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Fev
09
2024

Mecânica dos Sólidos

Aulas


Links de interesse:

Para praticar a execução de diagramas de esforços Resistência dos Materiais ISEP cossenos diretores de um vetor Componentes cartesianas do tensor de tensões

Para veres se o teu círculo de Mohr está bem desenhado

Círculos de Mohr (3d) no Wolfram Alpha - É preciso instalar um programa (de confiança) - CDF - para se poder ver.

Estudo

$$\vec{T} = [\sigma] \cdot \hat{n}$$

||{\vec \sigma}|| = [\sigma] \cdot \hat{n} \cdot \hat{n}

\vec{\sigma} = [\sigma] \cdot \hat{n} \cdot \hat{n} \cdot \hat{n}

\vec T = \vec{\tau} + \vec{\sigma}

\vec T^2 = ||\vec{\tau}||^2 + ||\vec{\sigma}||^2

\vec T = \vec{\tau} + \vec{\sigma}

Três maneiras de axar as tensões num plano rodado em relação ao original:


Num plano, para axar as tensões principais e o ângulo qe as direções das tensoes principais fazem com o referencial. A matriz de tensões será 2x2, calculam-se os vetores e valores próprios. Ordenam-se de maneira qe $\sigma_1 > \sigma_2$. Sabendo os cossenos diretores tira-se o ângulo qe estes fazem com as direções dos eixos do referencial.

Conselhos

Beams - Tension & compression

Stress

$\sigma = \frac{F}{A}$

Strain

$\varepsilon=\frac{\Delta l}{l}$

Young's modulus (modulus of elasticity)

$E=\frac{\sigma}{\varepsilon}$

Stiffness

C=\frac{F}{\Delta l}=\frac{EA}{l}

Poisson's ratio

\nu=\frac{\frac{\Delta h}{h}}{\frac{\Delta l}{l}}

For metals \nu \approx 0.3

Beams - Shear

Average shear stress

\tau=\frac{F_y}{A}=\frac{F_y}{w\cdot h}

Shear strain

\gamma=\frac{\delta}{l}=\tan{\alpha}

Shear modulus

G=\frac{\tau}{\gamma\ }=\frac{F_y}{A\cdot\gamma}=\frac{E}{2\left(1+\nu\right)}

Shear stiffness

C_y=\frac{F_y}{\delta}=\frac{G\cdot A}{l}

Beams - Bending

Beam: Bending - Load Case 1

Curvature $\theta$
Deflection $\delta y$
Reaction force $R$
Shear force $V$
Reaction moment $M_R$
Stress $\sigma$
Stiffness $C$
$\theta_A=0$

$\theta_B=\frac{{\mathrm{FL}}^2}{2EI_x}$

$\delta y_z=-\frac{\left({\rm Fz}^2\right)\left(3L-z\right)}{6EI_x}$

$\delta y_{max}=-\frac{{\rm FL}^3}{3{\rm EI}_x}@\ z=L$
$R_A=F$

$R_B=N.A.$

$V_z=F$

$M_{Rz}=F\left(z-L\right)$

$M_{Rmax}=-FL\ @\ z=0$
$\left|\sigma_z\right|=\frac{\left|M_{Rz}\right|}{I_x}u=-\frac{\left|F\right|u\left(z-L\right)}{I_x}$

$\left|\sigma_{max}\right|=\frac{\left|F\right|Lu}{I_x}@\ z=0$

$\left|C_y\right|=\left|\frac{F}{\delta y_{z\left(F\right)}}\right|=\frac{3{\rm EI}_x}{L^3}\ @\ z=L$


Beam: Bending - Load Case 2
Curvature $\theta$
Deflection $\delta y$
Reaction force $R$
Shear force $V$
Reaction moment $M_R$
Stress $\sigma$
Stiffness $C$
$\theta_A=0$

$\theta_B=\frac{ML}{{\rm EI}_x}$

$\delta y_z=-\frac{{\rm Mz}^2}{2{\rm EI}_x}$

$\delta y_{max}=-\frac{{\rm ML}^2}{2{\rm EI}_x}@\ z=L$
$R_A=0$

$R_B=N.A. $

$V_z=N.A. $

$M_{Rz}=M $

$M_{Rmax}=\ M\ @\ z=const. $
$\left|\sigma_z\right|=\frac{\left|M_{Rz}\right|}{I_x}u=\frac{\left|M\right|u}{I_x}$

