CLPT - Classical Laminated Plate Theory

Description

For the CLPT the displacement field components are:

u, v, w

And approximated as:

u = u_0(x, \theta) + z \phi_x(x, \theta) \\ v = v_0(x, \theta) + z \phi_\theta(x, \theta) \\ w = w_0(x, \theta)

where u_0, v_0, w_0 are the displacements of the shell mid-surface and \phi_x and \phi_\theta the shell rotations along x and \theta following the right-hand rule. For the CLPT the rotations are defined as:

\phi_x = - \frac{\partial w}{\partial x} = -w_{,x} \\ \phi_\theta = - \frac 1 r \frac{\partial w}{\partial \theta} = - \frac 1 r w_{,\theta}

For the ConeCyl implementations the displacement field is approximated and the approximated functions can be separated as:

u = u_0 + u_1 + u_2\\ v = v_0 + v_1 + v_2\\ w = w_0 + w_1 + w_2\\

where u_0 contains the approximation functions corresponding to the prescribed degrees of freedom, u_1 contains the functions independent of \theta and u_2 the functions that depend on both x and \theta.

The aim is to have models capable of simulating the displacement field of cones and cylinders. The approximation functions are the same for both the Donnell’s and the Sanders’ models.

Models

Below it follows a more detailed description of each of the implementations:

Each model can be accessed using the linear_kinematics parameter of the ConeCyl object. For linear static analysis the most general model is the clpt_donnell_bcn.

For linear buckling analysis the following models should be used for each type of boundary conditions:

clpt_donnell_bc1

SS1- and CC1-types of boundary conditions, or anything in between by using elastic restrained edges in w_{,x} and w_{,\theta}. The approximation functions are:

u = u_0 + \sum_{i_1=0}^{m_1} {c_{i_1}}^{u} \sin{{b_x}_1} + \sum_{i_2=0}^{m_2} \sum_{j_2=1}^{n_2} \left( {c_{i_2 j_2}}_a^{u} \sin{{b_x}_2} \sin{j_2 \theta} +{c_{i_2 j_2}}_b^{u} \sin{{b_x}_2} \cos{j_2 \theta} \right) \\ v = v_0 + \sum_{i_1=0}^{m_1} {c_{i_1}}^{v}\sin{{b_x}_1} + \sum_{i_2=0}^{m_2} \sum_{j_2=1}^{n_2} \left( {c_{i_2 j_2}}_a^{v} \sin{{b_x}_2} \sin{j_2 \theta} +{c_{i_2 j_2}}_b^{v} \sin{{b_x}_2} \cos{j_2 \theta} \right) \\ w = w_0 + \sum_{i_1=0}^{m_1} {c_{i_1}}^{w}\sin{{b_x}_1} + \sum_{i_2=0}^{m_2} \sum_{j_2=1}^{n_2} \left( {c_{i_2 j_2}}_a^{w} \sin{{b_x}_2} \sin{j_2 \theta} +{c_{i_2 j_2}}_b^{w} \sin{{b_x}_2} \cos{j_2 \theta} \right)

with:

{b_x}_1 = i_1 \pi \frac x L \\ {b_x}_2 = i_2 \pi \frac x L

The following general form of elastic constraints at the edges is used:

U_{springs} = \int_\theta r_1 \left( K_{Bot}^u u_{x=L}^2 + K_{Bot}^v v_{x=L}^2 + K_{Bot}^w w_{x=L}^2 + K_{Bot}^{\phi_x} {\phi_x}_{x=L}^2 + K_{Bot}^{\phi_\theta} {\phi_\theta}_{x=L}^2 \right) \\ + \int_\theta r_2 \left( K_{Top}^u u_{x=0}^2 + K_{Top}^v v_{x=0}^2 + K_{Top}^w w_{x=0}^2 + K_{Top}^{\phi_x} {\phi_x}_{x=0}^2 + K_{Top}^{\phi_\theta} {\phi_\theta}_{x=0}^2 \right)

Note that the stiffnesses: K_{Top}^u, K_{Top}^v and K_{Top}^w are not used in clpt_donnell_bc1, but since they are required in other implementations, it is convenient to present the general form using all the elastic terms.

