One-dimensional analysis of thin-walled beams with diaphragms and its application to optimization for stiffness reinforcement

Abstract

This paper presents a method to analyze thin-walled beams with quadrilateral cross sections reinforced with diaphragms using a one-dimensional higher-order beam theory. The effect of a diaphragm is reflected focusing on the increase of static stiffness. The deformations on the beam-interfacing boundary of a thin diaphragm are described by using deformation modes of the beam cross section while the deformations inside the diaphragm are approximated in the form of complete cubic polynomials. By using the principle of minimum potential energy, its stiffness that significantly affects distortional deformation of a thin-walled beam can be considered in the one-dimensional beam analysis. It is shown that the accuracy of the resulting one-dimensional analysis is comparable with that by a shell element based analysis. As a means to demonstrate the usefulness of the present approach for design, position optimization problems of diaphragms for stiffness reinforcement of an automotive side frame are solved.

Appendices

Appendix

Shape functions for rigid body cross-section deformations of a general quadrilateral cross section

$$\begin{aligned} \psi _n^{U_x } ( {s_j })= & {} \hbox {sin}({\alpha _j } ),\psi _s^{U_x } ( {s_j })=\cos ({\alpha _j }), \end{aligned}$$

(A1a)

$$\begin{aligned} \psi _n^{U_y } ( {s_j })= & {} -\cos \left( {\alpha _j } \right) ,\psi _s^{U_y } ( {s_j })=\hbox {sin}\left( {\alpha _j } \right) , \end{aligned}$$

(A1b)

$$\begin{aligned} \psi _z^{U_z } ( {s_j })= & {} 1, \end{aligned}$$

(A1c)

$$\begin{aligned} \psi _z^{\theta _x } ( {s_j })= & {} \frac{y_{j+1} -y_j }{l_j }s_j +y_j -y_c ,\quad \left( {y_5 =y_1 } \right) \end{aligned}$$

(A1d)

$$\begin{aligned} \psi _z^{\theta _y } ( {s_j })= & {} -\frac{x_{j+1} -x_j }{l_j }s_j +x_j -x_c ,\quad \left( {x_5 =x_1 } \right) \end{aligned}$$

(A1e)

$$\begin{aligned} \psi _n^{\theta _z } ( {s_j })= & {} s_j -l_j ,\psi _s^{\theta _z } ( {s_j })=r_j , \end{aligned}$$

(A1f)

where $\alpha _j$ angle between edge j and the x-axis, ($x_j ,y_j )$ coordinates of corner j, ($x_c ,y_c )$ coordinates of the geometric center, $r_j$ distance from the shear center to edge j, $l_j$ distance from corner j to the point $N_j$ ($N_j$: intersection between edge j and the normal line from the shear center to edge j).

In Eq. (A1), $U_q$ and $\theta _q ({q=x,y,z} )$ denote rigid-body translations and rotations of the cross section in the q-direction, respectively. Note that shape functions for bending deflection and bending/shear rotations are defined in terms of (x, y)-coordinates in Fig. 2, not necessarily in terms of principal coordinates.

Shape functions for torsional and Poisson distortions and corresponding warpings of a rectangular cross section

$$\begin{aligned} \psi _s^{\chi _1 } ({s_i })= & {} -\frac{bh}{b+h},\quad ( {i=1,3}); \nonumber \\ \psi _s^{\chi _1 } ( {s_j })= & {} \frac{bh}{b+h},\quad \left( {j=2,4} \right) , \end{aligned}$$

(A2a)

$$\begin{aligned} \psi _n^{\chi _1}(s_i)= & {} \frac{bh}{b+h}+\frac{2(b-h)}{b+h}s_i-\frac{6}{b+h}s_i^2\nonumber \\&+\frac{4}{b(b+h)}s_i^3, \quad (i=1,3),\nonumber \\ \psi _n^{\chi _1}(s_j)= & {} \frac{bh}{b+h}+\frac{2(b-h)}{b+h}s_j+\frac{6}{b+h}s_j^2\nonumber \\&-\frac{4}{h(b+h)}s_j^3, \quad (j=2,4), \end{aligned}$$

