In this section, theorems for elastic structures are specialized for beam structures.
7.2.1. Strong formulation of the problem
Indefinite equilibrium. External distributed loads are equilibrated by internal actions, resultant of stress field on beam sections.
\[\begin{split}\begin{aligned}
\mathbf{0} & = \mathbf{F}' + \mathbf{f} \\
\mathbf{0} & = \mathbf{M}' + \hat{\mathbf{z}} \times \mathbf{F} + \mathbf{m} \\
\end{aligned}\end{split}\]
Kinematics. Displacement
\[\mathbf{s}(x,y,z) = \mathbf{s}_P(z) - \mathbf{r}_P(x,y) \times \boldsymbol\theta(z) + \mathbf{w}(x,y,z)\]
Strain
\[\begin{split}\begin{aligned}
\varepsilon_{zz} & = s'_{Pz} - x \theta'_{y} + y \theta'_{x} + w_{z/z} \\
\varepsilon_{xx} & = w_{x/x} \\
\varepsilon_{yy} & = w_{y/y} \\
2 \varepsilon_{zx} & = s'_{Px} - \theta_y - y \theta'_z + w_{x/z} + w_{z/x} \\
2 \varepsilon_{zy} & = s'_{Py} + \theta_x - x \theta'_z + w_{y/z} + w_{z/y} \\
2 \varepsilon_{xy} & = w_{x/y} + w_{y/x} \\
\end{aligned}\end{split}\]
\[\begin{split}\begin{aligned}
\boldsymbol{\varepsilon}_z & := \begin{bmatrix} \gamma_{zx} \\ \gamma_{zy} \\ \varepsilon_{z} \end{bmatrix}
= \mathbf{s}'_{P} - \mathbf{r}_P \times \boldsymbol\theta' + \hat{\mathbf{z}} \times \boldsymbol\theta + \mathbf{v}_1(w_{i/j}) \\
\boldsymbol{\varepsilon}_2 & := \begin{bmatrix} \gamma_{xy} \\ \varepsilon_{xx} \\ \varepsilon_{yy} \end{bmatrix}
= \mathbf{v}_2(w_{i/j})
\end{aligned}\end{split}\]
Constitutive equations. Under the assumptions … (kinematic assumptions, decoupling,…),
\[\begin{split}\begin{aligned}
\mathbf{F}(z) & := \int_{A(z)} \hat{\mathbf{n}} \cdot \boldsymbol\sigma = \int_{A(z)} \boldsymbol{\sigma}_z \\
\mathbf{M}(z) & := \int_{A(z)} \mathbf{r}_P \times \hat{\mathbf{n}} \cdot \boldsymbol\sigma = \int_{A(z)} \mathbf{r}_P \times \boldsymbol\sigma_z =
\end{aligned}\end{split}\]
\[\begin{split}\begin{aligned}
\boldsymbol\sigma_z := \begin{bmatrix} \sigma_{zx} \\ \sigma_{zy} \\ \sigma_{zz} \end{bmatrix}
& = \mathbf{C}_1 \cdot \boldsymbol{\varepsilon}_z + \mathbf{C}_2 \cdot \boldsymbol{\varepsilon}_2 - \mathbf{b} \Delta T = \\
& = \mathbf{C}_1 \cdot \begin{bmatrix} \gamma_{zx} \\ \gamma_{zy} \\ \varepsilon_{zz} \end{bmatrix} + \mathbf{C}_2 \cdot \begin{bmatrix} \gamma_{xy} \\ \varepsilon_{xx} \\ \varepsilon_{yy} \end{bmatrix} - \mathbf{b} \Delta T = \\
& = \mathbf{C}_1 \cdot \left( \mathbf{s}'_{P} - \mathbf{r}_P \times \boldsymbol\theta' + \hat{\mathbf{z}} \times \boldsymbol\theta + \mathbf{v}_1(w_{i/j}) \right) + \mathbf{D}_2 \cdot \mathbf{v}_2(w_{i/j}) - \mathbf{b} \Delta T
\end{aligned}\end{split}\]
Neglecting (or evaluating and finding that their contribution is zero) the contribution of \(\boldsymbol{\varepsilon}_2\),
\[\begin{split}\begin{aligned}
\mathbf{F}
& = \int_{A} \left\{ \mathbf{C}_1 \boldsymbol\varepsilon_z - \mathbf{b} \Delta T \right\} = \\
& = \int_{A} \mathbf{C}_1 ( \mathbf{s}'_P + \hat{\mathbf{z}} \times \boldsymbol\theta ) - \int_A \mathbf{C}_1 \mathbf{r}_{\times} \boldsymbol\theta' - \int_A \mathbf{b} \Delta T = \\
