Beam Design Formulas

on . Posted in Structural Engineering

Structural beam design refers to the process of determining the appropriate size and shape of beams that will be used in construction to support and distribute loads within a building or structure.  Beams are horizontal or inclined structural members that carry vertical loads.  The design of structural beams involves several important considerations to ensure the safety, stability, and efficiency of a structure.

Beam Design Formulas Index

Key Points about Design of Structural Beams

  • Load Calculation  -  The first step is to calculate the various loads that the beams will need to support, including dead loads (permanent loads) and live loads (temporary loads).  Additionally, factors such as wind, snow, earthquakes, and other environmental loads must also be considered.
  • Material Selection  -  Beams can be constructed from various materials, such as steel, concrete, timber, or composite materials.  The choice of material depends on factors like the type of structure, load requirements, cost considerations, and desired aesthetics.
  • Bending and Shear Stress Analysis  -  Beams primarily experience bending stresses due to the applied loads, which cause the beam to bend or deflect.  Shear stresses also play a role, especially near supports.  Engineers analyze these stresses to ensure that the beam can safely withstand the loads without failure.
  • Beam Sizing  -  Based on the load calculations and stress analysis, engineers determine the appropriate size and shape (cross-sectional dimensions) of the beam. This includes selecting the appropriate width, depth, and other dimensions to distribute the loads effectively and avoid excessive deflection or failure.
  • Code Compliance  -  Designing beams involves adhering to building codes and standards set by relevant authorities.  These codes dictate the minimum safety requirements and design practices that must be followed to ensure structural integrity.
  • Connection Design  -  Beams are often connected to other structural members, such as columns, beams, and foundations.  The design of these connections is crucial to ensure load transfer and overall stability of the structure.
  • Deflection Control  -  Excessive deflection can lead to discomfort for occupants and potential damage to finishes.  Engineers design beams to limit deflection within acceptable limits.
  • Economic Considerations  -  Beam design also takes into account economic factors.  This includes optimizing beam sizes and material choices to achieve a balance between structural performance and cost effectiveness.
  • Construction Considerations  -  The practicality of constructing the designed beams is an important aspect.  Engineers need to ensure that the beams can be fabricated, transported, and installed effectively.
  • Safety Factors  -  Engineers apply safety factors to the calculated loads and stresses to provide an additional margin of safety, accounting for uncertainties in material properties, construction practices, and loading conditions.

diagram Symbols

  • 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.


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 \(lbf\;/\;in\) \(N\;/\;m\) \(N-m^{-1}\)  
\(x\) - horizontal distance from reaction to point on beam \(in\) \(mm\) \(mm\) -
\(w\) - load per unit length \(lbf\;/\;in\) \(N\;/\;m\) \(N-m^{-1}\)  -
\(M\) - maximum bending moment \(lbf-in\) \(N-mm\) \(N-mm\)   -
\(V\) - maximum shear force \(lbf\) \(N\) \(kg-m-s^{-2}\)  
\(\lambda\) lambda modulus of elasticity \(lbf\;/\;in^2\)  \(MPA\) \(N-mm^{-2}\)  -
\(R\) - 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 \(\left( \frac{w\;L}{2} \right)\) \(lbf\) \(N\) \(kg-m-s^{-2}\) -
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