AMME2301/AMME5301
Mohr’s Circle – Plane Stress
• Lecturer: Dr Li Chang
• Room s503, Building J07
• Tel: 9351-5572
• e-mail: [email protected]
Mohr’s Circle – Plane Stress
τsn
s
Ѳ
y
σnn
n
Ѳ
[
]2 [
]2
[ ]2 [
]2
x
[c, 0]
τsn
σnn
Mohr’s Circle – Principal Stresses & Maximum In-Plane Shear Stress
[σavg, -τmax]
[σyy, -τxy]
[σ11, 0]
[σ22, 0]
[c, 0]
2Ѳp
a) Principal stresses
2Ѳs
[σxx,τxy]
[σavg, τmax]
b) Maximum in-plane stress
Ѳp - Ѳs= 45o
τ
σ
Mohr’s Circle – Plane Stress Transformation
9.18 A point on a thin plate is subjected to two successive states of stress as shown. Determine
the resulting state of stress with reference to an oriented as shown on the bottom. (p460)
I
45 kPa
II
50 kPa
50o
30o
18 kPa
τxy
σxx
σyy
Mohr’s Circle – Principal Stresses & Maximum In-Plane Shear Stress
9-15 Determine (a) the principal stresses and (b) the maximum in-plane shear stress and
average normal stress at the point. Specify the orientation of the element in each case. (p. 459)
30MPa
45MPa
60MPa
State of Stresses Caused by Combined Loadings
Example: The 25 mm diameter rod is subjected to the loads shown. Determine the principal
stresses and the maximum shear stress at point B. (The representations of the Mohr circle and
the infinitesimal elements are required)
AMME2301/AMME5301
Applications of Plane Stresses
• Lecturer: Dr Li Chang
• Room s503, Building J07
• Tel: 9351-5572
• e-mail: [email protected]
Plane Stresses: Thin-Walled Pressure Vessels
Hoop Stress (or Circumferential Stress):
Plane Stresses: Thin-Walled Pressure Vessels
Axial Stress(or Longitudinal Stress):
Plane Stresses: Thin-Walled Pressure Vessels
Combined Load Conditions
AMME2301/AMME5301
Theories of Failure
• Lecturer: Dr Li Chang
• Room s503, Building J07
• Tel: 9351-5572
• e-mail: [email protected]
Stress & Strain: Mechanical Properties of Materials
σult
Maximum-Normal-Stress Theory: Brittle Materials
[σavg, -τmax]
[σyy, -τxy]
[σ22, 0]
[σ11, 0]
σ
[c, 0]
2Ѳ
[σxx,τxy]
τ
2Ѳs
lσ22l, lσ11l < σult
p
[σavg, τmax]
τ
Brittle materials
σult
σult
σY
y
0
x
Maximum-Normal-Stress Theory: Ductile Materials
Ductile materials
Maximum-Normal-Stress Theory: Ductile Materials
σY = 2 τmax
Ductile materials
Tresca’s Yield Criterion (Maximum-Shear-Stress Theory)
[σavg, -τmax]
σY
y
[0, 0]
[c, 0]
[σ11, 0]
σ
x
Lüders lines
σY = 2τmax
τ
[σavg, τmax]
Ductile materials
Tresca’s Yield Criterion (Maximum-Shear-Stress Theory)
[σavg, -τmax]
[σyy, -τxy]
[σ22, 0]
[σ11, 0]
σ
[c, 0]
2Ѳ
[σxx,τxy]
τ
z
2Ѳs
σY = 2τmax
p
[σavg, τmax]
σ11
σ22
y
x
σ11
z
x
von Mises Yield Criterion (Maximum-Distortion-Energy Theory)
z
σ33
σ11
x
σ22
y
z
=
σavg
σavg
x
σavg
y
z
+ σ -σ
avg 33
σavg- σ11
σavg- σ22
y
x
(textbook p.526)
Plane Stresses: Thin-Walled Pressure Vessels
10-89 The gas tank is made from A-36 steel and has an inner diameter of 1.5 m. If the tank is
designed to withstand a pressure of 5 MPa, determine the required minimum wall thickness to
the nearest milimeter using (a) the maximum-shear-stress theory, and (b) maximumdistortion-energy theory. Apply a factor of safety of 1.5 against yielding. (p.535)
Assignment Questions: Bending
Draw the shear force and bending moment diagrams for the beam.
Determine the maximum shearing stress in the beam.
Determine the maximum tensile and compressive stresses in the beam.
30 kN/m
45 kN m
A
1.5 m
𝜏𝑚𝑎𝑥
𝑉𝑚𝑎𝑥 𝑄
=−
𝐼𝑡
𝜎𝑚𝑎𝑥
𝜎𝑚𝑎𝑥
𝑀max −
=−
𝑐(±)
𝐼
1.5 m
1.5 m
RA = 41.25 kN
V (x)
M (x)
33.75
39.375
RB = 3.75 kN
B
A
-3.75
(41.25)
45
𝑀max +
=−
𝑐(±)
𝐼
B
A
5.625
x
B
(45)
x
Assignment Questions: Plane Stresses
Determine the principal stresses and the maximum shearing stress at the point;
Determine the orientations of principal stresses and maximum shear stress at the point
(the representations of the Mohr circle and the infinitesimal elements are required).
1) Mohr’s circle
[σyy, -τxy]
[σ22, 0]
[σ11, 0]
[c, 0]
2) Draw infinitesimal elements
2
1
2Ѳp
avg
θs1
[σxx,τxy]
max
τ
θp1
σ