Beam Wiki

Mechanics of Materials Toolbox for Maple

All presented Beam Internal Factor's functions are depend from z coordinate along Beam Axis. The order is from left to right. Positive direction for Reaction Forces is up, for Reaction Moments is by clockwise. All solutions created by coding within Mechanics of Materials Toolbox for Maple as is. Send us your own scheme (little picture) to beam@orlovsoft.com and we put it here. Next

Beam Bending Sample #1

Reaction Forces:

R_{A} = \frac{b F}{a + b}

R_{B} = \frac{a F}{a + b}

Shearing Forces:

Q_{1} = \frac{b F}{a + b}

Q_{2} = - \frac{a F}{a + b}

Bending Moments:

M_{1} = \frac{b F z}{a + b}

M_{2} = \frac{a F (- z + b)}{a + b}

Bending Angles:

ϕ_{1} = - \frac{1}{6} \frac{b F (- 3 z^{2} + a^{2} + 2 a b)}{(a + b) E J_{x}}

ϕ_{2} = \frac{1}{6} \frac{a F (- 3 z^{2} + 6 b z + 2 a b - 2 b^{2})}{(a + b) E J_{x}}

Deflections:

w_{1} = - \frac{1}{6} \frac{b F z (- z^{2} + a^{2} + 2 a b)}{(a + b) E J_{x}}

w_{2} = - \frac{1}{6} \frac{a F (z^{3} - 3 b z^{2} - 2 b z a + 2 b^{2} z + 2 b^{2} a)}{(a + b) E J_{x}}

Numeric Sample (Maple input data): a:=2; b:=1; F:=10000; E:=2*10^11; J[x]:=200*10^(-8);
Structural analysis software sample 1

Beam Bending Sample #2

Reaction Forces:

R_{A} = \frac{1}{2} \frac{q a (a + 2 b)}{a + b}

R_{B} = \frac{1}{2} \frac{q a^{2}}{a + b}

Shearing Forces:

Q_{1} = \frac{1}{2} \frac{q (- 2 z a - 2 z b + a^{2} + 2 a b)}{a + b}

Q_{2} = - \frac{1}{2} \frac{q a^{2}}{a + b}

Bending Moments:

M_{1} = \frac{1}{2} \frac{q z (- z a - z b + a^{2} + 2 a b)}{a + b}

M_{2} = \frac{1}{2} \frac{q a^{2} (- z + b)}{a + b}

Bending Angles:

ϕ_{1} = - \frac{1}{24} \frac{q (4 z^{3} a + 4 z^{3} b - 6 a^{2} z^{2} - 12 a z^{2} b + a^{4} + 4 b a^{3} + 4 a^{2} b^{2})}{(a + b) E J_{x}}

ϕ_{2} = \frac{1}{24} \frac{q a^{2} (- 6 z^{2} + 12 z b + a^{2} + 4 a b - 4 b^{2})}{(a + b) E J_{x}}

Deflections:

w_{1} = - \frac{1}{24} \frac{q z (z^{3} a + z^{3} b - 2 a^{2} z^{2} - 4 a z^{2} b + a^{4} + 4 b a^{3} + 4 a^{2} b^{2})}{(a + b) E J_{x}}

w_{2} = - \frac{1}{24} \frac{q a^{2} (2 z^{3} - 6 z^{2} b - z a^{2} - 4 z a b + 4 z b^{2} + b a^{2} + 4 a b^{2})}{(a + b) E J_{x}}

Numeric Sample (Maple input data): a:=2; b:=1; E:=2*10^11; J[x]:=200*10^(-8); q:=1000:
Structural analysis software sample 1

Beam Bending Sample #3

Reaction Forces:

R_{A} = - \frac{1}{2} \frac{- q a^{2} - 2 b q a + 2 b F}{a + b}

R_{B} = - \frac{1}{2} \frac{a (- q a + 2 F)}{a + b}

Shearing Forces:

Q_{1} = - \frac{1}{2} \frac{2 q z a + 2 q z b - q a^{2} - 2 b q a + 2 b F}{a + b}

Q_{2} = \frac{1}{2} \frac{a (- q a + 2 F)}{a + b}

Bending Moments:

M_{1} = - \frac{1}{2} \frac{z (q z a + q z b - q a^{2} - 2 b q a + 2 b F)}{a + b}

M_{2} = - \frac{1}{2} \frac{a (- q a + 2 F) (- z + b)}{a + b}

Bending Angles:

ϕ_{1} = \frac{1}{24} \frac{- 4 q z^{3} a - 4 q z^{3} b + 6 z^{2} q a^{2} + 12 z^{2} b q a - 12 z^{2} b F - q a^{4} - 4 b q a^{3} + 4 b a^{2} F - 4 a^{2} b^{2} q + 8 a b^{2} F}{(a + b) E J_{x}}

ϕ_{2} = - \frac{1}{24} \frac{a (6 z^{2} q a - 12 z^{2} F - 12 b z q a + 24 b z F - q a^{3} - 4 b q a^{2} + 8 b a F + 4 a b^{2} q - 8 b^{2} F)}{(a + b) E J_{x}}

Deflections:

w_{1} = \frac{1}{24} \frac{z (- q z^{3} a - q z^{3} b + 2 z^{2} q a^{2} + 4 z^{2} b q a - 4 z^{2} b F - q a^{4} - 4 b q a^{3} + 4 b a^{2} F - 4 a^{2} b^{2} q + 8 a b^{2} F)}{(a + b) E J_{x}}

w_{2} = \frac{1}{24} \frac{a (- 2 q z^{3} a + 4 z^{3} F + 6 z^{2} b q a - 12 z^{2} b F + z q a^{3} + 4 z b q a^{2} - 8 z b a F - 4 z a b^{2} q + 8 z b^{2} F - b q a^{3} - 4 a^{2} b^{2} q + 8 a b^{2} F)}{(a + b) E J_{x}}

Numeric Sample (Maple input data): a:=2; b:=1; F:=10000; E:=2*10^11; J[x]:=200*10^(-8); q:=1000;
Structural analysis software sample 1