Circle
The first case is for a circular bar of radius \(R\), centered in the origin
Geometric properties
Computing the integrals
\[I_{a, b} = \int_{\Omega} x^a \cdot y^b \ d\Omega\]
\[\begin{split}I = \begin{bmatrix}\pi R^2 & 0 & \frac{\pi R^4}{4} & 0 \\ 0 & 0 & 0 & 0 \\ \frac{\pi R^4}{4} & 0 & \frac{\pi R^6}{24} & 0 \\ 0 & 0 & 0 & 0\end{bmatrix}\end{split}\]
Meaning the basic properties are
\[A = \pi R^2\]
\[Q_x = 0\]
\[Q_y = 0\]
\[I_{xx} = \dfrac{\pi R^4}{4}\]
\[I_{xy} = 0\]
\[I_{yy} = \dfrac{\pi R^4}{4}\]
\[I_{xxx} = 0\]
\[I_{xxy} = 0\]
\[I_{xyy} = 0\]
\[I_{yyy} = 0\]
Torsion properties
The warping function is given by
\[\omega = 0\]
Leads to the torsional constant \(J\)
\[\mathbb{J}_{\omega} = 0\]
\[J = \dfrac{\pi R^4}{2}\]
The torsion center \(\mathbf{T}\)
\[\mathbf{T} = \left(0, \ 0\right)\]
Shear properties
The shear center \(\mathbf{S}\)
\[\mathbf{S} = \left(0, \ 0\right)\]
Strain and Stress
The strain and stresses are given by:
\[\sigma_{zz}(x, y) = \dfrac{F_z}{A}\]
\[\begin{split}\boldsymbol{\sigma}(x, \ y) = \dfrac{F_z}{A} \begin{bmatrix}0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1\end{bmatrix}\end{split}\]
\[\begin{split}\boldsymbol{\varepsilon}(x, \ y) = \dfrac{F_z}{EA}\begin{bmatrix}-\nu & 0 & 0 \\ 0 & -\nu & 0 \\ 0 & 0 & 1\end{bmatrix}\end{split}\]
\[\sigma_{zz}(x, \ y) = \dfrac{4\left(M_{y} \cdot x + M_{x} \cdot y\right)}{\pi R^4}\]
\[\begin{split}\mathbf{\sigma}(x, \ y) = \dfrac{4\left(M_{y} \cdot x + M_{x} \cdot y\right)}{\pi R^4} \begin{bmatrix}0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix}\end{split}\]
Torsion Moment
Shear Forces
Hollowed circle
Hollowed circle of external radius \(R_e\) and internal radius \(R_i\)
Geometric properties
Computing the integrals
\[I_{a, b} = \int_{\Omega} x^a \cdot y^b \ d\Omega\]
\[\begin{split}I = \begin{bmatrix}\pi \left(R_e^2-R_i^2\right) & 0 & \frac{\pi \left(R_e^4-R_i^4\right)}{4} & 0 \\ 0 & 0 & 0 & 0 \\ \frac{\pi \left(R_e^4-R_i^4\right)}{4} & 0 & \frac{\pi \left(R_e^6-R_i^6\right)}{24} & 0 \\ 0 & 0 & 0 & 0\end{bmatrix}\end{split}\]
Meaning the basic properties are
\[A = \pi \left(R_{e}^2 -R_{i}^2\right)\]
\[Q_x = 0\]
\[Q_y = 0\]
\[I_{xx} = \dfrac{\pi \left(R_{e}^4 -R_{i}^4\right) }{4}\]
\[I_{xy} = 0\]
\[I_{yy} = \dfrac{\pi \left(R_{e}^4 -R_{i}^4\right) }{4}\]
\[I_{xxx} = 0\]
\[I_{xxy} = 0\]
\[I_{xyy} = 0\]
\[I_{yyy} = 0\]
The bending center \(\mathbf{B}\)
\[\mathbf{B} = \left(0, \ 0\right)\]
Torsion properties
The warping function is given by
\[\omega = 0\]
Leads to the torsional constant \(J\)
\[\mathbb{J}_{\omega} = 0\]
\[J = \dfrac{\pi \left(R_{e}^4 -R_{i}^4\right)}{2}\]
The torsion center \(\mathbf{T}\)
\[\mathbf{T} = \left(0, \ 0\right)\]
Shear properties
The shear center \(\mathbf{S}\)
\[\mathbf{S} = \left(0, \ 0\right)\]
Strain and Stress
The strain and stresses are given by:
Normal Force
Bending Moments
Torsion Moment
Shear Forces
Ellipse
Ellipse of major axis \(a\) and minor axis \(b\), centered in origin
Geometric properties
