**2.3. Constitutive equations for orthotropy**

Let's consider orthotropic plate with principal axes of ortothropy *1* and *2* parallel to plate edges (Figure 2).

As same as in previous paragraph let's consider *i*-th plate or strip of structures under analysis. The stress – strain relationship for orthotropic plate can be written in following form:

$$
\begin{Bmatrix}
\sigma\_{i1} \\
\sigma\_{i2} \\
\sigma\_{i12}
\end{Bmatrix} = \begin{bmatrix}
Q\_{i11} & Q\_{i12} & 0 \\
Q\_{i21} & Q\_{i22} & 0 \\
0 & 0 & Q\_{66}
\end{bmatrix} \begin{Bmatrix}
\varepsilon\_{i1} \\
\varepsilon\_{i2} \\
\mathcal{V}\_{i12}
\end{Bmatrix} \prime \tag{3}
$$

where:

$$\begin{Bmatrix} \mathcal{E}\_{i1} \\ \mathcal{E}\_{i2} \\ \mathcal{V}\_{i12} \end{Bmatrix} = \begin{Bmatrix} \mathcal{E}\_{i1}^{m} \\ \mathcal{E}\_{i2}^{m} \\ \mathcal{V}\_{i12}^{m} \end{Bmatrix} + \mathcal{Z} \begin{Bmatrix} \mathcal{K}\_{i1} \\ \mathcal{K}\_{i2} \\ \mathcal{Q}\kappa\_{i12} \end{Bmatrix} \tag{4}$$

$$\begin{aligned} Q\_{i11} &= \frac{E\_{i1}}{1 - \nu\_{i12}\nu\_{i21}},\\ Q\_{i12} &= Q\_{21} = \nu\_{i21} \frac{E\_{i1}}{1 - \nu\_{i12}\nu\_{i21}} = \nu\_{i12} \frac{E\_{i2}}{1 - \nu\_{i12}\nu\_{i21}},\\ Q\_{i22} &= \frac{E\_{i2}}{1 - \nu\_{i12}\nu\_{i21}},\\ Q\_{i66} &= G\_{12}.\end{aligned} \tag{5}$$

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

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

224 Nonlinearity, Bifurcation and Chaos – Theory and Applications

plane and in-plane bending of the plate.

direction of deflections

edges (Figure 2).

where:

**2.3. Constitutive equations for orthotropy** 

, , ,

*ix i xx iy i yy ixy i xy*

The geometrical relationship given by equations (1) and (2) allow to consider both out-of-

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

Let's consider orthotropic plate with principal axes of ortothropy *1* and *2* parallel to plate

As same as in previous paragraph let's consider *i*-th plate or strip of structures under analysis.

1 11 12 1 2 21 22 2 12 66 12

*i ii i i ii i i i*

0 0

0

*Q*

0 ,

  (3)

 

 

The stress – strain relationship for orthotropic plate can be written in following form:

*Q Q Q Q*

 

*w w w* , , .

(2)

Young modulus and Poisson ratio occurring in (5) according to Betty-Maxwell theorem or according to symmetry condition of stress tensor should fulfil following relation:

$$E\_{i1}\nu\_{i21} = E\_{i2}\nu\_{i12} \,. \tag{6}$$

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

$$
\begin{Bmatrix}
\sigma\_{ix} \\ \sigma\_{iy} \\ \sigma\_{ixy} \\ \tau\_{ixy}
\end{Bmatrix} = \frac{E\_i}{1 - \nu\_i^2} \begin{vmatrix}
\mathbf{1} & \nu\_i & \mathbf{0} \\
\nu\_i & \mathbf{1} & \mathbf{0} \\
\mathbf{0} & \mathbf{0} & \frac{1 - \nu\_i}{\mathbf{2}}
\end{vmatrix} \left| \begin{Bmatrix} \mathcal{E}\_{ix}^m \\ \mathcal{E}\_{iy}^m \\ \mathcal{E}\_{iy}^m \end{vmatrix} + \mathbf{z} \begin{Bmatrix} \kappa\_{ix} \\ \kappa\_{iy} \\ \kappa\_{iy} \end{Bmatrix} \right| . \tag{7}
$$

