## 1INTRODUCTION

Columns with variable cross-section have been used in the architectural buildings around the world. Some illustrations of such columns are shown in Figs. 1 and 2. In addition, brief historical review of columns with entasis (a column with convexity in its middle) is given in the paper published by one of authors (Yoon et al., 2015).

In the design of column, the elastic buckling strength (or load) should be found. If the column is simply supported at its both ends and has uniform cross-section along the length (prismatic column) the elastic buckling load depends only on the elastic modulus *E* and the slenderness ratio *l/r* in which *l* is the length and *r* is the radius of gyration of the column (Timoshenko and Gere, 1961). However, if the column has variable cross-sectional area along the length of column there is no constant slenderness ratio. Hence, the Euler buckling analysis is no longer applicable.

In order to find the elastic buckling load of columns having various cross-sectional area along the length we may need other solution techniques such as the successive approximations technique, the finite element method, etc. If the column has the same length and volume but different cross-sectional area along its length it is interesting to find the buckling load in order to figure out which shape of columns results in the highest buckling load.

In this paper we present the result of investigations pertaining to the elastic buckling load of column with same length and volume but variable cross-sectional area along the length. In the investigations we adopted a column (for illustration) at the Muryangsujeon in Buseoksa-Temple, located in Youngju, Korea. The column is made of wood (Zelkova Serrata tree). Detailed informations on the mechanical properties and dimension of the actual wooden column are discussed in the paper (Yoon et al., 2015). In the study, although wood is one of the typical natural composites with orthotropic nature of material, we assumed the material as isotropic but the effect of orthotropic material is taken into account.

In this study, based on the actual dimension of column at the Muryangsujeon in Buseoksa-Temple, we modeled five columns having different cross-section dimension but approximately the same length and volume of each column.

In order to distinguish the difference between the columns visually, 1/25 scaled down model of columns are generated by the 3D printer as shown in Fig. 2.

## 2MODELING OF COLUMNS

The mechanical properties (i.e., *E _{wood}*= 4.067GPa,

*ν*= 0.490) and the dimension of actual circular wooden column are given in the paper (Yoon et al., 2015).

_{wood}In this study, we fixed the length of circular cross-section columns to 3000*mm*. Each column is divided into 10 segments but 12 segments for H/3 convex column. Each segment is assumed to be prismatic (i.e., stepped configuration along the length) with circular cross-section. In order to make the columns have identical volume, AutoCAD (2010) program is used to change the diameter of segment in each column and the column named as Actual, Prismatic, H/2 Convex, H/3 Convex, and Tapered (refer to Fig. 2). Along the length of column the diameter and the moment of inertia for each segment are given in Tables 1 to 5, respectively.Tables 2, 3 ,4

## 3PREDICTION OF BUCKLING LOAD

### 3.1Successive Approximations Technique

If the diameter varies along the length of column Euler buckling analysis can not be applied. In this case we may use other approximate analysis techniques such as the successive approximations or the finite element techniques.

It was known that the successive approximations technique was developed by Schwartz and the mathematical verification on the accuracy of the method was made by Trefftz.

It was also known that Engesser applied this method at first to the column buckling analysis (Timoshenko and Gere, 1961).

Detailed step by step calculation procedure is described by Lee (2015), Yoon (2015), and Timoshenko and Gere (1961), and complete results of calculation for five columns simply supported at both ends are given in Tables 8 to 12 attached in Appendix.Tables 9 ,10, 11

### 3.2The Finite Element Analysis

Length of columns are fixed to 3000*mm*. Volume of each column is equal to 0.524*m ^{3}* approximately.

Three co*mm*ercially available finite element analysis programs, i.e., MIDAS^{Ⓡ}, GTSTRUDL^{Ⓡ}, and ANSYS^{Ⓡ} are used in the study.

The variable cross-section columns are modeled by AutoCAD^{Ⓡ}. The boundary condition at both ends of the column is modeled as simply supported and the concentrical compressive unit load is applied.

For illustration, one of the analysis results, the analysis model and buckled mode shape of the column obtained by each program, is shown in Figs. 3 to 5, respectively. Complete results of the finite element analysis performed by each program and obtained in each shape of columns are given in Tables 6 to 7 and also given by Lee (2015). Fig 4

## 4COMPARISON OF RESULTS

### 4.1Accuracy of Results

In order to figure out the accuracy of buckling load calculation, buckling loads of prismatic column obtained by the successive approximations and three finite element methods are compared with the Euler buckling load as given in Table 6. Euler buckling load is calculated by Eq. (1). In Eq. (1) *E* is the elastic modulus, *I* is the moment of inertia, and *l* is the length of column.

As can be seen in Table 6, the buckling load obtained by the successive approximations method differs +0.03% compared with the Euler buckling load. Results obtained by the finite element analysis differ by –0.8% to +3.4%. It was found that the results obtained by either theoretical or numerical methods are accurate enough for the practical standpoints.

### 4.2Comparison of Results

The buckling load of all columns, i.e., Actual, Prismatic, H/2 convex, H/3 convex, and Tapered are given in Table 7. From the study it was found that the result obtained by the MIDAS program yields the lowest buckling load compared with the successive approximations result but the result is close enough for the practical purposes.

## 5CONCLUSION

The buckling analysis on the columns with variable cross-section and simply supported at both ends is performed by the successive approximations technique and the finite element method. To confirm accuracy of methods, Euler buckling load of simply supported column is compared with the buckling loads obtained by other methods. The buckling load of prismatic column obtained by the successive approximations technique is almost the same but +0.03% higher. Also, the buckling loads obtained by the finite element methods are in the range between –0.8% to +3.4% which are accurate enough for the practical standpoint.

As expected, the elastic buckling load for H/2 convex column is the highest while the buckling load for Tapered column is the lowest. If the buckling loads obtained by the successive approximations technique are compared with the buckling loads obtained by the numerical methods it was found that the results are in an acceptable range for the practical purposes.