Ctc mtc 222 strength of materials
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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

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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


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