**3. Solution method**

To determine the critical loads, natural frequencies and the coefficients of the equation describing the postbuckling equilibrium path, the analytical-numerical method has been employed. The proposed method also allows analysing dynamic response of the structure subjected to pulse loading. Taking the time courses of deflections and applying the relevant dynamic buckling criteria it is possible to determine the dynamic critical load.

### **3.1. Equilibrium equations**

The differential equations of equilibrium for orthotropic plate or strip directly from the equations (19) can be derived and have the form:

$$\begin{aligned} \left(N\_{x,x} + N\_{xy,y} + \{\left(N\_x \boldsymbol{u}\_{,x}\right)\_{,x} + \left(N\_y \boldsymbol{u}\_{,y}\right)\_{,y} + \left(N\_{xy} \boldsymbol{u}\_{,x}\right)\_{,y} + \left(N\_{xy} \boldsymbol{u}\_{,y}\right)\_{,x}\} - h\rho \stackrel{\cdots}{u} &= 0 \\ \left(N\_{xy,x} + N\_{y,y} + \{\left(N\_x \boldsymbol{v}\_{,x}\right)\_{,x} + \left(N\_y \boldsymbol{v}\_{,y}\right)\_{,y} + \left(N\_{xy} \boldsymbol{v}\_{,x}\right)\_{,y} + \left(N\_{xy} \boldsymbol{v}\_{,y}\right)\_{,x}\} - h\rho \stackrel{\cdots}{v} &= 0 \\ \left(N\_{x,xx} + M\_{y,yy} + 2M\_{xy,xy} + \left(N\_x \boldsymbol{w}\_{,x}\right)\_{,x} + \left(N\_y \boldsymbol{w}\_{,y}\right)\_{,y} + \left(N\_{xy} \boldsymbol{w}\_{,x}\right)\_{,y} + \left(N\_{xy} \boldsymbol{w}\_{,y}\right)\_{,x} - h\rho \stackrel{\cdots}{w} = 0\end{aligned} \tag{26}$$

Above equilibrium equations after omitting the inertia forces .. *h u* , .. *h v* and .. *h w* becomes the equilibrium equations for thin plates allowing analysis of both local and global buckling mode.

### **3.2. Boundary and initial condition**

As the wave propagation effects have been neglected, the boundary conditions referring to the simply supported columns at their both ends, i.e. *x* = 0 and *x* = *l*, according to (20), are assumed to be:

230 Nonlinearity, Bifurcation and Chaos – Theory and Applications

1

*t*

3

*x x xy y x*

*E h M M M dSdt*

 

 

*y yx x y y*

*M M M dSdt*

 

*xy xy xy*

 

To determine the critical loads, natural frequencies and the coefficients of the equation describing the postbuckling equilibrium path, the analytical-numerical method has been employed. The proposed method also allows analysing dynamic response of the structure subjected to pulse loading. Taking the time courses of deflections and applying the relevant dynamic buckling criteria it is possible to determine the dynamic critical

The differential equations of equilibrium for orthotropic plate or strip directly from the

{( ) ( ) ( ) ( ) } 0

{( ) ( ) ( ) ( ) } 0

the equilibrium equations for thin plates allowing analysis of both local and global buckling

As the wave propagation effects have been neglected, the boundary conditions referring to the simply supported columns at their both ends, i.e. *x* = 0 and *x* = *l*, according to (20), are

2 ( )( )( )( ) 0

*Gh M M dSdt*

12

*x*

12

*y*

*E h*

6

0

0

..

.. *h u* , ..

.. *h v* 

and

..

(26)

.. *h w*

becomes

(25)

0

3

3

, , ,, ,, ,, ,,

*N N Nu Nu N u N u h u*

*N N Nv Nv N v N v h v*

*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*

Above equilibrium equations after omitting the inertia forces

**3.2. Boundary and initial condition** 

, , , ,, ,, ,, ,,

*M M M Nw Nw N w N w h w*

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

0 1

*t S t*

0 1

*t S t*

0

**3. Solution method** 

**3.1. Equilibrium equations** 

equations (19) can be derived and have the form:

load.

mode.

assumed to be:

