Journal of Natural Disaster Science, Volume 28, Number 1, 2006, pp1-13

Bracing effect mechanism in the collapse of timber-framehouses due to earthquake and snow weight
Junji KIYONO1 and Aiko FURUKAWA2
Department of Urban Management, Kyoto University, Japan2
Department of Civil Engineering, Kyushu University, Japan

1(Received for 26 Apr., 2005 and in revised from 15 May, 2006)

ABSTRACTIn this study, a simulation procedure with which the collapse mechanism of a timber-frame house can be traced
was developed using the 3-dimensional distinct element method (3D-DEM). The collapse phenomenon of timber-frame houses due to both earthquake and the weight of snow on roofs was simulated, and the effect of diagonal bracing in preventing structures from collapsing was examined. Timber-frame houses with structural elements (columns, beams, floors, foundations, roofs, and diagonal braces) were modeled. The 3D-DEM program
was developed introducing rectangular parallelepiped, hexahedral and octahedral elements in order to model thesestructural elements, and the process of detecting contact between the elements was simplified and made computationally efficient. The difference in collapse behavior between structures that have no diagonal braces, few
braces, and many braces was compared using the developed program. The simulated results showed that
installing diagonal braces is effective in preventing structures from collapsing due to both earthquake motion andsnow weight on roofs. The proposed procedure contributes to providing people with a better understanding of thenecessity of repair and reinforcement to avoid catastrophic disaster in coming earthquakes.

1. INTRODUCTION
The Mid-Niigata Prefecture Earthquake, the magnitude ofwhich was 6.8, occurred on October 23rd in 2004. It is reported
that more than 10,000 houses collapsed due to this earthquake and,since Niigata is an area of very heavy snowfall, some structures
damaged by this earthquake collapsed afterward due to the weightof snow on their roofs (Ministry of Land, 2004). Therefore, wehave to consider not only the possibility of structures collapsing
due to earthquakes but also the possibility of structures that are
leaning or have been damaged by earthquakes collapsing due tothe weight of snow on their roofs, and the possibility of the snowon the roofs enlarging the degree of earthquake damage. Repairingand reinforcing the structures as well as removing snow from roofs
are important in order to prevent collapse.
We also learned from the 1995 Hyogo-ken Nanbu Earthquakethat an important factor that induces structural collapse during
earthquakes is the lack of bearing walls, such as diagonal braces
and structural walls. It is necessary to increase the yield strength
by introducing these diagonal braces and structural walls to prepare for coming great earthquakes. To meet these needs, motivating private users by showing how these bearing walls prevent
structural damage in a very understandable manner is important,because users of most timber-frame houses are not public but private.
In this study, the collapse phenomenon of timber houses due
to both earthquake and the weight of snow on roofs was simulated,

and the effect of diagonal bracing was examined using the 3dimensional distinct element method (3D-DEM). The DEM is anumerical analysis technique, in which the positions of elements
are calculated by solving equations of motion step by step
(Cundall, 1971). Both individual and group behavior can be simulated. The DEM as a tool for numerical simulation has been widely applied to various fields including not only geotechnical engineering (Ohnishi et al., 1986), structural engineering (Meguro etal., 1994) and coastal engineering (Harada et al, 2002), but alsosocial science such as crowd evacuation (Kiyono et al., 1998;
Kiyono et al., 2001) and train derailment (Kiyono et al., 2003).
The DEM is originally a method for dealing with discontinuum;
therefore, joints that connect elements were introduced to deal with
continuum. Now that the DEM can treat behavior from continuum
to discontinuum, it can trace structural behavior from elastic
behavior to large deformation such as half collapse and complete
collapse. Using the theoretical background of the DEM developedby Cundall (1971), we created a DEM code to calculate the behavior of timber-frame houses. We originally modeled the structuralcomponents such as columns, beams, bracings and roofs as distinct
elements, and applied the DEM to timber-frame houses.
In the DEM, a structure is modeled as an assembly of distinct
elements connected by virtual springs and dashpots where elements come into contact. Timber-frame houses with structural elements (columns, beams, floors, foundations, roofs, and diagonalbraces) were modeled. The 3D-DEM program was developed
introducing rectangular parallelepiped, hexahedral and octahedral

KEY WORDS: timber-frame house, collapse simulation, distinct element method, earthquake motion, snow weight, bracing effect2

J. KIYONO AND A. FURUKAWA

elements in order to model these structural elements, and theprocess of detecting contact between elements was simplified andmade computationally efficient.
The difference in collapse behavior between structures thathave no diagonal braces, few braces, and many braces was compared using the developed program. The simulated results showedthat installing diagonal braces is effective in preventing structuresfrom collapsing due to both earthquake motion and snow weighton roofs. The structures, with some of their column-beam connections and brace-beam connections broken, were also simulated andthe results were compared with those of intact structures. It was
found that intact brace-beam connections worked effectively toprevent collapse even though some joints broke. Our developed
3D-DEM program succeeded in showing visually the bracing
effects to prevent structural collapse, which will motivate users
initiative for reinforcement.
2. ANALYTICAL METHOD
2.1