\[\begin{split}I = \begin{bmatrix}\pi ab & 0 & \frac{\pi ab^3}{4} & 0 \\ 0 & 0 & 0 & 0 \\ \frac{\pi a^3b}{4} & 0 & \frac{\pi a^3b^3}{24} & 0 \\ 0 & 0 & 0 & 0\end{bmatrix}\end{split}\]
Meaning the basic properties are
\[A = \pi ab\]
\[Q_x = 0\]
\[Q_y = 0\]
\[I_{xx} = \dfrac{\pi ab^3 }{4}\]
\[I_{xy} = 0\]
\[I_{yy} = \dfrac{\pi a^3b }{4}\]
\[I_{xxx} = 0\]
\[I_{xxy} = 0\]
\[I_{xyy} = 0\]
\[I_{yyy} = 0\]
The bending center \(\mathbf{B}\)
\[\mathbf{B} = \left(0, \ 0\right)\]
Torsion properties
The warping function
\[\omega(x, y) = xy\]
The torsion center \(\mathbf{T}\)
\[\mathbf{T} = \left(0, \ 0\right)\]
Shear properties
The shear center \(\mathbf{S}\)
\[\mathbf{S} = \left(0, \ 0\right)\]
Strain and Stress
The strain and stresses are given by:
\[\sigma_{zz}(x, y) = \dfrac{F_z}{A}\]
\[\begin{split}\boldsymbol{\sigma}(x, \ y) = \dfrac{F_z}{A} \begin{bmatrix}0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1\end{bmatrix}\end{split}\]
\[\begin{split}\boldsymbol{\varepsilon}(x, \ y) = \dfrac{F_z}{EA}\begin{bmatrix}-\nu & 0 & 0 \\ 0 & -\nu & 0 \\ 0 & 0 & 1\end{bmatrix}\end{split}\]
\[\sigma_{zz}(x, \ y) = \dfrac{4\left(M_{y} \cdot x + M_{x} \cdot y\right)}{\pi R^4}\]
\[\begin{split}\mathbf{\sigma}(x, \ y) = \dfrac{4\left(M_{y} \cdot x + M_{x} \cdot y\right)}{\pi R^4} \begin{bmatrix}0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix}\end{split}\]
Torsion Moment
Shear Forces
Rectangle
The fourth is a rectangle of base \(b\) and height \(g\)
Geometric properties
\[\begin{split}I = \begin{bmatrix}LH & 0 & \dfrac{bh^3}{12} & 0 \\ 0 & 0 & 0 & 0 \\ \frac{b^3h}{12} & 0 & \frac{b^3h^3}{144} & 0 \\ 0 & 0 & 0 & 0\end{bmatrix}\end{split}\]
Meaning the basic properties are
\[A = bh\]
\[Q_x = 0\]
\[Q_y = 0\]
\[I_{xx} = \dfrac{\pi \left(R_{e}^4 -R_{i}^4\right) }{4}\]
\[I_{xy} = 0\]
\[I_{yy} = \dfrac{\pi \left(R_{e}^4 -R_{i}^4\right) }{4}\]
\[I_{xxx} = 0\]
\[I_{xxy} = 0\]
\[I_{xyy} = 0\]
\[I_{yyy} = 0\]
Torsion properties
The warping function
\[k_n = \dfrac{\pi\left(2n+1\right)}{2}\]
\[\omega = xy - \dfrac{8a^2}{\pi^3}\sum_{n=0}^{\infty} \dfrac{(-1)^n}{\left(2n+1\right)^3} \cdot \dfrac{\sin (k_n \cdot x)\sinh (k_n \cdot y)}{\cosh (k_n \cdot b)}\]
The torsion center \(\mathbf{T}\)
\[\mathbf{T} = \left(0, \ 0\right)\]
Shear properties
The shear center \(\mathbf{S}\)
\[\mathbf{S} = \left(0, \ 0\right)\]
Strain and Stress
The strain and stresses are given by:
\[\sigma_{zz}(x, y) = \dfrac{F_z}{A}\]
\[\begin{split}\boldsymbol{\sigma}(x, \ y) = \dfrac{F_z}{A} \begin{bmatrix}0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1\end{bmatrix}\end{split}\]
\[\begin{split}\boldsymbol{\varepsilon}(x, \ y) = \dfrac{F_z}{EA}\begin{bmatrix}-\nu & 0 & 0 \\ 0 & -\nu & 0 \\ 0 & 0 & 1\end{bmatrix}\end{split}\]
\[\sigma_{zz}(x, \ y) = \dfrac{4\left(M_{y} \cdot x + M_{x} \cdot y\right)}{\pi R^4}\]
\[\begin{split}\mathbf{\sigma}(x, \ y) = \dfrac{4\left(M_{y} \cdot x + M_{x} \cdot y\right)}{\pi R^4} \begin{bmatrix}0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix}\end{split}\]
Torsion Moment
Shear Forces