Theoretical model and finite element analysis for restoring moment at column foot during rocking
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Abstract
Column usually rests directly on top of a stone base without any physical connection in traditional timber structures. When timber structure is subjected to horizontal loads, rotation around the edge of column foot may occur with or without slippage of column. Columns rock with cyclic lift and reset. Compression deformation occurs to column foot due to the heavy vertical load of the structure and it is an important factor to the loading bearing capacity of timber structure. Accordingly, the rotation behavior of column foot during rocking has to be taken into account when carrying out the structural analysis. For this purpose, rotation behaviors of column foot during rocking is studied, and theoretical model of restoring moment at column foot joint is proposed with consideration of the compression effect; finite element model is established to verify the proposed model and good agreement of the two models is found. Result shows that column foot has contributions to resist horizontal loads with rocking under lateral load.
Keywords
Column foot Theoretical model Finite element analysis Restoring moment CompressionIntroduction
Timber as a structural material may be found in many building and structures, especially in heritage structures [1]. Historical timber constructions are parts of cultural heritage that need to be maintained for generations [2]. Traditional timber structures are constructed of timber frameworks which can undertake significant deformation and dissipate a large amount of energy when taking up seismic load [3]. Different components of the timber structure are connected into the joints at the column foot and column head and the beam–column joints. These joints are the main loadcarrying elements in a traditional wooden building and have top research priority [4]. Under horizontal loads, such as earthquake or wind loads, “column rocking” happened [5]. Restoring force characteristics of column rocking play major role in lateral resistance mechanism of the timber structure, and it may be indicated that as much as 70% of lateral strength of the building is given by the column rocking in a type of wooden ancient temple and shrine structures [6]. Experimental research on this subject was made by Ban [7]. Theoretical and experimental studies have been extensively made and clarified the static and dynamic features of the column rocking characteristics [8, 9, 10]. It is known that the restoring moment of a timber structure due to its deformation is a result of all bending moments at the top and bottom of columns during rocking, as well as those at beam–column joints due to resistance of tie beams [11]. Column foot joint at the bottom of the whole structure plays a crucial role on the restoring moment and structural behavior of timber structures, and it is without any physical connection and usually affects the internal force distributions in the whole structure. Hence, restoring moment and mechanical behavior of column foot joint during rocking are needed to be further studies.
More and more research interests have been devoted to the study of column in heritage buildings in recent years. Frictionsliding seismic isolation model has been proposed and the slippage criterion of a column foot is given [12]. It is assumed that the moment carried by the column foot is the same as that at the column head, and restoring moment at the column foot is determined according to the analytical moment at the top of column [13]. Lee et al. [14] proposes a technique for obtaining the skeleton curve of moment–rotation relation at the stone base of traditional timber buildings from local compression tests of wooden components. Those studies mainly focused on the frictionsliding characteristics, restoring moment of column head and stiffness characteristics of column. However, study on the whole process of column rocking under external loads is rare, and the rotational behaviors of column foot joint during rocking have not been studied. Furthermore, a recent trend in analyzing timber structures involves considering the timber connections as semirigid joints. But, few researches have focused on the semirigidity of column foot joint in traditional timer structure, and many studies take column foot as rigid or hinge joint during structural analysis. In fact, the rotation behavior of column foot needs to be better known to properly analyze the whole structure, and there is an urgent to study the mechanical behavior and propose a theoretical model for predicting the restoring moment and the rotational performance of column foot joint.
This paper tends to analyze the rotation behavior and performance of column foot joint in timber structure under horizontal load in detail. Resisting compression stress distributions at the interface are studied. Theoretical model of restoring moment for column foot joint is formulated with consideration of the compression effect and validated by finite element analysis.
Rotation behaviors of column foot
Working states of column
Columns in traditional timber structures are usually supported directly on the stone base. Example shown in Fig. 1 is the column foot in the Forbidden City in Beijing, China. The cross section of stone base is a little bigger than that of the column, which allows some slippage of the column foot [15] but not slipping off from the stone base. Under horizontal load, slip or rotation will happen, and column will become inclined with lifting and rotation about the edge of column foot, called “column rocking”, which can resist the external force acting on the timber structure.
When vertical and horizontal loads are acting on the column, stone base will provide vertical reaction force as well as horizontal friction force to the column foot. The joint between column and stone base transmits compression force but no tensile force. If the horizontal load exceeds the friction force between column foot and stone base, slippage will occur at the column foot.
