1. The principal stresses are σ1=164MPa and σ2=-24MPa. The maximum in-plane shear stress is τm,ip=-94MPa.
2. The normal and shear stresses in the wood grain direction are σx'=0MPa and τx'y'=63MPa.
3. Mohr's circle is used to determine the stresses by rotating the stress element to principal planes and the plane of maximum shear.
1. 1
Figure Q1 shows the stress condition on a piece of wooden structure at a critical point.
Determine:-
a) The principal stresses and sketch the stress element;
b) The maximum in-plane shearing stress and show the stress element in this condition;
c) The normal and shear stresses in the wood grain direction and sketch the stress element.
Figure Q1
x
y
σx=20MPa
τxy=
80MPa
σy=120MPa
50o
Wood grain
2. 2
Firstly, it is suggested to
place your graph paper in
the landscape orientation.
4. σ
σavg
4
• Calculate σavg = (σx + σy)/2.
• Locate σavg on σ–axis. It
should be somewhere at the
middle of the graph. It is
because you are going to draw a
circle with the centre (σavg,0).
9. σ
τ
σavg
σx, τxy
σy, τxy
9
Draw a straight line
that connects (σx, τxy)
and σavg. You will get
(σy, τxy) at the other end
of the line.
x
y
σx=20MPa
τxy=
80MPa
σy=120MPa
50o
Wood grain
19. 19
σ
τ
σavg
σ1σ2
σy’, τm,ip
σx’, τm,ip
σx, τxy
σy, τxy
2θp
2θs
• Label (σx’, τm,ip) and
(σy’, τm,ip) on the Mohr’s
circle.
• Note that σx’is on the
top, and σy’is at the
bottom of the circle.
24. 24
θs
x
y’
x’ σavg=
70MPa
σavg=70MPa σ
τ
σavg
σ1σ2
σx, τxy
σy, τxy
2θp
2θs
Note that x’–axis is located
on the top.
σy’, τm,ip
σx’, τm,ip
Hence, it means that x’–axis
is having the τm,ip component
that tends to rotate the
element in CW direction.
25. 25
θs
x
y’
x’ σavg=
70MPa
σavg=70MPa σ
τ
σavg
σ1σ2
σx, τxy
σy, τxy
2θp
2θs
At the same time, you could
also notice that y’–axis is
located at the bottom, …
σy’, τm,ip
σx’, τm,ip
and thus having τm,ip
component that tends to rotate
the element in CCW direction.
30. 30
σ
τ
σavg
σ1σ2
σx, τxy
σy, τxy
σx', τx’y’
σy', τx’y’
2θ
2θp
2θs
• Label (σx’, τx’y’) and
(σy’, τx’y’).
• Note that y’ is attained
after rotating 2θ from y.
Similarly, x’ comes from x.
x
y
σx=20MPa
τxy=
80MPa
σy=120MPa
50o
Wood grain
σy’, τm,ip
σx’, τm,ip
31. 31
θ
x
σ
τ
σavg
σ1σ2
σx, τxy
σy, τxy
σx', τx’y’
σy', τx’y’
2θ
2θp
2θs
• To draw the stress element, by
using x–axis as reference,
rotate the plane by an angle of
θin the CCW direction.
• Note that rotating y by 50o in
CCW is the same as rotating x
by 50o in CCW.
σy’, τm,ip
σx’, τm,ip
35. 35
θ
x
x'y'
σy'=140MPa
σ
τ
σavg
σ1σ2
σx, τxy
σy, τxy
σx', τx’y’
σy', τx’y’
2θ
2θp
2θs
Note that σx’ is located at
the bottom. Hence, it
means that x’–axis is
having the τx’y’ component
that tends to rotate the
element in CCW direction.
σy’, τm,ip
σx’, τm,ip
36. 36
θ
x
x'y'
σy'=140MPa
σ
τ
σavg
σ1σ2
σx, τxy
σy, τxy
σx', τx’y’
σy', τx’y’
2θ
2θp
2θs
At the same time, you
could also notice that y’–
axis is located at the top,
and thus having τx’y’
component that tends to
rotate the element in CW
direction.
σy’, τm,ip
σx’, τm,ip