Equation of Motion
The DEM, a numerical analysis method that computes the
position of individual elements by solving equations of motion stepby step, was used. All the elements are assumed to be rigid.
Virtual springs and dashpots in the normal and tangential directions are generated when an element comes into contact with otherelements, and the contact force acts through these generated virtual
springs and dashpots. By solving the equation of motion for eachelement step by step, the behavior of all the elements in a combination can be traced. The motion of a rigid body can be classified as
the translational motion of its center of gravity and rotationalmotion around the center of gravity.
(1) Translational motion of the center of gravity
The forces acting on an element are external force ( fx , fy , fz )such as gravity and earthquake motion, the sum of the contact
forces between the elements ( Fx , Fy , Fz ), and the resistance forcesof the joints ( Fpx , Fpy , Fpz ).
Therefore, accelerations of an element are calculated by:x˙˙ = ( fx + Â Fx + Â Fpx ) / m(1)˙˙
y = ( fy + Â Fy + Â Fpy ) / m(2)˙˙z = ( fz + Â Fz + Â Fpz ) / m(3)

in which x, y, z are the coordinates of the center of gravity, subscripts x, y, z, the parameters in the x-, y-, z- direction, m, the element s mass, and Δt, the time increment in the analysis.
Assuming that acceleration is constant for a short period, Δt,
the velocity and displacement of the center of gravity of each element can be calculated by means of the following difference
scheme, and the total structural behavior can be traced:Ï x˙t ¸ Ï x˙t - Dt ¸ Ï x˙˙t - Dt ¸Ô ˙ Ô Ô ˙ Ô Ô ˙˙ ÔÌ yt ˝ = Ì yt - Dt ˝ + Ì yt - Dt ˝DtÔz˙ Ô Ôz˙ Ô Ô˙˙ÔÓ t ˛ Ó t - Dt ˛ Ózt - Dt ˛(4)Ï xt ¸ Ï xt - Dt ¸ Ï x˙t - Dt ¸
Ô Ô Ô
Ô Ô
Ô
Ì yt ˝ = Ì yt - Dt ˝ + Ì y˙t - Dt ˝DtÔz Ô Ôz Ô Ôz˙ Ô
Ó t ˛ Ó t - Dt ˛ Ó t - Dt ˛(5)(2) Rotational motion around the center of gravity
Here, we consider two frames of reference, the absolute frameof reference and the element frame of reference. The absolute
frame of reference (x, y, z) stated here is a system that has its origin
at the center of gravity. The element frame of reference (ξ,η,ζ)stated here is a system whose coordinate axes are inertia axes andwhose origin is the center of gravity.
Rotation around its center of gravity is obtained from Euler's
equations of motion. The equations for the ξ-, η-, and ζ-directions areIx

dw x- ( Ih - Iz )whwz = ( Â ri ¥ Fi )x
i
dt

(6)

Ih

dwh
- ( Iz - Ix )wz wx = ( Â ri ¥ Fi )h
i
dt

(7)

Iz

dwz
- ( Ix - Ih )wxwh = ( Â ri ¥ Fi )z
i
dt

(8)

in which ξ,η, and ζ are the inertia axes of the coordinates, and Ii
and ωi (i= ξ,η,ζ), respectively, are the moments of inertia androtational velocities around the center of gravity in the element
frame of reference. ri denotes the coordinate vector of a vertex of arigid body, and Fi denotes the force vector affecting the vertex in
the absolute frame of reference.
The angular velocity in the element frame of reference, ωξ,ωη, ωζ, can be obtained using the angular velocity in the previous
step and the angular acceleration obtained from Eqs. (6), (7) and(8) asÏwx ,t ¸ Ïwx , t -Dt ¸ Ïw˙ x , t -Dt ¸
Ô Ô
Ô
Ô Ô Ô
Ìwh ,t ˝ = Ìwh , t -Dt ˝ + Ìw˙ h , t -Dt ˝ DtÔw Ô ÔwÔ Ô˙ÔÓ z ,t ˛ Ó z , t -Dt ˛ Ówz , t -Dt ˛(9)The angular velocity in the absolute frame of reference, ωx ,ωy , ωz , can be obtained by the following transformation

w x = l xw x + m xw h + n xw z

(10)w y = l yw x + m yw h + n yw z

(11)w x = l zw x + m zw h + n zw z

(12)where λx ,μx , n x are the direction cosine of the x-axis (cosine
between the x-axis and the ξ-,η-,ζ-axes), λy ,μy , n y , the direction cosine of the y-axis, and λz ,μz , n z , the direction cosine of thez-axis.
Velocity at an arbitrary point in the rigid body, r = ( rx , ry , rz ),is obtained as the outer product between the angular velocity vector and coordinate vector as

