CTC / MTC 222 Strength of Materials

1. CTC / MTC 222 Strength of Materials Chapter 9 Shear Stress in Beams

2. 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.

3. 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

4. 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 ̅y̅ , 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 • ̅y̅ = distance to the centroid of AP from the overall centroidal axis • Units of Q are length cubed; in3, mm3, m3,

5. Distribution of Shear Stress in Beams • The maximum shear stress, , at any point in a beam’s 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

6. 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

7. 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

8. 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, smax , can be calculated from: smax= Fsd / q