Mechanics of Materials Equation Sheet PDF

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Fundamental Equations of Mechanics of Materials Axial Load

Shear

Normal Stress

Average direct shear stress N A

s =

tavg =

Displacement

V A

Transverse shear stress L

N(x)dx L0 A(x )E

d =

t =

NL AE dT = a TL

VQ It

Shear flow

d = 

q = tt =

VQ I

Torsion Stress in Thin-Walled Pressure Vessel Shear stress in circular shaft t= where

Cylinder

Tr J

s1 =

p 4 solid cross section c 2 p tubular cross section J = (co4 - ci 4) 2

f =

L0

sx =

f = 

T(x )dx J(x)G

sx + sy

sx - sy

+ 2 sx - sy 2

tan 2up =

Average shear stress in a thin-walled tube s1,2 = T 2tA m

2

txy (sx - s y)>2 sx + sy 2

T 2A m

Normal stress

savg =

I

tabs = max

Iz

+

M yz Iy

,

sx - sy 2

2

2 b + txy

txy s x - sy

A 2 sx + s y a

2

b + t2xy

2 Absolute maximum shear stress

My

Unsymmetric bending Mz y

A

a

(sx - sy)>2

tan 2us = tmax =

s = -

{

Maximum in-plane shear stress

Bending

s=

cos 2u + txy sin 2u

sin 2u + txy cos 2u

Shear Flow q = tavg t =

pr 2t

Principal Stress

TL JG

tavg =

2t

Stress Transformation Equations

txy = L

t

s1 = s2 =

Power

Angle of twist

pr

s2 =

Sphere

J =

P = Tv = 2pf T

pr

tan a =

Iz Iy

tan u

tabs

max

smax

for smax, smin same sign 2 s - smin = max for smax, smin opposite signs 2

Geometric Properties of Area Elements Material Property Relations A = bh

y

Poisson’s ratio n = -

Plat Plong

Generalized Hooke’s Law 1 3sx - n(sy + sz ) 4 Px = E 1 Py = 3sy - n(sx + sz ) 4 E 1 Pz = 3sz - n(sx + sy) 4 E 1 1 1 gxy = txy , gyz = t g = t G G yz, zx G zx

h

x

C b

Ix =

1 12

Iy =

1 3 12 hb

Ix =

1 36

Ix = Iy =

4 1 8 pr 4 1 8 pr

Ix =

4 1 4 pr

Iy =

4 1 4 pr

bh3

Rectangular area A= h

–1 2

C

bh x 1–h 3

b

bh3

Triangular area

where E 2(1 + n)

G =

Elastic Curve

dM = V dx

x

2

r A = π—– 2

r 4— 3π

r

EI

C

x

Semicircular area y A= π r2

Buckling Critical axial load

r

Pcr = Critical stress scr = Secant formula

p 2EI

x

C

(KL )2

p 2E , r = 2I >A (KL >r)2

Circular area 2– a 5

P L P ec bd c1 + 2 sec a A 2r A EA r

Energy Methods Conservation of energy

b

A = –23 ab 3 –8 b

C a

Ue = Ui Strain energy

N 2L constant axial load 2A E L 2 M dx bending moment Ui = L0 2EI L 2 fs V dx transverse shear Ui = L0 2GA L 2 T dx torsional moment Ui = L0 2GJ

h

Trapezoidal area y

d 4v = w( x) dx 4 d3v EI 3 = V (x) dx d2v EI 2 = M(x) dx

1 2a + b –3 ——— a+b

b

1 M = EI r

smax =

A = 2–1h (a + b)

C

h

Relations Between w, V, M dV = w(x), dx

a

Semiparabolic area

Ui =

A= b C 3 –a 4

a

Exparabolic area

1 ab — 3

3 — 10b

Average Mechanical Properties of Typical Engineering Materialsa (SI Units) Yield Strength (MPa) SY Tens. Comp.b Shear

Ultimate Strength (MPa) Su Tens. Comp.b Shear

(Mg , m3)

Moduls of Elasticity E (GPa)

Modulus of Rigidity G (GPa)

2.79 2.71

73.1 68.9

27 26

414 255

414 255

172 131

469 290

469 290

7.19 7.28

67.0 172

27 68

– –

– –

– –

179 276

8.74

101

37

70.0

70.0

8.83

103

38

345

345

– –

1.83

44.7

18

152

152

–

Structural A-36

7.85

200

75

250

250

Structural A992

7.85

200

75

345

345

Stainless 304

7.86 8.16

193 200

75 75

207 703

207 703

– – – –

[Ti-6Al-4V]

4.43

120

44

924

924

Nonmetallic Low Strength

2.38

22.1

–

–

High Strength

2.37

29.0

–

–

Plastic

Kevlar 49

1.45

131

–

Reinforced

30% Glass

1.45

72.4

0.47

13.1

Materials

Density R

Coef. of Therm. Expansion A

%Elongation in 50 mm specimen

Poisson’s Ratio N

290 186

10 12

0.35 0.35

23

669 572

– –

0.6 5

0.28 0.28

12 12

241

241

655

655

– –

35 20

0.35 0.34

18 17

276

276

152

1

0.30

26

400

400

–

30

0.32

12

450

450

–

30

0.32

12

517 800

517 800

– –

40 22

0.27 0.32

17 12

–

1,000

1,000

–

16

0.36

9.4

–

12

–

–

–

–

0.15

11

–

38

–

–

–

–

0.15

11

–

–

–

717

483

20.3

2.8

0.34

–

–

–

–

–

90

131

–

–

0.34

–

–

–

–

–

2.1c

26d

6.2d

–

0.29e

–

–

2.5c

36d

6.7d

–

0.31e

–

(10–6) , C

Metallic Aluminum Wrought Alloys Cast Iron Alloys Copper Alloys

Gray ASTM 20 Malleable ASTM A-197 Red Brass C83400 Bronze C86100

Magnesium Alloy Steel Alloys

2014-T6 6061-T6

[Am 1004-T61]

Tool L2 Titanium Alloy

Concrete

Wood Select Structural Grade

Douglas Fir White Spruce

0.36

9.65

–

–

–

a

Specific values may vary for a particular material due to alloy or mineral composition,mechanical working of the specimen,or heat treatment. For a more exact value reference books for the material should be consulted. b The yield and ultimate strengths for ductile materials can be assumed equal for both tension and compression. c Measured perpendicular to the grain. d e

Measured parallel to the grain. Deformation measured perpendicular to the grain when the load is applied along the grain.

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