Fundamental Mechanics of Materials Equations Basic definitions

𝜎 𝑠𝑖𝑔𝑚𝑎 Average normal stress in an axial member 𝜀 𝑒𝑝𝑠𝑖𝑙𝑜𝑛 F 𝜏 𝑡𝑎𝑢 avg ⫽ A 𝛾 𝑔𝑎𝑚𝑚𝑎 Average direct shear stress 𝜈 𝑛𝑢 𝛿 𝛥 𝑑𝑒𝑙𝑡𝑎 V avg ⫽ 𝛼 𝑎𝑙𝑝ℎ𝑎 AV 𝜑 𝑝ℎ𝑖 Average bearing stress 𝜔 𝑜𝑚𝑒𝑔𝑎 F b ⫽ 𝜃 𝑡ℎ𝑒𝑡𝑎 Ab Average normal strain in an axial member 𝛥𝑑 𝛥𝑤 𝛥𝑡  ␧avg ⫽ 𝜀𝑡𝑟𝑎𝑛𝑠𝑣𝑒𝑟𝑠𝑒 = 𝑜𝑟 𝑜𝑟 L 𝑑 𝑤 𝑡 𝛾 = 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑎𝑛𝑔𝑙𝑒 𝑓𝑟𝑜𝑚 90°

Six rules for constructing shear-force and bending-moment diagrams Rule 1:

⌬V ⫽ P0

Rule 2:

⌬V ⫽ V2 ⫺ V1 ⫽

x2

∫x

w( x ) dx

1

Rule 3: Rule 4:

dV ⫽ w( x ) dx ⌬ M ⫽ M 2 ⫺ M1 ⫽

x2

∫x

V dx

1

Rule 5: Rule 6:

dM ⫽V dx ⌬M ⫽ ⫺M 0

Flexure

Average normal strain caused by temperature change ␧T ⫽ ⌬ T Hooke’s Law (one-dimensional)  ⫽ E ␧ and  ⫽ G  Poisson’s ratio ␧lat ␯ ⫽ ⫺␧ long

Flexure formula My Mc M x ⫽ ⫺ or max ⫽ where ⫽ I I S Unsymmetric bending of arbitrary cross sections

Relationship between E, G, and ν E G⫽ 2(1 ⫹  ) Definition of allowable stress   allow ⫽ failure or  allow ⫽ failure FS FS Factor of safety

Unsymmetric bending of symmetric cross sections M y z Mz y My I z tan  ⫽ x ⫽ ⫺ Iy Iz Mz I y

 FS ⫽ failure actual

or

 FS ⫽  failure actual

Axial deformation Deformation in axial members FL FL ⫽ or  ⫽ ∑ i i AE i Ai Ei Force-temperature-deformation relationship ⫽

FL ⫹ ⌬TL AE

Torsion Maximum torsion shear stress in a circular shaft Tc  max ⫽ J where the polar moment of inertia J is defined as   J ⫽ [ R4 ⫺ r 4 ] ⫽ [ D 4 ⫺ d 4 ] 2 32 𝑔𝑒𝑎𝑟𝑠 Angle of twist in a circular shaft 𝑟2 𝑇1 = 𝑟1 𝑇2 TL TL 𝑟1 𝜔1 = 𝑟2 𝜔2 ⫽ or ⫽ ∑ i i JG i Ji Gi Power transmission in a shaft 𝑤𝑎𝑡𝑡𝑠 = 𝑁𝑚/𝑠 P ⫽ T

ℎ𝑝 = 6600 𝑖𝑛 ∙ 𝑙𝑏/𝑠

⎡ I z z ⫺ I yz y ⎤ ⎡⫺ I y ⫹ I yz z ⎤ ⎥ My ⫹ ⎢ y ⎥M x ⫽ ⎢ 2 ⎢⎣ I y I z ⫺ I yz ⎥⎦ ⎢⎣ I y I z ⫺ I y2z ⎥⎦ z