### **2.4. Generalized sectional forces**

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

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

$$\begin{aligned} \begin{Bmatrix} N\_{ix} \\ N\_{iy} \\ N\_{ixy} \end{Bmatrix} = \frac{E\_i h\_i}{1 - \nu\_i^2} \begin{bmatrix} 1 & \nu\_i & 0 \\ \nu\_i & 1 & 0 \\ 0 & 0 & \frac{1 - \nu\_i}{2} \end{bmatrix} \begin{Bmatrix} \varepsilon\_{ix}^m \\ \varepsilon\_{iy}^m \\ \gamma\_{ixy}^m \end{Bmatrix}, \\\begin{Bmatrix} M\_{ix} \\ M\_{iy} \\ M\_{iy} \end{Bmatrix} = D\_i \begin{bmatrix} 1 & \nu\_i & 0 \\ \nu\_i & 1 & 0 \\ 0 & 0 & 1 - \nu\_i \end{bmatrix} \begin{Bmatrix} \kappa\_{ix} \\ \kappa\_{iy} \\ \kappa\_{iy} \end{Bmatrix}, \end{aligned} \tag{8}$$

( )0

*dt K dt* . (11)

, (13)

 

  (15)

*Q W* (12)

where is the Lagrangian function for the system, *K* is a kinetic energy of the system and

The subscript *i* denoting *i*-th plate or strip in all equations in this subchapter is omitted – all equations are presented for one plate, which could be *i*-th plate, wall or strip of considered

Taking the action functional in form (9) the Hamilton's principle can be written as:

1 1

 

The total potential energy variation δ for *i*-th thin plate (or strip) can be written in form:

( ) *Q d x x y y xy xy*

and is the volume of the plate and *S* is its area, the volume can be expressed as *l*·*b*·*h* or

The variation of internal elastic strain energy for *i*-th plate or strip could be expressed by

,, , ( ) ( 2 ).

*N N N dS M w M w M w dS*

(14)

*x x y y xy xy x xx y yy xy xy*

The work *W* of external forces (neglecting the out-of plane load) done on *i*-th plate can be

00 00

where: *p*0(*x*), *p*0(*y*), 0xy(*x*), 0xy(*y*) are the prebuckling load applied to the middle surface of the

For thin plates, it is assumed that the displacements *u* and *v* do not depend on rotation *w,x* and *w,y* and therefore do not depend on the coordinate *z*. This approach results in exclusion of rotational inertia [50] in the equation for kinetic energy, which for the *i*-th thin plate

*W h p y u y v dy h p x v x u dx*

[ ( ) ( )] [ ( ) ()] ,

*xy xy o*

*t t*

where δQ is a variation of internal elastic strain energy:

strain and sectional forces and moments in a following way:

*mm m*

 

*S S*

 

0

*b*

*m b*

 

*QQ Q*

considered plate (wall or strip)

(strip) can be written as:

written as follows:

0 0

*t t*

 

 

is a total potential energy of the system.

plate, beam-columns or girder (Figure 1).

*S*·*h*.

where: 3 <sup>2</sup> 12 1 *i i i i E h <sup>D</sup>* 

for *i*-th orthotropic strip or wall are:

$$
\begin{aligned}
\begin{Bmatrix} N\_{ix} \\ N\_{iy} \\ N\_{ixy} \end{Bmatrix} &= \frac{h\_i}{1 - \nu\_{ixy}\nu\_{iyx}} \begin{bmatrix} E\_{ix} & \nu\_{iyx}E\_{ix} & 0 \\ \nu\_{ixy}E\_{iy} & E\_{iy} & 0 \\ 0 & 0 & \left(1 - \nu\_{ixy}\nu\_{iyx}\right)G\_{ixy} \end{bmatrix} \begin{Bmatrix} \boldsymbol{\varepsilon}\_{ix}^m \\ \boldsymbol{\varepsilon}\_{iy}^m \\ \boldsymbol{\varepsilon}\_{iy}^m \end{Bmatrix}, \\\ \begin{Bmatrix} M\_{ix} \\ M\_{iy} \\ M\_{ixy} \end{Bmatrix} &= \begin{bmatrix} D\_{ix} & \nu\_{ixy}D\_{ix} & 0 \\ \nu\_{ixy}D\_{iy} & D\_{iy} & 0 \\ 0 & 0 & D\_{ixy} \end{bmatrix} \begin{Bmatrix} \boldsymbol{\kappa}\_{ix} \\ \kappa\_{iy} \\ \kappa\_{ixy} \end{Bmatrix}, \end{aligned} \tag{9}
$$

where: 3 3 3 , , . 12 1 12 1 <sup>6</sup> *iy i ixy i ix i ix iy ixy ixy iyx ixy iyx E h E h G h DDD* 

