Simply supported beam with bending moment calculator to find forces, moments, stresses, deflections and slopes of a simple beam which is subjected to a moment. The bending moment can be applied at any point of the beam. Formulas are given below the calculator.

Simply Supported Beam with Bending Moment Calculator:

Simply Supported Beam with Concentrated Moment
Parameter Value  
Moment [M] *
Beam Length [L] 
Distance a
Distance x
Modulus of Elasticity [E]
Distance from neutral axis to extreme fibers [c]
Second moment of area [I]**

Note : Use dot "." as decimal separator.

Note * : M is positive in clockwise direction as shown in the figure.

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


Parameter Value
Reaction Force 1 [R1] ---
Reaction Force 2 [R2] ---
Transverse Shear Force @ distance x [Vx] ---
Maximum Transverse Shear Force [Vmax] ---
Moment @ distance x [Mx] ---
Maximum Moment [Mmax] ---
Slope 1 [θ1] ---
Slope 2 [θ2] ---
Slope @ distance x [θx] ---
Maximum Slope [θmax] ---
Deflection @ distance x [yx] ---
Maximum Deflection [ymax] ---
Bending Stress @ distance x [σx] ---
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.




Shear Force


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.

Pin support: A pinned support resist both vertical and horizontal forces but not a moment. They will allow the structural member to rotate, but not to translate in any direction. A pinned connection could allow rotation in only one direction; providing resistance to rotation in any other direction.

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.


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

Simply Supported Beam with Bending Moment Formulas:

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

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