(A2b)

$$\begin{aligned} \psi _s^{\chi _2 } ({s_i })= & {} -\frac{b}{2}+s_i ,\quad ( {i=1,3}); \nonumber \\ \psi _s^{\chi _2 } ( {s_j })= & {} -\frac{h}{2}+s_j ,\quad \left( {j=2,4} \right) ,\end{aligned}$$

(A2c)

$$\begin{aligned} \psi _n^{\chi _2 } ({s_i })= & {} -\frac{h}{2},\quad \left( {i=1,3} \right) ; \nonumber \\ \psi _n^{\chi _2 } ( {s_j })= & {} -\frac{b}{2},\quad \left( {j=2,4} \right) , \end{aligned}$$

(A2d)

$$\begin{aligned} \psi _{s}^{{\chi _{3} }} \left( {s_{i} } \right)= & {} \left( { \!-\! 1} \right) ^{{\frac{{i \!-\! 1}}{2}}} \left( { \!-\! \frac{{b^{2} }}{{12}} \!-\! \frac{h}{2}s_{i} } \right) ,\quad \left( {i \!=\! 1,3} \right) ,\nonumber \\ \psi _{s}^{{\chi _{3} }} \left( {s_{j} } \right)= & {} \left( { - 1} \right) ^{{\frac{{j - 2}}{2}}} \left( {\frac{{bh}}{4} - \frac{h}{2}s_{j} + \frac{1}{2}s_{j}^{2} } \right) ,\quad \left( {j = 2,4} \right) , \nonumber \\ \end{aligned}$$

(A2e)

$$\begin{aligned} \psi _{n}^{{\chi _{3} }} \left( {s_{i} } \right)= & {} \left( { - 1} \right) ^{{\frac{{i - 1}}{2}}} \left( - \frac{{bh}}{4} + \frac{{3h^{2} }}{{2\left( {b + 3h} \right) }}s_{i}\right. \nonumber \\&\left. + \frac{{3h}}{{2\left( {b + 3h} \right) }}s_{i}^{2} - \frac{h}{{b\left( {b + 3h} \right) }}s_{i}^{3} \right) ,\quad \left( {i = 1,3} \right) ,\nonumber \\ \psi _{n}^{{\chi _{3} }} \left( {s_{j} } \right)= & {} \left( { - 1} \right) ^{{\frac{{j - 2}}{2}}} \left( \frac{{b^{2} }}{{12}} + \frac{{3h^{2} }}{{2\left( {b + 3h} \right) }}s_{j} \right. \nonumber \\&\left. - \frac{{3h}}{{2\left( {b + 3h} \right) }}s_{j}^{2} \right) ,\quad \left( {j = 2,4} \right) , \end{aligned}$$

(A2f)

$$\begin{aligned} \psi _{s}^{{\chi _{4} }} \left( {s_{i} } \right)= & {} \left( { - 1} \right) ^{{\frac{{i - 1}}{2}}} \left( {\frac{{bh}}{4} - \frac{h}{2}s_{i} } \right) ,\quad \left( {i = 1,3} \right) ,\nonumber \\ \psi _{s}^{{\chi _{4} }} \left( {s_{j} } \right)= & {} \left( { - 1} \right) ^{{\frac{{j - 2}}{2}}} \nonumber \\&\times \left( {\frac{{h^{2} }}{{12}} - \frac{h}{2}s_{j} + \frac{1}{2}s_{j}^{2} } \right) ,\quad \left( {j = 2,4} \right) , \end{aligned}$$

(A2g)

$$\begin{aligned} \psi _{n}^{{\chi _{4} }} \left( {s_{i} } \right)= & {} \left( { - 1} \right) ^{{\frac{{i - 1}}{2}}} \left( \frac{{h^{2} }}{{12}} + \frac{{3b^{2} }}{{2\left( {3b + h} \right) }}s_{i} \right. \nonumber \\&\left. - \frac{{3b}}{{2\left( {3b + h} \right) }}s_{i}^{2} \right) ,\quad \left( {i = 1,3} \right) , \nonumber \\ \psi _{n}^{{\chi _{4} }} \left( {s_{j} } \right)= & {} \left( { - 1} \right) ^{{\frac{{j - 2}}{2}}} \nonumber \\&\times \left( \frac{{bh}}{4} - \frac{{3b^{2} }}{{2\left( {3b + h} \right) }}s_{j} - \frac{{3b}}{{2\left( {3b + h} \right) }}s_{j}^{2} \right. \nonumber \\&\left. + \frac{b}{{h\left( {3b + h} \right) }}s_{j}^{3} \right) ,\quad \left( {j = 2,4} \right) ,\end{aligned}$$