BRACING EFFECT MECHANISM IN THE COLLAPSE OF TIMBER-FRAME HOUSES DUE TO EARTHQUAKE AND SNOW WEIGHTÏr˙x ¸ Ïw x ¸ Ïrx ¸Ô Ô Ô Ô Ô ÔÌr˙y ˝ = Ìw y ˝ ¥ Ìry ˝Ô˙ Ô Ô Ô Ô ÔÓrz ˛ Ów z ˛ Órz ˛(13)

By integrating the obtained velocity, the coordinate of thepoint is obtained from the following difference scheme.
Ïrx ,t ¸ Ïrx , t - Dt ¸ Ïr˙x , t - Dt ¸
ÔÔ
Ô ÔÔ
Ô Ô Ô
Ìry ,t ˝ = Ìry , t - Dt ˝ + Ìr˙y , t - Dt ˝Dt
Ô
Ô Ô Ô
Ô Ô
Órz ,t ˛ ÔÓrz , t - Dt Ô˛ ÔÓr˙z , t - Dt Ô˛contact between vertexes and contact between a vertex and anedge, can be ignored. These assumptions are also effective in solving the instability phenomenon that occurs in contact between a
sharp vertex and another sharp vertex or an edge. In this study, we
adopted 10% of the edge length of the least edge for the value ofthe radius, r.
2.3

(14)

Since distinct elements are modeled using rectangular parallelepipeds, hexahedrons and octahedrons in this paper, we take r as
the coordinate of the vertex.
As explained above, the 3-dimensional behavior of each distinct element can be calculated, and the structural behavior can beobtained as an assembly of the behavior of all elements.
2.2

3

Contact Evaluation
When elements come into contact, springs and dashpots are
generated in the calculation (Fig. 1). Evaluation of contact is madeby considering the collocation between the two elements concerned. To simplify and make time-efficient the process of detecting contact, we assume the vertex of the rectangular parallelepiped
to be 1/8 of a circle, and the edge to be 1/4 of a circular cylinder asshown in Fig. 2. A similar assumption was made regarding hexahedron and octahedron elements. By introducing this assumption,
the patterns of contact are classified into 4 patterns: contact
between vertexes, contact between a vertex and an edge, contact
between a vertex and a face, and contact between edges as shownin Fig. 3. Figure 3 shows the contact between the vertex and faceof two elements, and the virtual springs and dashpots generated
between these elements. By assuming small values for the radiusof the circle and the circle cylinder, the first two contact patterns,

Contact force
Increments in the restoring and damping forces in the normal
and tangential directions (Δen , Δes , Δdn , Δds ) for interval Δt
are expressed by the increments of relative displacement in these
directions, Δn and Δs:
Den = Kn Dn(15)

Des = Ks Ds(16)

Ddn = Cn Dn / Dt

(17)

Dds = Cs Ds / Dt

(18)

in which Kn and Ks , and Cn and Cs , respectively, are the spring
constants and damping coefficients in the normal and tangential
directions. The restoring and damping forces in each direction( [en]t , [es]t , [dn]t , [ds]t ) at arbitrary time t are obtained from thepreceding equations:[en ]t = [en ]t -Dt + Den(19)[es ]t = [es ]t -Dt + Des(20)[ dn ]t = Ddn(21)[ ds ]t = Dds(22)

When the restoring force exceeds the friction limit, the tangential force is governed by dynamic friction. The total contact
forces in both directions are:[ Fn ]t = [en ]t + [ d n ]t(23)

Fig. 1 Contact model of elements(a) contact between vertexes(b) contact between vertex and edge(c) contact between vertex and faceFig. 2 Rectangular parallelepiped modelFig. 3(d) contact between edges

Contact patterns

4

J. KIYONO AND A. FURUKAWA[ Fs ]t = [es ]t + [ d s ]t

(24)

Forces Fx, Fy and Fz, in Eqs. (1), (2) and (3) are obtained by
combining the above forces in the target direction.
2.4

Resistance force at joint
Increments in the resistance forces in the normal and tangential directions (Δepn ,Δeps ,Δdpn ,Δdps ) for interval Δt are expressedby the increments of relative displacement in these directions, Δnand Δs:De pn = K pn Dn(25)

De ps = K ps Ds(26)

Dd pn = Cpn Dn / Dt(27)

Dd ps = Cps Ds / Dt

(28)

in which Kpn and Kps , and Cpn and Cps , respectively, are the spring
constants and damping coefficients in the normal and tangential
directions of joints. The resistance forces in each direction ( [epn]t ,[eps]t , [dpn]t , [dps]t ) at arbitrary time t are obtained from the preceding equations:[e pn ]t = [e pn ]t -Dt + De pn(29)[e ps ]t = [e ps ]t -Dt + De ps(30)[ d pn ]t = Dd pn(31)[ d ps ]t = Dd ps(32)

When the resistance force exceeds the limit, Pmax, the joint isbroken and it can no longer sustain any force. The total resistanceforces in both directions are:[ Fpn ]t = [e pn ]t + [ d pn ]t(33)[ Fps ]t = [e ps ]t + [ d ps ]t

(34)

3.

