BEAM THEORY

Engineering Fundamentals

INTRODUCTION

Beam theory or beam deflection 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.

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From various load cases; the following beam theory equations are given:

  • beam deflection (sag)
  • beam curvature
  • reaction force
  • shear force
  • reaction moment
  • stress
  • stiffness
Beam theory Basic

Load case

Beam theory - Load Case 1

Curvature \theta
Sag \delta z

\theta _A=0

\theta _B=\frac{FL^2}{2EI}

\delta z_x=-\frac{(Fx^2)(3L-x)}{6EI}

\delta z_{max}=-\frac{FL^3}{3EI_y}\ @\ x=L

Reaction force R
Shear force D
Reaction moment M_R

R_A=F

R_B={N.A.}

D_x=F

M_{Rx}=F(x-L)

M_{Rmax}=-FL\ @\ x=0

Stress \sigma
Stiffness C

|\sigma _x|=\frac{|M_{Rx}|}{I_y}u=\frac{-|F|u(x-L)}{I_y}

|\sigma _{max}|=\frac{|F|Lu}{I_y}\ @\ x=0

|C_z|=\left|\frac {F}{\delta z_{x(F)}}\right|=\frac{3EI_y}{L^3}\ @\ x=L

Load case

Beam theory - Load Case 2

Curvature \theta
Sag \delta z

\theta _A= 0

\theta _B=\frac{ML}{EI_y}

\delta z_x=-\frac{Mx^2}{2EI_y}

\delta z_{max}=-\frac{ML^2}{2EI_y}\ @\ x=L

Reaction force R
Shear force D
Reaction moment M_R

R_A=0

R_B={N.A.}

D_x={N.A.}

M_{Rx}=M

M_{Rmax}=M\ @\ x= const.

Stress \sigma
Stiffness C

|\sigma _x|=\frac{|M_Rx|}{I_y}u=\frac{|M|u}{I_y}

|\sigma _{max}|=\frac{|M|u}{I_y}\ @\ x= const.

|C_z|=\left|\frac {M}{\theta _B}\right|=\frac{EI_y}{L}\ @\ x=L

Load case

Beam theory - Load Case 3

Curvature \theta
Sag \delta z

\theta _A=0

\theta _B=\frac{qL³}{6EI_y}

\delta z_x=-\frac{qx^2}{24EI_y}(6L^2-4Lx+x^2)

\delta z_{max}=-\frac{qL^4}{8EI_y}\ @\ x = L

Reaction force R
Shear force D
Reaction moment M_R

R_A=qL

R_B={N.A.}

D_x=q(L-x)

M_{Rx}=-\frac{q(L-x)²}{2}

M_{Rmax}=-\frac{qL^2}{2}\ @\ x=0

Stress \sigma
Stiffness C

|\sigma _x|=\frac{|M_{Rx}|}{I_y}u=\frac{|q|u(L-x)^2}{2I_y}

|\sigma _{max}|=\frac{|q|L^2u}{2I_y}\ @\ x=0

|C_z|=\left|\frac {q}{\delta z_{max}}\right|=\frac{8EI_y}{L^4}\ @\ x=L

Load case

Beam theory - Load Case 4

For: 0≤x≤a

Curvature \theta
Sag \delta z

\theta _A=\frac{Fab(L+b)}{6EI_yL}

\theta _B=\frac{Fab(L+a)}{6EI_yL}

\delta z_x=-\frac{Fab^2}{6EI_y}\left[\left(1+\frac {L}{b}\right)\frac{x}{L}-\frac{x^3}{abL}\right]

\delta z_{max}=-\frac{Fb\sqrt{(L^2-b^2)^3}}{9\sqrt 3EI_yL}\ @\ x=
\sqrt{(L^2-b^2)/3} only if a>b