The equation for U_{springs} can be written in matrix form, and it will result in an additional term [K_e] to the linear stiffness matrix [K_0]. The new stiffness matrix with the elastic constraints at the edges ([{K_0}_e]) becomes:

[{K_0}_e] = [K_0] + [K_e] [K_e] = \int_{\theta} { \left( r_1 [g_{new}]_{x=L}^T [K]_{Bot} [g_{new}]_{x=L}^. + r_2 [g_{new}]_{x=0}^T [K]_{Top} [g_{new}]_{x=0}^. \right) d\theta }

with :

[K_{Bot}] = \begin{bmatrix} K_{Bot}^u & 0 & 0 & 0 & 0 \\ 0 & K_{Bot}^v & 0 & 0 & 0 \\ 0 & 0 & K_{Bot}^w & 0 & 0 \\ 0 & 0 & 0 & K_{Bot}^{\phi_x} & 0 \\ 0 & 0 & 0 & 0 &K_{Bot}^{\phi_\theta} \end{bmatrix}

and:

[K_{Top}] = \begin{bmatrix} K_{Top}^u & 0 & 0 & 0 & 0 \\ 0 & K_{Top}^v & 0 & 0 & 0 \\ 0 & 0 & K_{Top}^w & 0 & 0 \\ 0 & 0 & 0 & K_{Top}^{\phi_x} & 0 \\ 0 & 0 & 0 & 0 &K_{Top}^{\phi_\theta} \end{bmatrix}

and the shape functions [g_{new}] contains two extra rows that are built from the relations:

\phi_x = - \frac{\partial w}{\partial x} = -w_{,x} \\ \phi_\theta = - \frac 1 r \frac{\partial w}{\partial \theta} = - \frac 1 r w_{,\theta}

and therefore:

[g^{\phi_x}] = - \frac {\partial [g^w]} {\partial x} \\ [g^{\phi_\theta}] = - \frac 1 r \frac {\partial [g^w]} {\partial \theta} \\ [g_{new}]^T = \left[ [g^u], [g^v], [g^w], [g^{\phi_x}], [g^{\phi_\theta}] \right]

Observations:

\checkmark linear buckling implemented

\checkmark linear static implemented

\checkmark non-linear analysis implemented

clpt_donnell_bc2

Planned to simulate the SS2- and CC2-types of boundary conditions (or anything in between). The flexibily in v is removed if compared to the clpt_donnell_bc4. Giving:

u = u_0 + \sum_{i_1=0}^{m_1} {c_{i_1}}^{u} \sin{{b_x}_1} + \sum_{i_2=0}^{m_2} \sum_{j_2=1}^{n_2} \left( {c_{i_2 j_2}}_a^{u} \cos{{b_x}_2} \sin{j_2 \theta} +{c_{i_2 j_2}}_b^{u} \cos{{b_x}_2} \cos{j_2 \theta} \right) \\ v = v_0 + \sum_{i_1=0}^{m_1} {c_{i_1}}^{v}\sin{{b_x}_1} + \sum_{i_2=0}^{m_2} \sum_{j_2=1}^{n_2} \left( {c_{i_2 j_2}}_a^{v} \sin{{b_x}_2} \sin{j_2 \theta} +{c_{i_2 j_2}}_b^{v} \sin{{b_x}_2} \cos{j_2 \theta} \right) \\ w = w_0 + \sum_{i_1=0}^{m_1} {c_{i_1}}^{w}\sin{{b_x}_1} + \sum_{i_2=0}^{m_2} \sum_{j_2=1}^{n_2} \left( {c_{i_2 j_2}}_a^{w} \sin{{b_x}_2} \sin{j_2 \theta} +{c_{i_2 j_2}}_b^{w} \sin{{b_x}_2} \cos{j_2 \theta} \right) \\

The linear stiffness matrix [K_0] is changed using the same elastic contraints used for the clpt_donnell_bc1.