$\left|\sigma_{max}\right|=\frac{\left|M\right|u}{I_x}\ \ @\ z=const.$

$\left|K_y\right|=\left|\frac{M}{\theta_B}\right|=\frac{{\rm EI}_x}{L}\ @\ z=L $


Beam: Bending - Load Case 3
Curvature $\theta$
Deflection $\delta y$
Reaction force $R$
Shear force $V$
Reaction moment $M_R$
Stress $\sigma$
Stiffness $C$
$\theta_A=0$

$\theta_B=\frac{{\rm qL}^3}{6{\rm EI}_x} $

$\delta y_z=-\frac{{\mathrm{qz}}^2\left(6L^2-4Lz+z^2\right)}{24{\mathrm{EI}}_x}$

$\delta y_{max}=-\frac{{\mathrm{qL}}^4}{8{\mathrm{EI}}_x}@\ z\ =\ L $
$R_A=qL $

$R_B=N.A. $

$V_z=q\left(L-z\right) $

$M_{Rz}=-\frac{q\left(L-z\right)^2}{2} $

$M_{Rmax}=-\frac{{\rm qL}^2}{2}\ @\ z=0 $
$\left|\sigma_z\right|=\frac{\left|M_{Rz}\right|}{I_x}u=\frac{\left|q\right|u\left(L-z\right)^2}{2I_x} $

$\left|\sigma_{max}\right|=\frac{\left|q\right|L^2u}{2I_x}@ z=0 $

$\left|C_y\right|=\left|\frac{q}{\delta y_{max}}\right|=\frac{8{\rm EI}_x}{L^4}@\ z=L $


Beam: Bending - Load Case 4
ConditionCurvature $\theta$
Deflection $\delta y$
Reaction force $R$
Shear force $V$
Reaction moment $M_R$
Stress $\sigma$
Stiffness $C$
$0\le z\le a$$\theta_A=\frac{Fab\left(L+b\right)}{6{\rm EI}_xL} $

$\theta_B=\frac{Fab\left(L+a\right)}{6{\rm EI}_xL} $

$\delta y_z=-\frac{{\rm Fab}^2}{6{\rm EI}_x}\left[\left(1+\frac{L}{b}\right)\frac{z}{L}-\frac{z^3}{abL}\right] $

$\delta y_{max}=-\frac{Fb\sqrt{\left(L^2-b^2\right)^3}}{9\sqrt3{\rm EI}_xL}\ @\ z=\sqrt{\frac{L^2-b^2}{3}}\ $ only if $a>b$
$R_A=\ \frac{Fb}{L} $

$R_B=\frac{Fa}{L} $

$V_z=\frac{-Fb}{L} $

$M_{Rz}=\frac{Fbz}{a+b} $

$M_{Rmax}=\frac{Fba}{a+b}\ @\ z=a $
$\left|\sigma_z\right|=\frac{\left|M_{Rz}\right|}{I_x}u=\frac{\left|F\right|bzu}{I_xL} $

$\left|\sigma_{max}\right|=\frac{\left|F\right|bau}{I_x\left(a+b\right)}\ @\ z=a $

$\left|C_y\right|=\left|\frac{F}{\delta y_{z\left(F\right)}}\right|=\frac{3{\rm EI}_xL}{a^2b^2}\ @\ z=a$
$a\le z\le L$$\theta_A=\frac{Fab\left(L+b\right)}{6{\rm EI}_xL} $

$\theta_B=\frac{Fab\left(L+a\right)}{6{\rm EI}_xL} $

$\delta y_z=-\frac{{\rm Fa}^2b}{6{\rm EI}_x}\left[\left(1+\frac{L}{a}\right)\frac{L-z}{L}-\frac{\left(L-z\right)^3}{abL}\right] $

$\delta y_{max}=-\frac{Fa\sqrt{\left(L^2-a^2\right)^3}}{9\sqrt3{\rm EI}_xL}\ @\ z=L-\sqrt{\frac{L^2-a^2}{3}}$ only if $b>a$
$R_A=\frac{Fb}{L} $

