CANTILEVER STRUCTURAL BEAM WITH PARTIALLY DISTRIBUTED LOAD

Following calculator has been developed to find forces, moments, stresses, deflections and slopes in structural beam for a specific case which is partially distributed loading of a cantilever structural beam.

Note: For more information on shear, moment, slope and deflection calculations for different end constraints, please refer to "Beams; Flexure of Straight Bars" chapter of Roark's Formulas for Stress and Strain.

Calculator:

Cantilever Beam with Partially Distributed Load
INPUT PARAMETERS
Parameter Symbol Value Unit
Distributed load magnitude at a * wa
Distributed load magnitude at L * wL
Beam Length  L
Distance a a
Distance x x
Modulus of Elasticity E
Distance from neutral axis to extreme fibers c
Second moment of area**

Note : Use dot "." as decimal separator.

Note * : wa and wL 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 Symbol Value Unit
Reaction Force 1  R1 ---
Reaction Force 2 R2 ---
Transverse Shear Force
@ distance x
Vx ---
Maximum Transverse
 Shear Force
Vmax ---
Reaction Moment 1 M1 ---
Reaction Moment 2  M2 ---
Moment @ distance x Mx ---
Maximum Moment Mmax ---
Slope 1  θ1 ---
Slope 2 θ2 ---
Slope
@ distance x
θx ---
Maximum Slope θmax ---
End Deflection 1 y1 ---
End Deflection 2 y2 ---
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.

   
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.

Cantilever beam: Cantilever is a beam which is fixed at only 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:

Parameter Symbol Equation
Reaction Force 1 R1 Equation for reaction force in cantilever beam with partially distributed load
Reaction Force 2 R2 Equation for reaction force in cantilever beam with partially distributed load
Shear force at distance x V Equation for shear force in cantilever beam with partially distributed load
Reaction Moment 1 M1 Equation for reaction moment in cantilever beam with distributed load
Reaction Moment 2 M2 Equation for reaction moment in cantilever beam with distributed load
Moment at distance x M Equation for moment in cantilever beam with partially distributed load
Bending stress at distance x σ Equation for bending stress in cantilever beam
End Deflection 1 y1 Equation for deflection in cantilever beam with distributed load
End Deflection 2 y2 Equation for deflection in cantilever beam with distributed load
Deflection at distance x y Equation for deflection in cantilever beam with partially distributed load
Shear force at distance x V Equation for shear force in cantilever beam with partially distributed load
Slope 1 θ1 Equation for slope in cantilever beam with partially distributed load
Slope 2 θ2 Equation for slope in cantilever beam with partially distributed load
Slope θ Equation for slope in cantilever beam with partially distributed load

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

Reference: