Structures: Bending and Shear Stress

ending and shear stresses in beams relate to bending moment and shear force similar to the way axial stress relates to axial force (f = P/A).  Bending and shear stresses are derived here for a rectangular beam of homogeneous material (beam of constant property).  A general derivation follows later with the Flexure Formula.

Simple wood beam with hatched area and square marked for inquiry
2  Shear diagram with hatched area marked for inquiry
3  Bending moment diagram with hatched area marked for inquiry 
4  Partial beam of length x, with stress blocks for bending fb and shear fv, where x is assumed a differential (very small) length

Reactions, found by equilibrium ΣM = 0 (clockwise +)

Shear V, found by vertical equilibrium, ΣV=0 (upward +).

Bending moment M, found by equilibrium ΣM=0 (clockwise +) 

Bending stress fb is derived, referring to 4.  Bending is resisted by the force couple C-T,  with lever arm 2/3 d = distance between centroids of triangular stress blocks.  C=T= fb bd/4, M= C(2d/3) = (fbbd/4)(2d/3) = fbbd^2/6, or  fb= M/(bd^2/6); where bd^2/6 = S= Section Modulus for rectangular beam; thus

 * multiplying by 1000 converts kips to pounds, by 12 converts feet to inches. Shear stress fv is derived, referring to 4.  Bending stress blocks pushing and pulling in opposite directions create horizontal shear stress.  The maximum shear stress is fv=C/bx, where b = width and x = length of resisting shear plane.  Shear at left support is V = R.

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