$R_B=\frac{Fa}{L}$

$V_z=\frac{Fb}{L}-F $

$M_{Rz}=\left(\frac{Fbz}{L}\right)-F\left(z-a\right) $

$M_{Rmax}=\frac{Fba}{L}\ @\ z=a$
$\left|\sigma_z\right|=\frac{\left|M_{Rz}\right|}{I_x}u=\ \frac{\left|\left(\frac{Fbz}{L}\right)-F\left(z-a\right)\right|}{I_x}u $

$\left|\sigma_{max}\right|=\frac{\left|F\right|bau}{I_xL}@\ z=a $

$\left|C_y\right|=\left|\frac{F}{\delta y_{z\left(F\right)}}\right|=\frac{3{\rm EI}_xL}{a^2b^2}@\ z=a $


Beam: Bending - Load Case 5
Curvature $\theta$
Deflection $\delta y$
Reaction force $R$
Shear force $V$
Reaction moment $M_R$
Stress $\sigma$
Stiffness $C$
$\theta_A=\frac{ML}{6{\rm EI}_x} $

$\theta_B=\frac{ML}{3{\rm EI}_x} $

$\delta y_z=-\frac{{\rm ML}^2}{6{\rm EI}_x}\left[\frac{z}{L}-\left(\frac{z}{L}\right)^3\right] $

$\delta y_{max}=-\frac{{\rm ML}^2}{9\sqrt3{\rm EI}_{x\ }}\ @\ z=\frac{L}{\sqrt3} $
$R_A=\frac{M}{L} $

$R_B=-\frac{M}{L}$

$V_z=\frac{M}{L} $

$M_{Rz}=\left(\frac{M}{L}\right)z $

$M_{Rmax}=M\ @\ z=L $
$\left|\sigma_z\right|=\frac{\left|M_{Rz}\right|}{I_x}u=\frac{\left|M\right|zu}{{\mathrm{LI}}_x}$

$\left|\sigma_{max}\right|=\frac{\left|M\right|u}{I_x}@\ z=L$

$\left|K_y\right|=\left|\frac{M}{\theta_B}\right|=\frac{3{\mathrm{EI}}_x}{L}\ @\ z=L$


Beam: Bending - Load Case 6
Curvature $\theta$
Deflection $\delta y$
Reaction force $R$
Shear force $V$
Reaction moment $M_R$
Stress $\sigma$
Stiffness $C$
$\theta_A=\frac{{\rm qL}^3}{24{\rm EI}_x} $

$\theta_B=\frac{{\rm qL}^3}{24{\rm EI}_x} $

$\delta y_z=-\frac{{\rm qL}^4}{24{\rm EI}_x}\left[\frac{z\ }{L}-2\left(\frac{z}{L}\right)^3+\left(\frac{z}{L}\right)^4\ \right] $

$\delta y_{max}=-\frac{5}{384}\frac{{\rm qL}^4}{{\rm EI}_x}\ @\frac{L}{2} $
$R_A=\frac{qL}{2} $

$R_B=\frac{qL}{2} $

$V_z=\frac{qL}{2}-qz $

$M_{Rz}=\frac{qz\left(L-z\right)}{2} $

$M_{Rmax}=\frac{{\rm qL}^2}{8}\ @\ z=\frac{L}{2} $
$\left|\sigma_z\right|=\frac{\left|M_{Rz}\right|}{I_x}u=\frac{\left|qz\left(L-z\right)u\right|}{2I_x}$

$\left|\sigma_{max}\right|=\frac{\left|q\right|L^2u}{8I_x}@\ z=\frac{L}{2}$

$\left|C_y\right|=\left|\frac{q}{\delta y_{max}}\right|=\frac{384EI_x}{5L^4}\ @\ z=\frac{L}{2}$


Beam: Bending - Load Case 7
ConditionCurvature $\theta$
Deflection $\delta y$
Reaction force $R$
Shear force $V$
Reaction moment $M_R$
Stress $\sigma$
Stiffness $C$
$0\le z\le a$$\theta_A=\frac{FbL}{6{\rm EI}_x}$

$\theta_B=\frac{FbL}{3{\rm EI}_x}$

$\delta y_z=\frac{{\rm Fba}^2}{6{\rm EI}_x}\left[\frac{z}{a}-\left(\frac{z}{a}\right)^3\right]$

$\delta y_{max}=\frac{{\rm Fba}^2}{9\sqrt3{\rm EI}_x}\ @\ z=\frac{a}{\sqrt3}$
$R_A=-\frac{Fb}{a}$