& = \mathbf{K}_{fs} \left( \mathbf{s}'_P + \hat{\mathbf{z}} \times \boldsymbol\theta \right) + \mathbf{K}_{f\theta} \boldsymbol\theta'-\mathbf{b}_f \Delta T \\
\mathbf{M}
& = \int_{A} \mathbf{r}_{\times} \left\{ \mathbf{C}_1 \boldsymbol\varepsilon_z - \mathbf{b} \Delta T \right\} = \\
& = \int_{A} \mathbf{r}_{\times} \mathbf{C}_1 ( \mathbf{s}'_P + \hat{\mathbf{z}} \times \boldsymbol\theta ) - \int_A \mathbf{r}_{\times} \mathbf{C}_1 \mathbf{r}_{\times} \boldsymbol\theta' - \int_A \mathbf{r}_{\times} \mathbf{b} \Delta T = \\
& = \mathbf{K}_{ms} \left( \mathbf{s}'_P + \hat{\mathbf{z}} \times \boldsymbol\theta \right) + \mathbf{K}_{m\theta} \boldsymbol\theta'- \mathbf{b}_m \Delta T \ ,
\end{aligned}\end{split}\]
with \(\mathbf{K}_{ms} = \mathbf{K}_{f\theta}\), as \(\mathbf{C}_1 = \mathbf{C}_1^T\) and \(\mathbf{r}_{\times} = - \mathbf{r}_{\times}^{\ T}\).
\[\begin{split}\begin{bmatrix} \mathbf{F} \\ \mathbf{M} \end{bmatrix} = \begin{bmatrix} \mathbf{K}_{11} & \mathbf{K}_{12} \\ \mathbf{K}_{21} & \mathbf{K}_{22} \end{bmatrix} \begin{bmatrix} \mathbf{s}'+\hat{\mathbf{z}} \times \boldsymbol\theta \\ \boldsymbol\theta' \end{bmatrix} - \begin{bmatrix} \mathbf{b}_1 \\ \mathbf{b}_2 \end{bmatrix} \Delta T\end{split}\]
\[\begin{split}\begin{bmatrix} \mathbf{s}'+\hat{\mathbf{z}} \times \boldsymbol\theta \\ \boldsymbol\theta' \end{bmatrix} = \begin{bmatrix} \mathbf{S}_{11} & \mathbf{S}_{12} \\ \mathbf{S}_{21} & \mathbf{S}_{22} \end{bmatrix} \begin{bmatrix} \mathbf{F} \\ \mathbf{M} \end{bmatrix} + \begin{bmatrix} \mathbf{a}_1 \\ \mathbf{a}_2 \end{bmatrix} \Delta T \ ,\end{split}\]
with
\[\mathbf{I} = \mathbf{K} \mathbf{S} = \mathbf{S} \mathbf{K} \quad , \quad \mathbf{a} = \mathbf{S} \mathbf{b}\]
\[\begin{split}
\begin{bmatrix} \mathbf{I} & \mathbf{0} \\ \mathbf{0} & \mathbf{I} \end{bmatrix}
= \begin{bmatrix} \mathbf{S}_{11} & \mathbf{S}_{12} \\ \mathbf{S}_{21} & \mathbf{S}_{22} \end{bmatrix}
\begin{bmatrix} \mathbf{K}_{11} & \mathbf{K}_{12} \\ \mathbf{K}_{21} & \mathbf{K}_{22} \end{bmatrix}
= \begin{bmatrix} \mathbf{S}_{11}\mathbf{K}_{11} + \mathbf{S}_{12} \mathbf{K}_{21} &
\mathbf{S}_{11}\mathbf{K}_{12} + \mathbf{S}_{12} \mathbf{K}_{22} \\
\mathbf{S}_{21}\mathbf{K}_{11} + \mathbf{S}_{22} \mathbf{K}_{21} &
\mathbf{S}_{21}\mathbf{K}_{12} + \mathbf{S}_{22} \mathbf{K}_{22} \end{bmatrix}
\end{split}\]
Thus, stress as a function of internal actions reads
\[\begin{split}\begin{aligned}
\boldsymbol\sigma_z
& = \mathbf{C}_1 \begin{bmatrix} \mathbf{I} & - \mathbf{r}_{\times} \end{bmatrix} \begin{bmatrix} ( \mathbf{s}'_P + \hat{\mathbf{z}} \times \boldsymbol\theta ) \\ \boldsymbol\theta' \end{bmatrix} - \mathbf{b} \Delta T = \\
& = \mathbf{C}_1 \begin{bmatrix} \mathbf{I} & - \mathbf{r}_{\times} \end{bmatrix} \left( \begin{bmatrix} \mathbf{S}_{11} & \mathbf{S}_{12} \\ \mathbf{S}_{21} & \mathbf{S}_{22} \end{bmatrix} \begin{bmatrix} \mathbf{F} \\ \mathbf{M} \end{bmatrix} + \begin{bmatrix} \mathbf{a}_1 \\ \mathbf{a}_2 \end{bmatrix} \Delta T \right) - \mathbf{b} \Delta T \ .
\end{aligned}\end{split}\]
Evaluation of the term \(\int_{V} \widetilde{\boldsymbol\sigma}_z^T \boldsymbol\varepsilon_z\), with \(\widetilde{\Delta T} = 0\)
\[\begin{split}\begin{aligned}
\int_V \widetilde{\boldsymbol\sigma}_z^T \boldsymbol\varepsilon_z
& = \int_{\ell} \begin{bmatrix} \widetilde{\mathbf{F}}^T & \widetilde{\mathbf{M}}^T \end{bmatrix} \mathbf{S} \int_{A} \begin{bmatrix} \mathbf{I} \\ \mathbf{r}_{\times} \end{bmatrix} \mathbf{C}_1 \begin{bmatrix} \mathbf{I} & - \mathbf{r}_{\times} \end{bmatrix} \, dA \, \left( \mathbf{S} \begin{bmatrix} \mathbf{F} \\ \mathbf{M} \end{bmatrix} + \mathbf{a} \Delta T \right) \, d \ell = \\
& = \int_{\ell} \begin{bmatrix} \widetilde{\mathbf{F}}^T & \widetilde{\mathbf{M}}^T \end{bmatrix} \underbrace{\mathbf{S} \mathbf{K}}_{= \mathbf{I}} \left( \mathbf{S} \begin{bmatrix} \mathbf{F} \\ \mathbf{M} \end{bmatrix} + \mathbf{a} \Delta T \right) \, d \ell = \\
& = \int_{\ell} \begin{bmatrix} \widetilde{\mathbf{F}}^T & \widetilde{\mathbf{M}}^T \end{bmatrix} \left( \mathbf{S} \begin{bmatrix} \mathbf{F} \\ \mathbf{M} \end{bmatrix} + \mathbf{a} \Delta T \right) \, d \ell = \\
\end{aligned}\end{split}\]
For an elastic isotropic medium with structural decoupling,
\[\begin{split}\mathbf{S} = \begin{bmatrix}
\frac{\chi_x}{GA} & \cdot & \cdot & \cdot & \cdot & \cdot \\
\cdot & \frac{\chi_y}{GA} & \cdot & \cdot & \cdot & \cdot \\
\cdot & \cdot & \frac{1}{EA} & \cdot & \cdot & \cdot \\
\cdot & \cdot & \cdot & \frac{1}{EJ_x} & \cdot & \cdot \\
\cdot & \cdot & \cdot & \cdot & \frac{1}{EJ_y} & \cdot \\
\cdot & \cdot & \cdot & \cdot & \cdot & \frac{1}{GJ_z} \\
\end{bmatrix} \ ,\end{split}\]
s.t.
