### BEAMS SHEAR STRESS

The distribution of shear stress over the cross-section of beams is derived, referring to a beam part of length x marked on diagrams. Even though horizontal and vertical shear are equal at any part of a beam, horizontal shear stress is derived here because it is much more critical in wood due to horizontal fiber direction.

1 Beam, shear and bending diagrams with marked part of length x

2 Beam part with bending stress pushing and pulling to cause shear

3 Beam part with bending stress above an arbitrary shear plane

Let M be the differential bending moment between m and n. M is equal to the shear area between m and n (area method), thus M = V x. Substituting V x for M in the flexure formula f= M c /I yields bending stress f= V x c/I in terms of shear. The differential bending stress between m and n pushes top and bottom fibers in opposite directions, causing shear stress. At any shear plane y1 from the neutral axis of the beam the sum of shear stress above this plane yields a force F that equals average stress fy times the cross section area A above the shear plane, F = A fy. The average stress fy is found from similar triangles; fy relates to y as f relates to c, i.e., fy/y = f/c; thus fy = f y/c. Since f= V x c/I, substituting V x c/I for f yields fy = (V x c/I) y/c = V x y/I. Since F = A fy, it follows that F = A V x y/I. The horizontal shear stress v equals the force F divided by the area of the shear plane;

V = F/(x b) = A V x y/(I x b) = V A y/(I b)

The term A y is defined as Q, the first static moment of the area above the shear plane times the lever arm from its centroid to the neutral axis of the entire cross-section. Substituting Q for A y yields the working formula

v = horizontal shear stress.

Q = static moment (area above shear plane times distance from centroid of that area to

the neutral axis of the entire cross-section

I = moment of Inertia of entire cross section

b = width of shear plane

The formula for shear stress can also be stated as shear flow q, measured in force per unit length (pound per linear inch, kip per linear inch, or similar metric units); hence

1 Beam, shear and bending diagrams with marked part of length x

2 Beam part with bending stress pushing and pulling to cause shear

3 Beam part with bending stress above an arbitrary shear plane

Let M be the differential bending moment between m and n. M is equal to the shear area between m and n (area method), thus M = V x. Substituting V x for M in the flexure formula f= M c /I yields bending stress f= V x c/I in terms of shear. The differential bending stress between m and n pushes top and bottom fibers in opposite directions, causing shear stress. At any shear plane y1 from the neutral axis of the beam the sum of shear stress above this plane yields a force F that equals average stress fy times the cross section area A above the shear plane, F = A fy. The average stress fy is found from similar triangles; fy relates to y as f relates to c, i.e., fy/y = f/c; thus fy = f y/c. Since f= V x c/I, substituting V x c/I for f yields fy = (V x c/I) y/c = V x y/I. Since F = A fy, it follows that F = A V x y/I. The horizontal shear stress v equals the force F divided by the area of the shear plane;

V = F/(x b) = A V x y/(I x b) = V A y/(I b)

The term A y is defined as Q, the first static moment of the area above the shear plane times the lever arm from its centroid to the neutral axis of the entire cross-section. Substituting Q for A y yields the working formula

v = horizontal shear stress.

Q = static moment (area above shear plane times distance from centroid of that area to

the neutral axis of the entire cross-section

I = moment of Inertia of entire cross section

b = width of shear plane

The formula for shear stress can also be stated as shear flow q, measured in force per unit length (pound per linear inch, kip per linear inch, or similar metric units); hence