# CANTILEVER BEAM DEFLECTION FORMULA

### Cantilever Beam with Moment at Free End Formulas:

 Parameter Equation Reaction Force 1 [R1] $${ R }_{ 1 }=0$$ Reaction Force 2 [R2] $${ R }_{ 2 }=0$$ Shear force at distance x [V] $$V={ R }_{ 1 }$$ Reaction Moment 1 [M1] $${ M }_{ 1 }=0$$ Reaction Moment 2 [M2] $${ M }_{ 2 }={ M }_{ o }$$ Moment at distance x [M] $$M={ M }_{ 1 }+{ R }_{ 1 }x+{ M }_{ o }{ \left< x-a \right> }^{ 0 }$$ Bending stress at distance x [σ] $$σ=\frac { M\cdot c }{ I }$$ End Deflection 1 [y1] $${ y }_{ 1 }=\frac { { M }_{ o }({ L }^{ 2 }-{ a }^{ 2 }) }{ 2EI }$$ End Deflection 2 [y2] $${ y }_{ 2 }=0$$ Deflection at distance x [y] $$y={ y }_{ 1 }+{ \theta }_{ 1 }x+\frac { { M }_{ 1 }{ x }^{ 2 } }{ 2EI } +\frac { { R }_{ 1 }{ x }^{ 3 } }{ 6EI } +\frac { { M }_{ o } }{ 2EI } { \left< x-a \right> }^{ 2 }$$ Slope 1 [θ1] $${ \theta }_{ 1 }=\frac { -{ M }_{ o }(L-a) }{ EI }$$ Slope 2 [θ2] $${ \theta }_{ 2 }=0$$ Slope [θ] $$\theta ={ \theta }_{ 1 }+\frac { { M }_{ 1 }x }{ EI } +\frac { { R }_{ 1 }{ x }^{ 2 } }{ 2EI } +\frac { { M }_{ o } }{ EI } \left< x-a \right>$$

Note: In these formulas,  equations in brackets "< >" are singularity functions.

### Cantilever Beam Distributed Load Formulas:

 Parameter Equation Reaction Force 1 [R1] $${ R }_{ 1 }=0$$ Reaction Force 2 [R2] $${ R }_{ 2 }=\frac { { w }_{ a }+{ w }_{ L } }{ 2 } \cdot \left( L-a \right)$$ Shear force at distance x [V] $$V={ R }_{ 1 }-{ w }_{ a }\left< x-a \right> -\frac { { w }_{ L }-{ w }_{ a } }{ 2\left( L-a \right) } { \left< x-a \right> }^{ 2 }$$ Reaction Moment 1 [M1] $${ M }_{ 1 }=0$$ Reaction Moment 2 [M2] $${ M }_{ 2 }=-\frac { { w }_{ a } }{ 2 } { (L-a) }^{ 2 }-\frac { { w }_{ L }-{ w }_{ a } }{ 6 } { (L-a) }^{ 2 }$$ Moment at distance x  [M] $$M={ M }_{ 1 }+{ R }_{ 1 }x-\frac { { w }_{ a } }{ 2 } { \left< x-a \right> }^{ 2 }-\frac { { w }_{ L }-{ w }_{ a } }{ 6(L-a) } { \left< x-a \right> }^{ 3 }$$ Bending stress at distance x [σ] $$σ=\frac { M\cdot c }{ I }$$ End Deflection 1 [y1] $${ y }_{ 1 }=-\frac { { w }_{ a } }{ 24EI } { (L-a) }^{ 3 }(3L+a)-\frac { { w }_{ L }-{ w }_{ a } }{ 120EI } { (L-a) }^{ 3 }(4L+a)$$ End Deflection 2 [y2] $${ y }_{ 2 }=0$$ Deflection at distance x [y] $$y={ y }_{ 1 }+{ \theta }_{ 1 }x+\frac { { M }_{ 1 }{ x }^{ 2 } }{ 2EI } +\frac { { R }_{ 1 }{ x }^{ 3 } }{ 6EI } -\frac { { w }_{ a } }{ 24EI } { \left< x-a \right> }^{ 4 }-\frac { { w }_{ L }-{ w }_{ a } }{ 120EI(L-a) } { \left< x-a \right> }^{ 5 }$$ Shear force at distance x [V] $$V={ R }_{ 1 }-{ w }_{ a }\left< x-a \right> -\frac { { w }_{ L }-{ w }_{ a } }{ 2(L-a) } { \left< x-a \right> }^{ 2 }$$ Slope 1 [θ1] $$V{ \theta }_{ 1 }=\frac { { w }_{ a } }{ 6EI } { \left( L-a \right) }^{ 3 }+\frac { { w }_{ L }-{ w }_{ a } }{ 24EI } { (L-a) }^{ 3 }$$ Slope 2 [θ2] $${ \theta }_{ 2 }=0$$ Slope [θ] $$\theta ={ \theta }_{ 1 }+\frac { { M }_{ 1 }x }{ EI } +\frac { { R }_{ 1 }{ x }^{ 2 } }{ 2EI } -\frac { { w }_{ a } }{ 6EI } { \left< x-a \right> }^{ 3 }-\frac { { w }_{ L }-{ w }_{ a } }{ 24EI(L-a) } { \left< x-a \right> }^{ 4 }$$

### Cantilever Beam With Point Load Formulas:

 Parameter Equation Reaction Force 1 [R1] $${ R }_{ 1 }=0$$ Reaction Force 2 [R2] $${ R }_{ 2 }=P$$ Shear force at distance x [V] $$V={ R }_{ 1 }-P{ \left< x-a \right> }^{ 0 }$$ Reaction Moment 1 [M1] $${ M }_{ 1 }=0$$ Reaction Moment 2 [M2] $${ M }_{ 2 }=-P(L-a)$$ Moment at distance x [M] $$M={ M }_{ 1 }+{ R }_{ 1 }x-P\left< x-a \right>$$ Bending stress at distance x [σ] $$σ=\frac { M\cdot c }{ I }$$ End Deflection 1 [y1] $${ y }_{ 1 }=-\frac { P }{ 6EI } (2{ L }^{ 3 }-3{ L }^{ 2 }a+{ a }^{ 3 })$$ End Deflection 2 [y2] $${ y }_{ 2 }=0$$ Deflection at distance x [y] $$y={ y }_{ 1 }+{ \theta }_{ 1 }x+\frac { { M }_{ 1 }{ x }^{ 2 } }{ 2EI } +\frac { { R }_{ 1 }{ x }^{ 3 } }{ 6EI } -\frac { P }{ 6EI } { \left< x-a \right> }^{ 3 }$$ Slope 1 [θ1] $${ \theta }_{ 1 }=\frac { P{ (L-a) }^{ 2 } }{ 2EI }$$ Slope 2 [θ2] $${ \theta }_{ 2 }=0$$ Slope [θ] $$\theta ={ \theta }_{ 1 }+\frac { { M }_{ 1 }x }{ EI } +\frac { { R }_{ 1 }{ x }^{ 2 } }{ 2EI } -\frac { P }{ 2EI } { \left< x-a \right> }^{ 2 }$$

Note: In these formulas,  equations in brackets "< >" are singularity functions.