On the static analysis of laminated composite frames having variable cross section

Abstract

The complementary functions method (CFM) is used to investigate the static behavior of laminated composite frames consisting of straight and/or curved members of variable cross section. The Timoshenko beam theory (TBT) is used to obtain the set of the governing equations. The fifth-order Runge–Kutta (RK5) algorithm is employed in the solution process of initial value problems via the CFM. A computer program is substantially coded in Fortran on rigidity matrix based on the CFM to acquire the rigidity matrices and load vectors of these structural elements. With the help of the suggested method, the influences of the symmetric layer stacking sequence, the ratio of E₁/E₂, and various boundary conditions on the nodal displacements and element end forces of the considered frames are investigated. Carrying out the static behavior of laminated composite frame systems which contain straight and circular axis elements for the first time by using the CFM is the novelty of this study. Verification and accuracy of the suggested scheme are intently performed through the comparison of the present results with those of the finite element method. The effectiveness of the method and good agreement of the results are observed.

Appendix A: Formulations for composite materials

The stress–strain and strain–stress relations for composite materials in 1, 2, 3 coordinate systems are written, respectively, as:

$$\sigma_{i} = C^{\prime}_{ij} \varepsilon_{j } \;\varepsilon_{i } = S^{\prime}_{ij} \sigma_{j} \quad \left( {i, \, j = \, 1,2, \ldots ,6} \right)$$

(47)

In the above relations the notation $C^{\prime}_{ij}$ and $S^{\prime}_{ij}$ shows the transformed stiffness and compliance matrices and can be calculated as follows [38, 43].

$$\left[ {C^{\prime}} \right] = \left[ {T^{^\circ } } \right]^{ - 1} \left[ C \right] \left[ \forall \right] \left[ {T^{^\circ } } \right]{ }\left[ \forall \right]^{ - 1} ,\;\left[ {S^{\prime}} \right] = \left[ \forall \right]\left[ {T^{^\circ } } \right]^{ - 1} \left[ \forall \right]^{ - 1} \left[ S \right] \left[ {T^{^\circ } } \right]{ }$$

(48)

The transformation matrices components are defined as

$$\left[ {T^{^\circ } } \right] = \left[ {\begin{array}{*{20}c} {m^{2} } & {n^{2} } & 0 & 0 & 0 & {2mn} \\ {n^{2} } & {m^{2} } & 0 & 0 & 0 & { - 2mn} \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & m & { - n} & 0 \\ 0 & 0 & 0 & n & m & 0 \\ { - mn} & {mn} & 0 & 0 & 0 & {m^{2} - n^{2} } \\ \end{array} } \right]$$

(49)

in which $m = \cos \left( \theta \right)$ and $n = \sin \left( \theta \right)$.

(50)

A, B,  F and D matrices are $\left( {3 \times 3} \right)$ in dimensions and are dependent to the material properties and geometry of the cross section and can be written as follows (see [43] and [50]):

$$A_{ij } = \mathop \smallint \limits_{A} \tilde{Q}_{ij} {\text{d}}A\quad B_{ij } = \varepsilon_{mjk} \mathop \smallint \limits_{A} \tilde{Q}_{im} x_{k} {\text{d}}A$$

(51)

$$F_{ij } = \varepsilon_{ikm} \mathop \smallint \limits_{A} x_{k} \tilde{Q}_{mj} {\text{d}}A\quad D_{ij } = \varepsilon_{ihk } \varepsilon_{mjp} \mathop \smallint \limits_{A} x_{h} x_{p} \tilde{Q}_{km} {\text{d}}A$$

(52)

In the above equations $\varepsilon_{ijk}$ is the permutation tensor and $\tilde{Q}_{ij}$ is reduced stiffness matrix [43].