Under horizontal load, column may be in different working states. If the external horizontal load is zero, column will not move, which is named as “full compression without slip”. Under cyclic horizontal load, such as earthquake and wind, if the external force is less than the friction force at column foot, there is no relative displacement between column and stone base. Column foot transfer seismic energy all from the bottom of structure to upper structure. Column will rotate under the effect of horizontal load, which is named as “column rocking without slip”. If the external force is larger than the friction force at column foot, slip will appear to the column foot. Column will rotate under the effect of horizontal load, which is named as “column rocking with slip”. At this working state, the shear force at column foot is equal to the friction force according to the horizontal force equilibrium equation. And, the load transferred from the bottom of column foot to upper structure is same with friction force, which plays an obvious seismic isolation effects. After seismic load, column may be in a new state with irreclaimable slip, which is named as “full compression with slip”. During column rocking, if the restoring moment is less than the external moment, column will overturn, which is named as “column overturning”. Therefore, with the combined effects of vertical and horizontal loads, column may be in five possible working states: (a) full compression without slip; (b) full compression with slip; (c) column rocking without slip; (d) column rocking with slip and (e) column overturning. The force systems of column in working states (a) and (b) are clear. While the force systems of column in working states (c) and (d) under seismic load are not clear, which should be studied further.
“Chinese earthquake chronology” indicates that traditional timber structures have exhibited good resistance to large seismic excitations with small damages incurred. In the earthquake record of China, there are statements like “the wall collapsed but the structure stands” or “column foot removed but the whole structure is safe” [15]. The survival of traditional timber structure in a large earthquake with long duration indicates that the traditional timber structures are capable of maintaining stability over many cycles of excitations [16]. Therefore, detailed analysis on the deformation and the mechanical characteristics of the structure is needed. A study on the loadcarrying mechanism of column foot in the state of column rocking without slip will lead to significant knowledge on the mechanical behaviors of column rocking with slip. This also provides a platform for developing the restoring moment model for the traditional timber structure under horizontal load.
Compression deformation of column foot
Joints in traditional timber structure resist external force by compression or embedment and friction to each member and show high ductility [17]. Such joints play an important role in the mechanical performance of traditional timber structures [18]. Compression deformation of timber at the bottom of the column is the main source of restoring force and moment in column foot joint [11]. During column rocking, compressive stress would occur at the contact interface between column and stone base due to the vertical load acting on the contact interface. Such distributions can be determined from the deformation and constitutive relation of timber at column foot as well as the changing of the contact interface as follows.
Compression effect
Periodic loads such as seismic actions would induce column rocking existing in both Chinese and Japanese traditional timber structures. Column foot rotates about the column edge under horizontal load as shown in Fig. 2. Pivotal points of column foot may switch between the two opposite edges of column foot. Under cyclic horizontal load, column foots have compression deformations due to the heavy upper structure, and contact area between column foot and stone base are changing during column rocking. If horizontal force is not big enough, there is no lifting occurring to column foot, shown in Fig. 3. Compression interface is equal to the whole cross section of column foot shown in Fig. 4. If horizontal force is big enough, lifting on one side and compression on the other side would happen to the column foot shown in Fig. 5, and compression interface is shown in Fig. 6. Repeated lifting and setting of column foot make the timber structure be a selfcentering structure through selfweight.
Compression deformation and compression interface between column foot and stone base are changing during column rocking. Working states of column rocking without slip can be described in six substates according to the shape of compression deformation and compression interface as shown in Figs. 3, 4, 5 and 6. \(xyz\) denotes the initial coordinate system. \(x^{\prime}y^{\prime}z^{\prime}\) is rotational coordinate system. \(O\) is rotational center, whose position coincides with the initial position of the center \(O^{\prime}\)of column foot. \(\theta\) is rotation angle. \(r\) is the radius of column. \(+\) is on behalf of positive loading.
Loading configurations of the six substates in the positive loading process are as follows:

\({S_1}\): Initial state with only vertical load acting at the top of column without horizontal load. Column foot and stone base contact with each other totally.

\(S_{2}^{+}\): Column foot and stone base contact with each other totally. Compression deformation at the right edge of column foot is larger than that of left edge.

\(S_{3}^{+}\): Column foot and stone base is not separated, but compression deformation is zero in the left edge of column foot. This is the critical state of lifting of column foot.

\(S_{4}^{+}\): Lifting at one side and compression at other side occur to column foot. Compression stress appears at large cross section of column foot.

\(S_{5}^{+}\): Compression stress appears at half cross section of column foot. This is the critical state of large and small cross section with compression stress.

\(S_{6}^{+}\): Compression stress appears at small cross section of column foot, and compression interface is mainly located in the edge of column foot.
Constitutive relation of timber parallel to grain
Column foot mainly depends on the timber parallel to grain to bear compression force under vertical load. Therefore, the derivation of theoretical model mainly uses the stress–strain relationship of timber parallel to grain. Due to that the stress on the weakening segment decreases little, the weakening segment can be taken as a horizontal line, and the stress–strain relationship can be simplified to the bilinear constitutive model [19]. Shown in Fig. 7, compressive stress of timber parallel to the grain increases with the increase of strain initially. After achieving the peak stress \({\sigma _{{\text{pc}}}},\) the compressive stress of timber will keep constant.
