# A simple procedure to approximate slip displacement of freestanding rigid body subjected to earthquake motions

## Summary (2 min read)

### INTRODUCTION

- In recent highly developed and complicated social system the damage of structures no longer represents total effects of earthquakes.
- The former investigators clearly pointed out a need for investigation into seismic behavior of nonstructural components in order to assess their vulnerability [1].
- The senior author pointed out that the period of horizontal base excitation makes an important contribution to elongation of the slip displacement of the body in addition to the friction coefficient and peak horizontal acceleration [7].
- The prediction accuracy of the slip displacement of the body on the building floor may deteriorate because filtering effects of structure enhance the advent of a certain wave component in the floor response.
- The use of the root mean square value of the vertical acceleration at the peak horizontal acceleration [11, 12] is proposed as the uniform downward acceleration.

### DESCRIPTION OF THE PROBLEM

- An equation of motion that governs the slip behavior of the body subjected to simultaneous horizontal and vertical acceleration is given as follows.
- Figure 1(a) and Figure 1(b) shows a mechanical model of the body placed on the ground and set on the building floor respectively.
- If the authors replace the shaking motion according to the problem, the subjects are essentially the same.
- Hz&& vz&& g , μ and ν are gravitational acceleration, static and kinetic friction coefficients, respectively.

### Simplification of the problem

- A simple treatment of the vertical acceleration is necessary to mathematically obtain the slip displacement of the body induced by the horizontal sinusoidal acceleration, i.e., the basic slip displacement of the body.
- Therefore, this study introduces a monotonous reduction in friction force while the body slips [13].
- The probability density of this ratio was modeled by the normal distribution and the product of σ and PVGA gave the root mean square value of the vertical ground acceleration.
- In case of the body set on the ground, this paper uses PVGA of each vertical accelerogram.
- In contrast, η is the magnification factor for the vertical floor response given by the quotient of the peak vertical floor response acceleration (VFRA) by PVGA.

### Horizontal sinusoidal acceleration

- In case of the body set on the ground, this paper uses PPHA determined by the period which maximizes a pseudo-acceleration spectrum of each horizontal accelerogram.
- In actual case scenario, PPHA is determined by the natural period of soil, since it will be predominant over other T wave period in the earthquake.
- Ref. [14] gives the calculation of the natural period of soil.
- In case of the body set on building floor, this paper uses PPHA determined by FFT analysis of each horizontal floor acceleration.
- For the convenience of actual case scenario, the calculation of the predominant period of the horizontal floor response acceleration is proposed latter.

### Mathematical solution

- Solve Eqs. (5) to (7) mathematically to obtain the basic slip displacement of the body.
- Figures 1(d) and 1(e) depict the typical time history of slip motion of the body subjected to the horizontal sinusoidal acceleration with =7m/sgAgx 2, T =6.28s and μ =0.4.
- Since Eq. (12) is, however, the transcendental function in terms of the cosine function, this study tries to find its approximate solution employing the Taylor’s series.
- The modification factor for the basic slip displacement to calculate the design slip displacement is statistically examined.

### Earthquake records

- This study uses 104 accelerograms observed around Japan [15].
- Maintaining the relation between PHGA and PVGA of each pair of accelerograms, the time history of vertical acceleration is also normalized and scaled in a same manner.
- The absolute maximum slip displacement of the body induced by the earthquake wave, Max Eq vhx , , is numerically computed and compared with the corresponding basic slip displacement, , induced by the horizontal sinusoidal acceleration with the nominal friction coefficient, μ Sin vhx , ′ .
- Figure 3(e) shows the probability density of the slip ratio, vh,β vh,β vh,β vh,β vh,β , of all results.
- The product of the modification factor, ob vhβ Pr , ob vhβ Pr , , and the basic slip displacement, , gives the design slip displacement, , as the maximum expected value.

### Methodologies

- The following section tries to apply the proposed procedure to estimate the slip displacement of the body set on the building floor.
- The modification factor for the basic slip displacement is statistically examined.
- Figures 6(a) to 6(c) show the predominant period of HFRA observed at the specified floor of linear building while Figures 6(d) to 6(f) show that of nonlinear building.
- The abscissa of these figures shows the predominant period of earthquake input to the building model determined by a pseudo-acceleration spectrum of each earthquake.
- Figure 7(g) is the probability density of the slip ratio, vh,β vh,β vh,β , of all results.

### CONCLUSION

- This paper proposes the use of the horizontal sinusoidal acceleration to approximate the slip displacement of the body wherever it is set on during an earthquake.
- A mathematical solution of the slip displacement of the body induced by the horizontal sinusoidal acceleration, i.e., the basic slip displacement, is presented in a practical form.
- The modification factor is quantified as an estimation error between the basic slip displacement and the absolute maximum slip displacement based on the probability of the nonexceedence of a certain threshold.
- It forms a certain probability density despite the acceleration intensity and the friction coefficient.
- The modification factors are slightly different according to the place of the body and linear/nonlinear state of building.

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##### Citations

171 citations

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### Cites background from "A simple procedure to approximate s..."

...There have been few studies on friction behavior, which have mainly focused on rigid body sliding under earthquake motions [8–12], whereas many other studies within structural engineering have focused on base isolation systems [13–18] and energy dissipation systems [19–22] that do not consider traditional structural materials....

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### Cites methods from "A simple procedure to approximate s..."

...…response of such objects using both analytical as well as numerical methods (Ishiyama, 1982; Shi et al., 1996; Zhang and Makris, 2001; Taniguchi and Miwa, 2007; Voyagaki et al., 2012, 2014) Purvance, Anooshehpoor, and Brune (2008) were the first to apply the rigid-body rocking…...

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### Cites background from "A simple procedure to approximate s..."

...Many researchers have analyzed the sliding response of the rectangular block subjected to earthquake ground motions both analytically and numerically [41, 94, 100, 105]....

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...[100] and estimated the range of PGA required to cause the observed sliding displacement to be 2....

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...[100] for the block’s maximum sliding displacement as a function of the sine wave parameters, Hough et al....

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##### References

2,071 citations

### "A simple procedure to approximate s..." refers background in this paper

...Historically, Newmark [9] presented a simple formula to determine the sliding distance of a freestanding body subjected to a single rectangular acceleration pulse at the base concerning earthquake response of embankments....

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...Choi and Tung [10] concluded that Newmark’s formula could be used if an adjustment factor consisting of the friction coefficient and peak horizontal base acceleration was applied....

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1,999 citations

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274 citations

### "A simple procedure to approximate s..." refers background in this paper

...They clarified five modes of the body (rest, slide, rock, slid-rock and jump), equations of motion and relation of wave properties of base excitation to these modes [2-6]....

[...]

195 citations

### "A simple procedure to approximate s..." refers background in this paper

...They clarified five modes of the body (rest, slide, rock, slid-rock and jump), equations of motion and relation of wave properties of base excitation to these modes [2-6]....

[...]