The transformed stress–strain relations can be written as

$$\tilde{\sigma }_{i} = \tilde{Q}^{^{\prime}}_{ij} \tilde{\varepsilon }_{j} \quad \left( {i \, , \, j = \, 1,2, \ldots ,6} \right)$$

(53)

in which the $\tilde{Q}^{^{\prime}}_{ij}$ show the reduced and transformed stiffness matrix. The nonzero elements of $\tilde{Q}^{^{\prime}}_{ij}$ are obtained in terms of the general three-dimensional transformed stiffness and compliance matrices.

$$\tilde{Q}^{^{\prime}}_{11} = C^{\prime}_{11} + \left( {C^{\prime}_{12} S^{\prime}_{12} + C^{\prime}_{13} S^{\prime}_{31} } \right)\alpha^{\prime}_{11} + \left( {C^{\prime}_{12} S^{\prime}_{26} + C^{\prime}_{13} S^{\prime}_{36} } \right)\alpha^{\prime}_{61}$$

(54)

$$\tilde{Q}^{^{\prime}}_{12} = C^{\prime}_{16} + \left( {C^{\prime}_{12} S^{\prime}_{21} + C^{\prime}_{13} S^{\prime}_{31} } \right)\alpha^{\prime}_{16} + \left( {C^{\prime}_{12} S^{\prime}_{26} + C^{\prime}_{13} S^{\prime}_{36} } \right)\alpha^{\prime}_{66}$$

(55)

$$\tilde{Q}^{^{\prime}}_{22} = C^{\prime}_{66} + \left( {C^{\prime}_{62} S^{\prime}_{21} + C^{\prime}_{63} S^{\prime}_{31} } \right)\alpha^{\prime}_{16} + \left( {C^{\prime}_{62} S^{\prime}_{26} + C^{\prime}_{63} S^{\prime}_{36} } \right)\alpha^{\prime}_{66}$$

(56)

$$\tilde{Q}^{^{\prime}}_{22} = C^{\prime}_{55}$$

(57)

$$\alpha^{\prime}_{11} = \frac{{S^{\prime}_{66} }}{{S^{\prime}_{11} S^{\prime}_{66} - S_{16}^{\prime 2} }}\;\alpha^{\prime}_{16} = \alpha^{\prime}_{61} = \frac{{ - S^{\prime}_{66} }}{{S^{\prime}_{11} S^{\prime}_{66} - S_{16}^{\prime 2} }} \;\alpha^{\prime}_{66} = \frac{{S^{\prime}_{11} }}{{S^{\prime}_{11} S^{\prime}_{66} - S_{16}^{\prime 2} }}$$

(58)

The transformed stiffness and compliance matrices for an orthotropic material are obtained in the following form.

$$\left[ {C^{\prime}} \right] = \left[ {\begin{array}{*{20}c} {C^{\prime}_{11} } & {C^{\prime}_{12} } & {C^{\prime}_{13} } & 0 & 0 & {C^{\prime}_{16} } \\ {C^{\prime}_{21} } & {C^{\prime}_{22} } & {C^{\prime}_{23} } & 0 & 0 & {C^{\prime}_{26} } \\ {C^{\prime}_{31} } & {C^{\prime}_{32} } & {C^{\prime}_{33} } & 0 & 0 & {C^{\prime}_{36} } \\ 0 & 0 & 0 & {C^{\prime}_{44} } & {C^{\prime}_{45} } & 0 \\ 0 & 0 & 0 & {C^{\prime}_{54} } & {C^{\prime}_{55} } & 0 \\ {C^{\prime}_{61} } & {C^{\prime}_{62} } & {C^{\prime}_{63} } & 0 & 0 & {C^{\prime}_{66} } \\ \end{array} } \right]$$

