# SIMPLY SUPPORTED STRUCTURAL BEAM STRESS AND DEFLECTION ANALYSIS

Simply supported beam deflection and stress calculator to find forces, moments, stresses, deflections and slopes of a simply supported beam under point, distributed and moment loads.

### Simply Supported Beam Deflection and Stress Calculator:

 INPUT PARAMETERS POINT LOADS Param. Magnitude Distance N kN lbf m mm inch ft Load 1 [P1] * Load 2 [P2] * Load 3 [P3] * Load 4 [P4] * Load 5 [P5] * CONCENTRATED MOMENTS Param. Magnitude Distance N*m kN*m lbf*in lbf*ft m mm inch ft Moment 1 [M1] * Moment 2 [M2] * Moment 3 [M3] * Moment 4 [M4] * Moment 5 [M5] * DISTRIBUTED LOADS Param. Magnitude Distance Pa-m N/m Pa-cm Pa-mm lbf/in psi-in psi-ft lbf/ft m mm inch ft wa wb a b Distributed Load 1 [w1] * Distributed Load 2 [w2] * Distributed Load 3 [w3] * Distributed Load 4 [w4] * Distributed Load 5 [w5] * STRUCTURAL BEAM PROPERTIES Param. Value Beam Length [L] m mm inch ft 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 * : P is positive in downward direction as shown in the figure and negative in upward direction. M is positive in clockwise direction as shown in the figure. 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".

CONCENTRATED MOMENTS
RESULTS Param. 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. 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.

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. Simply Supported Beam Deflection Calculation Example An example on calculation of max. deflection, max. shear force, max. bending moment and mid-span slope/deflection of a simply supported beam under multiple point loads and a distributed load.

### List of Equations:

"Simply Supported Beam with Concentrated Load at any Point", Simply Supported Structural Beam with Partially Distributed Load" and Simply Supported Beam with Concentrated Moment at any Point" calculators have been used for the calculation of forces, moments, stresses, deflections and slopes with superposition principal.

### 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