*t S*

$$\begin{aligned} \frac{1}{b\_i} \int \mathcal{N}\_{ix} \left( \mathbf{x}\_i = \mathbf{0}, y\_i, t \right) d\mathbf{y}\_i &= \frac{1}{b\_i} \int \mathcal{N}\_{ix} \left( \mathbf{x}\_i = \mathbf{l}, y\_i, t \right) d\mathbf{y}\_i = \mathcal{N}\_{ix}^{(0)}, \\ \mathcal{U}\_i \left( \mathbf{x}\_i = \mathbf{0}, y\_i, t \right) &= \boldsymbol{\upsilon}\_i \left( \mathbf{x}\_i = \mathbf{l}, y\_i, t \right) = \mathbf{0}, \\ \mathcal{U}\_i \left( \mathbf{x}\_i = \mathbf{0}, y\_i, t \right) &= \boldsymbol{\upsilon}\_i \left( \mathbf{x}\_i = \mathbf{l}, y\_i, t \right) = \mathbf{0}, \\ \mathcal{M}\_{ix} \left( \mathbf{x}\_i = \mathbf{0}, y\_i, t \right) &= \mathcal{M}\_{iy} \left( \mathbf{x}\_i = \mathbf{l}, y\_i, t \right) = \mathbf{0}, \end{aligned} \tag{27}$$

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:

$$\sum\_{i=1}^{l} \frac{1}{b\_i} \int\_0^{b\_i} N\_{ix}^{(2)} dy\_i \Big|\_{x=0;l} = 0 \tag{28}$$

Summation is performed only for the *J* number of the strips, between which the angle i,i+1 (Figure 3) is equal to zero.

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 equations (23) result the following initial conditions:

$$\begin{aligned} \stackrel{\bullet}{u}\_i(\mathbf{x}\_i, y\_i, t = t\_0) &= \stackrel{\bullet}{u}\_i(\mathbf{x}\_i, y\_i) \quad \text{and} \quad u\_i(\mathbf{x}\_i, y\_i, t = t\_0) = \overline{u\_i}(\mathbf{x}\_i, y\_i), \\ \stackrel{\bullet}{w}\_i(\mathbf{x}\_i, y\_i, t = t\_0) &= \stackrel{\bullet}{v}\_i(\mathbf{x}\_i, y\_i) \quad \text{and} \quad v\_i(\mathbf{x}\_i, y\_i, t = t\_0) = \overline{v\_i}(\mathbf{x}\_i, y\_i), \\ \stackrel{\bullet}{w}\_i(\mathbf{x}\_i, y\_i, t = t\_0) &= w\_i(\mathbf{x}\_i, y\_i) \quad \text{and} \quad w\_i(\mathbf{x}\_i, y\_i, t = t\_0) = \overline{w\_i}(\mathbf{x}\_i, y\_i), \end{aligned} \tag{29}$$

where the following functions , , , , , ~~~ *ii i ii i uvw uvw* are given for the initial moment *t* = *t0*.

### **3.3. Interaction condition between adjacent plates**

Static and kinematic junction conditions on the longitudinal edges of adjacent plates (Figure 3), according to (21), can be written as:

$$\begin{aligned} \left|u\_{i+1}\right|^{-} &= \left|u\_i\right|^{+}, \\ \left|w\_{i+1}\right|^{-} &= \left.w\_i\right|^{+} \cos(\phi) - \left.v\_i\right|^{+} \sin(\phi), \\ \left|v\_{i+1}\right|^{-} &= \left.w\_i\right|^{+} \sin(\phi) + \left.v\_i\right|^{+} \cos(\phi), \\ \left|w\_{i+1,y}\right|^{-} &= \left.w\_{i,y}\right|^{+}, \\ \left|M\_{(i+1)y}\right|^{-} &= M\_{iy}\Big|^{+}, \\ \left|N\_{(i+1)y}\right|^{-} &= \left.N\_{iy}^{+}\right|^{+} \cos(\phi) - \left.Q\_{iy}^{\*}\right|^{+} \sin(\phi) = 0, \\ \left|Q\_{(i+1)y}^{\*}\right|^{-} + \left.N\_{iy}^{+}\right|^{+} \sin(\phi) - \left.Q\_{iy}^{\*}\right|^{+} \cos(\phi) = 0, \\ \left|N\_{(i+1)xy}^{\*}\right|^{-} &= N\_{ixy}^{+}\Big|^{+} \end{aligned} \tag{30}$$

Nonlinear Plate Theory for Postbuckling Behaviour of Thin-Walled Structures Under Static and Dynamic Load 233

 

> 

*i ij i ij i ij i ij*

... ...