Stress distributions at the compression interface
Initially, column foot and stone base contact with each other fully. In this state, compression stress may be in elastic stage due to the large compression interface. When the column rotates and deforms to certain extent, column and stone base will be in partial contact with different compression deformations, and the edge of column foot may be into plastic stage due to the small compression interface. Thus, the material at column foot may be elastic or plastic [20] under vertical and horizontal load. And compression stress distributions of vertical load \(q_{i}^{+}(x)\;(i=1,2, \ldots ,6)\) of the six loading substates of the column at the compression interface may be in the elastic and elastoplastic stage.
Stress state \({S_1}\)
Stress state \(S_{2}^{{\text{+}}}\)
During column rocking with deviating from the upright position, the right side of column foot has maximum compression stress, and left side of column has minimum compression stress. If the stress of right point is smaller than \({\sigma _{{\text{pc}}}},\) column foot is in the elastic stage. If the stress of right point is equal to \({\sigma _{{\text{pc}}}},\) the edge of column foot has been yield. The stress of timber material in the compression interface has a bilinear distribution. The peak stress of stress distribution is \({\sigma _{{\text{pc}}}}.\) Shown in Fig. 9, \({L_\theta }\) is the compression length of column foot, \({L_{{\theta _{\text{e}}}}}\) is the elastic compression length of column foot and \({L_{{\theta _{\text{y}}}}}\) is the plastic compression length of column foot.
If compression stresses at the compression interface are known, forces and moments can be obtained by integrating the compression stress distributions at the compression interface. Thus, the material constitutive relation, geometric equation and geometrical relation of contact interface are used for establishment of the stress distributions as follows.
Stress state \(S_{3}^{{\text{+}}}\)
Compression deformation and stress distribution in elastic stage or elastoplastic stage are shown in Fig. 10. \(\Delta h_{{{\text{br}}}}^{+}\) and \(\Delta h_{{{\text{bl}}}}^{+}\) are the compression height at the right and left edge of column foot, respectively. \(\Delta h_{{{\text{bl}}}}^{+}\) is zero at this state, which is the critical state of \(S_{2}^{{\text{+}}}\) and \(S_{4}^{{\text{+}}}.\)
Stress state \(S_{4}^{{\text{+}}}\)
Compression deformation and stress distribution in elastic or elastoplastic stage are shown in Fig. 11. \(\Delta h_{{{\text{cr}}}}^{+}\) is the compression height at the right edge of column foot.
Stress state \(S_{5}^{{\text{+}}}\)
Compression deformation and stress distribution in elastic or elastoplastic stage are shown in Fig. 12. \(\Delta h_{{{\text{dr}}}}^{+}\) is the compression height at the right edge of column foot.
Stress state \(S_{6}^{{\text{+}}}\)
Compression deformation and stress distribution in elastic or elastoplastic stage are shown in Fig. 13. \(\Delta h_{{{\text{er}}}}^{+}\) is the compression height at the right edge of column foot.
Summary on the resisting compression stress distributions
And, resisting compression stress distributions \(w_{i}^{+}(x)\) at \(S_{i}^{{\text{+}}} (i=2, \ldots ,6)\) are summarized and shown in Table 1.
Resisting compression stress distributions
Substates  \(w_{i}^{+}(x)\) 

\(S_{2}^{{\text{+}}}\)  
Elastic  \(w_{{2{\text{e}}}}^{+}(x)=\frac{{\tan \theta {E_{\text{t}}}}}{h}(x  r)+q_{{{\text{ar}}}}^{+}\qquad (34)\) 
Elastoplastic  \(w_{{2{p_{\rm I}}}}^{+}(x)=\frac{{q_{{{\text{al}}}}^{+}  {\sigma _{{\text{pc}}}}}}{{  {L_{{\theta _{\text{e}}}}}}}\left( {x  {L_{{\theta _{\text{e}}}}}+r} \right)+{\sigma _{{\text{pc}}}},  r \leqslant x \leqslant {L_{{\theta _{\text{e}}}}}  r\qquad (35)\) \(w_{{2{p_\Pi }}}^{+}(x)={\sigma _{\text{pc}}},\quad {L_{{\theta _e}}}  r < x \leqslant r\qquad (36)\) 