### **2.5. Dynamic equations of stability for thin plate**

Differential equations of motion of the plate were derived basing on Hamilton's principle. It states that the dynamics of a physical system is determined by a variation problem for a functional based on a single function, the Lagrangian, which contains all physical information concerning the system and the forces acting on it. In dynamic buckling problem the motion should be understand as the time dependent deflection.

The Hamilton's principles for conservative systems states that the true evolution (compatible with constrains) of the system between two specific states in specific time range (*t0*, *t1*) is a stationary point (a point where the variation is zero) of the action functional . Action functional for *i*-th plate is described by following equation:

$$
\Psi \Psi = \int\_{t\_0}^{t\_1} \Lambda dt = \int\_{t\_0}^{t\_1} (K - \Gamma \Gamma) dt \tag{10}
$$

where is the Lagrangian function for the system, *K* is a kinetic energy of the system and is a total potential energy of the system.

The subscript *i* denoting *i*-th plate or strip in all equations in this subchapter is omitted – all equations are presented for one plate, which could be *i*-th plate, wall or strip of considered plate, beam-columns or girder (Figure 1).

Taking the action functional in form (9) the Hamilton's principle can be written as:

$$
\delta \Phi^{\mathbf{p}} = \delta \int\_{t\_0}^{t\_1} \Lambda dt = \delta \int\_{t\_0}^{t\_1} (K - \Pi) dt = 0 \tag{11}
$$

The total potential energy variation δ for *i*-th thin plate (or strip) can be written in form:

$$
\delta \mathbf{J} \mathbf{I} = \delta \mathbf{Q} - \delta \mathbf{W} \tag{12}
$$

where δQ is a variation of internal elastic strain energy:

226 Nonlinearity, Bifurcation and Chaos – Theory and Applications

*i*

for *i*-th orthotropic strip or wall are:

 

<sup>2</sup> 12 1 *i i*

3

where:

where:

*i*

*E h <sup>D</sup>*

*N*

*N*

*M*

*M*

*E h <sup>N</sup>*

*M D*

0 0

*M D*

the motion should be understand as the time dependent deflection.

Action functional for *i*-th plate is described by following equation:

*ix iy ixy ixy iyx ixy iyx*

*DDD*

**2.5. Dynamic equations of stability for thin plate** 

*M D D M DD*

 

*N E E <sup>h</sup> <sup>N</sup> E E*

*ix iyx ix ix ix iy ixy iy iy iy ixy ixy ixy*

2

<sup>1</sup> <sup>1</sup> 0 0

1 0

*ix i ix iy i i iy i ixy ixy*

0 01

1 0

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

10 ,

 

 

 

, 

0

*m*

 

(9)

*m*

 

(8)

2

10 ,

0 , <sup>1</sup>

 

*i m*

0 01

*N G*

3 3 3

 

Differential equations of motion of the plate were derived basing on Hamilton's principle. It states that the dynamics of a physical system is determined by a variation problem for a functional based on a single function, the Lagrangian, which contains all physical information concerning the system and the forces acting on it. In dynamic buckling problem

The Hamilton's principles for conservative systems states that the true evolution (compatible with constrains) of the system between two specific states in specific time range (*t0*, *t1*) is a stationary point (a point where the variation is zero) of the action functional .

1 1

*t t*

( )

*dt K dt* (10)

0 0

*t t*

, , . 12 1 12 1 <sup>6</sup> *iy i ixy i ix i*

*E h E h G h*

*ix iyx ix ix ix*

*iy ixy iy iy iy ixy iyx <sup>m</sup> ixy ixy iyx ixy ixy*

0 0

$$\delta \mathcal{Q} = \int\_{\Omega} (\sigma\_x \delta \varepsilon\_x + \sigma\_y \delta \varepsilon\_y + \tau\_{xy} \delta \gamma\_{xy}) \, d\Omega \tag{13}$$

and is the volume of the plate and *S* is its area, the volume can be expressed as *l*·*b*·*h* or *S*·*h*.

