**2.2. Geometrical equations for thin plate**

A plate model has been assumed for a thin plates and thin-walled beam-columns or girders. For easier explanation the plate (Figure 1a) or each *i*-th strip (Figure 1b) of the plate (or wall of the girder) or each *i*-th wall of the girder (Figure 1c) are called plate.

To describe the middle surface strains for each plate the following strain tensor have been assumed:

$$\begin{aligned} \boldsymbol{\varepsilon}\_{i\boldsymbol{x}}^{\rm m} &= \boldsymbol{u}\_{i,\boldsymbol{x}} + \frac{1}{2} (\boldsymbol{\varepsilon} \boldsymbol{\upsilon}\_{i,\boldsymbol{x}}^{2} + \boldsymbol{u}\_{i,\boldsymbol{x}}^{2} + \boldsymbol{\upsilon}\_{i,\boldsymbol{x}}^{2}) \,, \\ \boldsymbol{\varepsilon}\_{i\boldsymbol{y}}^{\rm m} &= \boldsymbol{\upsilon}\_{i,\boldsymbol{y}} + \frac{1}{2} (\boldsymbol{\varepsilon} \boldsymbol{\upsilon}\_{i,\boldsymbol{y}}^{2} + \boldsymbol{u}\_{i,\boldsymbol{y}}^{2} + \boldsymbol{\upsilon}\_{i,\boldsymbol{y}}^{2}) \,, \\ \boldsymbol{\nu}\_{i\boldsymbol{x}\boldsymbol{y}}^{\rm m} &= \boldsymbol{u}\_{i,\boldsymbol{y}} + \boldsymbol{\upsilon}\_{i,\boldsymbol{x}} + \boldsymbol{\upsilon}\_{i,\boldsymbol{x}} \boldsymbol{\varpi}\_{i,\boldsymbol{y}} + \boldsymbol{u}\_{i,\boldsymbol{x}} \boldsymbol{u}\_{i,\boldsymbol{y}} + \boldsymbol{\upsilon}\_{i,\boldsymbol{x}} \boldsymbol{\upsilon}\_{i,\boldsymbol{y}} \end{aligned} \tag{1}$$

where: *u*i, *v*i, *w*i - displacements parallel to the respective axes *x*i, *y*i, *z*i of the local Cartesian system of co-ordinates, whose plane *x*i*y*i coincides with the middle surface of the *i*-th plate before its buckling (Figure 1).

In the majority of publications devoted to structure stability, the terms 2 2 , , ( ) *ix ix u v* , 2 2 , , ( ) *iy iy u v* and , , ,, ( ) *ix iy ix iy uu vv* are in general neglected for , , *mm m ix iy ixy* correspondingly, in (1) in the strain tensor components.

The change of the bending and twisting curvatures of the middle surface are assumed according to [48, 49] as follows:

$$\begin{aligned} \kappa\_{i\chi} &= -\varpi v\_{i,\chi\chi'} \\ \kappa\_{ij\chi} &= -\varpi v\_{i,yy'} \\ \kappa\_{ixy} &= -\varpi v\_{i,xy} \end{aligned} \tag{2}$$

Nonlinear Plate Theory for Postbuckling Behaviour of Thin-Walled Structures Under Static and Dynamic Load 225

*z*

2

 

1 2

*i i i i*

, 1 1

*i i*

 

12 21 12 21

 

 

  (4)

. (6)

(7)

(5)

1 1 1 22 2 12 12 12

*m i i i m ii i <sup>m</sup> i i <sup>i</sup>*

1

*i i*

, <sup>1</sup>

 

*i*

2

*i i*

, <sup>1</sup>

*i*

.

direction *2*, *Gi12* is a shear modulus (Kirchhoff modulus) in *12* plane.

 

12 21

12 21

12 21 21 12

*E E Q Q*

and *Ei1*, *Ei2* are Young modulus in longitudinal *1* and transverse *2* direction respectively, *νi12* is a Poisson ratio for which strains are in longitudinal direction *1* and stress in transverse

Young modulus and Poisson ratio occurring in (5) according to Betty-Maxwell theorem or

*ii ii* 1 21 2 12 *E E* 

1 0 . <sup>1</sup> <sup>1</sup> <sup>2</sup> 0 0

*m*

*z*

 

> 

2

Substituting stress-strain relation from previous subchapter, the sectional moments and

according to symmetry condition of stress tensor should fulfil following relation:

For isotropic plate (wall of beam-columns) the constitutive equations are as follows:

1 0

*ix ix i ix i m iy i iy iy <sup>i</sup> <sup>m</sup> <sup>i</sup> ixy ixy ixy*

2

for *i*-th isotropic plate or wall of beam-column are expressed by:

 

*E*

*ii i*

11

*<sup>E</sup> <sup>Q</sup>*

*i*

22

*Q G*

*i*

*i*

**Figure 2.** Plates or walls with principal axes of orthotropy

**2.4. Generalized sectional forces** 

forces:

66 12

*<sup>E</sup> <sup>Q</sup>*

The geometrical relationship given by equations (1) and (2) allow to consider both out-ofplane and in-plane bending of the plate.

**Figure 1.** Possible models: plates, strips or wall with assumed dimension, coordinate systems and direction of deflections