PARAMETERS AND ANALYTICAL MODEL

3.1 Contact parameters(1) Contact parameters for structural elements
In the DEM, most of the force-displacement laws for elastic
contact between two spherical elements in the normal direction are
based on the Hertz theory (Johnson, 1985). Imagine that two
spherical elements, i and j , are in contact and subjected to a normal
contact force. Assume ri and rj to be the radii of the two elements.
Young s modulus for the elements are Ei and Ej , and Poisson sratios are vi and vj . The equivalent Young s modulus, E , is
Ê 1 - ni2 1 - n j2 ˆE=Á+E j ˜¯Ë Ei-1

and the relative radius of contact curvature, r , is

Ê 1 1ˆ
r=Á + ˜
Ë ri rj ¯

-1

(36)

According to the Hertz theory of the elastic contact of two
spheres in the normal direction, the force-displacement relationship is[en ]t =4
E r ◊ n33(37)

where [en]t and n , respectively, are the normal contact force andcontact displacement at time t. This force-displacement relationship can be transformed into the differential equation[en ]t = [en ]t -Dt + 2 E r ◊ n Dn

(38)

where Δn is the increment of contact displacement. The spring
constant in the normal direction at time t is therefore expressed as
Kn = 2 E r ◊ n(39)Because this study uses rectangular parallelepiped elements,
we utilized an equivalent radius that has the same volume.
The spring constant in the tangential direction was defined as
Ks =Kn
2(1 + n )(40)

where v is Poisson s ratio for the timber frame.
As for the damping coefficient, critical damping was adopted:
Cn = 2 mKn(41)

Cs = 2 mKs(42)

where m is the equivalent mass of two elements. When their masses are mi and mj ,
Ê 11ˆm=Á + ˜Ë mi m j ¯

-1

(43)

These parameters are calculated by the use of Young s modulus of timber, 90×10 3 kgf/cm 2, and a Poisson s ratio of 0.4(Architectural Institute of Japan, 1995).
In fact, Young s modulus of timber is different in two directions: the fibrous direction and its perpendicular direction.
Young s modulus in the fibrous direction is larger than that in its
perpendicular direction. However, we do not consider the
anisotropy of timber in this study, and used Young s modulus in
the fibrous direction because timber-frame houses are designed touse the strength of timber in the fibrous direction.

(35)(2) Contact parameters for snow
We assumed the contact parameters for snow ( Kn , Ks , Cn , Cs )to be proportional to the element mass because reaction force due
to contact becomes larger as the density of snow increases by

BRACING EFFECT MECHANISM IN THE COLLAPSE OF TIMBER-FRAME HOUSES DUE TO EARTHQUAKE AND SNOW WEIGHT

tamping. As for the damping coefficient, critical damping was
adopted.
In this study, the value of the contact parameters for snow is
the same in the normal and tangential directions, even though thosefor structures are different in two directions. It is based on theassumption that snow is a mass of isotropic crystals, and using different contact parameters in two directions is not appropriate
because we model snow using rectangular parallelepipeds for convenience.
3.2

Joint Parameters
In the usual DEM, connectivity between elements is not considered, with the elements acting independently as a discontinuum.
The columns, beams and braces of a real structure, however, areconnected by joints, and the structure acts as a continuum. Joints,
which allow a structure to behave as a continuum until its joints
are broken, are therefore introduced.
Increasing the amount of bearing wall is important but not
sufficient in strengthening a structure. Increasing the strength ofjoints is also necessary. Lessons from the 1995 Hyogo-ken Nanbu
Earthquake showed that a major factor in the collapse of many timber-frame houses is the weakness of joints.
As for old timber houses, clamps were used for connecting
columns and nails were used for connecting braces. As for recent
timber houses, auxiliary metal fittings such as corner metal fittings
and angle plates are used at the joint parts of tenons connecting
columns and beams, and bracing plates are used to connect braces
and beams. In this study, we assumed joints at column-beam connections to be modeled using corner metal fittings, and joints atbrace-beam connections to be modeled using bracing plates.
Joints resist not only tension force but also compression force.
When two elements connected by a joint are in contact with eachother, both contact force and resistance force work at the sametime.
(1) Joint parameters for column-beam connection
Joint parameters for column-beam connections ( Kpn , Kps , Cpn ,Cps , Pmax ) were obtained from the threshold intensity of the corner metal fitting shown in Fig. 4 (a). According to the Japan
Housing & Wood Technology Center (Japan Housing & Woodtechnology Center, 2001), the yield strength is published as
Pmax =600 kgf=600×9.8 N=5880 N. In this study, it is assumed
that metal fittings are attached to columns with a length ofabout 10 cm. It was also assumed that the metal fittings breakwhen the column and beam concerned are separated 10 cm from
Table 1.