(A2h)

$$\begin{aligned} \psi _z^{W_1 } ({s_i })= & {} \frac{bh}{4}-\frac{h}{2}s_i ,\quad ({i=1,3});\nonumber \\ \psi _z^{W_1 } ( {s_j })= & {} -\frac{bh}{4}+\frac{b}{2}s_j ,\quad ({j=2,4}), \end{aligned}$$

(A2i)

$$\begin{aligned} \psi _z^{W_2 } ({s_i})= & {} \frac{1}{12}( {b^{2}-bh+h^{2}})\nonumber \\&-\frac{b}{2}s_i +\frac{1}{2}s_i^2 ,\quad \left( {i=1,3} \right) , \nonumber \\ \psi _z^{W_2 } ( {s_j })= & {} \frac{1}{12}( {b^{2}-bh+h^{2}})-\frac{h}{2}s_j \nonumber \\&+\frac{1}{2}s_j^2 ,\quad \left( {j=2,4} \right) , \end{aligned}$$

(A2j)

$$\begin{aligned} \psi _z^{W_3 } ({s_i })= & {} \frac{b^{4}+15bh^{3}}{120\left( {b+3h} \right) }-\frac{2b^{3}+5b^{2}h+5h^{3}}{20({b+3h})}s_i\nonumber \\&+\frac{b}{4}s_i^2 -\frac{1}{6}s_i^3 ,\quad \left( {i=1,3} \right) , \nonumber \\ \psi _z^{W_3 } ( {s_j })= & {} -\frac{b^{4}+15bh^{3}}{120( {b+3h} )}+\frac{bh}{4}s_j \nonumber \\&-\frac{b}{4}s_j^2 ,\quad \left( {j=2,4} \right) , \end{aligned}$$

(A2k)

$$\begin{aligned} \psi _z^{W_4 } ({s_i } )= & {} -\frac{h^{4}+15b^{3}h}{120\left( {3b+h} \right) }+\frac{bh}{4}s_i -\frac{h}{4}s_i^2 ,\quad \left( {i=1,3} \right) , \nonumber \\ \psi _z^{W_4 } ( {s_j })= & {} -\frac{h^{4}+15b^{3}h}{120({3b+h} )}+\frac{5b^{3}+5bh^{2}+2h^{3}}{20({3b+h})}s_j\nonumber \\&-\frac{h}{4}s_j^2 +\frac{1}{6}s_j^3 ,\quad ({j=2,4}), \end{aligned}$$

(A2l)

where b and h denote the width and height of a cross section (see Fig. 1).

Element stiffness matrix of a thin-walled beam with a rectangular cross section

The stiffness matrix of element e is given as

where l is the length of the element and $s_i$ are cross-section coefficients defined as

$$\begin{aligned} s_{1}= & {} \frac{h^{2}t(h+3b)}{6},\nonumber \\ s_{2}= & {} \frac{{b^{2}h}^{2}t(b+h)}{24},\quad s_{3}=\frac{8t^{3}}{b+h},\quad s_{4}=2ht, \nonumber \\ s_{5}= & {} 2ht,\quad s_{6}=\frac{hbt(h+b)}{2},\quad s_{7}=\frac{hbt(h+b)}{2},\nonumber \\ s_{8}= & {} \frac{{{2b}^{2}h}^{2}t}{b+h}+\frac{{{2t}^{3}(b^{2}+4bh+h}^{2})}{15(b+h)}, \nonumber \\ s_{9}= & {} 2ht,\quad s_{10}=\frac{hbt(b-h)}{2},\nonumber \\ s_{11}= & {} -\frac{{{2b}^{2}h}^{2}t}{b+h},\quad s_{12}=0. \end{aligned}$$

(A4)

The associated degrees of freedom for the stiffness matrix in Eq. (A3) are

$$\begin{aligned} {\varvec{d}}=\{{U_y ,\theta _x ,\theta _z ,W_1 ,\chi _1 }\}^{T}. \end{aligned}$$