Reaction force R
Shear force D
Reaction moment M_R

R_A=\frac{Fb}{L}

R_B=\frac{Fa}{L}

D_x=-\frac{Fb}{L}

M_{Rx}=\frac{Fbx}{a+b}

M_{Rmax}=\frac{Fba}{a+b}\ @\ x=a

Stress \sigma
Stiffness C

|\sigma _x|=\frac{|M_{Rx}|}{I_y}u=\frac{|F|bxu}{I_yL}

|\sigma _{max}|=\frac{|F|bau}{I_y(a+b)}\ @\ x=a

|C_z|=\left|\frac {F}{\delta z_{x(F)}}\right|=\frac{3EI_yL}{a^2b^2}\ @\ x=a

For: a≤x≤L

Curvature \theta
Sag \delta z

\theta _A=\frac{Fab(L+b)}{6EI_yL}

\theta _B=\frac{Fab(L+a)}{6EI_yL}

\delta z_x=-\frac{Fa^2b}{6EI_y}\left[\left(1+\frac {L}{a}\right)\frac{L-x}{L}-\frac{(L-x)^3}{abL}\right]

\delta z_{max}=-\frac{Fa\sqrt{(L^2-a^2)^3}}{9\sqrt 3EI_yL}\ @\ x=
L - \sqrt{(L^2-a^2)/3} only if a<b

Reaction force R
Shear force D
Reaction moment M_R

R_A=\frac{Fb}{L}

R_B=\frac{Fa}{L}

D_x=-\frac{Fb}{L}-F

M_{Rx}=\frac{Fbx}{L} - F(x-a)

M_{Rmax}=\frac{Fba}{L}\ @\ x=a

Stress \sigma
Stiffness C

|\sigma _x|=\frac{|M_{Rx}|}{I_y}u=\frac{\left(\frac{Fbx}{L}\right) - F(x-a)}{I_yL}u

|\sigma _{max}|=\frac{|F|bau}{I_yL}\ @\ x=a

|C_z|=\left|\frac {F}{\delta z_{x(F)}}\right|=\frac{3EI_yL}{a^2b^2}\ @\ x=a

Load case

Beam theory - Load Case 5

Curvature \theta
Sag \delta z

\theta _A=\frac{ML}{6EI_y}

\theta _B=\frac{ML}{3EI_y}

\delta z_x=-\frac{ML^2}{6EI_y}\left[\frac {x}{L}-\left(\frac x L)^3\right)\right]

\delta z_{max}=-\frac{ML^2}{9\sqrt 3EI_y}\ @\ x=\frac {L}{\sqrt 3}

Reaction force R
Shear force D
Reaction moment M_R

R_A=\frac{M}{L}

R_B=-\frac{M}{L}

D_x=\frac{M}{L}

M_{Rx}=\left(\frac{M}{L}\right)x

M_{Rmax}=M\ @\ x=L

Stress \sigma
Stiffness C

|\sigma _x\|=\frac{|M_{Rx}|}{I_y}u=\frac{|M|xu}{LI_y}

|\sigma _{max}|=\frac{|M|u}{I_y}\ @\ x=L

|C_z|=\left|\frac {M}{\theta _B}\right|=\frac{3EI_y}{L}\ @\ x=L

Load case

Beam theory - Load Case 6

Curvature \theta
Sag \delta z

\theta _A=\frac{qL^3}{24EI_y}

\theta _B=\frac{qL^3}{24EI_y}

\delta z_x=-\frac{qL^4}{24EI_y}\left[\frac {x}{L}-2\left(\frac {x}{L}\right)^3+\left(\frac {x}{L}\right)^4\right]

\delta z_{max}=-\frac{5}{384}\frac{qL^4}{EI_y}\ @\ \frac{L}{2}

Reaction force R
Shear force D
Reaction moment M_R

R_A=\frac{qL}{2}

R_B=\frac{qL}{2}

D_x=\frac{qL}{2}-qx

M_{Rx}=\frac{qx(L-x)}{2}

M_{Rmax}=\frac{qL^2}{8}\ @\ x=\frac{L}{2}

Stress \sigma
Stiffness C

|\sigma _x|=\frac{|M_{Rx}|}{I_y}u=\frac{|qx(L-x)u|}{2I_y}

\|\sigma _{max}|=\frac{|q|L^2u}{8I_y}\ @\ x=\frac{L}{2}

|C_z|=\left|\frac {q}{\delta z_{max}}\right|=
\frac{5|q|L^4}{384EI_y}\ @\ x=\frac{L}{2}

Load case

Beam theory - Load Case 7

For: 0≤x≤a

Curvature \theta
Sag \delta z

\theta _A=\frac{FbL}{6EI_y}

\theta _B=\frac{FbL}{3EI_y}

\delta z_x=-\frac{Fba^2}{6EI_y}\left[\frac {x}{a}-\left(\frac{x}{a}\right)^3}\right]