\(S_{3}^{{\text{+}}}\)  
Elastic  \(w_{{3{\text{e}}}}^{+}(x)=\frac{{\tan \theta {E_{\text{t}}}}}{h}(x+r)\qquad (37)\) 
Elastoplastic  \(w_{{3{p_{\rm I}}}}^{+}(x)=\frac{{{\sigma _{{\text{pc}}}}}}{{{L_{{\theta _{\text{e}}}}}}}(x+r),\quad  r \leqslant x \leqslant {L_{{\theta _{\text{e}}}}}  r\qquad (38)\) \(w_{{3{p_\Pi }}}^{+}(x)={\sigma _{{\text{pc}}}},\quad {L_{{\theta _{\text{e}}}}}  r < x \leqslant r\qquad (39)\) 
\(S_{4}^{{\text{+}}}\)  
Elastic  \(w_{{4{\text{e}}}}^{+}(x)=\frac{{\tan \theta {E_{\text{t}}}}}{h}(x  r+{L_\theta })\qquad (40)\) 
Elastoplastic  \(w_{{4{p_{\rm I}}}}^{+}(x)=\frac{{{\sigma _{{\text{pc}}}}}}{x}(x  r+{L_\theta }),\quad r  {L_\theta } \leqslant x \leqslant r  {L_{{\theta _{\text{y}}}}}\qquad (41)\) \(w_{{4{p_\Pi }}}^{+}(x)={\sigma _{{\text{pc}}}},\quad r  {L_{{\theta _{\text{y}}}}} < x \leqslant r\qquad (42)\) 
\(S_{5}^{{\text{+}}}\)  
Elastic  \(w_{{5{\text{e}}}}^{+}(x)=\frac{{\tan \theta {E_{\text{t}}}}}{h}x\qquad (43)\) 
Elastoplastic  \(w_{{5{p_{\rm I}}}}^{+}(x)=\frac{{{\sigma _{{\text{pc}}}}}}{{{L_{{\theta _{\text{e}}}}}}}x,\quad 0 \leqslant x \leqslant {L_{{\theta _{\text{e}}}}}\qquad (44)\) \(w_{{5{p_\Pi }}}^{+}(x)={\sigma _{{\text{pc}}}},\quad {L_{{\theta _{\text{e}}}}} < x \leqslant r\qquad (45)\) 
\(S_{6}^{{\text{+}}}\)  
Elastic  \(w_{{6{\text{e}}}}^{+}(x)=\frac{{\tan \theta {E_{\text{t}}}}}{h}\left( {x  r+{L_\theta }} \right)\qquad (46)\) 
Elastoplastic  \(w_{{6{p_{\rm I}}}}^{+}(x)=\frac{{{\sigma _{{\text{pc}}}}}}{{{L_{{\theta _{\text{e}}}}}}}(x  r+{L_\theta }),\quad r  {L_\theta } \leqslant x \leqslant r  {L_{{\theta _{\text{y}}}}}\qquad (47)\) \(w_{{6{p_\Pi } }}^{ + } (x) = \sigma _{{{\text{pc}}}} ,\quad r  L_{{\theta _{{\text{y}}} }} < x \leqslant r \qquad (48)\) 
Theoretical model of restoring moment at column foot
Restoring moment of column foot at \(S_{i}^{+}\)
When column rocks under periodic load, compression reaction force \(R\) and compression friction force \(F\) are generated at the compression interface shown in Fig. 14. They resist the vertical and horizontal loads acting on the column. Moments caused by these forces are denoted as the restoring moment \(M_{{\text{bc}}}^{+}\) of the column foot joint. Compression reaction force, compression friction force, restoring moment at the stress state \(S_{i}^{{\text{+}}}\) are denoted by \(R_{i}^{+},\) \(F_{i}^{+}\) and \(M_{{\text{b}{{\text{c}}_i}}}^{+}\) with (i = 1, 2, 3, 4, 5, 6), respectively.
And, compression reaction forces \(R_{i}^{+}\) and restoring moments \(M_{{{\text{b}}{{\text{c}}_i}}}^{+}\) at \(S_{i}^{{\text{+}}}(i=2, \ldots ,6)\) can be obtained by integrating the resisting compression stress distribution on the compression interface [21], which is shown in the Table 2.
Restoring force and moment of column foot
Substates  \(R_{i}^{+}\)  \(M_{{{\text{b}}{{\text{c}}_i}}}^{+}\) 

\(S_{2}^{{\text{+}}}\)  
Elastic  \(R_{{2{\text{e}}}}^{+}=\int_{{  r}}^{r} {2w_{{2{\text{e}}}}^{+}(x)\sqrt {{r^2}  {x^2}} } {\text{d}}x\qquad (50)\)  \(M_{{{\text{b}}{{\text{c}}_{2{\text{e}}}}}}^{+}=\int_{{  r}}^{r} {2w_{{2{\text{e}}}}^{+}(x)\sqrt {{r^2}  {x^2}} } x{\text{d}}x\qquad (60)\) 
Elastoplastic  \(\begin{aligned} R_{{2p}}^{ + } = & \int_{{  r}}^{{L_{{\theta _{{\text{e}}} }}  r}} {2w_{{2p_{{\text{I}}} }}^{ + } (x)\sqrt {r^{2}  x^{2} } } {\text{d}}x \\ & + \int_{{L_{{\theta _{{\text{e}}} }}  r}}^{r} {2w_{{2{p_\Pi } }}^{ + } (x)\sqrt {r^{2}  x^{2} } } {\text{d}}x \\ \end{aligned} \qquad (51)\)  \(\begin{aligned} M_{{{\text{bc}}_{{2p}} }}^{ + } = & \int_{{  r}}^{{L_{{\theta _{{\text{e}}} }}  r}} {2w_{{2p_{{\rm I}} }}^{ + } (x)\sqrt {r^{2}  x^{2} } } x{\text{d}}x \\ & + \int_{{L_{{\theta _{{\text{e}}} }}  r}}^{r} {2w_{{2{p_\Pi } }}^{ + } (x)\sqrt {r^{2}  x^{2} } x} {\text{d}}x \\ \end{aligned} \qquad (61)\) 
