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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.

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ctc mtc 222 strength of materials

CTC / MTC 222 Strength of Materials

Chapter 9

Shear Stress in Beams

chapter objectives
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
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 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,
distribution of shear stress in beams
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
shear stress in common shapes
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 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
  • 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