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Fundamental Equations of Mechanics of Materials Axial Load
Normal Stress
s = P A
Displacement
d = L L
0 P(x)dx A (x)E
d = ! PL AE
dT = a "TL
Torsion
Shear stress in circular shaft
t = Tr
J
where
J =
p
2 c4 solid cross section
J =
p
2 (co
4 - ci 4) tubular cross section
Power
P = Tv = 2pf T
Angle of twist
f = L L
0 T(x)dx J(x)G
f = !
TL JG
Average shear stress in a thin-walled tube
tavg =
T 2tA m
Shear Flow
q = tavg t =
T 2Am
Bending
Normal stress
s =
My
I Unsymmetric bending
s = -
Mz y
Iz +
Myz
Iy , tan a =
Iz Iy
tan u
Shear
Average direct shear stress
tavg = V A
Transverse shear stress
t =
VQ It
Shear flow
q = tt =
VQ I
Stress in Thin-Walled Pressure Vessel
Cylinder
s1 =
pr
t s2 =
pr
2t
Sphere
s1 = s2 =
pr
2t
Stress Transformation Equations
sx# = sx + sy
2 +
sx - sy 2
cos 2u + txy sin 2u
tx#y# = - sx - sy
2 sin 2u + txy cos 2u
Principal Stress
tan 2up = txy
(sx - sy)>2 s1,2 =
sx + sy 2
{ A asx - sy2 b2 + txy2 Maximum in-plane shear stress
tan 2us = - (sx - sy)>2
txy
tmax = A asx - sy2 b2 + t2xy savg =
sx + sy 2
Absolute maximum shear stress
tabs max
= smax
2 for smax, smin same sign
tabs max
= smax - smin
2 for smax, smin opposite signs
Geometric Properties of Area Elements Material Property Relations
Poisson’s ratio
n = - Plat
Plong
Generalized Hooke’s Law
ex =
1 E
3sx - n(sy + sz)4
ey = 1 E
3sy - n(sx + sz)4
ez = 1 E
3sz - n(sx + sy)4
gxy = 1 G
txy, gyz = 1 G
tyz, gzx = 1 G
tzx
where
G =
E 2(1 + n)
Relations Between w, V, M
dV dx
= w(x), dM dx
= V
Elastic Curve
1 r
= M EI
EI
d4v
dx4 = w(x)
EI
d3v
dx3 = V (x)
EI
d2v
dx2 = M(x)
Buckling Critical axial load
Pcr =
p2EI
(KL)2
Critical stress
scr =
p2E
(KL >r)2 , r = 2I>A Secant formula
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