\[\begin{split}\begin{aligned}
s'_x - \theta_y & = \frac{\chi_x}{GA } F_x \\
s'_y + \theta_x & = \frac{\chi_y}{GA } F_y \\
s'_z & = \frac{1 }{EA } F_z \\
\theta'_x & = \frac{1 }{EJ_x} M_x \\
\theta'_y & = \frac{1 }{EJ_y} M_y \\
\theta'_z & = \frac{1 }{GJ_z} M_z \\
\end{aligned}\end{split}\]
No contribution of warping to internal actions
\[\int_A \mathbf{D}_1 \mathbf{v}_1 = \dots = \mathbf{0} \ .\]
Bernoulli beam. If Bernoulli kinematic assumption
\[\begin{aligned}
\left( \mathbb{I} - \hat{\mathbf{z}} \otimes \hat{\mathbf{z}} \right) \cdot \left( \mathbf{s}'_P + \hat{\mathbf{z}} \times \boldsymbol\theta \right) = \mathbf{0} \ ,
\end{aligned}\]
holds, …
7.2.2. Weak formulations of the problem
7.2.2.3. Principle of virtual work
…test functions: variations…the problem can be recast as the principle of stationariety of total potential energy
If test function \(\mathbf{u}\), \(\mathbf{v}\) in the weak form of the equilibrium conditions respectively represent the variation of a congruent displacement and a rotation field \(\mathbf{u} = \delta \widetilde{\mathbf{s}}\), \(\mathbf{v} = \delta \widetilde{\boldsymbol\theta}\), with \(\delta \widetilde{\mathbf{s}}|_{S_D} = \mathbf{0}\) and \(\delta \widetilde{\boldsymbol\theta}|_{S_D} = \mathbf{0}\) (these relations may hold just for the constrained components)
\[\begin{split}\begin{aligned}
0
& = - \int_{\ell} \left\{ ( \delta\widetilde{\mathbf{s}}' + \hat{\mathbf{z}} \times \delta\widetilde{\boldsymbol{\theta}} ) \cdot \mathbf{F} + \widetilde{\boldsymbol{\theta}}' \cdot \mathbf{M} \right\} + \int_{\ell} \left\{ \delta \widetilde{\mathbf{s}} \cdot \mathbf{f} + \delta \widetilde{\boldsymbol\theta} \cdot \mathbf{m} \right\} + \left. \left[ \delta \widetilde{\mathbf{s}} \cdot \mathbf{F} + \delta \widetilde{\boldsymbol{\theta}} \cdot \mathbf{M} \right] \right|_{\partial \ell / S_D} \\
\end{aligned}\end{split}\]
It looks like that this relation doesn’t add too much info beyond the weak form of the problem! Is that true? Spaces of test and basis functions and the prescription of essential boundary conditions need some words!
If the constitutive equation is introduced to use the kinematic variables as the independent variables the symmetry of this formulation of the elastic problem naturally arises, as already shown in the weak form of the equilibrium equations.
7.2.2.4. Principle of complementary virtual work
If the test function of the weak form of the congruence conditions is chosen to be the variation of an equilibrated stress field, \(\boldsymbol\Sigma_z = \delta \boldsymbol\sigma_z\), s.t.
\[\begin{split}\begin{aligned}
\mathbf{0} & = \delta \widetilde{\mathbf{F}}' \\
\mathbf{0} & = \delta \widetilde{\mathbf{M}}' + \hat{\mathbf{z}} \times \delta \widetilde{\mathbf{F}} \\
\end{aligned}\end{split}\]
and \(\left.\delta \widetilde{\mathbf{F}}\right|_{S_D} = \mathbf{0}\), \(\left.\delta \widetilde{\mathbf{M}}\right|_{S_D} = \mathbf{0}\), where essential boundary conditions are prescribed (some constraints prescribe only some components),
\[\begin{split}\begin{aligned}
0
& = \int_{V} \delta \widetilde{\boldsymbol{\sigma}}_z \cdot \boldsymbol\varepsilon_z - \left[ \widetilde{\delta \mathbf{F}} \cdot \mathbf{s}_P + \delta \widetilde{\mathbf{M}} \cdot \boldsymbol\theta \right]_{\partial \ell / S_N} = \\
& = \int_V \begin{bmatrix} \delta \widetilde{\mathbf{F}} \\ \delta \widetilde{\mathbf{M}} \end{bmatrix}^T \left( \mathbf{S} \begin{bmatrix} \mathbf{F} \\ \mathbf{M} \end{bmatrix} + \mathbf{a} \Delta T \right) - \left[ \widetilde{\delta \mathbf{F}} \cdot \mathbf{s}_P + \delta \widetilde{\mathbf{M}} \cdot \boldsymbol\theta \right]_{\partial \ell / S_N} = \\
