Shear Stress in Beams (6.1-6.4)

1
Shear Stress in Beams (6.1-6.4)

• MAE 314 Solid Mechanics
• Yun Jing

2
Review

• Previous two chapters only dealt with normal
  stresses caused by bending moments.
• Chapter 6 deals with shear stress caused by shear
  forces.

Line of failure
3
Shear Stress in Beams

• Consider the effects of shear force (V).
• Already know how to find resulting axial force
  and moment due to stress sx from Chapter 4.
• We have two more equations for shear stress
• Total shear force in the y-direction
• Total shear force in the z-direction

4
Shear Stress in Beams

• Consider a cantilever beam composed of separate
  planks clamped at one end
• Shear force causes tendency to slide.
• Stresses are equal in horizontal andvertical
  directions.

Pure bending
Shear force
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Shear Stress Horizontal

• Let us consider the horizontal component (tyx
  txy).
• Cut a section with cross-sectionalarea a at a
  distance y1 above thecentroid.

FBD ?
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Shear Stress Horizontal

• ?H is the horizontal shearing force.
• Element width is ?x.
• Sum forces in x-direction
• Recall from chapter 4
• Solve for ?H and use equation for s

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Shear Stress Horizontal

• Recall first moment, Q, is defined as
• The term MD-MC can be rewritten as
• Applying this to our equation for ?H
• We can rearrange this to define horizontal shear
  per unit length, q, called shear flow.

8
Side Note on Q

• Q is the definition of the first moment for the
  area above y1 with respect to the x-axis (see
  Appendix A in textbook),where y bar is the
  distance between the centroid of the shaded
  section and the centroid of beam cross-section.

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Example Problem

• A beam is made of three planks, 20 by 100 mm in
  cross-section, nailed together. Knowing that the
  spacing between nails is 25mm and that the
  vertical shear in the beam is V 500 N,
  determine the shearing force in each nail.

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Shear Stress Vertical

• Now, let us consider the vertical component (txy
  tyx).
• We can calculate the average vertical shear
  stress on the cross-section.

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Shear Stress Vertical

• So, where is tAVE maximum and minimum?
• Use Q to find out.
• Q 0 at top and bottom surfaces
• Q maximum somewhere in between

max normal stress
shear stress 0
max shear stress
normal stress 0
max normal stress
shear stress 0
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Shearing Stress in Common Shapes

• Rectangular cross-section

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Shearing Stress in Common Shapes

• Beams with flanges
• Vertical shear stresses are larger in the web
  than in the flange.
• Usually only calculate the values in the web.
• Ignore the effects of the small fillets at the
  corners.
• Flanges have large horizontal shear stresses,
  which we will learn how to calculate later on.

Flange
Web
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Example Problem

• For the beam and loading shown, consider section
  n-n and determine
• the shearing stress at (a) point a, (b) point b.

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Shear Stress in Thin Walled Members (6.7)

• MAE 314 Solid Mechanics
• Yun Jing

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Shear in Thin Walled Members

• May want to calculate horizontal or vertical
  shear stress, depending on the point of interest.
• Vertical cut tavg average txz
• Horizontal cut tavg average txy

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Shear in Thin Walled Members

• Why do we choose to cut the beam perpendicular
  to the cross-section wall?
• Want to cut across line of shear flow.

Shear flow in box-beam section.
Shear flow in wide-flange beam section.
18
Example Problem

• Knowing that the vertical shear is 50 kips in a
  W1068 rolled-steel beam, determine the
  horizontal shearing stress in the top flange at
  point a

19
Example Problem

• The built-up beam shown is made by gluing
  together two 20 x 250 mm
• plywood strips and two 50 x 100 mm planks.
  Knowing that the allowable
• average shearing stress in the glued joints is
  350 kPa, determine the
• largest permissible vertical shear in the beam.

20
Example Problem

• An extruded aluminum beam has the cross section
  shown. Knowing that
• the vertical shear in the beam is 150 kN,
  determine the shearing stress
• a (a) point a, and (b) point b.