5the contact point, meaning that the spring modeling joint islengthened by 10 cm. Therefore, the spring constant for the bothnormal and tangential directions was obtained by dividing the yieldstrength by a length of 10 cm, as Kpn , =Kps= 5880 N/0.1 m=58800
N/m. We assumed that the damping coefficient for the joint is 0.0
in both the normal and tangential directions. Joints break when therestoring force exceeds the joint intensity.
(2) Joint parameters for brace-beam connection
Joint parameters for brace-beam connections ( Kpn , Kps , Cpn ,Cps , Pmax ) were obtained from the threshold intensity of the bracing
plate shown in Fig. 4 (b). According to the Japan Housing &Wood Technology Center (2001), the yield strength is published as
Pmax =710 kgf=710×9.8 N=6958 N. In this study, it is assumed
that the bracing plates are attached to braces with a length of about
10 cm. It was also assumed that the bracing plates break when thecolumn and beam concerned are separated 10 cm from the contact
point, meaning that the spring modeling joint is lengthened by
10 cm. Therefore, the spring constant for both normal and tangential directions was obtained by dividing the yield strength by alength of 10 cm, as Kpn , =Kps= 6958 N/0.1 m=69580 N/m. We
assumed that the damping coefficient for the joint is 0.0 in both thenormal and tangential directions. Joints break when the restoring
force exceeds the joint intensity.
The contact parameters for structures and snow, and the joint
parameters used are shown in Table 1. When elements are in contact, it is assumed that spring contacts and dashpots are seriallyconnected. The P-δ effect is automatically taken into account in
the DEM.
3.3

Interval
The interval used for computation markedly affects the stability of the results. If it is too long, the results will diverge. Cundall(a) corner metal fittings

Contact and joint parameters(b) bracing plate

Fig. 4 auxiliary metal fittings

6

J. KIYONO AND A. FURUKAWA(1971) recommends the following interval:
Dt < 2 m / Kn

(44)

From the above equation, the limitation is <1.0×10 -3 (sec).
We chose to use an interval of =1.0×10 -5(sec).
3.4 Analytical model(1) Modeling of structures
The structures analyzed are two-story timber-frame houses.
Calculations of the mass and moments of inertia of these elements
are based on the density of the Japanese cypress (0.34 g/cm 3).
There are three models, A, B, and C. Their 3-D views, front
views, and side views are shown respectively in Figs. 5, 6, 7. The(a) Model A

difference in these three models is the number of diagonal braces.
The structures are 8.34 m wide, 7.43 m deep, and 7.5 m high.
Each floor is 2.73 m high. The entrance is 1.812 m wide and 3.76 m
deep. All columns are modeled by rectangular parallelepipeds.
The 18 columns with a width of 12 cm and a height of 2.73 m areplaced in total (9 columns in each floor). The 72 columns with awidth of 10.5 cm and a height of 2.73 m are placed in total (33columns in each floor, 6 in the entrance). Beams, floors and foundations are also modeled by rectangular parallelepipeds, and havea depth of 12 cm, 20 cm and 30 cm, respectively. Roofs are modeled using two hexahedrons and two octahedrons of 12 cm wide.
Diagonal braces are modeled using octahedrons that have a widthof 9 cm, a depth of 4.5 cm and a height of 2.73 m. The mass and(b) Model B(c) Model C

Fig. 5 Analytical models (3D view)(a) Model A(b) Model B(c) Model C

Fig. 6 Analytical models (front view)(a) Model A(b) Model B(c) Model C

Fig. 7 Analytical models (side view)(a) Model A(b) Model B
Fig. 8 Arrangement drawing of the first floor(c) Model C

BRACING EFFECT MECHANISM IN THE COLLAPSE OF TIMBER-FRAME HOUSES DUE TO EARTHQUAKE AND SNOW WEIGHT

inertia moment for each element are calculated using its volume
and a density of 0.34 g/cm3. The weight of the finishing material is
not considered.
The layout drawing of the columns and braces on the firstfloor for three models are shown in Fig. 8. The arrangement on
the second floor is the same as that of the first floor except that thesecond floor has no entrance. Model A has no diagonal braces,
Model B has 16 braces in the four corners (8 braces on each floor),and Model C has 44 braces (21 braces on each floor and 2 at theentrance).
(2) Modeling of snow
To simulate the collapse phenomenon of structures due to theweight of snow on their roofs, elements modeling snow were introduced. It is assumed that snow acts as clumps that are cumulated
and tamped. The snow was modeled using two hexahedrons andtwo octahedrons that have the same section area as that of the roofs
and with a width of 100 cm.
The density of the cumulated snow in Niigata is about
70-90 kg/m3, which is heavier than that in Hokkaido and Tohoku
District, 50-80 kg/m3. The condition of snow varies widely due totemperature and sunshine, and snow in areas where the tempera-ture in the afternoon exceeds 0℃ has a tendency to be greater,
being melted and re-tamped. Considering these facts, this study
adopted the upper value, 90 kg/m3, as the snow density. According
to the Japanese building code, the design load for snow is
300 kg/m3, which is heavier than the density used in the analysis.
We did not use the design load to model the realistic situation.
4.

