# FIXED BEAM DEFLECTION FORMULA

### Beam Fixed at Both Ends with Uniformly Distributed Load:

 Parameter Equation Reaction Force 1 [R1] $${ R }_{ 1 }=\frac { { w }_{ a } }{ 2{ L }^{ 3 } } { (L-a) }^{ 3 }(L+a)+\frac { { w }_{ L }-{ w }_{ a } }{ 20{ L }^{ 3 } } { (L-a) }^{ 3 }(3L+2a)$$ Reaction Force 2 [R2] $${ R }_{ 2 }=\frac { { w }_{ a }+{ w }_{ L } }{ 2 } \cdot (L-a)-{ R }_{ 1 }$$ Shear force at distance x [V] $$V={ R }_{ 1 }-{ w }_{ a }\left< x-a \right> -\frac { { w }_{ L }-{ w }_{ a } }{ 2\cdot (L-a) } { \left< x-a \right> }^{ 2 }$$ Reaction Moment 1 [M1] $${ M }_{ 1 }=-\frac { { w }_{ a } }{ 12{ L }^{ 2 } } { (L-a) }^{ 3 }(L+3a)-\frac { { w }_{ L }-{ w }_{ a } }{ 60{ L }^{ 2 } } { (L-a) }^{ 3 }(2L+3a)$$ Reaction Moment 2 [M2] $${ M }_{ 2 }={ R }_{ 1 }L+{ M }_{ 1 }-\frac { { w }_{ a } }{ 2 } { (L-a) }^{ 2 }-\frac { { w }_{ L }-{ w }_{ a } }{ 6 } { (L-a) }^{ 2 }$$ 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] $$\sigma =\frac { Mc }{ I }$$ End Deflection 1 [y1] $${ y }_{ 1 }=0$$ End Deflection 2 [y2] $${ y }_{ 2 }=0$$ Deflection at distance x [y] $$y={ y }_{ 1 }+{ \theta }_{ 1 }x+\frac { { M }_{ 1 }{ x }^{ 2 } }{ 2EI } +\frac { { R }_{ 1 }{ x }^{ 3 } }{ 6EI } -\frac { { w }_{ a } }{ 24EI } { \left< x-a \right> }^{ 4 }-\frac { { w }_{ L }-{ w }_{ a } }{ 120EI(L-a) } { \left< x-a \right> }^{ 5 }$$ Slope 1 [θ1] $${ \theta }_{ 1 }=0$$ Slope 2 [θ2] $${ \theta }_{ 2 }=0$$ Slope [θ] $$\theta ={ \theta }_{ 1 }+\frac { { M }_{ 1 }x }{ EI } +\frac { { R }_{ 1 }{ x }^{ 2 } }{ 2EI } -\frac { { w }_{ a } }{ 6EI } { \left< x-a \right> }^{ 3 }-\frac { { w }_{ L }-{ w }_{ a } }{ 24EI(L-a) } { \left< x-a \right> }^{ 4 }$$

### Fixed Beam with Point Load Formulas:

 Parameter Equation Reaction Force 1 [R1] $${ R }_{ 1 }=\frac { P }{ { L }^{ 3 } } { (L-a) }^{ 2 }(L+2a)$$ Reaction Force 2 [R2] $${ R }_{ 2 }=\frac { P{ a }^{ 2 } }{ { L }^{ 3 } } (3L-2a)$$ Shear force at distance x [V] $$VM={ R }_{ 1 }-P{ \left< x-a \right> }^{ 0 }$$ Reaction Moment 1 [M1] $${ M }_{ 1 }=\frac { -Pa }{ { L }^{ 2 } } { (L-a) }^{ 2 }$$ Reaction Moment 2 [M2] $${ M }_{ 2 }=\frac { -P{ a }^{ 2 } }{ { L }^{ 2 } } { (L-a) }$$ Moment at distance x [M] $$M={ M }_{ 1 }+{ R }_{ 1 }x-P\left< x-a \right>$$ Bending stress at distance x [σ] $$\sigma =\frac { Mc }{ I }$$ End Deflection 1 [y1] $${ y }_{ 1 }=0$$ 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] $${ \theta }_{ 1 }=0$$ 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.

### Fixed Beam Bending Moment Formulas:

 Parameter Equation Reaction Force 1 [R1] $${ R }_{ 1 }=-\frac { 6{ M }_{ o }a }{ { L }^{ 3 } } (L-a)$$ Reaction Force 2 [R2] $${ R }_{ 1 }=\frac { 6{ M }_{ o }a }{ { L }^{ 3 } } (L-a)$$ Shear force at distance x [V] $$V={ R }_{ 1 }$$ Reaction Moment 1 [M1] $${ M }_{ 1 }=-\frac { { M }_{ o } }{ { L }^{ 2 } } ({ L }^{ 2 }-4aL+3{ a }^{ 2 })$$ Reaction Moment 2 [M2] $${ M }_{ 2 }=\frac { { M }_{ o } }{ { L }^{ 2 } } (3{ a }^{ 2 }-2aL)$$ Moment at distance x [M] $$M={ M }_{ 1 }+{ R }_{ 1 }x+{ M }_{ o }{ \left< x-a \right> }^{ 0 }$$ Bending stress at distance x [σ] $$\sigma =\frac { Mc }{ I }$$ End Deflection 1 [y1] $${ y }_{ 1 }=0$$ 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] $${ \theta }_{ 1 }=0$$ 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.
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