\delta z_{max}=\frac{Fba^2}{9\sqrt 3EI_y}\ @\ x=\frac {a}{\sqrt 3}

Reaction force R
Shear force D
Reaction moment M_R

R_A=\frac{Fb}{a}

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

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

M_{Rx}=-\frac{Fbx}{a}

M_{Rmax}=-Fa\ @\ x=a

Stress \sigma
Stiffness C

|\sigma _x|=\frac{|M_{Rx}|}{I_y}u=\frac{|F|bxu}{aI_y}

|\sigma _{max}|=\frac{|F|au}{I_y}\ @\ x=a

|C_z|=\left|\frac {F}{\delta z_{x(F)}}\right|=\frac{3EI_y}{b^2L}\ @\ x=L

For: a≤x≤L

Curvature \theta
Sag \delta z

\theta _C=\frac{Fb(2a+3b)}{3EI_y}

\delta z_x=-\frac{F((-x)+a)}{6EI_y}[ab-3bx+x^2-2ax+a^2]

\delta z_{max}=-\frac{Fb^2L}{3EI_y}\ @\ x=L

Reaction force R
Shear force D
Reaction moment M_R

R_A=-\frac{Fb}{a}

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

D_x=F

M_{Rx}=-F[(-x)+a+b]

M_{Rmax}=-Fa\ @\ x=a

Stress \sigma
Stiffness C

|\sigma _x|=\frac{|M_{Rx}|}{I_y}u=\frac{|F|[(-x)+L)}{I_y}u

|\sigma _{max}|=\frac{|F|au}{I_y}\ @\ x=a

|C_z|=\left|\frac {F}{\delta z_{x(F)}}\right|=\frac{3EI_y}{b^2L}\ @\ x=L

Load case

Beam theory - Load Case 8

For: 0≤x≤a

Curvature \theta
Sag \delta z

\theta _A=\frac{qb^2a}{12EI_y}

\theta _B=\frac{qb^2a}{6EI_y}

\delta z_x=\frac{qb^2a^2}{12EI_y}\left[\frac {x}{a}-\left(\frac {x}{a}\right)^3\right]

\delta z_{max}=\frac{qb^2a^2}{18\sqrt 3EI_y}\ @\ x=\frac {a}{\sqrt 3}

Reaction force R
Shear force D
Reaction moment M_R

R_A=-\frac{qb^2}{2a}

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

D_x=-\frac{qb^2}{2a}

M_{Rx}=-\left(\frac{qb^2}{2a}\right)x

M_{Rmax}=-\frac{qb^2}{2}\ @\ x=a

Stress \sigma
Stiffness C

|\sigma _x|=\frac{|M_{Rx}|}{I_y}u=\frac{|q|b^2xu}{2aI_y}

|\sigma _{max}|=\frac{|q|b^2u}{2I_y}\ @\ x=a

|C_z|=\left|\frac {q}{\delta z_{max}}\right|=
\frac{18\sqrt 3EI_y}{b^2a^2}\ @\ x=\frac {a}{\sqrt 3}

For: a≤x≤L

Curvature \theta
Sag \delta z

\theta _C=\frac{qb^2L}{6EI_y}

\delta z_x=-\frac{qb^4}{24EI_y}\left[4\frac {a}{b}\frac{x-a}{b}+
6\left(\frac{x-a}{b}\right)^2-4\left(\frac{x-a}{b}\right)^3+\left(\frac{x-a}{b}\right)^4\right]

\delta z_{max}=-\frac{qb^3(4a+3b)}{24EI_y}\ @\ x=L

Reaction force R
Shear force D
Reaction moment M_R

R_A=-\frac{qb^2}{2a}

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

D_x=qb-q(x+a)

M_{Rx}=
-\frac{q(b^2-2bx+2ba+x^2-2xa+a^2)}{2}

M_{Rmax}=-\frac{qb^2}{2}\ @\ x=a

Stress \sigma
Stiffness C

|\sigma _x|=\frac{|M_{Rx}|}{I_y}u=
\frac{|qu(b^2-2bx+2ba+x^2-2xa+a^2)|}{2I_y}

|\sigma _{max}|=\frac{|q|b^2u}{2I_y}\ @\ x=a

|C_z|=\left|\frac {q}{\delta z_{max}}\right|=
\frac{24EI_y}{b^3(4a+3b)}\ @\ x=L

Load case

Beam theory - Load Case 9

For: 0≤x≤a

Curvature \theta
Sag \delta z

\theta _A=\frac{Fab^2}{4EI_yL}

\theta _B=0

\delta z_x=-\frac{FLb^2}{4EI_y}\left[\frac{ax}{L^2}-\frac {2}{3}\left(1+\frac {a}{2L}\right)\left(\frac {x}{L}\right)^3\right]

