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The stress in beam due to a bending moment can be derived with three principals: geometry, Hooke's Law, and equilibrium.



Geometry:

Hooke's Law:

Equilibrium:



 

Assumptions:
  • Beam is initially straight and unstressed
  • Beam material is homogeneous and isotropic
  • Material preperties are the same in tension and compression
  • Plane cross sections remain plane after bending
Therefore:



Second Moment of Area of Common Shapes
:

Rectangular cross-section



Circular cross-section