Stability and initial post-buckling of a standing sandwich beam under terminal force and self-weight

Abstract

Beams (columns) subjected to axially distributed load (e.g., self-weight) are commonly treated using the classical Euler–Bernoulli beam theory, which ignores the transverse shear effect. Adopting the Engesser and Haringx shear theories, respectively, we study in this article the stability and initial post-buckling of sandwich (or laminated composite) beams under terminal force and axially distributed load. Nonlinear governing equations are derived from geometrical compatibility, equilibrium of forces, and moments. The critical buckling load, modal shapes of deformation, and shear force together with bending moment at buckling can be obtained by using the Galerkin’s method in terms of trigonometric functions, and the initial post-buckled configuration of the beam is determined employing the shooting method. Predictions based on the Engesser theory agree with finite element simulation results, while the Haringx theory overestimates the buckling load. The effects of transverse shear and various different end constraint conditions on static buckling and initial post-buckling of sandwich beams are systematically explored.

Appendices

Appendix 1: Formulation of dimensionless parameter $\bar{{A}}$ for sandwich beams

A general asymmetric sandwich section (face sheets not having same thickness and/or material) is presented in Fig. 11. The sandwich beam consists of two face sheets having density $\rho _1$ and $\rho _2$, thickness $f_{1}$ and $f_{2}$, elastic modulus $E_{f1}$ and $E_{f2}$, and shear modulus $G_{f1 }$and $G_{f2}$, respectively, and a core of density $\rho _c$, thickness c, elastic modulus $E_{c}$, and shear modulus $G_{c}$. The beam width is uniform, W.

Fig. 11

Section of an asymmetric sandwich beam

It is assumed that the shear stresses are distributed uniformly over the entire beam section of area A. An equivalent shear angle can thence be defined based on the “effective” shear modulus of the section, $\bar{{G}}$, which is defined from the compliances of the constituent phases as:

$$\begin{aligned} {(f_1 +c+f_2 )}/{\bar{{G}}}={f_1 }/{G_{f1} }+c/{G_c }+{f_2 }/{G_{f2}} \end{aligned}$$

(43)

where $G_i ={E_i }/{2(1+\nu _i)}$ $(i =f_{1},\, f_{2}$, and c, denoting upper face sheet, bottom face sheet, and core, respectively).

With respect to the reference axis y through the center of the core (Fig. 11), the neutral axis (N. A.) of the section is defined at a distance e, as:

$$\begin{aligned} e=\frac{E_{f2} f_2 (f_2 +c)-E_{f1} f_1 (f_1 +c)}{2(E_{f1} f_1 +E_c c+E_{f2} f_2 )} \end{aligned}$$

(44)

As a result, the equivalent flexural rigidity of an asymmetric sandwich section $(EI)_\mathrm{eq}$ is given by:

$$\begin{aligned} (EI)_\mathrm{eq} =\left( {E_{f1} \frac{f_1^3 }{12}+E_{f1} f_1 \left( {\frac{f_1 }{2}+\frac{c}{2}+e} \right) ^{2}+E_{f2} \frac{f_2^3 }{12}+E_{f2} f_2 \left( {\frac{f_2 }{2}+\frac{c}{2}-e} \right) ^{2}+E_c \frac{c^{3}}{12}+E_c ce^{2}} \right) W\nonumber \\ \end{aligned}$$

(45)

The shear correction coefficient is calculated from strain energy considerations, accounting for the nonuniform distribution of shear stresses throughout the cross section and the contribution of face sheets [24], as:

$$\begin{aligned} \alpha= & {} \bar{{G}}AW\left( \sum _{i=1,2} \left\{ \frac{E_{fi}^2 }{4(EI)_\mathrm{eq}^2 G_{fi} }\left[ a_i^4 f_i -\frac{2}{3}a_i^2 \left( a_i^3 -b_i^3 \right) +\frac{1}{5}\left( a_i^5 -b_i^5 \right) \right] \right. \right. \nonumber \\&\left. \left. +\,\frac{E_{fi}^2}{(EI)_\mathrm{eq}^2 G_c }\left[ f_i^2 c_i^2 b_i +\frac{2}{15}\frac{E_c^2 }{E_{fi}^2 }b_i^5 +\frac{2}{3}\frac{E_c }{E_{fi} }f_i c_i b_i^3 \right] \right\} \right) \end{aligned}$$

(46)

where

$$\begin{aligned} a_i =f_i +c/2+(-1)^{i+1}e,\,b_i =c/2+(-1)^{i+1}e,\,c_i ={f_i }/2+c/2+(-1)^{i+1}e \end{aligned}$$

(47)

