Shear Stress in Beams
(6.1-6.4)
MAE 314 – Solid Mechanics
Yun Jing
Shear Stress in Beams
1
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
Shear Stress in Beams
2
Shear Stress in Beams



Consider the effects of shear force (V).
Already know how to find resulting axial force and
moment due to stress σx 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:

Shear Stress in Beams
xy
dA  V
xz
dA  0
3
Shear Stress in Beams

Consider a cantilever beam composed of separate planks clamped at
one end:
Pure bending

Shear force causes tendency to “slide.”

Stresses are equal in horizontal and
vertical directions.
Shear Stress in Beams
Shear force
4
Shear Stress: Horizontal

Let us consider the horizontal component (τyx = τxy).

Cut a section with cross-sectional
area a at a distance y1 above the
centroid.
FBD →
Shear Stress in Beams
5
Shear Stress: Horizontal



ΔH is the horizontal shearing force.
Element width is Δx.
Sum forces in x-direction:
F
x
 H    C   D dA  0
a
 
My
I

Recall from chapter 4:

Solve for ΔH and use equation for σ:
 M D  MC 
 M D  MC 
H    D   C dA   
 ydA 
 ydA
I
I


a
a
a
Shear Stress in Beams
6
Shear Stress: Horizontal

Recall first moment, Q, is defined as:
Q   yda
a

The term MD-MC can be rewritten as:
M D  M C  M 
dM
x  Vx
dx
H 
VQ
x
I

Applying this to our equation for ΔH:

We can rearrange this to define horizontal shear per unit length, q,
called shear flow.
q
H VQ

x
I
Shear Stress in Beams
7
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),
Q   ydA  a y
a
where y bar is the distance between the centroid of the shaded section
and the centroid of beam cross-section.
Shear Stress in Beams
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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.
Shear Stress in Beams
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Shear Stress: Vertical

Now, let us consider the vertical component (τxy= τyx).

We can calculate the average vertical shear stress on the cross-section.
 AVE 
H  VQ  1  VQ

x 
  AVE

A  I
 tx  It
 AVE
VQ

It
Shear Stress in Beams
10
Shear Stress: Vertical

So, where is τAVE 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
Shear Stress in Beams
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Shearing Stress in Common Shapes

Rectangular cross-section
Q  Ay  bc  y 

1
c  y   1 b c 2  y 2
2
2



VQ
V
b c2  y 2
3V 2
 xy 


c  y2
3
3
Ib
2b
4bc
b2c  / 12




3V  y 2 
1  2 
 xy 
2A  c 
 max 
Shear Stress in Beams
3V
2A
12
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.
V
 max 
Aweb
Flange
Web
Shear Stress in Beams
13
Example Problem
For the beam and loading shown, consider section n-n and determine
the shearing stress at (a) point a, (b) point b.
Shear Stress in Beams
14
Shear Stress in Thin Walled
Members (6.7)
MAE 314 – Solid Mechanics
Yun Jing
Shear Stress in Beams
15
Shear in Thin Walled Members



May want to calculate horizontal or vertical shear stress,
depending on the point of interest.
Vertical cut: τavg = average τxz
Horizontal cut: τavg = average τxy
Shear Stress in Beams
16
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.
Shear Stress in Beams
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Example Problem

Knowing that the vertical shear is 50 kips in a W10*68 rolled-steel
beam, determine the horizontal shearing stress in the top flange at
point a
Shear Stress in Beams
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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.
Shear Stress in Beams
19
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.
Shear Stress in Beams
20