\(S_{3}^{{\text{+}}}\)  
Elastic  \(R_{{3{\text{e}}}}^{+}=\int_{{  r}}^{r} {2w_{{3{\text{e}}}}^{+}(x)\sqrt {{r^2}  {x^2}} } {\text{d}}x\qquad (52)\)  \(M_{{{\text{b}}{{\text{c}}_{3{\text{e}}}}}}^{+}=\int_{{  r}}^{r} {2w_{{3{\text{e}}}}^{+}(x)\sqrt {{r^2}  {x^2}} } x{\text{d}}x\qquad (62)\) 
Elastoplastic  \(\begin{aligned} R_{{3{\text{e}}}}^{ + } = & \int_{{  r}}^{{L_{{\theta _{{\text{e}}} }}  r}} {2w_{{3p_{{\rm I}} }}^{ + } (x)\sqrt {r^{2}  x^{2} } } {\text{d}}x \\ & + \int_{{L_{{\theta _{{\text{e}}} }}  r}}^{r} {2w_{{3{p_\Pi } }}^{ + } (x)\sqrt {r^{2}  x^{2} } } {\text{d}}x \\ \end{aligned} \qquad (53)\)  \(\begin{aligned} M_{{{\text{bc}}_{{3p}} }}^{ + } = & \int_{{  r}}^{{L_{{\theta _{{\text{e}}} }}  r}} {2w_{{3p_{{\rm I}} }}^{ + } (x)\sqrt {r^{2}  x^{2} } } x{\text{d}}x \\ & + \int_{{L_{{\theta _{{\text{e}}} }}  r}}^{r} {2w_{{3{p_\Pi } }}^{ + } (x)\sqrt {r^{2}  x^{2} } x} {\text{d}}x \\ \end{aligned} \qquad (63)\) 
\(S_{4}^{{\text{+}}}\)  
Elastic  \(R_{{4{\text{e}}}}^{+}=\int_{{r  {L_\theta }}}^{r} {2w_{{4{\text{e}}}}^{+}(x)\sqrt {{r^2}  {x^2}} } {\text{d}}x\qquad (54)\)  \(M_{{{\text{b}}{{\text{c}}_{4{\text{e}}}}}}^{+}=\int_{{r  {L_\theta }}}^{r} {2w_{{4{\text{e}}}}^{+}(x)\sqrt {{r^2}  {x^2}} } x{\text{d}}x\qquad (64)\) 
Elastoplastic  \(\begin{aligned} R_{{4p}}^{ + } = \int_{{r  L_{\theta } }}^{{r  L_{{\theta _{{\text{y}}} }} }} {2w_{{4p_{{\rm I}} }}^{ + } (x)\sqrt {r^{2}  x^{2} } } {\text{d}}x \\ + \int_{{r  L_{{\theta _{{\text{y}}} }} }}^{r} {2w_{{4{p_\Pi } }}^{ + } (x)\sqrt {r^{2}  x^{2} } } {\text{d}}x \\ \end{aligned} \qquad (55)\)  \(\begin{aligned} M_{{bc_{{4p}} }}^{ + } = & \int_{{r  L_{\theta } }}^{{r  L_{{\theta _{{\text{y}}} }} }} {2w_{{4p_{{\rm I}} }}^{ + } (x)\sqrt {r^{2}  x^{2} } } x{\text{d}}x \\ & + \int_{{r  L_{{\theta _{{\text{y}}} }} }}^{r} {2w_{{4{p_\Pi } }}^{ + } (x)\sqrt {r^{2}  x^{2} } x} {\text{d}}x \\ \end{aligned} \qquad (65)\) 
\(S_{5}^{{\text{+}}}\)  
Elastic  \(R_{{5{\text{e}}}}^{+}=\int_{0}^{r} {2w_{{5{\text{e}}}}^{+}(x)\sqrt {{r^2}  {x^2}} } {\text{d}}x\qquad (56)\)  \(M_{{{\text{b}}{{\text{c}}_{5{\text{e}}}}}}^{+}=\int_{0}^{r} {2w_{{5{\text{e}}}}^{+}(x)\sqrt {{r^2}  {x^2}} } x{\text{d}}x\qquad (66)\) 
Elastoplastic  \(\begin{aligned} R_{{5p}}^{ + } = & \int_{0}^{{L_{{\theta _{{\text{e}}} }} }} {2w_{{5p_{{\rm I}} }}^{ + } (x)\sqrt {r^{2}  x^{2} } } {\text{d}}x \\ & + \int_{{L_{{\theta _{{\text{e}}} }} }}^{r} {2w_{{5{p_\Pi } }}^{ + } (x)\sqrt {r^{2}  x^{2} } } {\text{d}}x \\ \end{aligned} \qquad (57)\)  \(\begin{aligned} M_{{{\text{bc}}_{{5p}} }}^{ + } = & \int_{0}^{{L_{{\theta _{{\text{e}}} }} }} {2w_{{5p_{{\rm I}} }}^{ + } (x)\sqrt {r^{2}  x^{2} } } x{\text{d}}x \\ & + \int_{{L_{{\theta _{{\text{e}}} }} }}^{r} {2w_{{5{p_\Pi } }}^{ + } (x)\sqrt {r^{2}  x^{2} } x} {\text{d}}x \\ \end{aligned} \qquad (67)\) 
\(S_{6}^{{\text{+}}}\)  
Elastic  \(R_{{6{\text{e}}}}^{+}=\int_{{r  {L_\theta }}}^{r} {2w_{{6{\text{e}}}}^{+}(x)\sqrt {{r^2}  {x^2}} } {\text{d}}x\qquad (58)\)  \(M_{{{\text{b}}{{\text{c}}_{6{\text{e}}}}}}^{+}=\int_{{r  {L_\theta }}}^{r} {2w_{{6{\text{e}}}}^{+}(x)\sqrt {{r^2}  {x^2}} } x{\text{d}}x\qquad (68)\) 
Elastoplastic  \(\begin{aligned} R_{{6p}}^{ + } = & \int_{{r  L_{\theta } }}^{{r  L_{{\theta _{{\text{y}}} }} }} {2w_{{6p_{{\rm I}} }}^{ + } (x)\sqrt {r^{2}  x^{2} } } {\text{d}}x \\ & + \int_{{r  L_{{\theta _{{\text{y}}} }} }}^{r} {2w_{{6{p_\Pi } }}^{ + } (x)\sqrt {r^{2}  x^{2} } } {\text{d}}x \\ \end{aligned} \qquad (59)\)  \(\begin{aligned} M_{{{\text{bc}}_{{6p}} }}^{ + } = & \int_{{r  L_{\theta } }}^{{r  L_{{\theta _{{\text{y}}} }} }} {2w_{{6p_{{\rm I}} }}^{ + } (x)\sqrt {r^{2}  x^{2} } } x{\text{d}}x \\ & + \int_{{r  L_{{\theta _{{\text{y}}} }} }}^{r} {2w_{{6{p_\Pi } }}^{ + } (x)\sqrt {r^{2}  x^{2} } } x{\text{d}}x \\ \end{aligned} \qquad (69)\) 