RESULTS

Acceleration records obtained at Ojiya Observatory of
Kyoshin-Net managed by the National Research Institute forDisaster Prevention were used with the modeled structures(Fig. 9).
As for the collapse mechanism, only resistance force at joints
is considered. Buckling of columns and braces, and bending fracture of columns are ignored. A joint consists of four springs, andthese springs are located at four corners of the contact area. Eachspring can only resist compression and tension force and does nothave any resistance against bending moment. However, the joints
have resistance against the bending moment by a combination offour springs.
The findings were as follows.
4.1

Structural behavior due to earthquake
First, the collapse behavior of Models A, B, C, which are(a) NS component(a) Model B1(b) Model B2(c) Model C1(d) Model C2(b) EW component(c) UD component
Fig. 9 Input ground motion(a) 16 (sec)(b) 18 (sec)Fig. 10 Damaged joints(c) 20 (sec)(d) 22 (sec)(e) 24 (sec)Fig. 11 Collapse behavior of Model A due to earthquake (front view)(a) 16 (sec)(b) 18 (sec)7(c) 20 (sec)(d) 22 (sec)Fig. 12 Collapse behavior of Model A due to earthquake (side view)(e) 24 (sec)8

J. KIYONO AND A. FURUKAWA

affected by the input ground motion shown in Fig. 9, was simulated.
(1) Behavior of Model A
The collapse behavior of Model A is shown in Figs. 11, 12.
The structures started tilting in the south-north direction at about
18 seconds when the acceleration increased, and completely collapsed at 24 seconds. It was found that resistance to lateral force
due to the earthquake was very small because Model A has no
diagonal braces.
(2) Behavior of Model B
The seismic behavior of Model B is shown in Figs. 13, 14.
Model B did not collapse because it has 18 diagonal braces, which
resisted lateral force due to the earthquake. By comparing the sideview of Models A (Fig. 11) and B (Fig. 13), the effect of braces is(a) 20 (sec)(b) 22 (sec)clear; they prevent a structure from collapsing using its resistanceboth to compression and tension.
(3) Behavior of Model C
The seismic behavior of Model C is shown in Figs. 15, 16.
Model C did not collapse because it has 44 diagonal braces, which
resisted lateral force due to the earthquake.
4.2

Structural behavior due to the weight of snow
Secondly, the collapse behavior of Models A, B, C, which areaffected by the weight of snow on their roofs, was simulated.
(1) Behavior of Model A
The collapse behavior of Model A is shown in Figs. 17, 18.
Due to the heavy snow, Model A started tilting as soon as the(c) 24 (sec)(d) 26 (sec)(e) 28 (sec)Fig. 13 Collapse behavior of Model B due to earthquake (front view)(a) 20 (sec)(b) 22 (sec)(c) 24 (sec)(d) 26 (sec)(e) 28 (sec)Fig. 14 Collapse behavior of Model B due to earthquake (side view)(a) 20 (sec)(b) 22 (sec)(c) 24 (sec)(d) 26 (sec)(e) 28 (sec)Fig. 15 Collapse behavior of Model C due to earthquake (front view)(a) 20 (sec)(b) 22 (sec)(c) 24 (sec)(d) 26 (sec)(e) 28 (sec)Fig. 16 Collapse behavior of Model C due to earthquake (side view)(a) 4 (sec)(b) 6 (sec)(c) 8 (sec)(d) 10 (sec)Fig. 17 Collapse behavior of Model A due to snow weight (front view)(e) 12 (sec)BRACING EFFECT MECHANISM IN THE COLLAPSE OF TIMBER-FRAME HOUSES DUE TO EARTHQUAKE AND SNOW WEIGHT

analysis began and completely collapsed at 12 seconds. Model A
has less resistance both to vertical and lateral forces. Therefore,
once it tilted a little, it started to collapse due to the heavy snow.
(2) Behavior of Model B
Model B did not collapse due to the weight of snow because
the model is strengthened by bracings. This model stood still, andfigures describing the behavior of Model B are therefore omitted.
The effect of bracing is apparent.
(3) Behavior of Model C
Model C did not collapse due to the weight of snow because
the model is strengthened by bracings. This model stood still, andfigures describing the behavior of Model C are therefore omitted.
The effect of bracing is apparent.

(a) 4 (sec)(b) 6 (sec)

4.3

Behavior of damaged structures due to earthquake
Next, the seismic behavior of damaged structures was simulated. We assumed two damaged cases for both Models B and C, andnamed them B1, B2 and C1, C2.
(1) Damage modelsa) Model B1Model B1 is the damage model for Model B. Ten joints on
the first floor on the east side (the joints framed by the ellipse in
Fig. 10 (a)) are broken. Eight joints are column-beam connections
and 2 joints are brace-beam connections.
b) Model B2Model B2 is the damage model for Model B. Ten joints on
the first floor on the south side (joints framed by the ellipse in
Fig. 10 (b)) are broken. Eight joints are column-beam connections
and 2 joints are brace-beam connections.