The variation of internal elastic strain energy for *i*-th plate or strip could be expressed by strain and sectional forces and moments in a following way:

$$\begin{split} \delta \boldsymbol{\delta Q} &= \delta \boldsymbol{Q}^{m} + \delta \boldsymbol{Q}^{b} = \\ &= \int\_{\mathcal{S}} (\boldsymbol{N}\_{x} \delta \boldsymbol{\varepsilon}\_{x}^{m} + \boldsymbol{N}\_{y} \delta \boldsymbol{\varepsilon}\_{y}^{m} + \boldsymbol{N}\_{xy} \delta \boldsymbol{\gamma}\_{xy}^{m}) d\boldsymbol{S} - \int\_{\mathcal{S}} (\boldsymbol{M}\_{x} \delta \boldsymbol{w}\_{,xx} + \boldsymbol{M}\_{y} \delta \boldsymbol{w}\_{,yy} + 2 \boldsymbol{M}\_{xy} \delta \boldsymbol{w}\_{,xy}) d\boldsymbol{S}. \end{split} \tag{14}$$

The work *W* of external forces (neglecting the out-of plane load) done on *i*-th plate can be written as follows:

$$\mathcal{W} = \bigwedge\_{0}^{b} \mathbb{I}[p^{0}(y)\mathbb{I} + \tau\_{xy}^{0}(y)\upsilon] dy + \bigwedge\_{o}^{\ell} \mathbb{I}[p^{0}(\mathbf{x})\upsilon + \tau\_{xy}^{0}(\mathbf{x})\mu] d\mathbf{x},\tag{15}$$

where: *p*0(*x*), *p*0(*y*), 0xy(*x*), 0xy(*y*) are the prebuckling load applied to the middle surface of the considered plate (wall or strip)

For thin plates, it is assumed that the displacements *u* and *v* do not depend on rotation *w,x* and *w,y* and therefore do not depend on the coordinate *z*. This approach results in exclusion of rotational inertia [50] in the equation for kinetic energy, which for the *i*-th thin plate (strip) can be written as:

$$K = \frac{1}{2}\rho \int\_{\Omega} \left( \stackrel{\bullet}{\left(u\right)^2} + \stackrel{\bullet}{\left(v\right)^2} + \stackrel{\bullet}{\left(w\right)^2} \right) d\Omega \tag{16}$$

Nonlinear Plate Theory for Postbuckling Behaviour of Thin-Walled Structures Under Static and Dynamic Load 229

0

(22)

0

0

(23)

(24)

(21)

 

0

*t const*

*t const*

*t const*

Above conditions are fulfilled for the entire structure, so if one apply the restrictions in moment of the initial t0 and in moment of the final t1 that the displacement variations

are zero at all points of the structure. Then the system of equations (23) vanishes.

*x x x xy y x*

 

*E h N N N dSdt*

*y y yx x y y*

*E h N N N dSdt*

 

2 0

 

*xy xy xy*

*Gh N N dSdt*

already used the relationship between deformations and internal forces and moments

0

0

[ ( )] 0

*y y y xy x y const*

[ ( )] 0

*xy y y xy x xy y const*

0

*M M N w N w wdxdt*

0

( 2 ) 0

*y y xy x y y xy x y const*

2 0

*xy x const y const*

boundary conditions for longitudinal edges of the plate (*y* = const):

, ,

*N N v N v hp x vdxdt*

*N N u N u h x udxdt*

, ,

, ,, ,

*M wdt* 

.

 

 

 

*h u udS*

*h v vdS*

.

*S*

*S*

*S*

1

*t*

0 1

*t S t*

0 1

*t S t*

0

*t S*

.

