Beam theory: Bending

Engineering Fundamentals

Introduction

Beam theory is such a common engineering fundamental; it is impossible to be omitted from almost any engineering-specialism. However, this sheet incorporates stress and stiffness as well. For more information on, and calculations of the area moment of inertia I, see sheet: Area moment of inertia.

Validity

Euler-Bernoulli beam theory is only valid with the following assumptions:

  • Cross sections of the beam do not deform in a significant manner under the application of transverse or axial loads and can be assumed as rigid.
  • During deformation, the cross section of the beam is assumed to remain planar and normal to the deformed axis of the beam.

Generally, these criteria are met when the beam is a slender beam with small rotations. This means the slenderness of the beam (ratio L/h) should be larger than 10 and the rotation of the neutral axis (θ) should be smaller than 5°.

Beam theory: Bending validity
Load caseConditionCurvature $\theta$
Sag $\delta y$
Reaction force $R$
Shear force $D$
Reaction moment $M_R$
Stress $\sigma$
Stiffness $C$
Beam theory: Bending - Load Case 1$\theta_A=0$

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

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

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

$R_B=N.A.$

$D_z=F$

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

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

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

$\left|C_y\right|=\left|\frac{F}{\delta y_{z\left(F\right)}}\right|=\frac{3{\rm EI}_x}{L^3}\ @\ z=L$
Beam theory - Load Case 2$\theta_A=0$

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

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

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

$R_B=N.A. $

$D_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|C_y\right|=\left|\frac{M}{\theta_B}\right|=\frac{{\rm EI}_x}{L}\ @\ z=L $
Beam theory - Load Case 3$\theta_A=0$

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

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

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

$R_B=N.A. $

$D_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 theory - Load Case 4$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] $

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

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

$D_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] $

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

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

$D_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 theory - Load Case 5$\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}_{y\ }}\ @\ z=\frac{L}{\sqrt3} $
$R_A=\frac{M}{L} $

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

$D_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}{{\rm LI}_x} $

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

$\left|C_y\right|=\left|\frac{M}{\theta_B}\right|=\frac{3{\rm EI}_x}{L}\ @\ z=L $
Beam theory - Load Case 6$\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} $

$D_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|F\right|bzu}{{\rm aI}_x} $

$\left|\sigma_{max}\right|=\frac{\left|F\right|au}{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 theory - Load Case 7$0\le z\le a$$\theta_A=\frac{FbL}{6{\rm EI}_x}$

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

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

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

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

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

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

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

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

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

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

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

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

$D_z=F$

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

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

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

$\left|C_y\right|=\left|\frac{F}{\delta y_{z\left(F\right)}}\right|=\frac{3{\rm EI}_x}{b^2L}\ @z=L$
Beam theory - Load Case 8$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} $

$D_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} $

$D_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 theory - Load Case 9$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] $

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

$D_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\textsf{, 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} $

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

$\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} $

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

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

$D_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\textsf{, 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\textsf{, 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 theory - Load Case 10$0\le z\le a$
$b>a$
$\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) $

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

$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) $

$D_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 theory - Load Case 11$\theta_A=0$

$\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 $

$D_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|qu\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 theory - Load Case 12$\theta_A=0$

$\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 $

$D_z=F $

$M_z=\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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