# Structural Analysis Software - SUNY Polytechnic Institute CTC / MTC 222 Strength of Materials Chapter 9 Shear Stress in Beams Chapter Objectives List the situations where shear stress in a beam is likely to be critical. Compute the shear stress in a beam using the general shear formula. Compute the maximum shear stress in a solid rectangular or circular section using the appropriate formulas. Compute the approximate maximum shear stress in a hollow thin-walled tube or thin-webbed section using the appropriate formulas. Shear Stresses To determine shear stress at some point in a beam, first must determine shear force.

Construct V diagram to find distribution and maximum shear. Often calculate vertical shear at a section Horizontal shear at the section is equal. Shear stress is not usually critical in steel or aluminum beams Beam is designed or selected to resist bending stress. Section chosen is usually more than adequate for shear Shear stress may be critical in some cases: Wooden beams Wood is weaker along the grain, subject to failure from horizontal shear

Thin-webbed beams Short beams or beams with heavy concentrated loads Fasteners in built-up or composite beams Stressed skin structures The General Shear Formula The shear stress, , at any point within a beams cross- section can be calculated from the General Shear Formula: = VQ / I t, where V = Vertical shear force I = Moment of inertia of the entire cross-section about the centroidal axis t = thickness of the cross-section at the axis where shear stress is

to be calculated Q = Statical moment about the neutral axis of the area of the cross-section between the axis where the shear stress is calculated and the top (or bottom) of the beam Q is also called the first moment of the area Mathematically, Q = AP y , where: AP = area of theat part of the cross-section between the axis where the shear stress is calculated and the top (or bottom) of the beam y = distance to the centroid of AP from the overall centroidal axis Units of Q are length cubed; in3, mm3, m3, Distribution of Shear Stress in Beams The maximum shear stress, , at any point in a beams cross-section occurs at the centroidal axis, unless, the thickness of the cross-section is less at some other axis.

Other observations: Shear stress at the outside of the section is zero Within any area of the cross-section where the thickness is constant, the shear stress varies parabolically, decreasing as the distance from the centroid increases. Where an abrupt change in the thickness of the cross-section occurs, there is also an abrupt change in the shear stress Stress will be much higher in the thinner portion Shear Stress in Common Shapes The General Shear Formula can be used to develop formulas for the maximum shear stress in common shapes.

Rectangular Cross-section max = 3V / 2A Solid Circular Cross-section max = 4V / 3A Approximate Value for Thin-Walled Tubular Section max 2V / A Approximate Value for Thin-Webbed Shape max V / t h t = thickness of web, h = depth of beam Design Shear Stress, d Design stress, d , varies greatly depending on material Wood beams

Allowable shear stress ranges from 70 - 100 psi Allowable bending stress is 600 1800 psi Allowable tension stress is 400 1000 psi Failure is often by horizontal shear, parallel to grain Steel beams d = 0.40 SY Allowable stress is set low, because method of calculating stress (max V / t h ) underestimates the actual stress Shear Flow Shear flow A measure of the shear force per unit length at a given section of a member The shear flow q is calculated by multiplying thr

shear force at a given section by the thickness at that section: q = t By the General Shear Formula: = VQ / I t Then q = t = VQ / I Units of q are force per unit length, N / m, kips / inch, etc. Shear flow is useful in analyzing built-up sections If the allowable shear force on a fastener, Fsd , is known, the maximum allowable spacing of fasteners required to connect a component of a built-up section, s max , can be calculated from: smax= Fsd / q