# Base shear determination using response-spectrum modal analysis of multi-degree-of-freedom systems with soil–structure interaction

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## Abstract

Building codes and design guidelines, e.g. FEMA (NEHRP recommended seismic provisions for new buildings and other structures, FEMA P-1050, Washington, 2015) and ASCE (Minimum design loads for buildings and other structures ASCE/SEI 7-10/2010, Reston, 2010), describe the problem of multi-degree-of-freedom systems with soil-structure interaction (SSI). These systems are modeled like those having a fundamental degree of freedom on a foundation with lateral and rotational interactions and the other vibration modes isolated and supported on a fixed foundation. This model oversimplifies the problem, neglecting the effects of having all modes coupled in the foundation with SSI. A simple, easily programmable, SSI model in which all vibration modes are coupled an attached to an infinitely rigid shallow foundation subjected to soil excitation is introduced here. Initially, the total response of the coupled system is calculated. Then, using traditional procedures to combine modal responses, a simplified alternative methodology to find the total response of this coupled system is proposed. The new methodology is verified against a robust numerical technique, i.e. boundary elements method, using a wide variety of cases that combine several types of soils, building heights and two structural typologies: bending frames and shear walls. Finally, it is clear from the parametric study that current methodologies, based only on the interaction of the fundamental mode of vibration of the structure, in some cases has a significant influence on the total base shear of buildings, particularly in tall buildings founded in soft soils.

## Keywords

Base shear Soil-structure interaction Transfer function Modal analysis and impedance function## List of symbols

- \(A_{o}\)
Area of rigid circular surface foundation

*B*Dimension of rigid rectangular foundation measured in the same direction in which the soil particles move when a shear wave propagates vertically

- \(c_{j}\)
jth mode linear viscous damping

- \(c_{s}\)
Soil shear wave velocity

- \(g\)
Acceleration of gravity

*G*Shear elastic modulus

- \(G_{o}\)
Translation impedance function for soil-foundation system

- \(G_{\theta }\)
Rotation impedance function for soil-foundation system

- \(h_{j}\)
jth mode equivalent height

- \(H_{j}\)
Transfer function considering fixed-base and associated with j mode

- \(H_{SSIj}\)
Transfer function considering SSI effects associated with the fundamental mode in one degree-of-freedom-system

- \(H_{SSIj}^{*}\)
Transfer function with SSI effects associated with j mode and considering multiple degree of freedom systems coupled in the foundation

- \(I_{t}\)
Total rotational inertia of the system

- \(I_{o}\)
Rotational inertia of infinitely rigid foundation

- \(k_{j}\)
jth mode elastic stiffness

- \(k_{y}\)
Static displacement stiffness of the soil-foundation system

- \(k_{\theta }\)
Static rotation stiffness of the soil-foundation system

*L*Dimension of rigid rectangular foundation measured perpendicular to the direction in which the soil particles move when a shear wave propagates vertically

- \(M_{fj}\)
Overturning moment associated with the j mode

- \(M_{j}\)
jth mode mass

- \(m_{o}\)
Infinitely rigid foundation mass

- \(r_{o}\)
Equivalent radius of the infinitely rigid circular foundation

- \(S_{a}\)
Acceleration response spectra

- \(T_{SSIj}\)
jth mode vibration period considering soil-structure interaction

- \(V_{base}\)
Total base shear with the SSI and with all modes coupled on the foundation

- \(V_{baseEq15Mode1}\)
Base shear associated with the first mode of vibration considering SSI and with the upper modes coupled on the foundation

- \(V_{baseF.P - 1050Mode1}\)
Base shear associated with the first mode of vibration considering SSI and without the upper modes coupled on the foundation according to FEMA (2015)

- \(V_{baseFEMA P - 1050}\)
Total base shear calculated considering the fundamental mode considering SSI, plus the higher modes without SSI and according to FEMA (2015)

- \(V_{j}\)
Shear force associated with the j mode

- \(\alpha_{\theta 1j}\) and \(\alpha_{\theta 2j}\)
Constants used to calculate the transfer function

- \(\xi\)
Structural viscous damping

- \(\xi_{SSIj}\)
jth mode equivalent damping with soil-structure interaction

- \(\theta_{j}\)
Rotation of the rigid foundation associated with the j mode

- \(\lambda_{LIM}\)
Limit value to \(\lambda_{SSIij}\)

- \(\lambda_{SSIij}\)
Relative amplitude of the transfer function of mode

*i*in the frequency of mode*j,*with respect to the maximum amplitude of the transfer function of one-degree-of-freedom-system with fixed base- \(\lambda_{uo}\)
Constant associated with the total translation of the foundation

- \(\lambda_{\theta }\)
Constant associated with the total rotation of the foundation

- \(\nu\)
Poisson modulus

- \(\nu_{g}\)
Free field displacement

- \(\nu_{j}\)
j

^{th}mode relative displacement of the structural mass with respect to its foundation- \(\nu_{oj}\)
Displacement of the rigid foundation associated with the j mode

- \(\omega\)
Circular frequency

- \(\omega_{nj}\)
jth mode natural frequency

## Notes

## References

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