# 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:

\begin{align}\begin{aligned}\begin{split}u = u_0 + u_1 + u_2\\\end{split}\\\vdots\\{{\phi}_\theta} = {{\phi}_\theta}_0 + {{\phi}_\theta}_1 + {{\phi}_\theta}_2\end{aligned}\end{align}

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.

$\begin{split}u = u_0 + \sum_{i_1=0}^{m_1} {c_{i_1}}^{u} \sin{{b_x}_1} + \sum_{i_2=0}^{m_2} \sum_{j_2=1}^{n_2} \left( {c_{i_2 j_2}}_a^{u} \sin{{b_x}_2} \sin{j_2 \theta} +{c_{i_2 j_2}}_b^{u} \sin{{b_x}_2} \cos{j_2 \theta} \right) \\ v = v_0 + \sum_{i_1=0}^{m_1} {c_{i_1}}^{v} \sin{{b_x}_1} + \sum_{i_2=0}^{m_2} \sum_{j_2=1}^{n_2} \left( {c_{i_2 j_2}}_a^{v} \sin{{b_x}_2} \sin{j_2 \theta} +{c_{i_2 j_2}}_b^{v} \sin{{b_x}_2} \cos{j_2 \theta} \right) \\ w = w_0 + \sum_{i_1=0}^{m_1} {c_{i_1}}^{w} \sin{{b_x}_1} + \sum_{i_2=0}^{m_2} \sum_{j_2=1}^{n_2} \left( {c_{i_2 j_2}}_a^{w} \sin{{b_x}_2} \sin{j_2 \theta} +{c_{i_2 j_2}}_b^{w} \sin{{b_x}_2} \cos{j_2 \theta} \right) \\ \phi_x = {\phi_x}_0 + \sum_{i_1=0}^{m_1} {c_{i_1}}^{\phi_x} \cos{{b_x}_1} + \sum_{i_2=0}^{m_2} \sum_{j_2=1}^{n_2} \left( {c_{i_2 j_2}}_a^{\phi_x} \cos{{b_x}_2} \sin{j_2 \theta} +{c_{i_2 j_2}}_b^{\phi_x} \cos{{b_x}_2} \cos{j_2 \theta} \right) \\ {\phi}_\theta = {\phi_x}_0 + \sum_{i_1=0}^{m_1} {c_{i_1}}^{{\phi}_\theta} \sin{{b_x}_1} + \sum_{i_2=0}^{m_2} \sum_{j_2=1}^{n_2} \left( {c_{i_2 j_2}}_a^{{\phi}_\theta} \sin{{b_x}_2} \sin{j_2 \theta} +{c_{i_2 j_2}}_b^{{\phi}_\theta} \sin{{b_x}_2} \cos{j_2 \theta} \right)\end{split}$

Observations:

$$\checkmark$$ linear static implemented

$$\checkmark$$ linear buckling implemented

$$\checkmark$$ non-linear analysis implemented

### fsdt_donnell_bc2#

SS2- and CC2-types of boundary conditions.

$\begin{split}u = u_0 + \sum_{i_1=0}^{m_1} {c_{i_1}}^{u} \sin{{b_x}_1} + \sum_{i_2=0}^{m_2} \sum_{j_2=1}^{n_2} \left( {c_{i_2 j_2}}_a^{u} \cos{{b_x}_2} \sin{j_2 \theta} +{c_{i_2 j_2}}_b^{u} \cos{{b_x}_2} \cos{j_2 \theta} \right) \\ v = v_0 + \sum_{i_1=0}^{m_1} {c_{i_1}}^{v} \sin{{b_x}_1} + \sum_{i_2=0}^{m_2} \sum_{j_2=1}^{n_2} \left( {c_{i_2 j_2}}_a^{v} \sin{{b_x}_2} \sin{j_2 \theta} +{c_{i_2 j_2}}_b^{v} \sin{{b_x}_2} \cos{j_2 \theta} \right) \\ w = w_0 + \sum_{i_1=0}^{m_1} {c_{i_1}}^{w} \sin{{b_x}_1} + \sum_{i_2=0}^{m_2} \sum_{j_2=1}^{n_2} \left( {c_{i_2 j_2}}_a^{w} \sin{{b_x}_2} \sin{j_2 \theta} +{c_{i_2 j_2}}_b^{w} \sin{{b_x}_2} \cos{j_2 \theta} \right) \\ \phi_x = {\phi_x}_0 + \sum_{i_1=0}^{m_1} {c_{i_1}}^{\phi_x} \cos{{b_x}_1} + \sum_{i_2=0}^{m_2} \sum_{j_2=1}^{n_2} \left( {c_{i_2 j_2}}_a^{\phi_x} \cos{{b_x}_2} \sin{j_2 \theta} +{c_{i_2 j_2}}_b^{\phi_x} \cos{{b_x}_2} \cos{j_2 \theta} \right) \\ {\phi}_\theta = {\phi_x}_0 + \sum_{i_1=0}^{m_1} {c_{i_1}}^{{\phi}_\theta} \sin{{b_x}_1} + \sum_{i_2=0}^{m_2} \sum_{j_2=1}^{n_2} \left( {c_{i_2 j_2}}_a^{{\phi}_\theta} \sin{{b_x}_2} \sin{j_2 \theta} +{c_{i_2 j_2}}_b^{{\phi}_\theta} \sin{{b_x}_2} \cos{j_2 \theta} \right)\end{split}$