(c) 8 (sec)(d) 10 (sec)(e) 12 (sec)Fig. 18 Collapse behavior of Model A due to snow weight (side view)(a) 20 (sec)(b) 22 (sec)(c) 24 (sec)(d) 26 (sec)(e) 28 (sec)Fig. 19 Collapse behavior of Model B1 due to earthquake (front view)(a) 20 (sec)(b) 22 (sec)(c) 24 (sec)(d) 26 (sec)(e) 28 (sec)Fig. 20 Collapse behavior of Model B1 due to earthquake (side view)(a) 20 (sec)(b) 22 (sec)(c) 24 (sec)(d) 26 (sec)(e) 28 (sec)Fig. 21 Collapse behavior of Model B2 due to earthquake (front view)(a) 20 (sec)(b) 22 (sec)9(c) 24 (sec)(d) 26 (sec)Fig. 22 Collapse behavior of Model B2 due to earthquake (SIDE VIEW)(e) 28 (sec)10

J. KIYONO AND A. FURUKAWAc) Model C1Model C1 is the damage model for Model C. Twelve joints
on the first floor on the east side (joints framed by the ellipse in
Fig. 10 (c)) are broken. Eight joints are column-beam connections
and 4 joints are brace-beam connections.
d) Model C2Model C2 is the damage model for Model C. Eleven joints on
the first floor on the south side (joints framed by the ellipse in
Fig. 10 (c)) are broken. Eight joints are column-beam connections
and 3 joints are brace-beam connections.
(2) Behavior of Model B1
The seismic behavior of Model B1 is shown in Figs. 19, 20.
Because the joints on the first floor on the east side (the near sidein Fig. 19 and the left-hand side in Fig. 20) are broken, Model B1swayed greatly in the east-west direction (Fig. 20). As shown in
Fig. 19 (e), it was found that the column-beam connections on thefirst floor on the west side (the right-hand side in Fig. 19) broke
but the brace-beam connections did not break, and we succeeded insimulating these intact joints of braces supporting the structure andpreventing it from collapsing.
(3) Behavior of Model B2
The seismic behavior of Model B2 is shown in Figs. 21, 22.

(a) 20 (sec)(b) 22 (sec)

Because the joints on the first floor on the south side (the left-handside in Fig. 21 and the rear in Fig. 22) are broken, Model B2
swayed in the south-north direction (Fig. 21). As shown from thecomparison between Fig. 19 and Fig. 21, Model B2 whose joints
on the south side are broken are more significantly damaged in thesouth-north direction and less damaged in the east-west direction
than Model B1 whose joints on the east side are broken.
(4) Behavior of Model C1
The seismic behavior of Model C1 is shown in Figs. 23, 24.
Because the joints on the first floor on the east side (the near sidein Fig. 23 and the left-hand side in Fig. 24) are broken, Model C1
swayed in the east-west direction (Fig. 23). By comparing the
behavior of models B1 (Fig. 20) with that of Model C1 (Fig. 23), it
was found that the deformation of Model C1 with more braces wasless than that of Model B1. The effect of braces is clear.
(5) Behavior of Model C2
The seismic behavior of Model C2 is shown in Figs. 25, 26.
Although the joints on the first floor on the south side (the lefthand side in Fig. 25 and the rear in Fig. 26) are broken, the structure is very stable in both the south-north and east-west directions.
One reason for this is that the maximum peak acceleration in thesouth-north direction is smaller than that in the east-west direction,(c) 24 (sec)(d) 26 (sec)(e) 28 (sec)Fig. 23 Collapse behavior of Model C1 due to earthquake (front view)(a) 20 (sec)(b) 22 (sec)(c) 24 (sec)(d) 26 (sec)(e) 28 (sec)Fig. 24 Collapse behavior of Model C1 due to earthquake (side view)(a) 20 (sec)(b) 22 (sec)(c) 24 (sec)(d) 26 (sec)(e) 28 (sec)Fig. 25 Collapse behavior of Model C2 due to earthquake (front view)(a) 20 (sec)(b) 22 (sec)(c) 24 (sec)(d) 26 (sec)Fig. 26 Collapse behavior of Model C2 due to earthquake (side view)(e) 28 (sec)BRACING EFFECT MECHANISM IN THE COLLAPSE OF TIMBER-FRAME HOUSES DUE TO EARTHQUAKE AND SNOW WEIGHT

and another reason is that more braces are placed in the southnorth direction.
4.4

Behavior of damaged structures due to both earthquakeand snow weight
Lastly, the collapse behavior of damaged structures due toboth earthquake and snow weight was simulated.
(1) Behavior of Model B1
The collapse behavior of Model B1 is shown in Figs. 27, 28.
Because the joints on the first floor on the east side (the near sidein Fig. 27 and the left-hand side in Fig. 28) are broken, Model B1swayed largely in the east-west direction (Fig. 28). The structure
did not collapse, but more columns fell, the structure leaned more
due to the weight of snow on the roofs, and more damage occurredthan in the cases with no snow (Figs. 19, 20).