$R_B=F+\frac{Fb}{a}$

$V_z=-\left(\frac{Fb}{a}\right)$

$M_{Rz}=-\frac{Fbz}{a}$

$M_{Rmax}=-Fb\ @\ z=a$
$\left|\sigma_z\right|=\frac{\left|M_{Rz}\right|}{I_x}u=\frac{\left|F\right|bzu}{{\rm aI}_x}$

$\left|\sigma_{max}\right|=\frac{\left|F\right|bu}{I_x}@z=a$

$\left|C_y\right|=\left|\frac{F}{\delta y_{z\left(F\right)}}\right|=\frac{3{\rm EI}_x}{b^2L}\ @z=L$
$a\le z\le L$$\theta_C=\frac{Fb\left(2a+3b\right)}{3{\rm EI}_x}$

$\delta y_z=\ -\frac{F\left(\left(-z\right)+a\right)}{6{\rm EI}_x}\left[ab-3bz+z^2-2az+a^2\right]$

$\delta y_{max}=-\frac{{\rm Fb}^2L}{3{\rm EI}_x}\ @\ z=L$
$R_A=-\frac{Fb}{a}$

$R_B=F+\frac{Fb}{a}$

$V_z=F$

$M_{Rz}=-F\left[\left(-z\right)+a+b\right]$

$M_{Rmax}=-Fb\ @\ z=a$
$\left|\sigma_z\right|=\frac{\left|M_{Rz}\right|}{I_x}u=\frac{\left|F\right|\left[\left(-z\right)+L\right)}{I_x}u$

$\left|\sigma_{max}\right|=\frac{\left|F\right|bu}{I_x}\ @z=a$

$\left|C_y\right|=\left|\frac{F}{\delta y_{z\left(F\right)}}\right|=\frac{3{\rm EI}_x}{b^2L}\ @z=L$


Beam: Bending - Load Case 8
ConditionCurvature $\theta$
Deflection $\delta y$
Reaction force $R$
Shear force $V$
Reaction moment $M_R$
Stress $\sigma$
Stiffness $C$
$0\le z\le a$$\theta_A=\frac{{\rm qb}^2a}{12{\rm EI}_x} $

$\theta_B=\frac{{\rm qb}^2a}{6{\rm EI}_x} $

$\delta y_z=\frac{{{\rm qb}^2a}^2}{12{\rm EI}_x}\left[\frac{z}{a}-\left(\frac{z}{a}\right)^3\right] $

$\delta y_{max}=\frac{{{\rm qb}^2a}^2}{18\sqrt3{\rm EI}_x}\ @\ z=\frac{a}{\sqrt3}$
$R_A=-\frac{{\rm qb}^2}{2a} $

$R_B=\frac{qb\left(b+2a\right)}{2a} $

$V_z=-\frac{{\rm qb}^2}{2a} $

$M_{Rz}=-\left(\frac{{\rm qb}^2}{2a}\right)z $

$M_{Rmax}=-\frac{{\rm qb}^2}{2}\ \ @\ z=a $
$\left|\sigma_z\right|=\frac{\left|M_{Rz}\right|}{I_x}u=\frac{\left|q\right|b^2zu}{2{\rm aI}_x} $

$\left|\sigma_{max}\right|=\frac{\left|q\right|b^2u}{2I_x}\ \ @\ z=a $

$\left|C_y\right|=\left|\frac{q}{\delta y_{max}}\right|=\frac{18\sqrt3{\mathrm{EI}}_x}{b^2a^2}\ @\ z=\frac{a}{\sqrt3} $
$a\le z\le L$$\theta_C=\frac{{\rm qb}^2L}{6{\rm EI}_x} $

$\delta y_z=\ -\frac{{\rm qb}^4}{24{\rm EI}_x}\left[4\frac{a}{b}\frac{z-a}{b}+6\left(\frac{z-a}{b}\right)^2-4\left(\frac{z-a}{b}\right)^3+{\left(\frac{z-a}{b}\right)}^4\right]$

$\delta y_{max}=-\frac{{\rm qb}^3\left(4a+3b\right)}{24{\rm EI}_x}\ @\ \mathrm{z}=L $
$R_A=\ -\frac{{\rm qb}^2}{2a} $