Force system of column
Before rotation occurs, vertical load is applied to the top center of the column. Once rotation of the column occurs, the contact surface of the column and beam could become partial compression. However, according to the results of the shaking table tests and the static lateral loading tests on fullscale models of traditional wooden frames, the restoring force of a wooden structure consists of two parts: one is the resistant bending moments from the tie beams and another is the restoring forces at the column top and bottom due to rocking. When the deformation is small, the restoring force due to column rocking is the major part of the total restoring force. With the deformation increasing, the bending moments from the tie beams become more important [11]. In this study, the theoretical model and rotational behaviors of column foot is mainly analyzed under the small horizontal deformation, and the partial compression at the top of the column can be neglected. In addition, vertical load acting on the top center of the column is adopted on the local compression experiment of column conducted by Lee [14] and the numerical simulation conducted by Wan [22]. Thus vertical load can be assumed to be applied to the top center of column.
Horizontal force and restoring moment can be obtained by simultaneous Eqs. (50)–(59) with Eq. (70) and simultaneous Eqs. (60)–(69) with Eq. (71).
Due to the complex form of compression length, the expression of \({L_\theta }\) is not given.
In whole, the restoring moment at six substates can be obtained at the same way as the derivation process of Eq. (75). Because the expression of restoring moment is complex, the expression cannot be listed and can be got by numerical calculation in Matlab easily.
Finite element analysis
Finite element model of a typical column foot joint in traditional timber structure is established by ABAQUS to validate the proposed theoretical restoring moment model with horizontal loading.
Finite element model
The 7th ts’ai column recorded in Ying Zao Fa Shi [23] from the Song Dynasty of China is established. Diameter and height of column are 36f’en and 25f’en, respectively. Each f’en is equal to 10.815 mm, and the size of column under the international units system is as follows: column diameter = 389 mm and column height = 2758 mm. Stone base is modeled as a 1600 mm^{2} block with 200 mm height.
Stresses at the column foot will be changing with rocking of column. The reduction of contact area will lead to the increase of compression stress at the compression interface. Since the compression deformation mainly occurs to timber parallel to grain, the properties of timber perpendicular to grain have less effect on the longitudinal deformation of wood. The material may then be taken as isotropic and elastoplastic for this study, and the strength of timber parallel to grain can be taken as representative of the strength of timber. Based on the elastoplastic constitutive model with small deformation, material is assumed to be homogenous and continuous, and the isotropic elastoplastic model in ABAQUS is chosen as the material model. Material parameters are referred to results from experiments on Pinus koraiensis. Mechanical parameters of timber are as follows: material density = 420 kg/m^{3}; elastic modulus = 10 GPa; Poisson ratio = 0.3 and yield stress = 34.76 MPa. Elastic modulus of stone base is about ten times bigger than that of wood. Poisson ratio of stone base is 0.19.
