Fundamental Mechanics of Materials Equations Basic definitions
π π ππππ Average normal stress in an axial member π πππ ππππ F π π‘ππ’ avg β«½ A πΎ πππππ Av...
Fundamental Mechanics of Materials Equations Basic definitions
π π ππππ Average normal stress in an axial member π πππ ππππ F π π‘ππ’ avg β«½ A πΎ πππππ Average direct shear stress π ππ’ πΏ π₯ ππππ‘π V avg β«½ πΌ πππβπ AV π πβπ Average bearing stress π πππππ F b β«½ π π‘βππ‘π Ab Average normal strain in an axial member π₯π π₯π€ π₯π‘ β§avg β«½ ππ‘ππππ π£πππ π = ππ ππ L π π€ π‘ πΎ = πβππππ ππ πππππ ππππ 90Β°
Six rules for constructing shear-force and bending-moment diagrams Rule 1:
β¬V β«½ P0
Rule 2:
β¬V β«½ V2 β«Ί V1 β«½
x2
β«x
w( x ) dx
1
Rule 3: Rule 4:
dV β«½ w( x ) dx β¬ M β«½ M 2 β«Ί M1 β«½
x2
β«x
V dx
1
Rule 5: Rule 6:
dM β«½V dx β¬M β«½ β«ΊM 0
Flexure
Average normal strain caused by temperature change β§T β«½ β¬ T Hookeβs Law (one-dimensional) β«½ E β§ and β«½ G Poissonβs ratio β§lat β― β«½ β«Ίβ§ long
Flexure formula My Mc M x β«½ β«Ί or max β«½ where β«½ I I S Unsymmetric bending of arbitrary cross sections
Relationship between E, G, and Ξ½ E Gβ«½ 2(1 β«Ή ) Definition of allowable stress allow β«½ failure or allow β«½ failure FS FS Factor of safety
Unsymmetric bending of symmetric cross sections M y z Mz y My I z tan β«½ x β«½ β«Ί Iy Iz Mz I y
FS β«½ failure actual
or
FS β«½ failure actual
Axial deformation Deformation in axial members FL FL β«½ or β«½ β i i AE i Ai Ei Force-temperature-deformation relationship β«½
FL β«Ή β¬TL AE
Torsion Maximum torsion shear stress in a circular shaft Tc max β«½ J where the polar moment of inertia J is defined as J β«½ [ R4 β«Ί r 4 ] β«½ [ D 4 β«Ί d 4 ] 2 32 πππππ Angle of twist in a circular shaft π2 π1 = π1 π2 TL TL π1 π1 = π2 π2 β«½ or β«½ β i i JG i Ji Gi Power transmission in a shaft π€ππ‘π‘π = ππ/π P β«½ T
βπ = 6600 ππ β ππ/π
β‘ I z z β«Ί I yz y β€ β‘β«Ί I y β«Ή I yz z β€ β₯ My β«Ή β’ y β₯M x β«½ β’ 2 β’β£ I y I z β«Ί I yz β₯β¦ β’β£ I y I z β«Ί I y2z β₯β¦ z
Horizontal shear stress associated with bending
Sβ«½
I c ππππππ ππ‘π πππππ πΈπ΅ π= πΈπ΄ βππ¦ ππ΄ = π πΌ βπππ¦ ππ΅ = πΌπ
VQ π€βπππ π = βπ¦οΏ½π π΄π It Shear flow formula VQ qβ«½ I Shear flow, fastener spacing, and fastener shear relationship πππππ π πππππ π‘ππππ qs β± n f Vf β«½ n f f A f π= = or π πΌ For circular cross sections, 1 3 Qβ«½ d (solid sections) 12 2 1 Q β«½ [ R3 β«Ί r 3 ] β«½ [ D 3 β«Ί d 3 ] (hollow sections) 3 12 H β«½
Beam deflections Elastic curve relations between w, V, M, ΞΈ, and v for constant EI Deflection β«½ v dv Slope β«½ β«½ dx d 2v Moment M β«½ EI 2 dx dM d 3v Shear V β«½ β«½ EI 3 dx dx dV d4v Load w β«½ β«½ EI 4 dx dx
Fundamental Mechanics of Materials Equations Plane stress transformations
Generalized Hookeβs Law
Normal and shear stresses on an arbitrary plane
Normal stress/normal strain relationships 1 β§x β«½ [ β΄x β«Ί β― (β΄y β«Ή β΄z )] E 1 β§y β«½ [ β΄y β«Ί β― (β΄x β«Ή β΄z )] E 1 β§z β«½ [ β΄z β«Ί β― (β΄x β«Ή β΄y )] E Shear stress/shear strain relationships 1 1 1 β₯xy β«½ βΆ xy β₯ yz β«½ βΆ yz β₯zx β«½ βΆ zx G G G where