$R_B=\frac{qb\left(b+2a\right)}{2a} $

$V_z=qb-q\mathrm{(z+a)} $

$M_{Rz}=\frac{-q\left(b^2-2bz+{\mathrm{2ba+z}}^2-2za+a^2\right)}{2} $

$M_{Rmax}=-\frac{{\rm qb}^2}{2}\ \ @\ z=a $
$\left|\sigma_z\right|=\frac{\left|M_{Rz}\right|}{I_x}u= \frac{\left|qu\left(b^2-{2bz+{\mathrm{2ba+z}}^2-2za+a}^2\right)\right|}{2I_x} $

$\left|\sigma_{max}\right|=\frac{\left|q\right|b^2u}{2I_x}\ \ @\ z=a $

$\left|C_y\right|=\left|\frac{q}{\delta y_{max}}\right|=\frac{24{\rm EI}_x}{b^3\left(4a+3b\right)}\ @\ \mathrm{z}=L $


Beam: Bending - Load Case 9
ConditionCurvature $\theta$
Deflection $\delta y$
Reaction force $R$
Shear force $V$
Reaction moment $M_R$
Stress $\sigma$
Stiffness $C$
$0\le z\le a$$\theta_A=\frac{{\rm Fab}^2}{4{\rm EI}_xL} $

$\theta_B=0 $

$\delta y_z=-\frac{{\rm FLb}^2}{4{\rm EI}_x}\left[\frac{az}{L^2}-\frac{2}{3}\left(1+\frac{a}{2L}\right)\left(\frac{z}{L}\right)^3\right] $

$\delta y_{max}=-\frac{{\rm ab}^2F\sqrt{\frac{a}{2L+a}}}{6{\rm EI}_x}@z=L\cdot\sqrt{\frac{\frac{a}{2L}}{1+\frac{a}{2L}}}$
Only if $a \geq 0.414L$
$R_A=F\left(\frac{b}{L}\right)^2\left(1+\frac{a}{2L}\right) $

$R_B=F\left(\frac{a}{L}\right)^2\left(1+\frac{b}{2L}+\frac{3}{2}\frac{b}{a}\right) $

$V_z=F\left(\frac{b}{L}\right)^2\left(1+\frac{a}{2L}\right) $

$M_{Rz}=\ Fz\left(\frac{b}{L}\right)^2\left(1+\frac{a}{2L}\right)$

$M_{Rmax}=\frac{{\mathrm{Fab}}^2}{L^2}\left(1+\frac{a}{2L}\right)$
$@\ z=a$, with $a\le0.414L $
$\left|\sigma_z\right|=\frac{\left|M_{Rz}\right|}{I_x}u=\frac{\left|F\right|{\rm zb}^2u\left(2L+a\right)}{2L^3I_x} $

$\left|\sigma_{max}\right|=\frac{\left|F\right|{\rm ab}^2u\left(1+\frac{a}{2L}\right)}{L^2I_x}$
$@\ z=a$, with $a\le 0.414L$

$\left|C_y\right|=\left|\frac{F}{\delta y_{z\left(F\right)}}\right|=\left|\frac{12{\rm EI}_xL^3}{b^2a^2\left[3L^2-2aL-a^2\right]}\right|\ @\ z=a $
$a\le z\le L$$\theta_A=\frac{{\rm Fab}^2}{4{\rm EI}_xL} $

$\theta_B=0 $

$\delta y_z=\frac{Fa\left(L-z\right)^2}{12L^3}\frac{2{\rm La}^2-3L^2z+a^2z}{{\rm EI}_x} $

$\delta y_{max}=\delta\ y_z\left(z_{max}\right)\ @\ z_{max}=L\cdot\left[\frac{2aL+bL-ba}{2aL+3bL+ba}\right]$
Only if $a \le 0.414L$
$R_A=\ F\left(\frac{b}{L}\right)^2\left(1+\frac{a}{2L}\right)$

$R_B=F\left(\frac{a}{L}\right)^2\left(1+\frac{b}{2L}+\frac{3}{2}\frac{b}{a}\right) $

$V_z=-F\left(\frac{a}{L}\right)^2\left(1+\frac{b}{2L}+\frac{3}{2}\frac{b}{a}\right) $

$M_{Rz}=\ Fz\left(\frac{b}{L}\right)^2\left(1+\frac{a}{2L}\right)-F\left(z-a\right) $

