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)

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