Figure 7.**Story shear envelopes (

**left**) and lateral force distributions (

**right**) obtained from the NLTH analysis compared with the equivalent distribution employed for the linear static analysis (LSA) (uniform and inverted triangular).

**Figure 9.**Comparison between the several lateral force distributions available in the literature for the prototype BI buildings of three-story (

**a**), five-story (

**b**), seven-story (

**c**) equipped with different isolation systems.

**Figure 10.**Effective height of the different lateral distribution of shear force available in literature for three different base-isolated configurations: three-story (

**a**), five-story (

**b**) and seven-story building (

**c**).

**Figure 11.**Relationship between relative error of shear force at the varying levels and the equivalent damping ratio for three-story (

**a**), five-story (

**b**), and seven-story (

**c**) buildings.

Building | Beam | Girder | Column |
---|---|---|---|

three stories | 40 cm × 60 cm | 40 cm × 50 cm | 40 cm × 40 cm |

five stories | 40 cm × 60 cm | 40 cm × 50 cm | 50 cm × 50 cm |

seven stories | 40 cm × 60 cm | 40 cm × 50 cm | 60 cm × 60 cm |

Superstructure | Height | Total Mass | M_{s}/M_{b} | Period T_{bf} |
---|---|---|---|---|

three stories | 9.3 m | 1142 t | 3.40 | 0.41 s |

five stories | 15.5 m | 1817 t | 5.33 | 0.55 s |

seven stories | 21.7 m | 2555 t | 7.34 | 0.70 s |

Effective Period [s] | 1.5; 2.0; 2.5 |

Ductility [-] | 15; 20; 25 |

Initial Stiffness [N/mm] | 2500; 5000; 7500; 10,000 |

Damping [-] | 0.006–0.55 |

Stiffness ratio [-] | 0.005–0.78 |

Limit State | T_{r} [years] | a_{g}/g [-] | F_{0} [-] | T_{c}* [s] |
---|---|---|---|---|

SLO | 30 | 0.062 | 2.356 | 0.280 |

SLD | 50 | 0.084 | 2.330 | 0.297 |

SLV | 475 | 0.270 | 2.278 | 0.379 |

SLC | 975 | 0.368 | 2.281 | 0.410 |

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## Share and Cite

**MDPI and ACS Style**

Zinco, A.; Fraternali, F.; Benzoni, G.; Martinelli, E.
On the Distribution in Height of Base Shear Forces in Linear Static Analysis of Base-Isolated Structures. *Buildings* **2020**, *10*, 197.
https://doi.org/10.3390/buildings10110197

**AMA Style**

Zinco A, Fraternali F, Benzoni G, Martinelli E.
On the Distribution in Height of Base Shear Forces in Linear Static Analysis of Base-Isolated Structures. *Buildings*. 2020; 10(11):197.
https://doi.org/10.3390/buildings10110197

**Chicago/Turabian Style**

Zinco, Adamo, Fernando Fraternali, Gianmario Benzoni, and Enzo Martinelli.
2020. "On the Distribution in Height of Base Shear Forces in Linear Static Analysis of Base-Isolated Structures" *Buildings* 10, no. 11: 197.
https://doi.org/10.3390/buildings10110197