β΄n β«½ β΄x
cos 2 βͺ
β«Ή β΄y
sin 2 βͺ
β«Ή 2 βΆ xy sin βͺ cos βͺ
βΆ nt β«½ β«Ί(β΄ x β«Ί β΄y )sin βͺ cos βͺ β«Ή βΆ xy ( cos 2 βͺ β«Ί sin 2 βͺ) or ππ₯ + ππ¦ ππ₯ β ππ¦ + cos 2π + ππ₯π¦ sin 2π 2 2 ππ₯ + ππ¦ ππ₯ β ππ¦ ππ‘ = β cos 2π β ππ₯π¦ sin 2π 2 2 ππ₯ β ππ¦ sin 2π + ππ₯π¦ cos 2π πππ‘ = β 2 Principal stress magnitudes ππ =
2 β΄x β«Ή β΄y β β΄x β«Ί β΄y ββ 2 ββ β«Ή βΆ xy β«Ύ βββ β 2 2 β Orientation of principal planes βΆ xy tan 2βͺp β«½ (β΄x β«Ί β΄y ) 2 Maximum in-plane shear stress magnitude
I = Ξ£ ( I c + d 2 A) Table A.1 Properties of Plane Figures 1. Rectangle yβ²
6. Circle y
y β x
A = bh bh3 12 hb3 Iy = 12 hb3 I yβ² = 3
h 2 b x = 2 bh3 Ixβ² = 3 y =
x
h
C
β y
xβ² b
Ix =
Οd 2 4 Οr 4 Οd 4 Ix = Iy = = 4 64
r
A = Οr 2 =
x
C
d
2. Right Triangle
7. Hollow Circle y
yβ²
y
bh 2 h y = 3 b x = 3 bh3 Ixβ² = 12 A=
β x
h β y
C
x xβ²
b
bh3 36 hb3 Iy = 36 hb3 I yβ² = 12 Ix =
Ο 2 (D β d 2 ) 4 Ο I x = I y = ( R4 β r 4 ) 4 Ο (D4 β d 4 ) = 64
A = Ο ( R2 β r 2 ) =
R
r
x
C
d
D
3. Triangle
8. Parabola
yβ²
bh 2 h y = 3 (a + b) x = 3 bh3 Ixβ² = 12
yβ²
A=
a
y
β x
h β y
C
x xβ²
y β x
bh3 36 bh 2 (a β ab + b 2 ) Iy = 36
h 2 xβ² b2 2bh A= 3 3b x = 8 yβ² =
Ix =
x
h
C
β y xβ²
b
y =
3h 5
y =
3h 10
Zero slope
b
4. Trapezoid
9. Parabolic Spandrel a
h x
β 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)
(
yβ²
)
y
β x
h 2 xβ² b2 bh A= 3 3b x = 4
yβ² =
Zero slope
h β y
C b
x xβ²
b
5. Semicircle
10. General Spandrel A=
y, yβ²
r
C β y
x xβ²
696
yβ²
Οr 2
2 4r y = 3Ο
I x β² = I yβ² =
Ix = Οr 4 8
( Ο8 β 98Ο )r
4
y
β x
h n xβ² bn bh h A= x n +1 β y xβ² n +1 x = b n+2 yβ² =
Zero slope
C b
y =
n +1 h 4n + 2
SIMPLY SUPPORTED BEAMS Beam
Slope
Deflection 3
2
1
ΞΈ1 = βΞΈ 2 = β
2
PL 16 EI
Pb( L2 β b 2 ) ΞΈ1 = β 6 LEI
4
vmax = β 5
ML ΞΈ1 = β 3EI ML ΞΈ2 = + 6 EI
3
PL 48 EI
Pa 2b 2 v=β 3LEI
Pa ( L2 β a 2 ) ΞΈ2 = + 6 LEI 7
vmax = β
ML2 9 3 EI
β 3β at x = L ββ1 β β 3 ββ β 11
wL3 ΞΈ1 = βΞΈ 2 = β 24 EI 13
wa 2 ΞΈ1 = β (2 L β a ) 2 24 LEI
ΞΈ2 = + 16
wa 2 24 LEI
Px (3L2 β 4 x 2 ) 48 EI for 0 β€ x β€ L
6
v=β
2
Pbx 2 2 (L β b β x2 ) 6 LEI for 0 β€ x β€ a
9
v=β
Mx (2 L2 β 3Lx + x 2 ) 6 LEI
12
vmax
5wL4 =β 384 EI
14
v=β
v=β
wx ( L3 β 2 Lx 2 + x3 ) 24 EI
wx ( Lx3 β 4aLx 2 + 2a 2 x 2 + 4a 2 L2 24 LEI
wa 3 v=β (4 L2 β 7 aL + 3a 2 ) β4a 3 L + a 4 ) for 0 β€ x β€ a 24 LEI wa 2 2 2 (2 x3 β 6 Lx 2 + a 2 x + 4 L2 x β a 2 L) v = β at x = a (2 L β a ) 24 LEI 3
7 w0 L 360 EI w0 L3 ΞΈ2 = + 45 EI
ΞΈ1 = β
v=β
at x = a 8
10
Elastic Curve
17
w0 L4 vmax = β0.00652 EI at x = 0.5193L
15 18
v=β
for a β€ x β€ L
w0 x (7 L4 β 10 L2 x 2 + 3 x 4 ) 360 LEI
CANTILEVER BEAMS Beam
Slope
Deflection
Elastic Curve
20
19
ΞΈ max
PL2 =β 2 EI
22
21
vmax
PL3 =β 3EI
23
ΞΈ max = β
2
PL 8 EI
24
vmax = β
3
5 PL 48 EI
26
25
ΞΈ max = β
ML EI
28
ML2 =β 2 EI
29
ΞΈ max
31
ΞΈ max
for 0 β€ x β€ L
2
for L β€ x β€ L 2
Mx 2 v=β 2 EI 30
vmax
wL4 =β 8 EI
32
w0 L3 =β 24 EI
Px 2 v=β (3L β 2 x ) 12 EI PL2 (6 x β L ) v=β 48 EI
27
vmax
wL3 =β 6 EI
Px 2 v=β (3L β x ) 6 EI
vmax
wx 2 (6 L2 β 4 Lx + x 2 ) v=β 24 EI 33
w0 L4 =β 30 EI
w0 x 2 v=β (10 L3 β 10 L2 x + 5 Lx 2 β x 3 ) 120 LEI