(32)

(34)

(35)

(33)

(0) ( ) ( ) (0) ( ) ( )

 

It was assumed that the dimensionless amplitude of the initial deflections (imperfections)

\* ( ). *<sup>i</sup> U Us* 

By substituting expansions (32) into equations of equilibrium (26) with neglected inertia terms (static buckling problem), junction conditions (30) and boundary conditions (27), the boundary problem of the zero (superscript (0) in Equations (32) and further), first (superscript (i)) and second (superscript (ij)) order has been obtained [18, 50, 52, 53]. The zero approximation describes the prebuckling state, whereas the first order approximation allows for determination of critical loads and the buckling modes corresponding to them, taking into account minimisation with respect to the number of half-waves *m* in the lengthwise direction. The second order approximation is reduced to a linear system of differential heterogeneous equations, which right-hand sides depend on the force field and

The most important advantage of this method is that it enables us to describe a complete range of behaviour of thin-walled structures from all global (i.e. flexural, flexural–torsional, lateral, distortional buckling and their combinations) to the local dynamic stability. In the solution obtained, the shear lag phenomenon, the effect of cross-sectional distortions and

Having found the solutions to the first and second order of the boundary problem, the

\* \*

*L L*

**σ U U σ U U σ U**

*ijs s s*

*ijks s s*

*s s*

 

*L L <sup>b</sup>*

\* \*

( ) ( ) () () () ( ) 11 11 (0) ( ) 2

*i ssi j j*

( , ) 0.5 ( , ) , ( )

\*

( ) ( ) ( ) ( ) () () 11 11 (0) ( ) 2

*L*

*i jk s ij k s*

2 ( ,) (,) , ( )

*L*

\*

where: s – is the critical load corresponding to the *s*-th mode, L11 is the bilinear operator, L2 is the quadratic operator and **σ**(i), **σ**(ij) are the stress field tensors in the first and second order. The postbuckling static equilibrium paths for coupled buckling can be described by the

\* <sup>1</sup> ; s 1, , , *s ijs i j ijks i j k s*

 

*a b N*

 

 

 

**σ U U σ U U σ U**

*UU U U NN N N*

 

 

correspond to the considered buckling mode (for *s*-th buckling mode) is:

also the interaction between all the walls of structures are included.

coefficients aijs, bijks have been determined [18, 50, 52, 53]:

*a*

which for the uncoupled problem have the form:

equation:

the first order displacements only.

where:

$$\begin{aligned} \mathbf{N}\_{\text{ixy}}^{\*} &= \mathbf{N}\_{\dot{\eta}\mathbf{y}} + \mathbf{N}\_{\dot{\eta}\mathbf{y}}\boldsymbol{\upsilon}\_{i,\mathbf{y}} + \mathbf{N}\_{\dot{\alpha}\mathbf{y}}\boldsymbol{\upsilon}\_{i,\mathbf{x}} \\ \mathbf{N}\_{\text{ixy}}^{\*} &= \mathbf{N}\_{\dot{\mathbf{x}}\mathbf{y}} + \mathbf{N}\_{\dot{\mathbf{x}}\mathbf{y}}\boldsymbol{\mu}\_{i,\mathbf{x}} + \mathbf{N}\_{\dot{\eta}\mathbf{y}}\boldsymbol{\upsilon}\_{i,\mathbf{y}} \\ \mathbf{M}\_{\text{iy}} &= -\eta\_{i}\mathbf{D}\_{i}(\mathbf{t}\boldsymbol{w}\_{i,\mathbf{y}\mathbf{y}} + \boldsymbol{\nu}\_{i}\boldsymbol{\varpi}\_{i,\mathbf{x}\mathbf{x}})\_{\star} \\ \mathbf{Q}\_{\text{iy}}^{\*} &= -\eta\_{i}\mathbf{D}\_{i}\boldsymbol{\varpi}\_{i,\mathbf{y}\mathbf{y}\mathbf{y}} - (\nu\_{i}\eta\_{i}\mathbf{D}\_{i} + 2\mathbf{D}\_{i1})\mathbf{w}\_{i,\mathbf{x}\mathbf{x}\mathbf{y}} + \mathbf{N}\_{\dot{\mathbf{y}}\mathbf{y}}\boldsymbol{\varpi}\_{i,\mathbf{y}} + \mathbf{N}\_{\dot{\mathbf{x}}\mathbf{y}}\boldsymbol{\varpi}\_{i,\mathbf{x}'} \\ \boldsymbol{\phi} &= \boldsymbol{\phi}\_{i,i+1}\mathbf{1} \end{aligned} \tag{31}$$

