Motion analysis of furniture under sine wave excitations
D. Isobe & Z.H. He
Univ. of Tsukuba, Ibaraki, Japan

M. Kaneko
Shimizu Co., Tokyo, Japan

M. Hori
Univ. of Tokyo, Tokyo, Japan

ABSTRACT: Improperly secured furniture, especially on the upper floors of high-rise buildings under longperiod ground motion, can become dangerous objects for human life. Many tumbled furniture such as chairs
and tables in schools could become fatal obstacles that prevent children from evacuating. The objective of thisresearch is to develop an effective numerical code to analyze the motion behaviors of furniture subjected toseismic excitations. The numerical code is constituted based upon the adaptively shifted integration (ASI) –Gauss technique, which is a finite element scheme that provides higher computational efficiency than theconventional code. This code is able to analyze dynamic behavior with strong nonlinearities which are related
to phenomena such as fracture and contact. In this paper, the frictional contact between objects was fully considered by employing a sophisticated penalty method. A basic evaluation was carried out by comparing theresults of certain experiments with analyses of the motion behaviors of steel cabinets excited by sine waves. It
should be emphasized that the present code is able to evaluate the motion behaviors of the furniture as well as
the deformations and stress distributions, which is one of the advantages of using a finite element scheme.
1 INTRODUCTION

cal results are validated by comparing the motion
behaviors of steel cabinets with experiments performed on an excitation table.

Improperly secured furniture, especially on the upper floors of high-rise buildings under long-period
ground motion, can become dangerous objects forpeople. Many tumbled furniture such as chairs andtables in schools could become fatal obstacles thatprevent children from evacuating. To investigate themotion behaviors of furniture under various conditions, the E-defense of NIED carried out several experiments using a three-dimensional full-scale shaking table and obtained useful information (E-defense2009). In parallel, the E-simulator project is aiming
to develop an effective simulator for investigating
the motion behaviors of such furniture under abroader set of conditions. The discontinuous element
method (DEM) (Cundall 1971) may be useful forthis purpose; however, the mechanical behaviors of
the constituent materials of the structural components may not be well examined.
The main objective of this research is to develop
a low-cost, highly accurate numerical code to analyze the motion behaviors of furniture, without neglecting the elasto-plastic and damage behaviors ofstructural components. To model the motion behavior of furniture, a contact algorithm using the penalty
method is implemented into a finite element code
based upon the adaptively shifted integration (ASI) –Gauss technique (Lynn & Isobe 2007). The numeri-Figure 1. Linear Timoshenko beam element and its physicalequivalent.

2 NUMERICAL METHODS
2.1 ASI-Gauss techniqueIn this paper, the numerical analyses are carried out
using a finite element code based upon the ASIGauss technique with linear Timoshenko beam elements. Figure 1 shows a linear Timoshenko beam
element and its physical equivalence to the RBSM.
As shown in the figure, the relationship between the230

elastically deformed member are placed such thatthe stress evaluation points coincide with the Gaussian integration points of the member. This meansthat stresses and strains are evaluated at the Gaussian integration points of elastically deformed members. Gaussian integration points are optimal for
two-point integration, and the accuracy of the bending deformation is mathematically guaranteed (Presset al. 1992). The scheme takes advantage of twopoint integration while using one-point integration inactual calculations. More details on the code can befound in previously published papers (Lynn & Isobe
2007, Toi & Isobe 1993).

locations of the numerical integration point and thestress evaluation point where a plastic hinge isformed is expressed as (Toi 1991):r  s(1)Here, s is the location of the numerical integration
point and r is the location where stresses and strainsare evaluated. The variables s and r are nondimensional quantities that take values between -1 and 1.

2.2 Contact algorithm considering friction
To realize various contact phenomena during seismic excitation, frictional contact between objects
was considered in this paper with the penalty method and by considering dynamic friction. Figure 3
shows the subjected forces and the geometrical relations between two elements when they approach one
another with a relative velocity v. Once the current
distance l between the central lines of two elements
becomes shorter than the mean value L of the member widths of the elements, the penalty force vector
FP shown below is assumed to act in the normal direction of the contact plane.

Figure 2. Locations of numerical integration and stress evaluation points.