*h w wdS*

1

*t*

0 1

*t t* 0

0 1

*t t* 0

,

*M w dxdt*

*y y y const*

boundary condition for the plate corners (*x* = const and *y* = const):

1

*t*

0

*t*

0 1

*t t* 0

0

*t*

initial conditions for *t* = const:

(8) or (9):

0

The Hamilton's principle, it is the variation of the action functional (10) for *i*-th thin plate (strip or wall) which after taking into consideration equations from (11) to (15) can be written as:

$$
\delta \Phi \Psi = \int\_{t\_0}^{t\_1} (\delta K - \delta \mathbb{Q}^m - \delta \mathbb{Q}^b + \delta \mathcal{W}) dt = 0 \tag{17}
$$

The Lagrangian function for the whole system is equal to the sum of the Lagrangian functions of all *n* plates of which the system was composed. To determine the variation of action for *i*-th plate, the following identity:

$$X \,\,\delta Y = \delta(XY) - Y\delta X \tag{18}$$

was used.

In the obtained equation, terms with the same variations were grouped, and then each of the obtained groups of terms (due to the mutual independence of variations) were equated to zero, giving:

equilibrium equations:

$$\begin{aligned} &\int\_{t}^{t} \int \left[ \left( \mathbf{N}\_{x,x} + \mathbf{N}\_{xy,y} + (\mathbf{N}\_{x}\boldsymbol{\mu}\_{,x})\_{,x} + (\mathbf{N}\_{y}\boldsymbol{\mu}\_{,y})\_{,y} + (\mathbf{N}\_{xy}\boldsymbol{\mu}\_{,x})\_{,x} + (\mathbf{N}\_{xy}\boldsymbol{\mu}\_{,y})\_{,x} \right) - h\rho \overset{\circ}{u} \delta u dS dt = 0 \\ &\int\_{t}^{t} \int \left[ \left( \mathbf{N}\_{xy,x} + \mathbf{N}\_{y,y} + (\mathbf{N}\_{x}\boldsymbol{\nu}\_{,x})\_{,x} + (\mathbf{N}\_{y}\boldsymbol{\nu}\_{,y})\_{,y} + (\mathbf{N}\_{xy}\boldsymbol{\nu}\_{,x})\_{,y} + (\mathbf{N}\_{xy}\boldsymbol{\nu}\_{,y})\_{,x} \right) - h\rho \overset{\circ}{v} \right] \delta v dS dt = 0 \\ &\int\_{t}^{t} \int \left[ \left( \mathbf{M}\_{x,xx} + \mathbf{M}\_{y,yy} + 2\mathbf{M}\_{xy,xy} + (\mathbf{N}\_{x}\boldsymbol{w}\_{,x})\_{,x} + (\mathbf{N}\_{y}\boldsymbol{w}\_{,y})\_{,y} + (\mathbf{N}\_{xy}\boldsymbol{w}\_{,x})\_{,y} + (\mathbf{N}\_{xy}\boldsymbol{w}\_{,y})\_{,x} \right) + \\ &-h\rho \overset{\circ}{w} \right] \delta w dS dt = 0 \end{aligned} \tag{19}$$

boundary conditions for lateral edges of the plate (*x* = const):

$$\begin{aligned} \left. \int\_{t\_0}^{t\_1} [\mathbf{N}\_x + \mathbf{N}\_x \boldsymbol{u}\_{,x} + \mathbf{N}\_{xy} \boldsymbol{u}\_{,y} - hp^0(y)] \delta t \boldsymbol{u} dy dt \Big|\_{\mathbf{x}=const} = \mathbf{0} \\ \int\_{t\_0}^{t\_1} \left. \int\_0^t [\mathbf{N}\_{xy} + \mathbf{N}\_x \boldsymbol{v}\_{,x} + \mathbf{N}\_{xy} \boldsymbol{v}\_{,y} - hr\_{xy}^0(y)] \delta v dy dt \right|\_{\mathbf{x}=const} = \mathbf{0} \\ \int\_{t\_0}^{t\_1} \left. \int\_0^b \mathbf{M}\_x \delta w\_{,x} dy dt \right|\_{\mathbf{x}=const} = \mathbf{0} \\ \int\_{t\_0}^{t\_1} \left. \int\_0^b (\mathbf{M}\_{x,x} + 2 \mathbf{M}\_{xy,y} + \mathbf{N}\_x \mathbf{w}\_{,x} + \mathbf{N}\_{xy} \mathbf{w}\_{,y}) \delta v dy dt \right|\_{\mathbf{x}=const} = \mathbf{0} \end{aligned} \tag{20}$$