$M_{Rmax}=-F\frac{ab}{L}\left(1-\frac{b}{2L}\right)$
$@\ z=L$, with$\ a\geq0.414L $
$\left|\sigma_z\right|=\frac{\left|M_{Rz}\right|}{I_x}u=\frac{\left|Fu\left(2{\rm zLb}^2+b^2za-2L^3z+2L^3a\right)\right|}{2L^3I_x}$

$\left|\sigma_{max}\right|=\frac{\left|Fabu\left(1-\frac{b}{2L}\right)\right|}{{\rm LI}_x}$
$@\ z=L$, with$\ a\geq0.414L $

$\left|C_y\right|=\left|\frac{F}{\delta y_{z\left(F\right)}}\right|=\left|\frac{12{\rm EI}_xL^3}{b^2a^2\left[3L^2-2aL-a^2\right]}\right|\ @\ z=a $


Beam: Bending - Load Case 10
ConditionCurvature $\theta$
Deflection $\delta y$
Reaction force $R$
Shear force $V$
Reaction moment $M_R$
Stress $\sigma$
Stiffness $C$
$0\le z\le a$
$b>a$
$\theta_A=0 $

$\theta_B=0 $

$\delta y_z=-\frac{{\rm FLa}^2}{6{\rm EI}_x}\left[3\frac{b}{L^3}\left(L-z\right)^2-\left(1+\frac{2b}{L}\right)\frac{\left(L-z\right)^3}{L^3}\right] $

$\delta y_{max}=-\frac{2}{3}\frac{{\rm Fa}^3b^2}{{\rm EI}_xL^2}\left(\frac{1}{1+\frac{2a}{L}}\right)^2\ @\ z=L\frac{1}{1+\frac{L}{2a}}$
$R_A=F\left(\frac{b}{L}\right)^2\left(1+\frac{2a}{L}\right) $

$R_B=F\left(\frac{a}{L}\right)^2\left(1+\frac{2b}{L}\right) $

$V_z=F\left(\frac{b}{L}\right)^2\left(1+\frac{2a}{L}\right)$

$M_{Rz}=\frac{{\rm Fb}^2\left(zL+2za+aL\right)}{L^3} $

$M_{Rmax}=-\frac{{\rm Fab}^2}{L^2}\ @\ z=0 $
$\left|\sigma_z\right|=\frac{\left|M_{Rz}\right|}{I_x}u=\frac{\left|Fu\left[b^2zL+2b^2za+b^2aL-L^3z+L^3a\right]\right|}{L^3I_x} $

$\left|\sigma_{max}\right|=\frac{\left|F\right|{\rm uba}^2}{L^2I_x}@\ z=L $

$\left|C_y\right|=\left|\frac{F}{\delta y_{z\left(F\right)}}\right|=\frac{3{\rm EI}_xL^3}{b^2a^3\left(L-a\right)} @\ z=a$
$a\le z\le L$
$a>b$
$\theta_A=0 $

$\theta_B=0 $

$\delta y_z=-\frac{{\mathrm{FLa}}^2}{6{\mathrm{EI}}_x}\left[3\frac{b}{L^3}\left(L-z\right)^2- \left(1+\frac{2b}{L}\right)\frac{\left(L-z\right)^3}{L^3}\right]$

$\delta y_{max}=-\frac{2}{3}\frac{{\mathrm{Fa}}^3b^2}{{\mathrm{EI}}_xL^2}\left(\frac{1}{1+\frac{2a}{L}}\right)^2@\ z=L\frac{1}{1+\frac{L}{2a}} $
$R_A=F\left(\frac{b}{L}\right)^2\left(1+\frac{2a}{L}\right) $

$R_B=F\left(\frac{a}{L}\right)^2\left(1+\frac{2b}{L}\right) $

$V_z=-F\left(\frac{a}{L}\right)^2\left(1+\frac{2b}{L}\right) $

$M_{Rz}=\frac{F\left[b^2zL+2b^2za+b^2aL-L^3z+L^3a\right]}{L^3} $

$M_{Rmax}=-\frac{{\rm Fba}^2}{L^2}@\ z=L$
$\left|\sigma_z\right|=\frac{\left|M_{Rz}\right|}{I_x}u=\frac{\left|Fu\left[b^2zL+2b^2za+b^2aL-L^3z+L^3a\right]\right|}{L^3I_x} $

$\left|\sigma_{max}\right|=\frac{\left|F\right|{\mathrm{uba}}^2}{L^2I_x}\ @\ z=L $