Horizontal shear stress associated with bending

S⫽

I c 𝑐𝑜𝑚𝑝𝑜𝑠𝑖𝑡𝑒 𝑏𝑒𝑎𝑚𝑠 𝐸𝐵 𝑛= 𝐸𝐴 −𝑀𝑦 𝜎𝐴 = 𝑇 𝐼 −𝑛𝑀𝑦 𝜎𝐵 = 𝐼𝑇

VQ 𝑤ℎ𝑒𝑟𝑒 𝑄 = ∑𝑦�𝑖 𝐴𝑖 It Shear flow formula VQ q⫽ I Shear flow, fastener spacing, and fastener shear relationship 𝑉𝑏𝑒𝑎𝑚 𝑄 𝑛𝑉𝑓𝑎𝑠𝑡𝑒𝑛𝑒𝑟 qs ⱕ n f Vf ⫽ n f  f A f 𝑞= = or 𝑠 𝐼 For circular cross sections, 1 3 Q⫽ d (solid sections) 12 2 1 Q ⫽ [ R3 ⫺ r 3 ] ⫽ [ D 3 ⫺ d 3 ] (hollow sections) 3 12 H ⫽

Beam deflections Elastic curve relations between w, V, M, θ, and v for constant EI Deflection ⫽ v dv Slope ⫽ ⫽ dx d 2v Moment M ⫽ EI 2 dx dM d 3v Shear V ⫽ ⫽ EI 3 dx dx dV d4v Load w ⫽ ⫽ EI 4 dx dx

Fundamental Mechanics of Materials Equations Plane stress transformations

Generalized Hooke’s Law

Normal and shear stresses on an arbitrary plane

Normal stress/normal strain relationships 1 ␧x ⫽ [ ␴x ⫺ ␯ (␴y ⫹ ␴z )] E 1 ␧y ⫽ [ ␴y ⫺ ␯ (␴x ⫹ ␴z )] E 1 ␧z ⫽ [ ␴z ⫺ ␯ (␴x ⫹ ␴y )] E Shear stress/shear strain relationships 1 1 1 ␥xy ⫽ ␶ xy ␥ yz ⫽ ␶ yz ␥zx ⫽ ␶ zx G G G where

␴n ⫽ ␴x

cos 2 ␪

⫹ ␴y

sin 2 ␪

⫹ 2 ␶ xy sin ␪ cos ␪

␶ nt ⫽ ⫺(␴ x ⫺ ␴y )sin ␪ cos ␪ ⫹ ␶ xy ( cos 2 ␪ ⫺ sin 2 ␪) or 𝜎𝑥 + 𝜎𝑦 𝜎𝑥 − 𝜎𝑦 + cos 2𝜃 + 𝜏𝑥𝑦 sin 2𝜃 2 2 𝜎𝑥 + 𝜎𝑦 𝜎𝑥 − 𝜎𝑦 𝜎𝑡 = − cos 2𝜃 − 𝜏𝑥𝑦 sin 2𝜃 2 2 𝜎𝑥 − 𝜎𝑦 sin 2𝜃 + 𝜏𝑥𝑦 cos 2𝜃 𝜏𝑛𝑡 = − 2 Principal stress magnitudes 𝜎𝑛 =

2 ␴x ⫹ ␴y ⎛ ␴x ⫺ ␴y ⎞⎟ 2 ⎟⎟ ⫹ ␶ xy ⫾ ⎜⎜⎜ ⎝ 2 2 ⎠ Orientation of principal planes ␶ xy tan 2␪p ⫽ (␴x ⫺ ␴y ) 2 Maximum in-plane shear stress magnitude

␴ p1, p 2 ⫽

2 ⎛ ␴x ⫺ ␴y ⎞⎟ 2 ␶ max ⫽ ⫾ ⎜⎜ ⫹ ␶ xy ⎟ ⎜⎝ 2 ⎟⎠

␴avg ⫽

or

G⫽

␶ max ⫽

␴p1 ⫺ ␴p 2 2

␴x ⫹ ␴y

Pressure vessels Axial stress in spherical pressure vessel pr pd ␴a ⫽ ⫽ 2t 4t Longitudinal and hoop stresses in cylindrical pressure vessels pr pd pr pd ␴long ⫽ ␴hoop ⫽ ⫽ ⫽ 2t 4t t 2t

Plane strain transformations Normal and shear strain in arbitrary directions ␧n ⫽ ␧x cos 2 ␪ ⫹ ␧y sin 2 ␪ ⫹ ␥xy sin ␪ cos ␪ ␥nt ⫽ ⫺2( ␧x ⫺ ␧y )sin ␪ cos ␪ ⫹ ␥xy (cos 2 ␪ ⫺ sin 2 ␪)

