# SIMPLY SUPPORTED BEAM WITH DISTRIBUTED LOAD

Simply Supported Beam with Distributed Load Calculator to find forces, moments, stresses, deflections and slopes of a simply supported beam with uniformly, uniformly varying, trapezoidal, triangular and partially distributed load.

### Simply Supported Beam with Distributed Load Calculator:

 INPUT PARAMETERS Parameter Value Distributed load magnitude at a [wa] * Pa-m N/m Pa-cm Pa-mm lbf/in psi-in psi-ft lbf/ft Distributed load magnitude at b [wb] * Beam Length [L] mm m inch ft Distance a Distance b Distance x Modulus of Elasticity [E] GPa ksi Distance from neutral axis to extreme fibers [c] mm m inch ft Second moment of area [I ]** mm^4 cm^4 inch^4 ft^4

Note : Use dot "." as decimal separator.

Note * : wa and wb are positive in downward direction as shown in the figure and negative in upward direction.

Note ** : For second moment of area calculations of structural beams, visit " Sectional Properties Calculators".

 RESULTS Parameter Value Reaction Force 1 [R1] --- N kN lbf Reaction Force 2 [R2] --- Transverse Shear Force @ distance x [Vx] --- Maximum Transverse Shear Force [Vmax] --- Moment @ distance x [Mx] --- N*m kN*m lbf*in lbf*ft Maximum Moment [Mmax] --- Slope 1 [θ1] --- radian degree arcmin arcsec Slope 2 [θ2] --- Slope @ distance x [θx] --- Maximum Slope [θmax] --- Deflection @ distance x [yx] --- mm m inch ft Maximum Deflection [ymax] --- Bending Stress @ distance x [σx] --- MPa psi ksi Maximum Bending Stress [σmax] ---

Note * : R1 and R2 are vertical end reactions at the left and right, respectively, and are positive upward. Shear forces and deflections are positive in upward direction and negative in downward direction. All moments are positive when producing compression on the upper portion of the beam cross section. All slopes are positive when up and to the right.

Note: Stresses are positive numbers, and these are stress magnitudes in the beam. It does not distinguish between tension or compression of the structural beam. This distinction depends on which side of the beam's neutral plane c input corresponds.

Slope

Deflection

Moment

Shear Force

### Definitions:

Distributed load: A load which acts evenly over a structural member or over a surface that supports the load.

Fixed support: Fixed supports can resist vertical and horizontal forces as well as a moment. Since they restrain both rotation and translation, they are also known as rigid supports.

Roller support: Roller supports are free to rotate and translate along the surface upon which the roller rests. The resulting reaction force is always a single force that is perpendicular to the surface. Roller supports are commonly located at one end of long bridges to allow the expansion and contraction of the structure due to temperature changes.

Simply supported beam: A beam which is free to rotate at its supports, and also to expand longitudinally at one end.

Structural beam: A structural element that withstands loads and moments. General shapes are rectangular sections, I beams, wide flange beams and C channels.

### Supplements:

 Link Usage Sectional Properties Calculator of Profiles Sectional properties needed for the structural beam stress analysis can be calculated with sectional properties calculator.

### List of Equations:

Following simply supported beam distributed load formulas are used for the calculations. Superposition principle is used if needed.

 Parameter Equation Reaction Force 1 [R1] $${ R }_{ 1 }=\frac { { w }_{ a } }{ 2L } { (L-a) }^{ 2 }+\frac { { w }_{ L }-{ w }_{ a } }{ 6L } { (L-a) }^{ 2 }$$ Reaction Force 2 [R2] $${ R }_{ 2 }=\frac { { w }_{ a }+{ w }_{ L } }{ 2 } { (L-a) }-{ R }_{ 1 }$$ Shear force at distance x [V] $$V={ R }_{ 1 }-{ w }_{ a }\left< x-a \right> -\frac { { w }_{ L }-{ w }_{ a } }{ 2\left( L-a \right) } \left< x-a \right> ^{ 2 }$$ Moment at distance x  [M] $$M={ R }_{ 1 }x-\frac { { w }_{ a } }{ 2 } { \left< x-a \right> }^{ 2 }-\frac { { w }_{ L }-{ w }_{ a } }{ 6(L-a) } { \left< x-a \right> }^{ 3 }$$ Bending stress at distance x [σ] $$σ=\frac { M\cdot c }{ I }$$ Deflection at distance x [y] $$y={ \theta }_{ 1 }x+\frac { { R }_{ 1 }{ x }^{ 3 } }{ 6EI } -\frac { { w }_{ a } }{ 24EI } { \left< x-a \right> }^{ 4 }-\frac { { w }_{ L }-{ w }_{ a } }{ 120EI(L-a) } { \left< x-a \right> }^{ 5 }$$ Shear force at distance x [V] $$V={ R }_{ 1 }-{ w }_{ a }\left< x-a \right> -\frac { { w }_{ L }-{ w }_{ a } }{ 2\left( L-a \right) } { \left< x-a \right> }^{ 2 }$$ Slope 1 [θ1] $${ \theta }_{ 1 }=\frac { -{ w }_{ a } }{ 24EIL } { (L-a) }^{ 2 }\cdot ({ L }^{ 2 }+2aL-{ a }^{ 2 })-\frac { { w }_{ L }-{ w }_{ a } }{ 360EIL } \cdot { (L-a) }^{ 2 }\cdot (7{ L }^{ 2 }-6al-3{ a }^{ 2 })$$ Slope 2 [θ2] $${ \theta }_{ 2 }=\frac { { w }_{ a } }{ 24EIL } { ({ L }^{ 2 }-{ a }^{ 2 }) }^{ 2 }+\frac { { w }_{ L }-{ w }_{ a } }{ 360EIL } { (L-a) }^{ 2 }(8{ L }^{ 2 }+9aL+3{ a }^{ 2 })$$ Slope [θ] $$\theta ={ \theta }_{ 1 }+\frac { { R }_{ 1 }{ x }^{ 2 } }{ 2EI } -\frac { { w }_{ a } }{ 6EI } { \left< x-a \right> }^{ 3 }-\frac { { w }_{ L }-{ w }_{ a } }{ 24EI(L-a) } { \left< x-a \right> }^{ 4 }$$