Mechanics of Materials Formula and Data Sheet

Some Engineering Material Properties:

Property Steel Aluminum Brass

Young’s Modulus, E 210 GPa (30 psi x 106) 70 GPa (10 psi x 10

6) 105 GPa (15 psi x 10

6)

Shear Modulus, G 81 GPa (11.6 psi x 106) 26 GPa (3.7 psi x 10

6) 39 GPa (5.6 psi x 10

6)

Poisson’s Ratio, Ȟ 0.30 0.33 0.35

Thermal Expansion, Į 11x10-6

/ºC (6x10-6

º/F) 22x10-6

/ºC (12x10-6

º/F) 20x10-6

/ºC (11x10-6

º/F)

Density, ȡ 7850 kg/m3 (0.28 lb/in

3) 2720 kg/m

3 (0.10 lb/in

3) 8410 kg/m

3 (0.30 lb/in

3)

Moments of Area for Common Shapes:

Shape Cross-section A Ix Iy J = Ix + Iy

solid circle ʌ c2 ʌ c

4 / 4 ʌ c

4 / 4 ʌ c

4 / 2

thin-wall circle 2 ʌ t c ʌ t c3 ʌ t c

32 ʌ t c

3

solid rectangle b h b h3 / 12 h b

3 / 12 b h (b

2 + h

2) / 12

Elastic relationships:

T E

E

E

zyx

x 'D�V

Q�V

Q�V

H G

W

J )2(1

E G

Q�

)2-3(1

E K

Q

Failure Criteria: (for principal stresses ı1 � ı2 � ı3)

Tresca: ı1 – ı3 = ıY Mises: (ı1 – ı2)2 + (ı2 – ı3)

2 + (ı3 – ı1)

2 = 2ıY

2 Mohr: ı1/ıUT – ı3/ıUC = 1

Thin-Walled Pressure Vessels:

0 t2

R p

t

R p :Cylinder ra |V V VT 0

t2

R p :Sphere ra |V V VT

Shaft Torsion: Beam Bending: Parallel Axis Theorem:

L

G

J

T

r

I

W

R

E

I

M

Y

�V

2

0 dA I I(d) �

Power Transmission:

c

J f 2ʌ P

maxW watts (metric units)

c

J rpm 1015.87 P

max-6 Wu

h.p. (lb.in units)

Beam Formulas:

xd

M d V

xd

V d w

EI

M

dx

yd

2

2

³ dAy Q whereI

Q V

I

yVA q

< x – a > = ( x – a ) if x – a > 0, = 0 if x – a � 0

Some Typical I-Beams:

Mohr’s Circle:

ı

IJ ıy, IJyx

ıx, IJxy

ıy

IJxy

ıx

IJyx

Use sign convention “in the kitchen, the clock is above and the counter is below” for the shear stresses.

Then, rotations in the Mohr’s circle have the same direction and double the rotation angle of the physical

stresses. For Mohr’s circle of strain, use İ and Ȗ/2 in place of ı and IJ.

Column Buckling:

2

e

2

2

e

2

CR)r/(L

EA

L

EI P

S

S

where: Le = L (pinned-pinned) Le = 2 L (free-fixed)

Le = 0.7 L (pinned-fixed) Le = 0.5 L (sliding-fixed)

¸¸¹

·¨¨©

§�¸

¸¹

·¨¨©

§ S 1

P

P

2 sec e y :load Eccentric

CR

max

P P

P a y :curvature Initial

CR

max �

For “short” columns:

2

e

2

2

Y

YCR

Y

2

e

r

L

E4

E2

r

L¸¹·

¨©§¸¸¹

·¨¨©

§

SV

�V VVS

�

Strain Energy:

dx 2EA

P U:rod

2

³ dx 2EI

M U:beam

2

³ dx 2GJ

T U:shaft

2

³

Castigliano’s Theorem:

j

jP

U ww

G

j

jM

U ww

T dx P

P

EA

P :rod

j

j ww

G ³ dx P

M

EI

M :beam

j

j ww

G ³

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