␧x ⫹ ␧y ⎛ ␧x ⫺ ␧y ⎞⎟2 ⎛ ␥xy ⎞⎟2 ⎟ ⫹ ⎜⎜⎜ ⎟⎟ ⫽ ⫾ ⎜⎜⎜ ⎝ 2 ⎠ ⎝ 2 ⎟⎠ 2

𝜀𝑧 =

𝛾𝑥𝑦 𝜀𝑥 + 𝜀𝑦 𝜀𝑥 − 𝜀𝑦 sin 2𝜃 + cos 2𝜃 + 2 2 2 𝛾𝑥𝑦 𝜀𝑥 + 𝜀𝑦 𝜀𝑥 − 𝜀𝑦 sin 2𝜃 𝜀𝑡 = − cos 2𝜃 − 2 2 2 𝜀𝑥 − 𝜀𝑦 𝛾𝑥𝑦 𝛾𝑛𝑡 =− sin 2𝜃 + cos 2𝜃 2 2 2 Principal strain magnitudes 𝜀𝑛 =

−𝜈 (𝜀 + 𝜀𝑦 ) 1−𝜈 𝑥

or

Orientation of principal strains ␥xy tan 2␪p ⫽ ␧x ⫺ ␧y ⎛ ␧x ⫺ ␧y ⎞⎟2 ⎛ ␥xy ⎞⎟2 ⎜ ⎜⎜ ⎜⎝ 2 ⎟⎟⎠ ⫹ ⎜⎜⎝ 2 ⎟⎟⎠ ⫹ ␧y

2 Normal strain invariance ␧x ⫹ ␧y ⫽ ␧n ⫹ ␧t ⫽ ␧p1 ⫹ ␧p 2

Mises equivalent stress for plane stress 1/ 2

␴M ⫽ [ ␴2p1 ⫺ ␴ p1 ␴p 2 ⫹ ␴ 2p 2 ]

Column buckling Euler buckling load Pcr ⫽

␴cr ⫽ or

␥max ⫽ ␧p1 ⫺ ␧p 2

𝜎𝑟𝑎𝑑𝑖𝑎𝑙−𝑜𝑢𝑡𝑠𝑖𝑑𝑒 = 0 𝜎𝑟𝑎𝑑𝑖𝑎𝑙−𝑖𝑛𝑠𝑖𝑑𝑒 = −𝑝

Failure theories

␲ 2 EI ( KL )2

Euler buckling stress

Maximum in-plane shear strain ␥max ⫽⫾ 2 ␧x ␧avg ⫽

Normal stress/normal strain relationships for plane stress 1 ␧x ⫽ ( ␴x ⫺ ␯␴y ) E E ␴x ⫽ (␧x ⫹ ␯␧y ) 1 ⫺ ␯2 1 ␧y ⫽ ( ␴y ⫺ ␯␴x ) or E E ␴y ⫽ (␧y ⫹ ␯␧x ) 1 ⫺ ␯2 ␯ ␧z ⫽ ⫺ (␴x ⫹ ␴y ) E Shear stress/shear strain relationships for plane stress 1 or ␥xy ⫽ ␶ xy ␶ xy ⫽ G␥ xy G

2 Absolute maximum shear stress magnitude ␴ ⫺ ␴min ␶abs max ⫽ max 2 Normal, stress invariance ␴x ⫹ ␴y ⫽ ␴n ⫹ ␴t ⫽ ␴ p1 ⫹ ␴p 2

␧p1, p 2

E 2(1 ⫹ ␯ )

