FIXED STRUCTURAL BEAM WITH CONCENTRATED LOAD AT ANY POINT

Following calculator has been developed to find forces, moments, stresses, deflections and slopes in structural beam for a specific case which is concentrated loading of a structural beam whose both ends are fixed. Concentrated load can be applied at any point of the 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:


INPUT PARAMETERS
Parameter	Symbol	Value	Unit
Load *	P
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 **	I

Note : Use dot "." as decimal separator.

Note * : P is 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	V_x	---
Maximum Transverse Shear Force	V_max	---
Reaction Moment 1	M₁	---
Reaction Moment 2	M₂	---
Moment @ distance x	M_x	---
Maximum Moment	M_max	---
Slope 1	θ₁	---
Slope 2	θ₂	---
Slope @ distance x	θ_x	---
Maximum Slope	θ_max	---
End Deflection 1	y₁	---
End Deflection 2	y₂	---
Deflection @ distance x	y_x	---
Maximum Deflection	y_max	---
Bending Stress @ distance x	σ_x	---
Maximum Bending Stress	σ_max	---

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. M₁ and M₂ are the reaction end moments at the left and right, respectively. 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.

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.

Fixed beam: A beam which is fixed at both ends.

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/Condition	Symbol	Equation
Reaction Force 1	R₁
Reaction Force 2	R₂
Shear force at distance x	V
Reaction Moment 1	M₁
Reaction Moment 2	M₂
Moment at distance x	M
Bending stress at distance x	σ
End Deflection 1	y₁
End Deflection 2	y₂
Deflection at distance x	y
Slope 1	θ₁
Slope 2	θ₂
Slope	θ

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

Reference:

Young, W. C., Budynas, R. G.(2002). Roark's Formulas for Stress and Strain . 7nd Edition, McGraw-Hill, Chapter 8 , pp 125 - 267
Oberg.E , Jones.D.J., Holbrook L.H, Ryffel H.H., (2012) . Machinery's Handbook . 29th edition. Industrial Press Inc. , pp 236 - 261
Beer.F.P. , Johnston.E.R. (1992). Mechanics of Materials , 2nd edition. McGraw-Hill, Chapter 4-5-7-8-9