Transverse Shear - Mechanics of Materials

Straight Members

Shear and Moment loadings are typically supported by beams.

A straight beams cross section will have a shear-stress distribution that acts on it, called the transverse shear-stress. The shear, V, is the result of the transverse shear-stress.

These beams will also have longitudinal shear stress which will be found acting in the length of beam.

If you were to analyze a small piece from the interior cross section, you'll find that the piece will have longitudinal shear stress as well as transverse shear stress.

These shear stresses will create a complex nonuniform distortion in the cross section which will cause warping. When both of these stresses act on the beam it will violate the flexure formula, which states that a cross section will be perpendicular to the longitudinal axis and the cross section must be plane.

The Shear Formula is found through relating the flexure formula and the bending and shear relationship.

Shear Formula

= (VQ) / (It) This is the shear stress at a point in the cross section a distance y' from the
neutral axis.

• V = Internal shear.
• Q = y' A' -- Statical moment of area. y' is the distance from the neutral axis (centroid of cross section) to the centroid of A'. A' is the area of the top portion where t is measured.
  
• I = Moment of inertia of the whole cross section.
• t = thickness of the cross section.

will be zero at the top and bottom (because Q = 0 at top/bottom) of the cross section and change linearly along cross section to a max which will occur when Q is a max, all other variables will be constant. Qmax will occur at the neutral axis of a cross section because the largest area will be obtained above the neutral axis.



Limitations with the shear formula:

• Using this formula we assume that the shear stress is uniformly distributed over t (width).
• If there is a sudden change in the cross section this formula will not be accurate.
• This formula will have greater accuacy is the width/depth ratio is small. Therefore, a cross section that is tall and thin will yield more accurate results.

Shear Flow

q = (VQ) / I -- Shear flow measured as a force per unit of length. This flow will act along the longitudinal axis of the member.

• V = Internal shear force.
• Q = y' A' A' is the area of the cross section that is connected to the beam. y' is the distance between the neutal axis and the centroid of area A'.
• I = Moment of inertia.
  

Multiple members that are connected by nail, bolt, screw, glue, etc. Another formula is used that relates the shear stress in the fastener (nail, bolt, etc) and the spacing of these fasteners.

qs = NF

• q = shear flow, found from in the above equation.
• s = spacing of fasteners.
• N = number of fasteners through the member.
• F = shear force supported by the fastener.

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