(59)

$$\left[ {S^{\prime}} \right] = \left[ {\begin{array}{*{20}c} {S^{\prime}_{11} } & {S^{\prime}_{12} } & {S^{\prime}_{13} } & 0 & 0 & {S^{\prime}_{16} } \\ {S^{\prime}_{21} } & {S^{\prime}_{22} } & {S^{\prime}_{23} } & 0 & 0 & {S^{\prime}_{26} } \\ {S^{\prime}_{31} } & {S^{\prime}_{32} } & {S^{\prime}_{33} } & 0 & 0 & {S^{\prime}_{36} } \\ 0 & 0 & 0 & {S^{\prime}_{44} } & {S^{\prime}_{45} } & 0 \\ 0 & 0 & 0 & {S^{\prime}_{54} } & {S^{\prime}_{55} } & 0 \\ {S^{\prime}_{61} } & {S^{\prime}_{62} } & {S^{\prime}_{63} } & 0 & 0 & {S^{\prime}_{66} } \\ \end{array} } \right]$$

(60)

The nonzero components of the transformed stiffness matrix are obtained as follows

$$C^{\prime}_{11} = m^{4} C_{11} + 2n^{2} m^{2} C_{12} + n^{4} C_{22} + 4n^{2} m^{2} C_{66}$$

(61)

$$C^{\prime}_{12} = n^{2} m^{2} C_{11} + \left( {n^{4} + m^{4} } \right)C_{12} + n^{2} m^{2} C_{22} - 4n^{2} m^{2} C_{66}$$

(62)

$$C^{\prime}_{13} = m^{2} C_{13} + n^{2} C_{23}$$

(63)

$$C^{\prime}_{16} = nm^{3} C_{11} + nm\left( {n^{2} - m^{2} } \right)C_{12} - n^{3} mC_{22} + 2nm\left( {n^{2} - m^{2} } \right)C_{66}$$

(64)

$$C^{\prime}_{22} = n^{4} C_{11} + 2n^{2} m^{2} C_{12} + m^{4} C_{22} + 4n^{2} m^{2} C_{66}$$

(65)

$$C^{\prime}_{23} = n^{2} C_{13} + m^{2} C_{23}$$

(66)

$$C^{\prime}_{26} = nm^{3} C_{11} + nm\left( {m^{2} - n^{2} } \right)C_{12} - nm^{3} C_{22} - 2nm\left( {m^{2} - n^{2} } \right)C_{66}$$

(67)

$$C^{\prime}_{33} = C_{33}$$

(68)

$$C^{\prime}_{36} = nmC_{13} - nmC_{23}$$

(69)

$$C^{\prime}_{44} = m^{2} C_{44} + n^{2} C_{55}$$

(70)

$$C^{\prime}_{45} = - nmC_{44} + nmC_{55}$$

(71)

$$C^{\prime}_{55} = n^{2} C_{44} + m^{2} C_{55}$$

(72)

$$C^{\prime}_{66} = n^{2} m^{2} C_{11} - 2 n^{2} m^{2} C_{12} + n^{2} m^{2} C_{22} + \left( {m^{2} - n^{2} } \right)^{2} C_{66}$$

(73)

The nonzero components of the compliance matrix are obtained in the following form.

$$S^{\prime}_{11} = m^{4} S_{11} + 2n^{2} m^{2} S_{12} + n^{4} S_{22} + n^{2} m^{2} S_{66}$$

(74)

$$S^{\prime}_{12} = n^{2} m^{2} S_{11} + \left( {n^{4} + m^{4} } \right)S_{12} + n^{2} m^{2} S_{22} - n^{2} m^{2} S_{66}$$

(75)

$$S^{\prime}_{13} = m^{2} S_{13} + n^{2} S_{23}$$

(76)

$$S^{\prime}_{16} = 2 nm^{3} S_{11} + 2nm\left( {n^{2} - m^{2} } \right)S_{12} - 2n^{3} mS_{22} + nm\left( {n^{2} - m^{2} } \right)S_{66}$$

(77)

$$S^{\prime}_{22} = n^{4} S_{11} + 2n^{2} m^{2} S_{12} + m^{4} S_{22} + n^{2} m^{2} S_{66}$$