Observations:

$$\checkmark$$ linear static implemented

$$\checkmark$$ linear buckling implemented

$$\checkmark$$ non-linear analysis implemented

### fsdt_donnell_bc3#

SS3- and CC3-types of boundary conditions.

$\begin{split}u = u_0 + \sum_{i_1=0}^{m_1} {c_{i_1}}^{u} \sin{{b_x}_1} + \sum_{i_2=0}^{m_2} \sum_{j_2=1}^{n_2} \left( {c_{i_2 j_2}}_a^{u} \sin{{b_x}_2} \sin{j_2 \theta} +{c_{i_2 j_2}}_b^{u} \sin{{b_x}_2} \cos{j_2 \theta} \right) \\ v = v_0 + \sum_{i_1=0}^{m_1} {c_{i_1}}^{v} \sin{{b_x}_1} + \sum_{i_2=0}^{m_2} \sum_{j_2=1}^{n_2} \left( {c_{i_2 j_2}}_a^{v} \cos{{b_x}_2} \sin{j_2 \theta} +{c_{i_2 j_2}}_b^{v} \cos{{b_x}_2} \cos{j_2 \theta} \right) \\ w = w_0 + \sum_{i_1=0}^{m_1} {c_{i_1}}^{w} \sin{{b_x}_1} + \sum_{i_2=0}^{m_2} \sum_{j_2=1}^{n_2} \left( {c_{i_2 j_2}}_a^{w} \sin{{b_x}_2} \sin{j_2 \theta} +{c_{i_2 j_2}}_b^{w} \sin{{b_x}_2} \cos{j_2 \theta} \right) \\ \phi_x = {\phi_x}_0 + \sum_{i_1=0}^{m_1} {c_{i_1}}^{\phi_x} \cos{{b_x}_1} + \sum_{i_2=0}^{m_2} \sum_{j_2=1}^{n_2} \left( {c_{i_2 j_2}}_a^{\phi_x} \cos{{b_x}_2} \sin{j_2 \theta} +{c_{i_2 j_2}}_b^{\phi_x} \cos{{b_x}_2} \cos{j_2 \theta} \right) \\ {\phi}_\theta = {\phi_x}_0 + \sum_{i_1=0}^{m_1} {c_{i_1}}^{{\phi}_\theta} \sin{{b_x}_1} + \sum_{i_2=0}^{m_2} \sum_{j_2=1}^{n_2} \left( {c_{i_2 j_2}}_a^{{\phi}_\theta} \sin{{b_x}_2} \sin{j_2 \theta} +{c_{i_2 j_2}}_b^{{\phi}_\theta} \sin{{b_x}_2} \cos{j_2 \theta} \right)\end{split}$

Observations:

$$\checkmark$$ linear static implemented

$$\checkmark$$ linear buckling implemented

$$\checkmark$$ non-linear analysis implemented

### fsdt_donnell_bc4#

SS4- and CC4-types of boundary conditions.