Finally, the nondimensional parameter $\bar{{A}}$, in which the small but nonnegligible shear stiffness of the face sheets and the bending stiffness of the core are taken into account, is obtained as:

$$\begin{aligned} \bar{{A}}= & {} \frac{\alpha \left( {EI} \right) _\mathrm{eq} }{\left( {GA} \right) _\mathrm{eq} l^{2}}=\frac{W}{l^{2}}\left( \sum _{i=1,2} \left\{ \frac{E_{fi}^2 }{4(EI)_\mathrm{eq} G_{fi} }\left[ {a_i^4 f_i -\frac{2}{3}a_i^2 \left( a_i^3 -b_i^3 \right) +\frac{1}{5}\left( a_i^5 -b_i^5\right) } \right] \right. \right. \nonumber \\&\left. \left. +\,\frac{E_{fi}^2 }{(EI)_\mathrm{eq} G_c }\left[ {f_i^2 c_i^2 b_i +\frac{2}{15}\frac{E_c^2 }{E_{fi}^2 }b_i^5 +\frac{2}{3}\frac{E_c }{E_{fi} }f_i c_i b_i^3 } \right] \right\} \right) \end{aligned}$$

(48)

It should be mentioned that with respect to the reference axis y, the axis of mass center is defined at a distance g, as:

$$\begin{aligned} g=\frac{\rho _2 f_2 (c+f_2 )-\rho _1 f_1 (c+f_1 )}{2(\rho _1 f_1 +\rho _c c+\rho _2 f_2 )} \end{aligned}$$

(49)

which is analogous to Eq. (43). Generally, for sandwich beams considered in the present study, the axis of gravity center (G. A.) is coincident with that of mass center, if the gravity field is uniformly distributed.

It is worth noting that for an asymmetric sandwich beam subjected to axially distributed force, only when its neutral axis coincides with its gravity center axis (i.e., $e=g$) and the terminal force is exerted on the line of the neutral axis, can the relevant stability and initial post-buckling problems be solved using the present approach. Otherwise, they belong to problems of eccentric loading, which may be treated as imperfection problems.

For symmetric sandwich beams satisfying $E_{f1} =E_{f2},\, f_1 =f_2$ and $\rho _1 =\rho _2$, one has $e=g=0$.

Appendix 2: Governing equations of buckling in terms of lateral displacement

As a supplementary, buckling governing equations in terms of lateral displacement for a Timoshenko beam are derived. Let the dimensionless lateral displacement be defined as $\bar{{y}}=y/l$. According to Eq. (1), one has:

$$\begin{aligned} \frac{\mathrm{d}\bar{{y}}}{\mathrm{d}\xi }=\frac{\mathrm{d}y}{\mathrm{d}s}=\sin \beta \end{aligned}$$

(50)

Rewriting Eq. (6) and substituting Eqs. (8)–(9) into Eq. (50), we obtain the following coupled system of ordinary differential equations as the governing equations of a Timoshenko beam:

$$\begin{aligned}&\frac{\mathrm{d}^{2}\theta }{\mathrm{d}\xi ^{2}}+\left[ {\bar{{P}}+\bar{{q}}(1-\xi )} \right] \frac{\mathrm{d}\bar{{y}}}{\mathrm{d}\xi }-\bar{{Q}}\cos \beta =0 \end{aligned}$$

(51a)

$$\begin{aligned}&\frac{\mathrm{d}\bar{{y}}}{\mathrm{d}\xi }=\sin \theta +\gamma _\mathrm{eq} \cos \theta \end{aligned}$$

(51b)

For buckling analysis, based upon the assumption of small deformation, we have $\sin \theta \cong \theta ,\, \cos \theta \cong 1,\, \sin \beta \cong \beta ,\, \cos \beta \cong 1$, and thus ${\mathrm{d}\bar{{y}}}/{\mathrm{d}\xi }=\beta$. Then, the coupled Eqs. (51a) and (51b) may be rewritten as:

$$\begin{aligned}&\frac{\mathrm{d}^{2}\theta }{\mathrm{d}\xi ^{2}}+\left[ {\bar{{P}}+\bar{{q}}(1-\xi )} \right] \frac{\mathrm{d}\bar{{y}}}{\mathrm{d}\xi }-\bar{{Q}}=0 \end{aligned}$$

(52a)

$$\begin{aligned}&\frac{\mathrm{d}\bar{{y}}}{\mathrm{d}\xi }=\theta +\gamma _\mathrm{eq} \end{aligned}$$

(52b)

which are the governing equations for the buckling of Timoshenko beams.