boundary conditions for longitudinal edges of the plate (*y* = const):

$$\begin{cases} \int\_{t\_0}^{t\_f} \left[ N\_y + N\_y \upsilon\_{,y} + N\_{xy} \upsilon\_{,x} - hp^0(\mathbf{x}) \right] \delta v \mathbf{dx} dt \Big|\_{y=const} = 0\\ \int\_{t\_0}^{t\_f} \left[ \left[ N\_{xy} + N\_y \mu\_{,y} + N\_{xy} \mu\_{,x} - h \tau\_{xy}^0(\mathbf{x}) \right] \delta u \mathbf{dx} dt \right|\_{y=const} = 0\\ \int\_{t\_0}^{t\_f} \int\_{0}^{\ell} M\_y \delta w\_{,y} d\mathbf{x} dt \Big|\_{y=const} = 0\\ \int\_{t\_0}^{t\_f} \left( M\_{y,y} + 2M\_{xy,x} + N\_y \upsilon\_{,y} + N\_{xy} \upsilon\_{,x} \right) \delta u \mathbf{dx} dt \Big|\_{y=const} = 0 \end{cases} \tag{21}$$

boundary condition for the plate corners (*x* = const and *y* = const):

$$\int\_{t\_0}^{t\_1} \mathcal{D}M\_{xy} \delta v \, dt \Big|\_{x=const} \Big|\_{y=const} = 0 \tag{22}$$

initial conditions for *t* = const:

228 Nonlinearity, Bifurcation and Chaos – Theory and Applications

action for *i*-th plate, the following identity:

written as:

was used.

zero, giving:

1

*t*

0 1

*t S t*

0

*t*

*t S*

*S*

..

*h w wdSdt* 

} 0

1

*t b*

0 1

*t t b*

0

0 1

*t t b*

0

0 1

*t t b*

0

0

*t*

0

0

*t*

 1

equilibrium equations:

<sup>1</sup> 22 2 () () ( ) <sup>2</sup> ... *K u v wd*

The Hamilton's principle, it is the variation of the action functional (10) for *i*-th thin plate (strip or wall) which after taking into consideration equations from (11) to (15) can be

The Lagrangian function for the whole system is equal to the sum of the Lagrangian functions of all *n* plates of which the system was composed. To determine the variation of

*X Y XY Y X*

In the obtained equation, terms with the same variations were grouped, and then each of the obtained groups of terms (due to the mutual independence of variations) were equated to

{[ ( ) ( ) ( ) ( )] } 0

*N N N u N u N u N u h u udSdt*

{[ ( ) ( ) ( ) ( )] } 0

0

 

[ ( )] 0

*x x x xy y x const*

[ ( )] 0

*xy x x xy y xy x const*

0

*M M N w N w wdydt*

0

( 2 ) 0

*x x xy y x x xy y x const*

*N N N v N v N v N v h v vdSdt*

*m b*

   

 

 

(16)

0

( ) (18)

..

 

 

,, ,, ) ( )]

(20)

(19)

*xy x y xy y x*

*w Nw*

..

*K Q Q W dt* (17)

1

 

> 

, , ,, ,, ,, ,,

*x x xy y x x x y y y xy x y xy y x*

, , ,, ,, ,, ,,

*xy x y y x x x y y y xy x y xy y x*

, , , ,, ,,

*M M M Nw Nw N*

, ,

*N N u N u hp y udydt*

*N N v N v h y vdydt*

, ,

, ,, ,

*x xx y yy xy xy x x x y y y*

{[ 2 ( )( )(

boundary conditions for lateral edges of the plate (*x* = const):

,

*M w dydt*

*x x x const*

*t*

0

*t*

$$\begin{cases} h\rho \stackrel{\circ}{u} \delta u dS \Big|\_{t=const} = 0 \\\\ \int\_{S} h\rho \stackrel{\circ}{v} \delta v dS \Big|\_{t=const} = 0 \\\\ \int\_{S} h\rho \stackrel{\circ}{w} \delta w dS \Big|\_{t=const} = 0 \end{cases} \tag{23}$$