**Figure 3.** The geometrical dimensions and local coordinate systems adjacent plates

### **3.4. Buckling and postbuckling equilibrium paths**

A non-linear stability problem has been solved by means of the Koiter's asymptotic theory. The displacement field *U* , and sectional force field *N* have been expanded into the power series with respect to the parameter , - the buckling linear eigenvector amplitude (normalised with the equality condition between the maximum deflection and the thickness of the first plate h1).

232 Nonlinearity, Bifurcation and Chaos – Theory and Applications

1 1 1

*i i*

*u u*

1, ,

*i y iy*

*i y iy*

*M M*

*w w*

,

*ii i ii i*

 

*ww v vw v*

cos( ) sin( ), sin( ) cos( ),

cos( ) sin( ) 0,

  (30)

(31)

 

 

 

sin( ) cos( ) 0,

, 1, , ,

*iy i i i yyy i i i i i xxy iy i y ixy i x*

A non-linear stability problem has been solved by means of the Koiter's asymptotic theory. The displacement field *U* , and sectional force field *N* have been expanded into the power series with respect to the parameter , - the buckling linear eigenvector amplitude (normalised with the equality condition between the maximum deflection and the thickness

*Q Dw D D w N w N w*

( 2) ,

\* \* ( 1)

*N N*

*ixy iy iy i y ixy i x ixy ixy ixy i x iy i y iy i i i yy i i xx*

*N N Nv N v N N N u Nu M Dw w*

 

*i xy ixy*

\*\* \*

*NN Q*

*i y iy iy*

,

,

,

\*\* \*

*QN Q*

, ,

, ,

 

 

**Figure 3.** The geometrical dimensions and local coordinate systems adjacent plates

**3.4. Buckling and postbuckling equilibrium paths** 

( ),

, ,

*i y iy iy*

( 1)

( 1)

( 1)

\*

\*

\*

 

of the first plate h1).

; 1

*i i*

.

where:

$$\begin{aligned} \lambda \overline{\mathcal{U}} &= \lambda \overline{\mathcal{U}}^{(0)} + \xi\_i^{\varepsilon} \overline{\mathcal{U}}^{(i)} + \xi\_i^{\varepsilon} \xi\_j^{\varepsilon} \overline{\mathcal{U}}^{(ij)} + \dots \\ \overline{\mathcal{N}} &= \lambda \overline{\mathcal{N}}^{(0)} + \xi\_i^{\varepsilon} \overline{\mathcal{N}}^{(i)} + \xi\_i^{\varepsilon} \xi\_j^{\varepsilon} \overline{\mathcal{N}}^{(ij)} + \dots \end{aligned} \tag{32}$$

It was assumed that the dimensionless amplitude of the initial deflections (imperfections) correspond to the considered buckling mode (for *s*-th buckling mode) is:

$$
\overline{\mathcal{U}} = \underline{\mathcal{E}}\_s^\* \overline{\mathcal{U}}^{(i)}.\tag{33}
$$

By substituting expansions (32) into equations of equilibrium (26) with neglected inertia terms (static buckling problem), junction conditions (30) and boundary conditions (27), the boundary problem of the zero (superscript (0) in Equations (32) and further), first (superscript (i)) and second (superscript (ij)) order has been obtained [18, 50, 52, 53]. The zero approximation describes the prebuckling state, whereas the first order approximation allows for determination of critical loads and the buckling modes corresponding to them, taking into account minimisation with respect to the number of half-waves *m* in the lengthwise direction. The second order approximation is reduced to a linear system of differential heterogeneous equations, which right-hand sides depend on the force field and the first order displacements only.

The most important advantage of this method is that it enables us to describe a complete range of behaviour of thin-walled structures from all global (i.e. flexural, flexural–torsional, lateral, distortional buckling and their combinations) to the local dynamic stability. In the solution obtained, the shear lag phenomenon, the effect of cross-sectional distortions and also the interaction between all the walls of structures are included.