$\left|C_y\right|=\left|\frac{F}{\delta y_{z\left(F\right)}}\right|=\frac{3{\mathrm{EI}}_xL^3}{b^2a^3\left(L-a\right)}\ @\ z=a $


Beam: Bending - Load Case 11
Curvature $\theta$
Deflection $\delta y$
Reaction force $R$
Shear force $V$
Reaction moment $M_R$
Stress $\sigma$
Stiffness $C$
$\theta_A=0$

$\theta_B=0 $

$\delta y_z=-\frac{{\rm qL}^4}{24{\rm EI}_x}\left[\left(\frac{z}{L}\right)^2-2\left(\frac{z}{L}\right)^3+\left(\frac{z}{L}\right)^4\right] $

$\delta y_{max}=-\frac{{\rm qL}^4}{384{\rm EI}_x}\ @\ z=\frac{L}{2} $
$R_A=\frac{1}{2}qL$

$R_B=\frac{1}{2}qL $

$V_z=\frac{qL}{2}-qz $

$M_{Rz}=\frac{q\left({-6z}^2+6Lz-L^2\right)}{12} $

$M_{Rmax}=-\frac{{\rm qL}^2}{12}\ @\ \frac{z=0}{z=L} $
$\left|\sigma_z\right|=\frac{\left|M_{Rz}\right|}{I_x}u=\frac{\left|uq\left({-6z}^2+6Lz-L^2\right)\right|}{12I_x} $

$\left|\sigma_{max}\right|=\frac{\left|q\right|L^2u}{12I_x}\ @\ \frac{z=0}{z=L} $

$\left|C_y\right|=\left|\frac{q}{\delta y_{max}}\right|=\frac{384{\rm EI}_x}{L^4}\ @\ \mathrm{z}=\frac{L}{2} $


Beam: Bending - Load Case 12
Curvature $\theta$
Deflection $\delta y$
Reaction force $R$
Shear force $V$
Reaction moment $M_R$
Stress $\sigma$
Stiffness $C$
$\theta_A=0$

$\theta_B=0 $

$\delta y_z=-\frac{{\rm Fz}^2\left(3L-2z\right)}{12{\rm EI}_x} $

$\delta y_{max}=-\frac{{\rm FL}^3}{12{\rm EI}_x}\ @\ z=L $
$R_A=F $

$R_B=0 $

$V_z=F $

$M_{Rz}=\frac{FL-2Fz}{2} $

$M_{Rmax}=\frac{FL}{2}\ resp.\ -\frac{FL}{2}$
$@\ z=0\ resp.\ L $
$\left|\sigma_z\right|=\frac{\left|M_{Rz}\right|}{I_x}u=\frac{\left|u\left(FL-2Fz\right)\right|}{2I_x} $

$\left|\sigma_{max}\right|=\frac{\left|F\right|Lu}{2I_x}\ @\ \frac{z=0}{z=L} $

$\left|C_y\right|=\left|\frac{F}{\delta y_{max}}\right|=\frac{12{\rm EI}_x}{L^3}\ @\ z=L $
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Beams - Torsion

Beam theory - Torsion
Angular twist

For a torsionally loaded beam, the angular twist is described by:

$\large \theta=\frac{T\cdot L}{G\cdot J_T}$

When a straight beam is subjected to an axial moment, each cross-section twists around its torsional center. Shear stresses occur within the cross-sectional planes of the beam. $J_T$ is the torsion constant. It is equal to the polar moment of inertia $I_z$ if the cross section is circular. For non-circular cross sections warping occurs which reduces the effective torsion constant. For these shapes, approximate solutions of the torsion constant and maximum stress are given in the table below.

$G$ is the shear modulus. The relation between the shear modulus $G$ and the elastic modulus $E$ is defined by the following formula:

$G=\frac E{2\left(1+v\right)}\approx 0.38E$ (For most metals)

Rotational stiffness

The rotational stiffness of a torsionally loaded beam is:

$K_z=\frac{T}{\theta}=\frac{G\cdot J_T}{L}$

Maximum shear stress

For a torsionally loaded beam with a circular cross-section, the maximum shear stress can be calculated with:

$\tau_{max}=\frac{Tr}{J_T}$

Beam theory - Torsion

For pure shear, the Von Mises stress is then equal to:

$$\sigma_V=\sqrt3\tau_{max}$$

For non-circular cross-sections the equations below can be used.