Observations:

\checkmark linear buckling implemented

\checkmark linear static implemented

\checkmark non-linear analysis implemented

clpt_donnell_bc3

Planned for SS3- and CC3-types of boundary conditions (or anything in between). The approximation functions are:

u = u_0 + \sum_{i_1=0}^{m_1} {c_{i_1}}^{u} \sin{{b_x}_1} + \sum_{i_2=0}^{m_2} \sum_{j_2=1}^{n_2} \left( {c_{i_2 j_2}}_a^{u} \sin{{b_x}_2} \sin{j_2 \theta} +{c_{i_2 j_2}}_b^{u} \sin{{b_x}_2} \cos{j_2 \theta} \right) \\ v = v_0 + \sum_{i_1=0}^{m_1} {c_{i_1}}^{v}\sin{{b_x}_1} + \sum_{i_2=0}^{m_2} \sum_{j_2=1}^{n_2} \left( {c_{i_2 j_2}}_a^{v} \cos{{b_x}_2} \sin{j_2 \theta} +{c_{i_2 j_2}}_b^{v} \cos{{b_x}_2} \cos{j_2 \theta} \right) \\ w = w_0 + \sum_{i_1=0}^{m_1} {c_{i_1}}^{w}\sin{{b_x}_1} + \sum_{i_2=0}^{m_2} \sum_{j_2=1}^{n_2} \left( {c_{i_2 j_2}}_a^{w} \sin{{b_x}_2} \sin{j_2 \theta} +{c_{i_2 j_2}}_b^{w} \sin{{b_x}_2} \cos{j_2 \theta} \right) \\

The linear stiffness matrix [K_0] is changed using the same elastic contraints used for the clpt_donnell_bc1.

Observations:

\checkmark linear buckling implemented

\checkmark linear static implemented

\checkmark non-linear analysis implemented

clpt_donnell_bc4

SS4- or CC4-types of boundary conditions (or anything in between).

u = u_0 + \sum_{i_1=0}^{m_1} {c_{i_1}}^{u} \sin{{b_x}_1} + \sum_{i_2=0}^{m_2} \sum_{j_2=1}^{n_2} \left( {c_{i_2 j_2}}_a^{u} \cos{{b_x}_2} \sin{j_2 \theta} +{c_{i_2 j_2}}_b^{u} \cos{{b_x}_2} \cos{j_2 \theta} \right) \\ v = v_0 + \sum_{i_1=0}^{m_1} {c_{i_1}}^{v}\sin{{b_x}_1} + \sum_{i_2=0}^{m_2} \sum_{j_2=1}^{n_2} \left( {c_{i_2 j_2}}_a^{v} \cos{{b_x}_2} \sin{j_2 \theta} +{c_{i_2 j_2}}_b^{v} \cos{{b_x}_2} \cos{j_2 \theta} \right) \\ w = w_0 + \sum_{i_1=0}^{m_1} {c_{i_1}}^{w}\sin{{b_x}_1} + \sum_{i_2=0}^{m_2} \sum_{j_2=1}^{n_2} \left( {c_{i_2 j_2}}_a^{w} \sin{{b_x}_2} \sin{j_2 \theta} +{c_{i_2 j_2}}_b^{w} \sin{{b_x}_2} \cos{j_2 \theta} \right) \\

The linear stiffness matrix [K_0] is changed using the same elastic contraints used for the clpt_donnell_bc1.