$\begin{split}u = u_0 + \sum_{i_1=0}^{m_1} {c_{i_1}}^{u} \sin{{b_x}_1} + \sum_{i_2=0}^{m_2} \sum_{j_2=1}^{n_2} \left( {c_{i_2 j_2}}_a^{u} \cos{{b_x}_2} \sin{j_2 \theta} +{c_{i_2 j_2}}_b^{u} \cos{{b_x}_2} \cos{j_2 \theta} \right) \\ v = v_0 + \sum_{i_1=0}^{m_1} {c_{i_1}}^{v} \sin{{b_x}_1} + \sum_{i_2=0}^{m_2} \sum_{j_2=1}^{n_2} \left( {c_{i_2 j_2}}_a^{v} \cos{{b_x}_2} \sin{j_2 \theta} +{c_{i_2 j_2}}_b^{v} \cos{{b_x}_2} \cos{j_2 \theta} \right) \\ w = w_0 + \sum_{i_1=0}^{m_1} {c_{i_1}}^{w} \sin{{b_x}_1} + \sum_{i_2=0}^{m_2} \sum_{j_2=1}^{n_2} \left( {c_{i_2 j_2}}_a^{w} \sin{{b_x}_2} \sin{j_2 \theta} +{c_{i_2 j_2}}_b^{w} \sin{{b_x}_2} \cos{j_2 \theta} \right) \\ \phi_x = {\phi_x}_0 + \sum_{i_1=0}^{m_1} {c_{i_1}}^{\phi_x} \cos{{b_x}_1} + \sum_{i_2=0}^{m_2} \sum_{j_2=1}^{n_2} \left( {c_{i_2 j_2}}_a^{\phi_x} \cos{{b_x}_2} \sin{j_2 \theta} +{c_{i_2 j_2}}_b^{\phi_x} \cos{{b_x}_2} \cos{j_2 \theta} \right) \\ {\phi}_\theta = {\phi_x}_0 + \sum_{i_1=0}^{m_1} {c_{i_1}}^{{\phi}_\theta} \sin{{b_x}_1} + \sum_{i_2=0}^{m_2} \sum_{j_2=1}^{n_2} \left( {c_{i_2 j_2}}_a^{{\phi}_\theta} \sin{{b_x}_2} \sin{j_2 \theta} +{c_{i_2 j_2}}_b^{{\phi}_\theta} \sin{{b_x}_2} \cos{j_2 \theta} \right)\end{split}$

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:

$\begin{split}u = u_0 + \sum_{i_1=0}^{m_1} {c_{i_1}}^{u} \sin{{b_x}_1} + \sum_{i_2=0}^{m_2} \sum_{j_2=1}^{n_2} \left( {c_{i_2 j_2}}_a^{u} \cos{{b_x}_2} \sin{j_2 \theta} +{c_{i_2 j_2}}_b^{u} \cos{{b_x}_2} \cos{j_2 \theta} \right) \\ v = v_0 + \sum_{i_1=0}^{m_1} {c_{i_1}}^{v} \sin{{b_x}_1} + \sum_{i_2=0}^{m_2} \sum_{j_2=1}^{n_2} \left( {c_{i_2 j_2}}_a^{v} \cos{{b_x}_2} \sin{j_2 \theta} +{c_{i_2 j_2}}_b^{v} \cos{{b_x}_2} \cos{j_2 \theta} \right) \\ w = w_0 + \sum_{i_1=0}^{m_1} {c_{i_1}}^{w} \sin{{b_x}_1} + \sum_{i_2=0}^{m_2} \sum_{j_2=1}^{n_2} \left( {c_{i_2 j_2}}_a^{w} \cos{{b_x}_2} \sin{j_2 \theta} +{c_{i_2 j_2}}_b^{w} \cos{{b_x}_2} \cos{j_2 \theta} \right) \\ \phi_x = {\phi_x}_0 + \sum_{i_1=0}^{m_1} {c_{i_1}}^{\phi_x} \cos{{b_x}_1} + \sum_{i_2=0}^{m_2} \sum_{j_2=1}^{n_2} \left( {c_{i_2 j_2}}_a^{\phi_x} \cos{{b_x}_2} \sin{j_2 \theta} +{c_{i_2 j_2}}_b^{\phi_x} \cos{{b_x}_2} \cos{j_2 \theta} \right) \\ {\phi}_\theta = {\phi_x}_0 + \sum_{i_1=0}^{m_1} {c_{i_1}}^{{\phi}_\theta} \sin{{b_x}_1} + \sum_{i_2=0}^{m_2} \sum_{j_2=1}^{n_2} \left( {c_{i_2 j_2}}_a^{{\phi}_\theta} \cos{{b_x}_2} \sin{j_2 \theta} +{c_{i_2 j_2}}_b^{{\phi}_\theta} \cos{{b_x}_2} \cos{j_2 \theta} \right)\end{split}$

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

Note

NOT RECOMMENDED, implemented for comparative purposes only.

