FSDT - First-order Shear Deformation Theory

Description

For the FSDT the displacement field components are:

u, v, w, \phi_x, \phi_t

The displacement field is approximated similarly to the CLPT, but here \phi_x and \phi_\theta are independent variables, relaxing the approximation.

For the ConeCyl implementations, the approximation functions are separated into three components:

u = u_0 + u_1 + u_2\\ \vdots {{\phi}_\theta} = {{\phi}_\theta}_0 + {{\phi}_\theta}_1 + {{\phi}_\theta}_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 a model capable to simulate non-rigid supports, and where the displacement components u, \phi_x can habe a non-costant value along the edges.

It is of special importance to allow \phi_x to be between zero (clamped) and another value (up to simply supported), by using elastic stiffnesses for the corresponding degrees of freedom. The elastic stiffnesses are implemented for the FSDT in the same manner described for the CLPT.

Models

The recommended models, according to the desired boundary condition , are:

Note that the fsdt_donnell_bc4 model can be used to simulate all the other types of boundary condition, which is allowed by the use of elastic constraints.

A more general model, the fsdt_donnell_bcn (or the counterpart fsdt_sanders_bcn) has been proposed and despite it has the largest simulation capabilities, it can be unstable for high stiffeness applied to the elastic constraints. Moreover, this model cannot simulate the linear buckling analysis of the 4^{th} type of boundary conditions.

The models below were kept for future reference only and have been used in comparative studies:

fsdt_donnell_bc1

SS1- and CC1-types of boundary conditions.

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) \\ \phi_x = {\phi_x}_0 + \sum_{i_1=0}^{m_1} {c_{i_1}}^{\phi_x} \cos{{b_x}_1} + \sum_{i_2=0}^{m_2} \sum_{j_2=1}^{n_2} \left( {c_{i_2 j_2}}_a^{\phi_x} \cos{{b_x}_2} \sin{j_2 \theta} +{c_{i_2 j_2}}_b^{\phi_x} \cos{{b_x}_2} \cos{j_2 \theta} \right) \\ {\phi}_\theta = {\phi_x}_0 + \sum_{i_1=0}^{m_1} {c_{i_1}}^{{\phi}_\theta} \sin{{b_x}_1} + \sum_{i_2=0}^{m_2} \sum_{j_2=1}^{n_2} \left( {c_{i_2 j_2}}_a^{{\phi}_\theta} \sin{{b_x}_2} \sin{j_2 \theta} +{c_{i_2 j_2}}_b^{{\phi}_\theta} \sin{{b_x}_2} \cos{j_2 \theta} \right)

Observations:

\checkmark linear static implemented

\checkmark linear buckling implemented

\checkmark non-linear analysis implemented

fsdt_donnell_bc2

SS2- and CC2-types of boundary conditions.

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) \\ \phi_x = {\phi_x}_0 + \sum_{i_1=0}^{m_1} {c_{i_1}}^{\phi_x} \cos{{b_x}_1} + \sum_{i_2=0}^{m_2} \sum_{j_2=1}^{n_2} \left( {c_{i_2 j_2}}_a^{\phi_x} \cos{{b_x}_2} \sin{j_2 \theta} +{c_{i_2 j_2}}_b^{\phi_x} \cos{{b_x}_2} \cos{j_2 \theta} \right) \\ {\phi}_\theta = {\phi_x}_0 + \sum_{i_1=0}^{m_1} {c_{i_1}}^{{\phi}_\theta} \sin{{b_x}_1} + \sum_{i_2=0}^{m_2} \sum_{j_2=1}^{n_2} \left( {c_{i_2 j_2}}_a^{{\phi}_\theta} \sin{{b_x}_2} \sin{j_2 \theta} +{c_{i_2 j_2}}_b^{{\phi}_\theta} \sin{{b_x}_2} \cos{j_2 \theta} \right)

Observations:

\checkmark linear static implemented

\checkmark linear buckling implemented

\checkmark non-linear analysis implemented

fsdt_donnell_bc3

SS3- and CC3-types of boundary conditions.

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) \\ \phi_x = {\phi_x}_0 + \sum_{i_1=0}^{m_1} {c_{i_1}}^{\phi_x} \cos{{b_x}_1} + \sum_{i_2=0}^{m_2} \sum_{j_2=1}^{n_2} \left( {c_{i_2 j_2}}_a^{\phi_x} \cos{{b_x}_2} \sin{j_2 \theta} +{c_{i_2 j_2}}_b^{\phi_x} \cos{{b_x}_2} \cos{j_2 \theta} \right) \\ {\phi}_\theta = {\phi_x}_0 + \sum_{i_1=0}^{m_1} {c_{i_1}}^{{\phi}_\theta} \sin{{b_x}_1} + \sum_{i_2=0}^{m_2} \sum_{j_2=1}^{n_2} \left( {c_{i_2 j_2}}_a^{{\phi}_\theta} \sin{{b_x}_2} \sin{j_2 \theta} +{c_{i_2 j_2}}_b^{{\phi}_\theta} \sin{{b_x}_2} \cos{j_2 \theta} \right)

Observations:

\checkmark linear static implemented

\checkmark linear buckling implemented

\checkmark non-linear analysis implemented

fsdt_donnell_bc4

SS4- and CC4-types of boundary conditions.

