Bending, buckling and free vibration analyses of functionally graded curved beams with variable curvatures using isogeometric approach

Abstract

A study on the bending, buckling and free vibration of functionally graded curved beams with variable curvatures using isogeometric analysis is presented here. Non-uniform rational B-splines, known from computer aided geometric design, are employed to describe the exact geometry and approximate the unknown fields of a curved beam element based on Timoshenko model. Material properties of the beam are assumed to vary continuously through the thickness direction according to the power law form. The numerical examples investigated in this paper deal with circular, elliptic, parabolic and cycloid curved beams. Results have been verified with the previously published works in both cases of straight functionally graded beam and isotropic curved beam. The effects of material distribution, aspect ratio and slenderness ratio on the response of the beam with different boundary conditions are numerically studied. Furthermore, an interesting phenomenon of changing mode shapes for both buckling and free vibration characteristics corresponding to the variation in the parameters mentioned above is also examined.

Appendices

Appendix 1

Expressions of stiffnesses $A_{11},B_{11},D_{11},A_{55}$

$$\begin{aligned} A_{11}=\frac{b_wh(E_{t}+kE_{b})}{(1+k)} , \end{aligned}$$

(30a)

$$\begin{aligned} B_{11}=\frac{b_wh^2(E_{t}-E_{b})k}{2(1+k)(2+k)} , \end{aligned}$$

(30b)

$$\begin{aligned} D_{11}= \frac{b_wh^3(k(k^2+3k+8)E_{b}+3(k^2+k+2)E_{t})}{12(1+k)(2+k)(3+k)} , \end{aligned}$$

(30c)

$$\begin{aligned} A_{55}= \frac{b_wh(E_{t}+kE_{b})}{2(1+\nu )(1+k)} . \end{aligned}$$

(30d)

Expressions of $I_{0},I_{1},I_{2}$

$$\begin{aligned}I_{0}&= \frac{b_wh(\rho _{t}+k\rho _{b})}{(1+k)} , \\ I_{1}&= \frac{b_wh^2(\rho _{t}-\rho _{b})k}{2(1+k)(2+k)} , \\ I_{2}&= \frac{b_wh^3(k(k^2+3k+8)\rho _{b}+3(k^2+k+2)\rho _{t})}{12(1+k)(2+k)(3+k)} . \\ \end{aligned}$$

Appendix 2

Explicit forms of element stiffness matrix

$$\begin{aligned} {\mathbf{K}} =&\left[ \begin{array}{ccc} {\mathbf {K}}^{11} &{\mathbf {K}}^{12} & {\mathbf {K}}^{13} \\ {\mathbf {K}}^{21} &{\mathbf {K}}^{22} & {\mathbf {K}}^{23} \\ {\mathbf {K}}^{31} &{}{\mathbf {K}}^{32} &{} {\mathbf {K}}^{33} \end{array} \right] , \nonumber \\ K_{ij}^{11}&= \int _{L^e}^{L^{e+1}} (A_{11}N_i'N_j'+A_{55}\kappa ^2 N_i N_j)dx , \end{aligned}$$

(31a)

$$\begin{aligned} K_{ij}^{22}= \int _{L^e}^{L^{e+1}} (A_{11}\kappa ^2N_iN_j+A_{55} N_i' N_j')dx , \end{aligned}$$

(31b)

$$\begin{aligned} K_{ij}^{33}= \int _{L^e}^{L^{e+1}} (D_{11}N_i'N_j'+A_{55} N_i N_j)dx , \end{aligned}$$

(31c)

$$\begin{aligned} K_{ij}^{12}= \int _{L^e}^{L^{e+1}} (A_{11}\kappa N_i'N_j-A_{55} \kappa N_i N_j')dx , \end{aligned}$$

(31d)

$$\begin{aligned} K_{ij}^{13}= \int _{L^e}^{L^{e+1}} (-A_{55} \kappa N_iN_j+B_{11} N_i' N_j')dx , \end{aligned}$$

(31e)

$$\begin{aligned} K_{ij}^{23}= \int _{L^e}^{L^{e+1}} (A_{55}N_i'N_j+B_{11} \kappa N_i N_j')dx , \end{aligned}$$

(31f)

$$\begin{aligned} K_{ij}^{33}= \int _{L^e}^{L^{e+1}} (A_{55}N_i'N_j+B_{11} \kappa N_i N_j')dx . \end{aligned}$$

(31g)

Explicit forms of element geometric stiffness matrix

$$\begin{aligned} {\mathbf{ G }}=&\left[ \begin{array}{ccc} {\mathbf {G}}^{11} &{\mathbf {G}}^{12} &{}0 \\ {\mathbf {G}}^{21} & {\mathbf{G}}^{22} & 0 \\ 0 & 0 & 0 \end{array} \right] , \nonumber \\ G_{ij}^{11}&= \int _{L^e}^{L^{e+1}} ( \kappa ^2 N_iN_j )dx , \end{aligned}$$

(32a)

$$\begin{aligned} G_{ij}^{22}= \int _{L^e}^{L^{e+1}} N_i'N_j' dx , \end{aligned}$$

(32b)

$$\begin{aligned} G_{ij}^{12}= \int _{L^e}^{L^{e+1}} (- \kappa N_iN_j' )dx . \end{aligned}$$

(32c)

Explicit forms of element mass matrix

$$\begin{aligned}{ \mathbf{ M}} =&\left[ \begin{array}{ccc} {\mathbf {M}}^{11} &{}0 &{}{\mathbf {M}}^{13} \\ 0 &{}{\mathbf {M}}^{22} &{} 0 \\ {\mathbf {M}}^{31} &{} 0 &{} {\mathbf {M}}^{33} \end{array} \right] , \nonumber \\ M_{ij}^{11}&= \int _{L^e}^{L^{e+1}} (I_0+2\kappa I_1 +\kappa ^2 I_2)N_iN_jdx , \end{aligned}$$

(33a)

$$\begin{aligned} M_{ij}^{22}= \int _{L^e}^{L^{e+1}} I_0N_iN_jdx , \end{aligned}$$

(33b)

$$\begin{aligned} M_{ij}^{33}= \int _{L^e}^{L^{e+1}} I_2N_iN_jdx , \end{aligned}$$

(33c)

$$\begin{aligned} M_{13}^{1}= \int _{L^e}^{L^{e+1}} (I_1+\kappa I_2 )N_iN_jdx . \end{aligned}$$

(33d)

Explicit forms of element load vector

$$\begin{aligned} {\mathbf{F}} =&\left\{ \begin{array}{ccc} {\mathbf {F}}^1 \\ {\mathbf {F}}^2 \\ {\mathbf {F}}^3 \end{array} \right\} , \nonumber \\ F_{ij}^{1}&= \int _{L^e}^{L^{e+1}} p_x N_i dx , \end{aligned}$$

(34a)

$$\begin{aligned} F_{ij}^{2}= \int _{L^e}^{L^{e+1}} p_z N_i dx , \end{aligned}$$

(34b)

$$\begin{aligned} F_{ij}^{3}= \int _{L^e}^{L^{e+1}} p_m N_i dx . \end{aligned}$$

(34c)