In the ASI-Gauss technique, the numerical integration point is shifted adaptively according to Eq.
(1) when a fully plastic section is formed within anelement, to form a plastic hinge exactly at that section. By doing so, the plastic behavior of the element
is simulated appropriately, and the converged solution is achieved with a minimum number of elements per member. When the plastic hinge is unloaded, the corresponding numerical integration
point is shifted back to its normal position. Here, thenormal position means the location where the numerical integration point is placed when the element
acts elastically. Two consecutive elements forming amember are considered as a subset, as shown in Figure 2, and the numerical integration points of an

1 v
FP   (1  ) q N , where (l  L)
L vN(2)Here,is a penalty coefficient, q is a penalty index, and vN is the normal direction component of therelative velocity v, respectively. When the value ofis 50 kgf and q takes on various values, the penalty forces along the contact depth l/L are distributed
as shown in Figure 4.

Figure 3. Subjected forces and the geometrical relations between the elements.

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Here,
is a dynamic friction coefficient. The
component in the normal direction acts as a damping
force along the contact depth and contributes to increasing the numerical stability.
3 MOTION ANALYSIS OF OVERTURNINGFURNITURE
3.1 Numerical model and analytical conditions
Some motion analyses were carried out on furnitureand compared with the experimental results (Kanekoet al. 2004). Two steel cabinets, with properties asshown in Fig. 5, are subjected to two sine waves ofdifferent frequencies and maximum acceleration
values. The selected input waves are (a) a longperiod wave of 0.5 [Hz]–190 [gal], and (b) a shortperiod wave of 2 [Hz]–700 [gal], as shown in Fig. 6.
In case (a), one cabinet is set upright at the center ofan excitation table, and the other one is also set upright, beside a wall. In case (b), one cabinet is laid
down on the floor at the center of the excitation table,
and the other one is set upright, beside a wall. We
fixed the parameters as =50 kgf, q=1 and =0.6
for the dynamic friction coefficient between the cabinets and the floor or the wall. The cabinet frame,
the floor and the wall are modeled with two linearTimoshenko beam elements per member, and thecenter of gravity of the cabinet model is adjusted byvarying the density of the elements.

Figure 4. Penalty force as a function of contact depth.

Figure 5. General sizes and properties of steel cabinets.

3.2Results

Figure 7 and Fig. 8 show the comparison of the motions obtained by the experiments and the analyses.
The overturning durations and the behaviors of thecabinets agree well in both cases. Each analysis required approximately 6 hours of simulation time using a personal computer (CPU: 2.0 GHz Xeon).
4 CONCLUSIONSThe numerical code and the frictional contact algorithm developed in this paper succeeded in simulating the motion behaviors of furniture subjected tosine wave excitations. However, the motion behaviors of the models in this paper did not contain any
yielding or damage phenomena in the structural elements. Our next plan is to investigate the motion
behaviors of colliding furniture that may require
more detailed modeling of the constituted materials.

Figure 6. Input sine waves

Then, in this paper, a frictional force vector FD isassumed to act not only in the tangential direction ofthe contact plane but also in the normal direction, asshown below.

1 v
FD   (1  ) q , where (l  L)
L v

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E-defense, National Research Institute for Earth Science and(3)232

Disaster Prevention. 2009. http://www.bosai.go.jp/hyogo/index.html.
Kaneko, M., Nakamura, Y., Tamura, K. & Kambara, H. 2004.
Difference of overturning behavior of office furniture depending on its placement, Journal of Structural Engineering 50B: 679-685 (in Japanese).
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Collapse Problems of Framed Structures. Int. J. for Numerical Methods in Engineering 69(12): 2538-2563.
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Toi, Y. 1991. Shifted Integration Technique in OneDimensional Plastic Collapse Analysis using Linear andCubic Finite Elements, International Journal for Numerical
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0 [s]

4 [s]
(a)Experimental result

5 [s]

6 [s]

0 [s]

4 [s]

5 [s]

6 [s]

(b) Numerical result
Figure 7. Comparison of experimental and numerical results (0.5 [Hz]–190 [gal]).

0 [s]

4 [s]

6 [s]

16 [s](a)Experimental result

0 [s]

4 [s]

6 [s]
16 [s](b) Numerical result
Figure 8. Comparison of experimental and numerical results (2 [Hz]–700 [gal]).

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