(78)

$$S^{\prime}_{23} = n^{2} S_{13} + m^{2} S_{23}$$

(79)

$$S^{\prime}_{26} = 2 n^{3} mS_{11} + 2nm\left( {m^{2} - n^{2} } \right)S_{12} - 2nm^{3} S_{22} + nm\left( {m^{2} - n^{2} } \right)S_{66}$$

(80)

$$S^{\prime}_{33} = S_{33}$$

(81)

$$S^{\prime}_{36} = 2nmS_{13} - 2nmS_{23}$$

(82)

$$S^{\prime}_{44} = m^{2} S_{44} + n^{2} S_{55}$$

(83)

$$S^{\prime}_{45} = - nmS_{44} + nmS_{55}$$

(84)

$$S^{\prime}_{55} = n^{2} S_{44} + m^{2} S_{55}$$

(85)

$$S^{\prime}_{66} = 4n^{2} m^{2} S_{11} - 8 n^{2} m^{2} S_{12} + 4n^{2} m^{2} S_{22} + \left( {m^{2} - n^{2} } \right)^{2} S_{66}$$

(86)

${\varvec{B}} = {\varvec{F}} = 0$ relation is valid for symmetric laminates.

The nonzero elements of A and D matrices are given in the following equations.

$$A_{11} = \sum\limits_{k = 1}^{N} {\tilde{Q}_{11}^{\prime \left( k \right)} } A^{\left( k \right)} \quad A_{12} = \sum\limits_{k = 1}^{N} {\tilde{Q}_{12}^{\prime \left( k \right)} } A^{\left( k \right)}$$

(87)

$$A_{22 } = \mathop \sum \limits_{k = 1}^{N} \tilde{Q}_{22}^{\prime \left( k \right)} A^{\left( k \right)} \quad A_{33 } = \mathop \sum \limits_{k = 1}^{N} \tilde{Q}_{33}^{\prime \left( k \right)} A^{\left( k \right)}$$

(88)

$$D_{11} = \sum\limits_{k = 1}^{N} {\tilde{Q}_{33}^{\prime \left( k \right)} } I_{3}^{\left( k \right)} + \sum\limits_{k = 1}^{N} {\tilde{Q}_{22}^{\prime \left( k \right)} } I_{2}^{\left( k \right)}$$

(89)

$$D_{12} = - \sum\limits_{k = 1}^{N} {\tilde{Q}_{21}^{\prime \left( k \right)} } I_{2}^{\left( k \right)}$$

(90)

$$D_{22} = \sum\limits_{k = 1}^{N} {\tilde{Q}_{11}^{\prime \left( k \right)} } I_{2}^{\left( k \right)} \quad D_{33} = \sum\limits_{k = 1}^{N} {\tilde{Q}_{11}^{\prime \left( k \right)} } I_{3}^{\left( k \right)}$$

(91)

The nonzero elements of the transformed ${\varvec{A}}^{\prime}$ and ${\varvec{D}}^{\prime}$ matrices are calculated by using the following equations.

$$A_{11}^{^{\prime}} = \frac{{A_{22 } }}{{A_{11 } A_{22 } - A_{12}^{2} }} \quad A_{12}^{^{\prime}} = - \frac{{A_{12 } }}{{A_{11 } A_{22 } - A_{12}^{2} }}$$

(92)

$$A_{22}^{^{\prime}} = \frac{{A_{11 } }}{{A_{11 } A_{22 } - A_{12}^{2} }} \quad A_{33}^{^{\prime}} = \frac{1}{{A_{33 } }}$$

(93)

$$D_{11}^{^{\prime}} = \frac{{D_{22 } }}{{D_{11 } D_{22 } - D_{12}^{2} }} \quad D_{12}^{^{\prime}} = - \frac{{D_{12 } }}{{D_{11 } D_{22 } - D_{12}^{2} }}$$

(94)

$$D_{22}^{^{\prime}} = \frac{{D_{11 } }}{{D_{11 } D_{22 } - D_{12}^{2} }} \quad D_{33}^{^{\prime}} = \frac{1}{{D_{33 } }}$$

(95)