Based on the forgoing general derivation of shear stress, the formulas for shear stress in rectangular wood beams and flanged steel beams is derived here. The maximum stress in those beams is customarily defined as fv instead of v in the general shear formula.
1 Shear at neutral axis of rectangular beam (maximum stress),
Note: this is the same formula derived for maximum shear stress before
2 Shear stress at the bottom of rectangular beam. Note that y= 0 since the centroid of the area above the shear plane (bottom) coincides with the neutral axis of the entire section. Thus Q= Ay = (bd/2) 0 = 0, hence
v = V 0/(I b) = 0 = fv, thus
fv = 0
Note: this confirms an intuitive interpretation that suggests zero stress since no fibers below the beam could resist shear
3 Shear stress at top of rectangular beam. Note A = 0b = 0 since the depth of the shear area above the top of the beam is zero. Thus
Q = Ay = 0 d/2 = 0, hence v = V 0/(I b) = 0 = fv, thus
fv = 0
Note: this, too, confirms an intuitive interpretation that suggests zero stress since no fibers above the beam top could resist shear.
4 Shear stress distribution over a rectangular section is parabolic as implied by the formula Q=b(d^2)/8 derived above.
5 Shear stress in a steel beam is minimal in the flanges and parabolic over the web.
The formula v = VQ/(I b) results in a small stress in the flanges since the width b of flanges is much greater than the web thickness. However, for convenience, shear stress in steel beams is computed as “average” by the simplified formula:
1 Shear at neutral axis of rectangular beam (maximum stress),
2 Shear stress at the bottom of rectangular beam. Note that y= 0 since the centroid of the area above the shear plane (bottom) coincides with the neutral axis of the entire section. Thus Q= Ay = (bd/2) 0 = 0, hence
v = V 0/(I b) = 0 = fv, thus
fv = 0
Note: this confirms an intuitive interpretation that suggests zero stress since no fibers below the beam could resist shear
3 Shear stress at top of rectangular beam. Note A = 0b = 0 since the depth of the shear area above the top of the beam is zero. Thus
Q = Ay = 0 d/2 = 0, hence v = V 0/(I b) = 0 = fv, thus
fv = 0
Note: this, too, confirms an intuitive interpretation that suggests zero stress since no fibers above the beam top could resist shear.
4 Shear stress distribution over a rectangular section is parabolic as implied by the formula Q=b(d^2)/8 derived above.
5 Shear stress in a steel beam is minimal in the flanges and parabolic over the web.
The formula v = VQ/(I b) results in a small stress in the flanges since the width b of flanges is much greater than the web thickness. However, for convenience, shear stress in steel beams is computed as “average” by the simplified formula:
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