CTC / MTC 222 Strength of Materials

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CTC / MTC 222 Strength of Materials

• Chapter 9
• Shear Stress in Beams

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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.

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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

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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,

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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

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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

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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

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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