For the buckling of Engesser-type shear beams, making use of Eq. (11), we can rewrite the above coupled differential governing equations as:

$$\begin{aligned}&\frac{\mathrm{d}^{2}\theta }{\mathrm{d}\xi ^{2}}+\left[ {\bar{{P}}+\bar{{q}}(1-\xi )} \right] \frac{\mathrm{d}\bar{{y}}}{\mathrm{d}\xi }-\bar{{Q}}=0 \end{aligned}$$

(53a)

$$\begin{aligned}&\frac{\mathrm{d}\bar{{y}}}{\mathrm{d}\xi }=\theta +\bar{{A}}\left( {\left[ {\bar{{P}}+\bar{{q}}(1-\xi )} \right] \frac{\mathrm{d}\bar{{y}}}{\mathrm{d}\xi }-\bar{{Q}}} \right) \end{aligned}$$

(53b)

Similarly, for the buckling of Haringx-type shear theory, by rewriting Eq. (17), the coupled differential governing equations are obtained as:

$$\begin{aligned}&\frac{\mathrm{d}^{2}\theta }{\mathrm{d}\xi ^{2}}+\left[ {\bar{{P}}+\bar{{q}}(1-\xi )} \right] \frac{\mathrm{d}\bar{{y}}}{\mathrm{d}\xi }-\bar{{Q}}=0 \end{aligned}$$

(54a)

$$\begin{aligned}&\frac{\mathrm{d}\bar{{y}}}{\mathrm{d}\xi }=\theta +\bar{{A}}\left( {\left[ {\bar{{P}}+\bar{{q}}(1-\xi )} \right] \theta -\bar{{Q}}} \right) \end{aligned}$$

(54b)

Finally, through a series of differential operations and rearranging of Eqs. (52b) and Eqs. (B5), we obtain the uncoupled governing equations for buckling of Engesser-type shear beams in terms of lateral displacement $\bar{{y}}$, as:

$$\begin{aligned}&\displaystyle \left( {1-\bar{{A}}\left[ {\bar{{P}}+\bar{{q}}(1-\xi )} \right] } \right) \frac{\mathrm{d}^{3}\bar{{y}}}{\mathrm{d}\xi ^{3}}+2\bar{{A}}\bar{{q}}\frac{\mathrm{d}^{2}\bar{{y}}}{\mathrm{d}\xi ^{2}}+\left[ {\bar{{P}}+\bar{{q}}(1-\xi )} \right] \frac{\mathrm{d}\bar{{y}}}{\mathrm{d}\xi }-\bar{{Q}}=0 \end{aligned}$$

(55a)

$$\begin{aligned}&\displaystyle \theta =\left( {1-\bar{{A}}\left[ {\bar{{P}}+\bar{{q}}(1-\xi )} \right] } \right) \frac{\mathrm{d}\bar{{y}}}{\mathrm{d}\xi }+\bar{{A}}\bar{{Q}} \end{aligned}$$

(55b)

and the uncoupled governing equations for buckling of Haringx-type shear beams in terms of $\bar{{y}}$, as:

$$\begin{aligned}&\left( {1+\bar{{A}}\left[ {\bar{{P}}+\bar{{q}}(1-\xi )} \right] } \right) ^{2}\frac{\mathrm{d}^{3}\bar{{y}}}{\mathrm{d}\xi ^{3}}+2\bar{{A}}\bar{{q}}\left( {1+\bar{{A}}\left[ {\bar{{P}}+\bar{{q}}(1-\xi )} \right] } \right) \frac{\mathrm{d}^{2}\bar{{y}}}{\mathrm{d}\xi ^{2}}\nonumber \\&\quad \quad +\left\{ 2\bar{{A}}^{2}\bar{{q}}^{2}+\left[ {\bar{{P}}+\bar{{q}}(1-\xi )} \right] \left( {1+\bar{{A}}\left[ {\bar{{P}}+\bar{{q}}(1-\xi )} \right] } \right) ^{3}\right\} \frac{\mathrm{d}\bar{{y}}}{\mathrm{d}\xi } \nonumber \\&\quad \quad +\left\{ {2\bar{{A}}^{3}\bar{{q}}^{2}-\left( {1+\bar{{A}}\left[ {\bar{{P}}+\bar{{q}}(1-\xi )} \right] } \right) ^{3}} \right\} \bar{{Q}}=0 \end{aligned}$$

(56a)

$$\begin{aligned}&\theta =\frac{\frac{\mathrm{d}\bar{{y}}}{\mathrm{d}\xi }+\bar{{A}}\bar{{Q}}}{1+\bar{{A}}\left[ {\bar{{P}}+\bar{{q}}(1-\xi )} \right] } \end{aligned}$$

(56b)

Equations (55a) and (56a) are the “condensed” buckling governing equations in terms of $\bar{{y}}$ for the Engesser and Haringx shear beams, respectively.

Compared with the governing equation in terms of rotation angle $\theta$, i.e., Eq. (14) or Eq. (20), the governing equation in terms of lateral displacement $\bar{{y}}$, i.e., Eq. (55a) or Eq. (56a), appears to be more complicated. With $\theta$ chosen as the principal unknown function and trigonometric functions employed as the base function which automatically satisfy the boundary conditions, it would be more convenient to obtain convergent solutions. Further, since the post-buckling studied in this paper is a highly nonlinear geometric problem, it may be more appropriate to employ $\theta$ as the unknown function.