(a) 20 (sec)(b) 22 (sec)(2) Behavior of Model B2
The collapse behavior of Model B2 is shown in Figs. 29, 30.
Because the joints on the first floor on the south side (the left-handside in Fig. 29 and the rear in Fig. 30) are broken, Model B2
swayed in the south-north direction (Fig. 29). More damage
occurred than in the cases with no snow (Figs. 21, 22).
(3) Behavior of Model C1
The seismic behavior of Model C1 is shown in Figs. 31, 32.
Because the joints on the first floor on the east side (the near sidein Fig. 31 and the left-hand side in Fig. 32) are broken, Model C1
swayed in the east-west direction (Fig. 32). It was found that moredamage occurs due to snow through comparison with cases withno snow (Figs. 23, 24), and it was also found that the severity ofthe damage to Model C1 was smaller than that of Model B1 fromthe comparison with Model B1 (Figs. 27, 28), showing that braces(c) 24 (sec)(d) 26 (sec)(e) 28 (sec)Fig. 27 Collapse behavior of Model B1 due to earthquake and snow weight (front view)(a) 20 (sec)(b) 22 (sec)(c) 24 (sec)(d) 26 (sec)(e) 28 (sec)Fig. 28 Collapse behavior of Model B1 due to earthquake and snow weight (side view)(a) 20 (sec)(b) 22 (sec)(c) 24 (sec)(d) 26 (sec)(e) 28 (sec)Fig. 29 Collapse behavior of Model B2 due to earthquake and snow weight (front view)(a) 20 (sec)(b) 22 (sec)(c) 24 (sec)(d) 26 sec)(e) 28 (sec)Fig. 30 Collapse behavior of Model B2 due to earthquake and snow weight (side view)(a) 20 (sec)(b) 22 (sec)11(c) 24 (sec)(d) 26 (sec)(e) 28 (sec)Fig. 31 Collapse behavior of Model C1 due to earthquake and snow weight (front view)12

J. KIYONO AND A. FURUKAWA(a) 20 (sec)(b) 22 (sec)(c) 24 (sec)(d) 26 (sec)(e) 28 (sec)Fig. 32 Collapse behavior of Model C1 due to earthquake and snow weight (side view)(a) 20 (sec)(b) 22 (sec)(c) 24 (sec)(d) 26 (sec)(e) 28 (sec)Fig. 33 Collapse behavior of Model C2 due to earthquake and snow weight (front view)(a) 20 (sec)(b) 22 (sec)(c) 24 (sec)(d) 26 (sec)(e) 28 (sec)Fig. 34 Collapse behavior of Model C2 due to earthquake and snow weight (side view)

work effectively.
(4) Behavior of Model C2
The collapse behavior of Model C2 is shown in Figs. 33, 34.
Due to the weight of snow, the second floor leaned more than thecases with no snow (Figs. 25, 26). The magnitude of damage toModel C2 is apparently smaller than that of Model B2 from thecomparison with Model B2 (Figs. 29, 30), showing that braces
work effectively.
5.

CONCLUSIONSIn this study, a simulation procedure with which the collapse
mechanism and the bracing effect of a timber-frame house can betraced was proposed. An analytical program of the proposed procedure is developed based on the 3-dimensional distinct element
method (3D-DEM), which is an analytical method for discontinuum. The developed program can deal with rectangular parallelepipeds, hexahedrons, and octahedrons to model structural components such as columns, beams, floors, foundations, roofs, anddiagonal braces. The procedure of detecting contact between elements was simplified and made time-efficient. To deal with structural behavior from continuum to discontinuum, a structural joint
model was introduced.
We applied the developed program to two-story timber-frame
structures with no braces, with few braces, and with many braces,and demonstrated visually the collapse mechanism and the effectof braces against earthquake motion and the weight of snow on
roofs.
Collapse patterns vary with the number of diagonal braces thatcan resist the lateral loading of the earthquake. If the number ofbraces is sufficient, a structure will resist an earthquake, even one
as severe as the 2004 Mid-Niigata Prefecture Earthquake.
We succeeded in visually showing that installing diagonalbraces in a structure is very effective in preventing the structurefrom collapsing due to both earthquake and snow.
We also were successful in demonstrating that snow on roofs
potentially causes more damage to a structure during an earthquake, and pointed out the importance of installing braces as well
as removing snow.
By using the developed procedure, we can provide information on whether a damaged or aged structure can avoid collapse
just by installing braces or increasing the strength of the joints in avery easily understandable manner. This information gives people
the initiative to repair and reinforce their own houses.
Case studies alone are insufficient, more relevant analyses
being required, but the methodology presented here should contribute to providing people with a better understanding of thenecessity of repair and reinforcement to avoid catastrophic disasterin coming earthquakes.
ACKNOWLEDGMENTS
Partial financial support for this study was provided by theJapan Science and Technology Corporation.
We used the strong motion data by K-net operated by the
National Research Institute for Earth Science and Disaster
Prevention.

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