\delta z_{max}=-\frac{ab^2F\sqrt{\frac {a}{2L+a}}}{6EI_y}\ @\ x=
L \cdot \sqrt{\frac{\frac {a}{2L}}{1+\frac {a}{2L}}}

Only if a {\geq} 0,414L

Reaction force R
Shear force D
Reaction moment M_R

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

M_{Rx}=Fx\left(\frac {b}{L}\right)^2\left(1+\frac {a}{2L}\right)

M_{Rmax}=\frac{Fab^2}{L^2}\left(1+\frac {a}{2L}\right)
@\ x=a \ \ \ \ a {\leq} 0,414L

Stress \sigma
Stiffness C

|\sigma _x|=\frac{|M_{Rx}|}{I_y}u=\frac{|F|xb^2u(2L+a)}{2L^3I_y}

|\sigma _{max}|=\frac{|F|ab^2u(1+\frac {a}{2L})}{L^2I_y}

@x=a \ \ \ \ a {\leq} 0,414L

|C_z|=\left|\frac {F}{\delta z_{x(F)}}\right|=
\left|\frac{12EI_yL^3}{b^2a^2[3L^2-2aL-a^2]}\right|\ @\ x=a

For: a≤x≤L

Curvature \theta
Sag \delta z

\theta _A=\frac{Fab^2}{4EI_yL}

\theta _B=0

\delta z_x=\frac{Fa(L-x)^2}{12L^3}\frac{2La^2-3L^2x+a^2x}{EI_y}

\delta z_{max}=\delta z_x(x_{max})

@\ x_{max}=L \cdot\left[\frac{2aL+bL-ba}{2aL+3bL+ba}\right]

Only if a {\leq} 0,414L

Reaction force R
Shear force D
Reaction moment M_R

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

M_{Rx}=
Fx\left(\frac {b}{L}\right)^2\left(1+\frac {a}{2L}\right)-F(x-a)

M_{Rmax}=-F\frac{ab}{L}\left(1-\frac {b}{2L}\right)

@\ x=L\ \ \ \ a{\geq}0,414L

Stress \sigma
Stiffness C

|\sigma _x|=\frac{|M_{Rx}|}{I_y}u=
\frac{|Fu(2xLb^2+b^2xa-2L^3x+2L^3a)|}{2L^3I_y}

|\sigma _{max}|=\frac{\left|Fabu\left(1-\frac {b}{2L}\right)\right|}{LI_y}

@\ x=L\ \ \ \ a{\geq}0,414L

|C_z|=\left|\frac {F}{\delta z_{x(F)}}\right|=
\left|\frac{12EI_yL^3}{b^2a^2[3L^2-2aL-a^2]}\right|\ @\ x=a

Load case

Beam theory - Load Case 10

For: 0≤x≤a

a<b

Curvature \theta
Sag \delta z

\theta _A=0

\theta _B=0

\delta z_x=-\frac{FLb^2}{6EI_y}
\left[3\frac {a}{L}\left(\frac {x}{L}\right)^2-\left(1+\frac{2a}{L}\right)\left(\frac {x}{L}\right)^3\right]

\delta z_{max}=-\frac{2}{3}\frac{Fa^2b^3}{EI_yL^2}\left(\frac {1}{1+\frac{2b}{L}}\right)^2