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) \\ \phi_x = {\phi_x}_0 + \sum_{i_1=0}^{m_1} {c_{i_1}}^{\phi_x} \cos{{b_x}_1} + \sum_{i_2=0}^{m_2} \sum_{j_2=1}^{n_2} \left( {c_{i_2 j_2}}_a^{\phi_x} \cos{{b_x}_2} \sin{j_2 \theta} +{c_{i_2 j_2}}_b^{\phi_x} \cos{{b_x}_2} \cos{j_2 \theta} \right) \\ {\phi}_\theta = {\phi_x}_0 + \sum_{i_1=0}^{m_1} {c_{i_1}}^{{\phi}_\theta} \sin{{b_x}_1} + \sum_{i_2=0}^{m_2} \sum_{j_2=1}^{n_2} \left( {c_{i_2 j_2}}_a^{{\phi}_\theta} \sin{{b_x}_2} \sin{j_2 \theta} +{c_{i_2 j_2}}_b^{{\phi}_\theta} \sin{{b_x}_2} \cos{j_2 \theta} \right)

Observations:

\checkmark linear static implemented

\checkmark linear buckling implemented

\checkmark non-linear analysis implemented

fsdt_donnell_bcn

The current attempt adds more flexibility in v,w,\phi_\theta. The resulting 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} \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} \cos{{b_x}_2} \sin{j_2 \theta} +{c_{i_2 j_2}}_b^{w} \cos{{b_x}_2} \cos{j_2 \theta} \right) \\ \phi_x = {\phi_x}_0 + \sum_{i_1=0}^{m_1} {c_{i_1}}^{\phi_x} \cos{{b_x}_1} + \sum_{i_2=0}^{m_2} \sum_{j_2=1}^{n_2} \left( {c_{i_2 j_2}}_a^{\phi_x} \cos{{b_x}_2} \sin{j_2 \theta} +{c_{i_2 j_2}}_b^{\phi_x} \cos{{b_x}_2} \cos{j_2 \theta} \right) \\ {\phi}_\theta = {\phi_x}_0 + \sum_{i_1=0}^{m_1} {c_{i_1}}^{{\phi}_\theta} \sin{{b_x}_1} + \sum_{i_2=0}^{m_2} \sum_{j_2=1}^{n_2} \left( {c_{i_2 j_2}}_a^{{\phi}_\theta} \cos{{b_x}_2} \sin{j_2 \theta} +{c_{i_2 j_2}}_b^{{\phi}_\theta} \cos{{b_x}_2} \cos{j_2 \theta} \right)

Observations:

\checkmark linear static implemented

\times linear buckling not working

\checkmark non-linear analysis implemented

fsdt_sanders_bcn

Counterpart of fsdt_donnell_bcn using the Sanders non-linear equations.

Observations:

\checkmark linear static implemented

\times linear buckling not working

\rightarrow non-linear analysis not implemented

fsdt_shadmehri2012_bc2

Note

NOT RECOMMENDED, implemented for comparative purposes only.

Adapted from the model published by Shadmehri (2012) ([shadmehri2012], [shadmehri2012thesis]) for the SS2- and CC2-types of boundary conditions. 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} \sin{j_2 \theta} \right) \\ \phi_x = \sum_{i_2=0}^{m_2} \sum_{j_2=0}^{n_2} \left( {c_{i_2 j_2}}^{\phi_x} \cos{{b_x}_2} \sin{j_2 \theta} \right) \\ {\phi}_\theta = \sum_{i_2=0}^{m_2} \sum_{j_2=0}^{n_2} \left( {c_{i_2 j_2}}^{{\phi}_\theta} \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

fsdt_shadmehri2012_bc3

Note

NOT RECOMMENDED, implemented for comparative purposes only.

Published by Shadmehri (2012) (see [shadmehri2012] or [shadmehri2012thesis] for more details). This model was developed to simulate the SS3- and CC3-types of boundary condition. 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} \sin{{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} \cos{{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} \sin{j_2 \theta} \right) \\ \phi_x = \sum_{i_2=0}^{m_2} \sum_{j_2=0}^{n_2} \left( {c_{i_2 j_2}}^{\phi_x} \cos{{b_x}_2} \sin{j_2 \theta} \right) \\ {\phi}_\theta = \sum_{i_2=0}^{m_2} \sum_{j_2=0}^{n_2} \left( {c_{i_2 j_2}}^{{\phi}_\theta} \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

fsdt_geier1997_bc2

Note

NOT RECOMMENDED, implemented for comparative purposes only.

Published by Geier and Singh (1997) (see [geier1997] for more details). This model was developed to simulate the SS2- and CC2-types of boundary condition. Such models seem to be 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) \\ \phi_x = \sum_{i_2=0}^{m_2} \sum_{j_2=0}^{n_2} \left( {c_{i_2 j_2}}^{\phi_x} \cos{{b_x}_2} \cos{j_2 \theta} \right) \\ {\phi}_\theta = \sum_{i_2=0}^{m_2} \sum_{j_2=0}^{n_2} \left( {c_{i_2 j_2}}^{{\phi}_\theta} \sin{{b_x}_2} \sin{j_2 \theta} \right)

Observations:

\checkmark linear buckling implemented

\rightarrow linear static not implemented

\rightarrow non-linear analysis not implemented