Chen had measured the friction coefficients among the different contact interface and different directions of timber, and results showed the contact interface and the directions of timber play little effects to the friction coefficients. The mean value of static friction coefficient of Pinus koraiensis is 0.35, and the mean value of dynamic friction coefficient is 0.28 [24]. In this simulation, the friction coefficient is taken as 0.35 without considering the difference of dynamic and static friction coefficient due to the slow loading process. That is, coulomb friction model, which has the same static and dynamic friction coefficients, is selected as the interface model to represent the tangential behavior at the column foot in ABAQUS [22].
Vertical load acting on the column is estimated from all the dead loads it supports. Experiences with many historical buildings show that 35 kN is a reasonable estimate including own weight of column. Vertical load is applied to the top center of the column [14]. Oneway horizontal displacement is applied at the top of column. The maximum displacement is 190 mm. The bottom of stone base is assumed fixed to the ground.
Column and stone base are divided into different zones in establishing the finite element model. The gridline density at the bottom of column and the top of stone base is denser than those at other parts of column and stone base as shown in Fig. 16. The linear hexahedral element (element type C3D8R) is selected as element for both materials, as this type of element can avoid the “hourglass” phenomenon in the finite element computation [25].
Stress state of column foot
When the horizontal load is zero, compression stress distribution on the compression surface is shown in Fig. 17a. It is found that the compression stress distribution is uniform, which is coincident with the conceptual analysis in the previous section. With both horizontal and vertical loads acting on the column, compression stress distribution at the critical state before lifting is shown in Fig. 17b. It is found that the compressive stress has the same pattern with nearly zero compression at the side to be lifted and maximum compression value at the opposite side that agrees with the conceptual analysis. The compression distribution at \(S_{2}^{+}\) is not shown in this section because it is similar to that of \(S_{3}^{+}.\) Clearly, compression stress increases significantly on one side and decreases on the other side due to the horizontal force.
Column rocking with lifting would happen if horizontal load is big enough to make the edge of column foot lift from the stone base. Compression stress distributions of the three substates with lifting are shown in Fig. 18, which is corresponding to the concept analysis shown in Fig. 6. Compression stress distribution of column foot with large cross section is shown in Fig. 18a. Compression deformation is mainly located in the large area of column foot at the initial stage of column lifting. The maximum compression stress occurs to the edge of column foot. Compressions stress distribution of the column foot with halfcross section of column foot is shown in Fig. 18b, and compression stress distributions of column foot with small cross section at the maximum controlled displacement is shown in Fig. 18c and the maximum compression stress is equal to the ultimate compression stress parallel to the grain of column at this stage.
Shown in Figs. 17 and 18, compression interface decreases with the increase of rotation angle, while compression stresses at the edge of column foot show an obvious rising trend with the increase of rotation angle due to the decrease of compression interface. There is low risk of local damage in the column foot by the full contact interface during rocking. However, owing to small contact areas caused by rotation angle, the compression stresses between the column foot and stone base could reach high levels that up to the ultimate compressive strength of timber. Thus local damages on the column likely occur. Besides, the boundary line between the contact and noncontact areas is almost straight and the stress increment is nearly uniform. Compression interfaces of the numerical model are consistent with those of the conceptual analysis in the section of Compression effect. It is noted that compression stress distributions under negative loading are symmetrically consistent with those under positive loading shown in Figs. 17 and 18, since the absolute value of rotation angles are same.
Comparison between numerical simulation and the proposed model
The curve of restoring moment and rotation angle can illustrate the rotational capacity of column foot under seismic load. In order to clarify the rotation behaviors of column foot, the curve of restoring momentrotation angle with six substates is shown in Fig. 19. Both the simulated and theoretical curves are close to each other. The trend of variation of curves from both the proposed model and the finite element modeling is noted consistent.
The simulated model is drawn by red line, which can be obtained and extracted from the results of finite element analysis directly. The theoretical model is drawn by black line, which can be calculated by the previous theoretical derivation and drawn with feature points and scatter points. The feature points include the restoring moment and rotation angle at \({S_1},\) \(S_{3}^{+}\) and \(S_{5}^{+}.\) Such as critical rotation angle and restoring moment in elastic stage at \(S_{3}^{+}\) can be obtained as \(\tan \theta _{{3e}}^{+}=\frac{{Nh}}{{{E_{\text{t}}}\pi {r^3}}}\) and \(M_{{3e}}^{+}=\frac{{Nr}}{4}\) by simultaneous Eq. (52) with Eq. (70) and simultaneous Eq. (62) with Eq. (71). Similarly, critical rotation angle in elastic stage at \(S_{5}^{+}\) can be obtained as \(\tan \theta _{{5e}}^{+}=\frac{{3Nh}}{{2{E_{\text{t}}}{r^3}}}\) and \(M_{{5e}}^{+}=\frac{{3\pi Nr}}{{16}}\) by simultaneous Eq. (56) with Eq. (70) and simultaneous Eq. (66) with Eq. (71). In addition, scatter points include the restoring moment and rotation angle can be got by substituting horizontal displacement \(\Delta =10,20, \ldots ,190\) mm to the proposed model to get the theoretical curve. The elastic and elastoplastic stage have been noted in the Fig. 19. There is no plastic deformation at \({S_1}\) to \(S_{5}^{+},\) and plastic deformation mainly occurs to the column foot at \(S_{6}^{+}\) due to the small compression interface.
