Mechanics of Solids

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Oct 29, 2013 (3 years and 9 months ago)

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Mechanics of Solids
Stress
Stress
1. External loadings BC
F
BC
= 41.57 kN
2. Internal resultant loadings: BC
N
max
= 41.57 kN;
3. Stress analysis (member BC)
≤ σ
allow

cross-sectional area of member BC

A
B
C
60
o
2.4 m
40 mm
30 kN/m
1-103 Determine the required thickness of member BC and diameter of the pins at A and B.
Allowable normal stress for member BC σ
allow
= 200 MPa, and the allowable shear stress for
the pins τ
allow
= 70 MPa, (p. 62)
V = F
V = F/2
F
F
Single-Shear Double-Shear
t
BC
≥ 5.2 mm
Stress
1. External loadings BC
F
BC
= 41.57 kN
2. Internal resultant loadings: BC
N
max
= 41.57 kN;
3. Stress analysis
Pin at B

≤ τ
allow

d
B
≥ 19.44 mm
A
B
C
60
o
2.4 m
40 mm
30 kN/m
V = F
V = F/2
F
F
Single-Shear Double-Shear
1-103 Determine the required thickness of member BC and diameter of the pins at A and B.
Allowable normal stress for member BC σ
allow
= 200 MPa, and the allowable shear stress for
the pins τ
allow
= 70 MPa, (p. 62)
= 296.92 mm
2
B
Stress
1. External loadings A
F
BC
= 41.57 kN
Free-body diagram
Equations of equilibrium
F
Ax
+ F
Bx
cos 60
o
= 0
F
Ay
+ F
Bx
sin60
o
– 72 = 0
=41.57 kN
1-103 Determine the required thickness of member BC and diameter of the pins at A and B.
Allowable normal stress for member BC σ
allow
= 200 MPa, and the allowable shear stress for
the pins τ
allow
= 70 MPa, (p. 62)
A
B
C
60
o
2.4 m
40 mm
30 kN/m
V = F
V = F/2
F
F
Single-Shear Double-Shear
A
B
F
Ay
F
Ax
F
BC
30 x 2.4=72 kN
C
60
o
F
A y
= 36 kN
F
A x
= -20.79 kN
Stress
1. External loadings A
F
A
= 41.57 kN
2. Stress analysis
Pin at A

≤ τ
allow

d
A
≥ 27.50 mm
1-103 Determine the required thickness of member BC and diameter of the pins at A and B.
Allowable normal stress for member BC σ
allow
= 200 MPa, and the allowable shear stress for
the pins τ
allow
= 70 MPa, (p. 62)
A
B
C
60
o
2.4 m
40 mm
30 kN/m
V = F
V = F/2
F
F
Single-Shear Double-Shear
= 593.86 mm
2
B
c
F
Ay
F
Az
F

M
F
Ax
c
General State of Stress
ơ
zz
ơ‘
zz
B
A
F

z
y
x
o
τ
zy
τ’
zy
τ
zx
τ'
zx
Stress: the intensity of the internal force on a specific plane passing through a point
ơ
yy
τ
yz
τ
yx
ơ
xx
τ
xy
τ
xz
c
c
N
V
F’

z
y
x
o
Stress
Stress: the intensity of the internal force on a specific plane passing through a point
Shear Stress, τ,
Normal Stress, σ
[MPa]
Average Normal Stress
Average Shear Stress
Mechanics of Solids
Strain
Strain
Normal Strain
The change in length of the line is ΔS-ΔS’. We consequently define the
generalized strain mathematically as
[MPa]
Normal Stress
Mechanics of materials: a branch of mechanics that studies the internal effects of stress
and
strain
in a solid body that is subjected to an external loading
.
Strain
Average Normal Strain
If the stress in the body is everywhere constant, in other words, the deformation is
uniform in the material (e.g. uniform uniaxial tension or compression), the strain can
be computed by
Usually, for most engineering applications ε is very small, so measurements of strain are
in micrometers per meter (μm/m) or (μ/m).
Sometimes for experiment work, strain is expressed as a percent, e.g. 0.001m/m = 0.1%.
Unit of Strain
Strain
2-9 If a force is applied to the end D of the rigid member CBD and causes a normal strain in
the cable of 0.0035 mm/mm, determine the displacement of point D. (p76)
A
C
B
D
F
B’
D’
300 mm

300 mm
400 mm
A
B
B’
E
AB = AE
EB’ = AB’-AE = AB’-AB = ∆AB
C
D
D’
DD’ = 2 x BB’
EB’ = ∆AB = AB x ε = 500 x 0.0035 = 1.75 mm
α
β
α = β
BB’ = EB’/cosα = 1.75 x (5/4) = 2.1875 mm
DD’ = 2 x EB’ = 4.38 mm
Mechanics of Solids
Stress & Strain:
Mechanical Properties of
Materials

Stress & Strain: Mechanical Properties of Materials
The Stress-Strain diagram normally consists of 4 stages during the whole process, elastic,
yielding, hardening and necking stages respectively
Ductile materials
Stress & Strain: Mechanical Properties of Materials
The Stress-Strain diagram normally consists of 4 stages during the whole process, elastic,
yielding, hardening and necking stages respectively
Brittle materials
Stress & Strain: Mechanical Properties of Materials
The Stress-Strain diagram normally consists of 4 stages during the whole process, elastic,
yielding, hardening and necking stages respectively
Brittle materials
Ductile materials
Strain Hardening
Stress & Strain: Mechanical Properties of Materials
The Stress-Strain diagram normally consists of 4 stages during the whole process, elastic,
yielding, hardening and necking stages respectively
True strain-stress diagram
Conventional strain-stress diagram
Stress & Strain: Hooke’s Law
where E is terms as the Modulus of Elasticity
or Young's Modulus with units of N/m
2
or Pa.
For most of engineering metal material, GPa is
used, e.g. mild steel is about 200GPa ~ 210GPa
σ = Eε
~ 0.1% (steel)