Having found the solutions to the first and second order of the boundary problem, the coefficients aijs, bijks have been determined [18, 50, 52, 53]:

$$\begin{split} a\_{ijs} &= \frac{\mathbf{o}^{(i)} \ast L\_{11}(\mathbf{U}^{(j)}, \mathbf{U}^{(s)}) + 0.5\mathbf{o}^{(s)} \ast L\_{11}(\mathbf{U}^{(i)}, \mathbf{U}^{(j)})}{-\lambda\_{s}\mathbf{o}^{(0)} \ast L\_{2}(\mathbf{U}^{(s)})}, \\\\ b\_{ijks} &= \frac{2\mathbf{o}^{(i)} \ast L\_{11}(\mathbf{U}^{(k)}, \mathbf{U}^{(s)}) + \mathbf{o}^{(ij)} \ast L\_{11}(\mathbf{U}^{(k)}, \mathbf{U}^{(s)})}{-\lambda\_{s}\mathbf{o}^{(0)} \ast L\_{2}(\mathbf{U}^{(s)})}, \end{split} \tag{34}$$

where: s – is the critical load corresponding to the *s*-th mode, L11 is the bilinear operator, L2 is the quadratic operator and **σ**(i), **σ**(ij) are the stress field tensors in the first and second order.

The postbuckling static equilibrium paths for coupled buckling can be described by the equation:

$$\left(1 - \frac{\lambda}{\lambda\_s}\right)\tilde{\varphi}\_s + a\_{ijs}\tilde{\varphi}\_i\tilde{\varphi}\_j + b\_{ijks}\tilde{\varphi}\_i\tilde{\varphi}\_j\tilde{\varphi}\_k = \frac{\lambda}{\lambda\_s}\tilde{\varphi}\_s^\*; \quad \left(\text{s } = 1, \dots, N\right), \tag{35}$$

which for the uncoupled problem have the form:

$$\left(1 - \frac{\lambda}{\lambda\_{cr}}\right)\xi + a\_{111}\xi^2 + b\_{1111}\xi^3 = \xi^\* \frac{\lambda}{\lambda\_{cr}}\tag{36}$$

  *t t* . (40)

 

> 

 

[kg/m3]

, as

(41)

(39)

.. 2 3\* 2 1 1 111 1 1111 1 1 1 1 1 <sup>1</sup> 1 ; *a b*

> 0 0 and ( 0) 0 .

The Runge-Kutta method [57] for solving the equation (39) requires the following

 

2 2 22 32 \* 1 1 111 1 1 1111 1 1

 

which lead to the system of two differential equations. ''Complete'' equations of motion (41) are solved with the numerical Runge–Kutta method of order 8 (5,3), thanks to Dormand and

The exemplary results of numerical calculation are presented in this sub-chapter. All results are obtained using explained above proposed analytical-numerical method (ANM) based on

The material properties (E – Young modulus, ν – Poisson ratio, G=E/[2(1+ν)] – Kirchhoff

[GPa]

modulus; – density) for materials taken into account are presented in Table 1.

*E*

steel 200 0.3 7850 aluminium 70 0.33 2950 epoxy resin 3.5 0.33 1249 glass fibre 71 0.22 2450

The fibre composite material was modelled as orthotropic but for components (resin and fibre) the isotropic material properties (Table 1) was assumed. Necessary equations for material properties homogenization based on theory of mixture [57, 58] are as

1 1

( ) ( ) 1 ,

*t t b b*

 

It is assumed that in the initial moment of time *t* = 0 the non-dimensional displacement

 

  

well as the velocity of displacement are equal to zero, i.e.:

( ),

. *<sup>t</sup>*

Price (with step-size control and density output).

**4. Exemplary results of calculations** 

material type:

**Table 1.** Assumed material properties

follows:

the nonlinear orthotropic plate theory.

.

substitutions:

where cr is the critical load value.

In a special case, i.e. for the so-called ideal structure without initial imperfections (\*=0) and when the equilibrium path (*a*111) is symmetrical, the postbuckling equilibrium path is defined by the equation:

$$\frac{\lambda}{\lambda\_{cr}} = 1 + b\_{1111} \xi^2 \tag{37}$$
