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Beam Design Formulas

Written by Jerry Ratzlaff on . Posted in Structural Engineering

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Simple Supported Beam

sb 1EUniformly Distributed Load sb 2DLoad Increasing Uniformly to One End sb 3DLoad Increasing Uniformly to Center sb 4DUniform Load Partially Distributed at One End

 

sb 5DUniform Load Partially Distributed at Any Point sb 6DUniform Load Partially Distributed at Each End sb 7DConcentrated Load at Center sb 8DConcentrated Load at Any Point

 

sb 13DCentral Point Load and Variable End Moments

 
 

Beam Fixed at One End

Beam Fixed at Both Ends

febe 1AUniformaly Distributed Loadfebe 2AConcentrated Load at Centerfebe 3BConcentrated Load at Any Point

 

 

 

 

 

 

 

 

 

 

Cantilever Beam

cb 1AUniformaly Distributed Loadcb 2ALoad Increasing Uniformly to One End cb 3AUniformly Distributed Load and Variable End Moments cb 4AConcentrated Load at Any Point

 

cb 5AConcentrated Load at Free Endcb 6ALoad at Free End Deflection Vertically with No Rotation

 
 

Overhanging Beam

ob 1AUniformly Distributed Load ob 2AUniformly Distributed Load on Overhang ob 3AUniformly Distributed Load Over Supported Span ob 4AUniformly Distributed Load Overhanging Both Supports

 

ob 5APoint Load on Beam Endob 6APoint Load Between Supports at Any Point

 
 

Two Span Continuous Beam

cb3s 1AEqual Spans, Uniformly Distributed Load cb3s 2AEqual Spans, Uniform Load on One Span cb3s 3AUnequal Spans, Uniformly Distributed Load cb3s 4AEqual Spans, Concentrated Load at Center of One Span

 

cb3s 5AEqual Spans, Two Equal Concentrated Loads Symmetrically Placedcb3s 6AEqual Spans, Concentrated Load at Any Pointcb3s 7AUnequal Spans, Concentrated Load on Each Span Symmetrically Placed

 
 

 

Nomenclature, Symbols, and Units for Beam Supports

Symbol Greek Symbol Definition English Metric SI Value
\(\Delta\) Delta deflection or deformation \(in\) \(mm\) \(mm\) -
\(a, b\) - distance to point load \(in\) \(mm\) \(mm\) -
\(w\) - highest load per unit length \(\large{\frac{lbf}{in}}\) \(\large{\frac{N}{m}}\) \(N-m^{-1}\)  
\(x\) - horizontal distance from reaction to point on beam \(in\) \(mm\) \(mm\) -
\(w\) - load per unit length \(\large{\frac{lbf}{in}}\) \(\large{\frac{N}{m}}\) \(N-m^{-1}\)  -
\(M\) - maximum bending moment \(lbf-in\) \(N-mm\) \(N-mm\)   -
\(V\) - maximum shear force \(lbf\) \(N\) \(kg-m-s^{-2}\)  
\(\lambda\) lambda modulus of elasticity \(\large{\frac{lbf}{in^2}}\)  \(MPA\) \(N-mm^{-2}\)  -
\(R\) - reaction load at bearing point \(lbf\) \(N\) \(kg-m-s^{-2}\) -
\(I\) - second moment of area (moment of inertia) \(in^4\) \(mm^4\) \(mm^4\) -
\(L\) - span length of the bending member \(in\) \(mm\) \(mm\) -
\(P\) - total concentrated load \(lbf\) \(N\) \(kg-m-s^{-2}\) -
\(W\) - total load \(\left( \frac{w\;L}{2} \right)\) \(lbf\) \(N\) \(kg-m-s^{-2}\) -
 

diagrams

  • Bending moment diagram (BMD)  -  Used to determine the bending moment at a given point of a structural element.  The diagram can help determine the type, size, and material of a member in a structure so that a given set of loads can be supported without structural failure.
  • Free body diagram (FBD)  -  Used to visualize the applied forces, moments, and resulting reactions on a structure in a given condition.
  • Shear force diagram (SFD)  -  Used to determine the shear force at a given point of a structural element.  The diagram can help determine the type, size, and material of a member in a structure so that a given set of loads can be supported without structural failure.
  • Uniformly distributed load (UDL)  -  A load that is distributed evenly across the entire length of the support area.
 
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