Above conditions are fulfilled for the entire structure, so if one apply the restrictions in moment of the initial t0 and in moment of the final t1 that the displacement variations are zero at all points of the structure. Then the system of equations (23) vanishes.

 already used the relationship between deformations and internal forces and moments (8) or (9):

$$\begin{aligned} \int\_{t\_0}^{t\_1} \left( E\_x h \varepsilon\_x - N\_x + \nu\_{xy} N\_y \right) \delta N\_x dS dt &= 0 \\ \int\_{t\_0}^{t\_1} \left( E\_y h \varepsilon\_y + \nu\_{yx} N\_x - N\_y \right) \delta N\_y dS dt &= 0 \\ \int\_{t\_0}^{t\_1} \left( 2G h \varepsilon\_{xy} - N\_{xy} \right) \delta N\_{xy} dS dt &= 0 \end{aligned} \tag{24}$$

$$\begin{aligned} \prod\_{t\_0 \in S}^{t\_1} \left( \frac{E\_x h^3}{12} \kappa\_x - \mathcal{M}\_x + \nu\_{xy} \mathcal{M}\_y \right) \delta \mathcal{M}\_x dS dt &= 0 \\ \prod\_{t\_0 \in S}^{t\_1} \left( \frac{E\_y h^3}{12} \kappa\_y + \nu\_{yx} \mathcal{M}\_x - \mathcal{M}\_y \right) \delta \mathcal{M}\_y dS dt &= 0 \\ \prod\_{t\_0 \in S}^{t\_1} \left( \frac{G h^3}{6} \kappa\_{xy} - \mathcal{M}\_{xy} \right) \delta \mathcal{M}\_{xy} dS dt &= 0 \end{aligned} \tag{25}$$

*N x y t dy N x y t dy N b b*

l l

1 1 (0) 0, , ,, ,

*ix i i i ix i i i ix*

l

The condition written as a first of equations of (27) is satisfied for the prebuckling state and first-order approximation, the condition for deflection *v* (27) is satisfied for the first and second order of approximations, while the other two conditions are met for prebuckling state as well as for the first and second order of approximation. The condition of displacement in the *y* direction in the prebuckling state can be found for example in [51]. This approach allows to take into account the Poisson effect on the edges of the walls of the column. The boundary conditions described by equations (27) assume the lack of displacement possibility of points lying at the loaded edges in the transverse *v* and normal *w* directions to the surface in a wall or column. Furthermore, it is assumed that the moments *M*ix (as a vector parallel to the edge of the plate or end edge of the column walls) are zero.

For structures with material properties varying widthwise the strip model was adopted what forces the boundary conditions modification in the second order approximations [51].

Modification consists of changing the first condition of (27) onto the following form:

*<sup>i</sup> b J*

(Figure 3) is equal to zero.

.

.

.

3), according to (21), can be written as:

equations (23) result the following initial conditions:

where the following functions , , , , ,

**3.3. Interaction condition between adjacent plates** 

~

~

~

(2) 0; <sup>1</sup> <sup>0</sup> <sup>1</sup> <sup>0</sup>

Summation is performed only for the *J* number of the strips, between which the angle i,i+1

To determine the boundary conditions on the longitudinal edges of plates or free edges of columns with open cross-sections the equations (20) were used. Whereas, directly from

0 0

*iii iii iii iii*

*uxyt t uxy uxyt t uxy*

( , , ) ( , ) and ( , , ) ( , ),

0 0

*wxyt t wxy wxyt t wxy*

*iii iii iii iii*

*vxyt t vxy vxyt t vxy*

( , , ) ( , ) and ( , , ) ( , ),

0 0

*iii iii iii iii*

~~~

Static and kinematic junction conditions on the longitudinal edges of adjacent plates (Figure

( , , ) ( , ) and ( , , ) ( , ),

*<sup>l</sup>* (28)

*ii i ii i uvw uvw* are given for the initial moment *t* = *t0*.

*ix i <sup>x</sup> <sup>i</sup> <sup>i</sup> N dy <sup>b</sup>*

(27)

(29)

 

*i i ii i ii i ii i ii i ix i i iy i i*

*v x yt v x yt w x yt w x yt M x yt M x yt*

0, , , , 0, 0, , , , 0, 0, , , , 0,

<sup>l</sup>

 