\end{aligned}\end{split}\]
7.2.2.5. Principle of stationariety of total potential energy
Principle of virtual work can be recast as a principle of stationariety of total potential energy, being \(\widetilde{\mathbf{s}} = \mathbf{s}\), \(\widetilde{\boldsymbol\theta} = \boldsymbol\theta\),
\[\begin{split}\begin{aligned}
0
= & - \int_{\ell} ( \delta \mathbf{s}' + \hat{\mathbf{z}} \times \delta \boldsymbol{\theta} ) \cdot \left[ \mathbf{K}_{fs} \left( \mathbf{s}'_P + \hat{\mathbf{z}} \times \boldsymbol\theta \right) + \mathbf{K}_{f\theta} \boldsymbol\theta' - \mathbf{b}_f \Delta T \right] + \\
& - \int_{\ell} \delta \boldsymbol{\theta}' \cdot \left[ \mathbf{K}_{ms} \left( \mathbf{s}'_P + \hat{\mathbf{z}} \times \boldsymbol{\theta} \right) + \mathbf{K}_{m\theta} \boldsymbol{\theta}' - \mathbf{b}_m \Delta T \right] + \\
& + \int_{\ell} \left\{ \delta \mathbf{s} \cdot \mathbf{f} + \delta \boldsymbol{\theta} \cdot \mathbf{m} \right\} + \left. \left[ \delta \mathbf{s} \cdot \mathbf{F} + \delta \boldsymbol{\theta} \cdot \mathbf{M} \right] \right|_{\partial \ell / \partial S_D} = \\
= & \delta \left\{ - \dfrac{1}{2} \int_{\ell} \begin{bmatrix} \mathbf{s}'_P + \hat{\mathbf{z}} \times \boldsymbol{\theta} \\ \boldsymbol{\theta}' \end{bmatrix} \cdot \begin{bmatrix} \mathbf{K}_{fs} & \mathbf{K}_{f\theta} \\ \mathbf{K}_{ms} & \mathbf{K}_{m \theta} \end{bmatrix} \cdot \begin{bmatrix} \mathbf{s}'_P + \hat{\mathbf{z}} \times \boldsymbol{\theta} \\ \boldsymbol{\theta}' \end{bmatrix} \, d \ell - \dfrac{1}{2} \left.\begin{bmatrix} \mathbf{s} \\ \boldsymbol{\theta} \end{bmatrix} \cdot \mathbf{K}_{R} \cdot \begin{bmatrix} \mathbf{s}_P \\ \boldsymbol{\theta} \end{bmatrix}\right|_{S_R} \right. + \\
& + \left. \int_{\ell} \begin{bmatrix} \mathbf{s}_P \\ \boldsymbol{\theta} \end{bmatrix} \cdot \begin{bmatrix} \mathbf{f} \\ \mathbf{m} \end{bmatrix} \, d \ell + \left. \begin{bmatrix} \mathbf{s}_P \\ \boldsymbol{\theta} \end{bmatrix} \cdot \begin{bmatrix} \mathbf{F} \\ \mathbf{M} \end{bmatrix} \right|_{S_N} \right\} \ ,
\end{aligned}\end{split}\]
with some notation abuse, and prescribed temperature \(\Delta T\), s.t. \(\delta \Delta T = 0\), and loads.
7.2.2.6. Principle of stationariety of total complementary potential energy
If test functions of the congruence conditions are internal actions of an equilibrated and congruent solution (and thus the only solution of the elastic problem, if well-posed), and thus \(\delta \widetilde{\mathbf{F}} = \delta \mathbf{F}\) and \(\delta \widetilde{\mathbf{M}} = \delta \mathbf{M}\), the principle of complementary virtual work can be recast as the principle of stationariety of the total complementary potential energy, for given \(\mathbf{s}_P\), \(\boldsymbol\theta\) on \(S_N\),
\[\begin{split}\begin{aligned}
0
& = \delta \left\{ \int_{\ell} \left( \dfrac{1}{2} \begin{bmatrix} \mathbf{F} \\ \mathbf{M} \end{bmatrix}^T \mathbf{S} \begin{bmatrix} \mathbf{F} \\ \mathbf{M} \end{bmatrix} + \begin{bmatrix} \mathbf{F} \\ \mathbf{M} \end{bmatrix}^T \mathbf{a} \Delta T \right) \, d \ell - \left. \begin{bmatrix} \mathbf{F} \\ \mathbf{M} \end{bmatrix}^T \begin{bmatrix} \mathbf{s}_P \\ \boldsymbol{\theta} \end{bmatrix} \right|_{S_N} + \left.\dfrac{1}{2} \begin{bmatrix} \mathbf{F} \\ \mathbf{M} \end{bmatrix}^T \mathbf{S}_R \begin{bmatrix} \mathbf{F} \\ \mathbf{M} \end{bmatrix}\right|_{S_R} \right\} = \\
& = \delta \Pi^* \ ,
\end{aligned}\end{split}\]
if the b.c. on Robin boundaries reads…