$\begin{split}u = \sum_{i_2=0}^{m_2} \sum_{j_2=0}^{n_2} \left( {c_{i_2 j_2}}^{u} \cos{{b_x}_2} \cos{j_2 \theta} \right) \\ v = \sum_{i_2=0}^{m_2} \sum_{j_2=0}^{n_2} \left( {c_{i_2 j_2}}^{v} \sin{{b_x}_2} \sin{j_2 \theta} \right) \\ w = \sum_{i_2=0}^{m_2} \sum_{j_2=0}^{n_2} \left( {c_{i_2 j_2}}^{w} \sin{{b_x}_2} \sin{j_2 \theta} \right) \\ \phi_x = \sum_{i_2=0}^{m_2} \sum_{j_2=0}^{n_2} \left( {c_{i_2 j_2}}^{\phi_x} \cos{{b_x}_2} \sin{j_2 \theta} \right) \\ {\phi}_\theta = \sum_{i_2=0}^{m_2} \sum_{j_2=0}^{n_2} \left( {c_{i_2 j_2}}^{{\phi}_\theta} \sin{{b_x}_2} \cos{j_2 \theta} \right)\end{split}$

Observations:

$$\checkmark$$ linear buckling implemented

$$\rightarrow$$ linear static not implemented

$$\rightarrow$$ non-linear analysis not implemented

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:

$\begin{split}u = \sum_{i_2=0}^{m_2} \sum_{j_2=0}^{n_2} \left( {c_{i_2 j_2}}^{u} \sin{{b_x}_2} \cos{j_2 \theta} \right) \\ v = \sum_{i_2=0}^{m_2} \sum_{j_2=0}^{n_2} \left( {c_{i_2 j_2}}^{v} \cos{{b_x}_2} \sin{j_2 \theta} \right) \\ w = \sum_{i_2=0}^{m_2} \sum_{j_2=0}^{n_2} \left( {c_{i_2 j_2}}^{w} \sin{{b_x}_2} \sin{j_2 \theta} \right) \\ \phi_x = \sum_{i_2=0}^{m_2} \sum_{j_2=0}^{n_2} \left( {c_{i_2 j_2}}^{\phi_x} \cos{{b_x}_2} \sin{j_2 \theta} \right) \\ {\phi}_\theta = \sum_{i_2=0}^{m_2} \sum_{j_2=0}^{n_2} \left( {c_{i_2 j_2}}^{{\phi}_\theta} \sin{{b_x}_2} \cos{j_2 \theta} \right)\end{split}$

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:

$\begin{split}u = \sum_{i_2=0}^{m_2} \sum_{j_2=0}^{n_2} \left( {c_{i_2 j_2}}^{u} \cos{{b_x}_2} \cos{j_2 \theta} \right) \\ v = \sum_{i_2=0}^{m_2} \sum_{j_2=0}^{n_2} \left( {c_{i_2 j_2}}^{v} \sin{{b_x}_2} \sin{j_2 \theta} \right) \\ w = \sum_{i_2=0}^{m_2} \sum_{j_2=0}^{n_2} \left( {c_{i_2 j_2}}^{w} \sin{{b_x}_2} \cos{j_2 \theta} \right) \\ \phi_x = \sum_{i_2=0}^{m_2} \sum_{j_2=0}^{n_2} \left( {c_{i_2 j_2}}^{\phi_x} \cos{{b_x}_2} \cos{j_2 \theta} \right) \\ {\phi}_\theta = \sum_{i_2=0}^{m_2} \sum_{j_2=0}^{n_2} \left( {c_{i_2 j_2}}^{{\phi}_\theta} \sin{{b_x}_2} \sin{j_2 \theta} \right)\end{split}$

Observations:

$$\checkmark$$ linear buckling implemented

$$\rightarrow$$ linear static not implemented

$$\rightarrow$$ non-linear analysis not implemented