# FIXED BEAM AT BOTH ENDS - BENDING MOMENT AT ANY POINT

Fixed - Fixed Beam with Bending Moment Calculator for stress and deflection calculations .

### Fixed Beam Bending Moment Calculator: INPUT PARAMETERS Parameter Value Moment [Mo] * N*m kN*m lbf*in lbf*ft Beam Length [L] mm m inch ft Distance a 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 * : 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".

 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] --- Reaction Moment 1 [M1] --- N*m kN*m lbf*in lbf*ft Reaction Moment 2 [M2] --- Moment @ distance x [Mx] --- Maximum Moment [Mmax] --- Slope 1 [θ1] --- radian degree arcmin arcsec Slope 2 [θ2] --- Slope @ distance x [θx] --- Maximum Slope [θmax] --- End Deflection 1 [y1] --- mm m inch ft End Deflection 2 [y2] --- Deflection @ distance x [yx] --- 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. 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.

### 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.

### 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 ,F.D. , Horton H.L. , Ryffel H.H., (2016) . Machinery's Handbook . 30th edition.  Industrial Press Inc. , pp 248 - 272
• Oberg.E , Jones.D.J., Holbrook L.H, Ryffel H.H., (2012) . Machinery's Handbook, 29th . 29th edition.  Industrial Press Inc. , pp 236 - 261
• Beer.F.P. , Johnston.E.R. (1992). Mechanics of Materials, 7th Edition , 2nd edition. McGraw-Hill, Chapter 4-5-7-8-9