Observations:

\checkmark linear buckling implemented

\checkmark linear static implemented

\checkmark non-linear analysis implemented

clpt_donnell_bcn

General approximation function for the CLPT. It allows any type of boundary condition by setting the proper values for the elastic constants.

u = u_0 + \sum_{i_1=0}^{m_1} {c_{i_1}}^{u} \sin{{b_x}_1} + \sum_{i_2=0}^{m_2} \sum_{j_2=1}^{n_2} \left( {c_{i_2 j_2}}_a^{u} \cos{{b_x}_2} \sin{j_2 \theta} +{c_{i_2 j_2}}_b^{u} \cos{{b_x}_2} \cos{j_2 \theta} \right) \\ v = v_0 + \sum_{i_1=0}^{m_1} {c_{i_1}}^{v}\sin{{b_x}_1} + \sum_{i_2=0}^{m_2} \sum_{j_2=1}^{n_2} \left( {c_{i_2 j_2}}_a^{v} \cos{{b_x}_2} \sin{j_2 \theta} +{c_{i_2 j_2}}_b^{v} \cos{{b_x}_2} \cos{j_2 \theta} \right) \\ w = w_0 + \sum_{i_1=0}^{m_1} {c_{i_1}}^{w}\sin{{b_x}_1} + \sum_{i_2=0}^{m_2} \sum_{j_2=1}^{n_2} \left( {c_{i_2 j_2}}_a^{w} \sin{{b_x}_2} \sin{j_2 \theta} +{c_{i_2 j_2}}_b^{w} \sin{{b_x}_2} \cos{j_2 \theta} +{c_{i_2 j_2}}_c^{w} \cos{{b_x}_2} \sin{j_2 \theta} +{c_{i_2 j_2}}_d^{w} \cos{{b_x}_2} \cos{j_2 \theta} \right)

The linear stiffness matrix [K_0] is changed using the same elastic contraints used for the clpt_donnell_bc1.

Observations:

\checkmark linear static implemented

\times not working for linear buckling

\rightarrow non-linear analysis not implemented

clpt_sanders_bc1

Counterpart of clpt_donnell_bc1 using the Sanders non-linear equations.

Observations:

\checkmark linear static implemented

\checkmark linear buckling implemented

\rightarrow non-linear analysis not implemented

clpt_sanders_bc2

Counterpart of clpt_donnell_bc2 using the Sanders non-linear equations.

Observations:

\checkmark linear static implemented

\checkmark linear buckling implemented

\rightarrow non-linear analysis not implemented

clpt_sanders_bc3

Counterpart of clpt_donnell_bc3 using the Sanders non-linear equations.

Observations:

\checkmark linear static implemented

\checkmark linear buckling implemented

\rightarrow non-linear analysis not implemented

clpt_sanders_bc4

Counterpart of clpt_donnell_bc4 using the Sanders non-linear equations.

Observations:

\checkmark linear static implemented

\checkmark linear buckling implemented

\rightarrow non-linear analysis not implemented

clpt_geier1997_bc2

Note

NOT RECOMMENDED, implemented for comparative purposes only.

Analogous to the model published by Geier and Singh (1997) (see [geier1997] for more details) for the SS2- and CC2-types of boundary condition. Originally proposed by Khdeir et al. (1989) (see [khdeir1989]). Uses the Donnell’s equations and the approximation functions are:

u = \sum_{i_2=0}^{m_2} \sum_{j_2=0}^{n_2} \left( {c_{i_2 j_2}}^{u} \cos{{b_x}_2} \cos{j_2 \theta} \right) \\ v = \sum_{i_2=0}^{m_2} \sum_{j_2=0}^{n_2} \left( {c_{i_2 j_2}}^{v} \sin{{b_x}_2} \sin{j_2 \theta} \right) \\ w = \sum_{i_2=0}^{m_2} \sum_{j_2=0}^{n_2} \left( {c_{i_2 j_2}}^{w} \sin{{b_x}_2} \cos{j_2 \theta} \right)

Observations:

\checkmark linear buckling implemented

\rightarrow linear static not implemented

\rightarrow non-linear analysis not implemented