Beam theory - Torsion
Torsion constant $J_T$ Maximum shear stress $\tau_{max}$
$J_T=I_Z=\frac{\pi}{2}r^4$ $\tau_{max}=\frac{2T_{max}}{{\pi}r^3}$


Beam theory - Torsion
Torsion constant $J_T$ Maximum shear stress $\tau_{max}$
$J_T=I_Z=\frac{\pi}{2}(r_o^4-r_i^4)$ $$\tau_{max}=\frac{2T_{max}r_o}{\pi\left(r_o^4-r_i^4\right)}$$


Beam theory - Torsion
Condition Torsion constant $J_T$ Maximum shear stress $\tau_{max}$
$h=w$ $J_T=\frac{9}{64}w^4$ $\tau_{max}=\frac{4.808T}{w^3}$
$w\geq\ h$ $\small J_T=\frac{1}{16}wh^3\left(\frac{16}{3}-3.36\frac{h}{w}\left(1-\frac{h^4}{12w^4}\right)\right)$

$\approx\frac{1}{3}wh^3$ for $w\gg h$
$\small \tau_{max}=\frac{3T}{wh^2}\left(1+0.6095\frac{h}{w}+0.8865\left(\frac{h}{w}\right)^2-1.8023\left(\frac{h}{w}\right)^3+0.9100\left(\frac{h}{w}\right)^4\right)$

$\approx\frac{3T}{wh^2}$ for $ w\gg\ h$


Beam theory - Torsion
Condition Torsion constant $J_T$ Maximum shear stress $\tau_{max}$
For thin walled structures $$J_t=\frac{2t^2\left(w-t\right)^2\left(h-t\right)^2}{\left(w+h\right)t-2t^2}$$ $$\tau_{average}=\frac{T}{2t(w-t)(h-t)}$$ Note: stress is nearly uniform if t is small. There will be higher stresses at inner corners unless fillets of fairly large radius are provided.

Beams - Buckling

Moment of inertia - area

The area moment of inertia (also referred to as second moment of area) is a geometrical property of a shape describing the distribution of points around an axis. In classical mechanics it is used as a measure of a body’s resistance against bending. Note that next to the area moment of inertia, the polar moment of inertia is used as a measure of a body’s resistance to torsion (see Beam: Torsion above). Also, the area moment of inertia should not be confused with the mass moment of inertia, which is used as a measure of how an object resists rotational acceleration about a particular axis (see Mass moment of inertia). The area moment of inertia is typically denoted with an I and has an axis that lies in the plane, the polar second moment of inertia is typically denoted with a J and has an axis perpendicular to the plane.

Cross-section Area moment of inertia
Beam theory - Torsion $$I_x=I_y=\frac{\pi}{4}r^4$$
Beam theory - Torsion $$I_x=I_y=\frac{\pi}{4}\left(r_o^4-r_i^4\right)$$
Beam theory - Torsion $$I_x=\frac{{wh}^3}{12}$$
$$I_y=\frac{w^3h}{12}$$
Beam theory - Torsion $$I_x=\frac{w_oh_o^3-w_ih_i^3}{12}$$
$$I_y=\frac{w_o^3h_o-w_i^3h_i}{12}$$

Moment of inertia - mass

The mass moment of inertia measures the extent to which an object resists rotational acceleration about a particular axis, and is the rotational analogue to mass. The torque needed to achieve an angular acceleration is defined by:

$$T=I\dot{\omega}$$ The mass moment of inertia for several shapes is shown in the table below.
FigureDescriptionMass moment of inertia
Area moment of inertia shapePoint mass$$I=mr^2$$
Area moment of inertia shapeRod of length l and mass mWith l>>r
$$I_{z1}=\frac{1}{12}ml^2$$
$$I_{z2}=\frac{1}{3}mL^2$$
Area moment of inertia shapeSolid cylinder$$I=\frac{1}{2}mr^2$$
Area moment of inertia shapeCylindrical shellWith r>>t
$$I=mr^2$$
Area moment of inertia shapeSphere$$I=\frac{2}{5}mr^2$$
Area moment of inertia shapeBlock$$I_{z1}=\frac{1}{12}m\left(w^2+h^2\right)$$
$$I_{z2}=\frac{1}{3}m\left(w^2+h^2\right)$$