␲2 E ( KL r )2

Radius of gyration r2 ⫽

I A

2 ] ⫽ [ ␴x2 ⫺ ␴x ␴y ⫹ ␴y2 ⫹ 3 ␶ xy

1/2 2

x =

Σxi Ai ΣAi

Σyi Ai ΣAi

y =

I = Σ ( I c + d 2 A) Table A.1 Properties of Plane Figures 1. Rectangle y′

6. Circle y

y − x

A = bh bh3 12 hb3 Iy = 12 hb3 I y′ = 3

h 2 b x = 2 bh3 Ix′ = 3 y =

x

h

C

− y

x′ b

Ix =

πd 2 4 πr 4 πd 4 Ix = Iy = = 4 64

r

A = πr 2 =

x

C

d

2. Right Triangle

7. Hollow Circle y

y′

y

bh 2 h y = 3 b x = 3 bh3 Ix′ = 12 A=

− x

h − y

C

x x′

b

bh3 36 hb3 Iy = 36 hb3 I y′ = 12 Ix =

π 2 (D − d 2 ) 4 π I x = I y = ( R4 − r 4 ) 4 π (D4 − d 4 ) = 64

A = π ( R2 − r 2 ) =

R

r

x

C

d

D

3. Triangle

8. Parabola

y′

bh 2 h y = 3 (a + b) x = 3 bh3 Ix′ = 12

y′

A=

a

y

− x

h − y

C

x x′

y − x

bh3 36 bh 2 (a − ab + b 2 ) Iy = 36

h 2 x′ b2 2bh A= 3 3b x = 8 y′ =

Ix =

x

h

C

− y x′

b

y =

3h 5

y =

3h 10

Zero slope

b

4. Trapezoid

9. Parabolic Spandrel a

h x

− y

C

(a + b)h A= 2 1 2a + b y = h 3 a+b h3 (a 2 + 4 ab + b 2 ) Ix = 36 (a + b)

(

y′

)

y

− x

h 2 x′ b2 bh A= 3 3b x = 4

y′ =

Zero slope

h − y

C b

x x′

b

5. Semicircle

10. General Spandrel A=

y, y′

r

C − y

x x′

696

y′

πr 2

2 4r y = 3π

I x ′ = I y′ =

Ix = πr 4 8

( π8 − 98π )r

4

y

− x

h n x′ bn bh h A= x n +1 − y x′ n +1 x = b n+2 y′ =

Zero slope

C b

y =

n +1 h 4n + 2

SIMPLY SUPPORTED BEAMS Beam

Slope

Deflection 3

2

1

θ1 = −θ 2 = −

2

PL 16 EI

Pb( L2 − b 2 ) θ1 = − 6 LEI

4

vmax = − 5

ML θ1 = − 3EI ML θ2 = + 6 EI

3

PL 48 EI

Pa 2b 2 v=− 3LEI

Pa ( L2 − a 2 ) θ2 = + 6 LEI 7

vmax = −

ML2 9 3 EI

⎛ 3⎞ at x = L ⎜⎜1 − ⎟ 3 ⎟⎠ ⎝ 11

wL3 θ1 = −θ 2 = − 24 EI 13

wa 2 θ1 = − (2 L − a ) 2 24 LEI

θ2 = + 16

wa 2 24 LEI

Px (3L2 − 4 x 2 ) 48 EI for 0 ≤ x ≤ L

6

v=−

2

Pbx 2 2 (L − b − x2 ) 6 LEI for 0 ≤ x ≤ a

9

v=−

Mx (2 L2 − 3Lx + x 2 ) 6 LEI

12

vmax

5wL4 =− 384 EI

14

v=−

v=−

wx ( L3 − 2 Lx 2 + x3 ) 24 EI

wx ( Lx3 − 4aLx 2 + 2a 2 x 2 + 4a 2 L2 24 LEI

wa 3 v=− (4 L2 − 7 aL + 3a 2 ) −4a 3 L + a 4 ) for 0 ≤ x ≤ a 24 LEI wa 2 2 2 (2 x3 − 6 Lx 2 + a 2 x + 4 L2 x − a 2 L) v = − at x = a (2 L − a ) 24 LEI 3

7 w0 L 360 EI w0 L3 θ2 = + 45 EI

θ1 = −

v=−

at x = a 8

10

Elastic Curve

17

w0 L4 vmax = −0.00652 EI at x = 0.5193L

15 18

v=−

for a ≤ x ≤ L

w0 x (7 L4 − 10 L2 x 2 + 3 x 4 ) 360 LEI

CANTILEVER BEAMS Beam

Slope

Deflection

Elastic Curve

20

19

θ max

PL2 =− 2 EI

22

21

vmax

PL3 =− 3EI

23

θ max = −

2

PL 8 EI

24

vmax = −

3

5 PL 48 EI

26

25

θ max = −

ML EI

28

ML2 =− 2 EI

29

θ max

31

θ max

for 0 ≤ x ≤ L

2

for L ≤ x ≤ L 2

Mx 2 v=− 2 EI 30

vmax

wL4 =− 8 EI

32

w0 L3 =− 24 EI

Px 2 v=− (3L − 2 x ) 12 EI PL2 (6 x − L ) v=− 48 EI

27

vmax

wL3 =− 6 EI

Px 2 v=− (3L − x ) 6 EI

vmax

wx 2 (6 L2 − 4 Lx + x 2 ) v=− 24 EI 33

w0 L4 =− 30 EI

w0 x 2 v=− (10 L3 − 10 L2 x + 5 Lx 2 − x 3 ) 120 LEI