@\ x=L-L \cdot \frac {1}{1+L/2b}

Reaction force R
Shear force D
Reaction moment M_R

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

M_{Rx}=\frac{Fb^2(xL+2xa+aL)}{L^3}

M_{Rmax}=-\frac{Fab^2}{L^2}\ @\ x=0

Stress \sigma
Stiffness C

|\sigma _x|=\frac{|M_{Rx}|}{I_y}u=
\frac{|F|b^2u(xL+2xa+aL)}{I_yL^3}

|\sigma _{max}|=\frac{|F|uab^2}{L^2I_y}\ @\ x=0

|C_z|=\left|\frac{F}{\delta z_{x(F)}}\right|=\frac{3EI_yL^3}{b^2a^3(L-a)}

@\ x=a

For: a≤x≤L

a>b

Curvature \theta
Sag \delta z

\theta _A=0

\theta _B=0

\delta z_x=-\frac{FLa^2}{6EI_y}
\left[3\frac {b}{L^3}(L-x)^2-\left(1+\frac{2b}{L}\right)\frac{(L-x)^3}{L^3}\right]

\delta z_{max}=-\frac{2}{3}\frac{Fa^3b^2}{EI_yL^2}\left(\frac {1}{1+\frac{2a}{L}}\right)^2

@\ x=L \cdot \frac {1}{1+L/2a}

Reaction force R
Shear force D
Reaction moment M_R

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

M_{Rx}=
\frac{F[b^2xL+2b^2xa+b^2aL-L^3x+L^3a]}{L^3}

M_{Rmax}=-\frac{Fba^2}{L^2}\ @\ x=L

Stress \sigma
Stiffness C

|\sigma _x|=\frac{|M_{Rx}|}{I_y}u=
\frac{|Fu[b^2xL+2b^2xa+b^2aL-L^3x+L^3a]|}{L^3I_y}

|\sigma _{max}|=\frac{|F|uba^2}{L^2I_y}\ @\ x=L

\left|C_z\right|=\left|\frac F{\delta z_{x\left(F\right)}}\right|=\frac{3EI_yL^3}{b^2a^3\left(L-a\right)} }

@\ x=a

Load case

Beam theory - Load Case 11

Curvature \theta
Sag \delta z

\theta _A=0

\theta _B=0

\delta z_x=-\frac{qL^4}{24EI_y}\left[\left(\frac {x}{L}\right)^2-2\left(\frac {x}{L}\right)^3+\left(\frac {x}{L}\right)^4\right]

\delta z_{max}=-\frac{qL^4}{384EI_y}\ @\ x=\frac {L}{2}

Reaction force R
Shear force D
Reaction moment M_R

R_A=\frac {1}{2}qL

R_B=\frac {1}{2}qL

D_x=\frac{qL}{2}-qx

M_{Rx}=\frac{q(-6x^2+6Lx-L^2)}{12}

M_{Rmax}=-\frac{qL^2}{12}\ @\ \frac{x=0}{x=L}

Stress \sigma
Stiffness C

|\sigma _x|=\frac{|M_{Rx}|}{I_y}u=
\frac{|qu(-6x^2+6Lx-L^2)|}{12I_y}

|\sigma _{max}|=\frac{|q|L^2u}{12I_y}\ @\ \frac{x=0}{x=L}

|C_z|=\left|\frac {q}{\delta z_{max}}\right|=\frac{384EI_y}{L^4}

@\ x=\frac L 2

Load case

Beam theory - Load Case 12

Curvature \theta
Sag \delta z

\theta _A=0

\theta _B=0

\delta z_x=-\frac{Fx^2(3L-2x)}{12EI_y}

\delta z_{max}=-\frac{FL^3}{12EI_y}\ @\ x=L

Reaction force R
Shear force D
Reaction moment M_R

R_A=F

R_B=0

D_x=F

M_x=\frac{FL-2Fx}{2}

M_{Rmax}=\frac{FL}{2} resp. -\frac{FL}{2}

@\ x=0 resp. L

Stress \sigma
Stiffness C

|\sigma _x|=\frac{|M_{Rx}|}{I_y}u=
\frac{|u(FL+2Fx)|}{2I_y}

|\sigma _{max}|=\frac{|F|Lu}{2I_y}\ @\ \frac{x=0}{x=L}

|C_z|=\left|\frac {F}{\delta z_{max}}\right|=\frac{12EI_y}{L^3}
@\ x=L

Area moment of inertia

Cross section

Area moment of inertia I

I_x=\frac{1}{12}wh^3

Cross section

Area moment of inertia I

I_x=\frac{1}{12}({w_oh_o}^3-{w_ih_i}^3)

Cross section

Area moment of inertia I

I_x=\frac{\Pi}{4}r^4

Cross section

Area moment of inertia I

I_x=\frac{\Pi}{4}({r_o}^4-{r_i}^4)

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