The curve of restoring moment and rotation angle of column foot joint has a reverse Z shape and antisymmetric at positive and negative loading in Fig. 19, which has the similar tendency with the results of local compression of test [14]. The initial restoring moment is noted linear and increases rapidly, which shows column foot has a high resistance to small rotation angle. The reason is that compression interface decreases, which can lead to the compression reaction force move to the column edge quickly. The rapid movement of the compression reaction force makes the restoring moment increase rapidly. Besides, the compression stress is the elastic stage and isn’t up to the ultimate compression due to large compression interface. The curve is no longer linear after 0.01 rad and restoring moment shows a slow rising trend. The main reason is that the compression interface and the action point of the compression reaction force are close to the edge of column. Changing of the compression interface leads to small changing of the action point of compression reaction force, which causes small changing of restoring moment. Then, the restoring moment almost remains the same value after 0.015 rad, that is because the compression interface is located at the edge of column foot and the changing of compression interface cannot cause changing of action point of the compression force any more. Restoring moment of column foot remains to be positive and active in preventing the column from collapse at the maximum rotation angle, which shows column foot joint has certain amount of restoring moment during column rocking to resist the overturning of structure under lateral load.
Discussions
Good agreement can be found between simulated and theoretical model, and thus the theoretical model and assumptions can be regarded as valid to some extent. However, the accuracy of the proposed model is not sufficient. It should be noted that there are some assumptions in the derivation of the theoretical model and the simulated model including the simplified constitutive model of timber parallel to the grain and the position of vertical load, which is convenient for the analysis.
For the first assumption, the bilinear constitutive model made little difference to the compression stress distributions in the elastic stage. But, the simplified model has certain effects to the compression stress distribution in the elastoplastic stage. The diagrams of real and simplified stress distributions are shown in Fig. 20 at the \(S_{3}^{+}\) stage. Compression stresses at the edge of column foot was overestimated due to the horizontal line instead of the weakening segment. In addition, the elastic compression length \({L_{{\theta _{\text{e}}}}}\) was underestimated and the plastic compression length \({L_{{\theta _{\text{y}}}}}\) was overestimated according to the balance of the vertical load. The theoretical moment obtained by the simplified model was overestimated compared with that of real model due to the action point of compression reaction force is closer to the edge of column foot than that of real model. The theoretical moments in the other substates are the similar with that of \(S_{3}^{+}\).
For the second assumption, the position of vertical load is assumed to be applied at the top center of column. Before column rocking, the assumption is reasonable. But once rotation of the column occurs, the position of vertical load deviates from the center of column head. Because the theoretical model is mainly obtained based on the small rotation angle, the offset distance of vertical load is small. While the position of vertical load makes a little effects on the restoring moment. Equation (71) changes into \(Ph+N\left( {\Delta  {\Delta _{\text{d}}}} \right)={M^{+}_{{\text{bc}}}}\) with taking the partial compression into consideration. \({\Delta _{\text{d}}}\) is the offset distance of vertical load. The theoretical moment may be overestimated without considering the partial compression based on the above formula. All in all, it should be noted that the above assumptions should be improved in the future study in order to make the theoretical model more accurate.
Conclusions
The rotational behavior of column foot in traditional timber structure is analyzed in this paper. Under cyclic load, column is subjected to lift and reset called “column rocking”. Column rotates around the edge of column foot. During rocking, column foot has compression deformation under vertical load, and the material at column foot may be in elastic stage or elastoplastic stage. Compression stress occurs at the contact interface between column and stone base. Compression reaction force and friction force are generated at the interface between column foot and stone base, as well restoring moment occurs to column foot.
The theoretical model of the restoring moment at column foot is proposed for the different substates of column foot with consideration the compression effect, and derived based on constitutive equations, geometric equation, geometrical relationship and equilibrium equation. The proposed model generally shows good agreement with results of numerical simulation, indicating the validity of the theoretical method, and can be considered as a suitable basis for mechanical analyses to predict the structural performance of traditional timber structure. The results of theoretical model and numerical simulation show that column foot joint has certain amount of restoring moment during rocking to resist the overturning of structure under lateral load.
Notes
Acknowledgements
This work is supported by the National Natural Science Foundation of China (nos. 51408038 and 51338001). The authors would also like to thank the research project funding of the 111 Project of China (no. B13002), National Key R&D Program of China (nos. 2015BAK01B02 and 2017YFC0703505) and the Beijing Natural Science Foundation (no. 8151003).
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