Timothy A. Philpot - Mechanics of materials _ an integrated learning system-John Wiley (2017)

Gear relationships between gears A and B
TA
TB
= R φ = − R φ R ω = R ω
RA
RB
RA
DA
N A
Gear ratio = = =
R D N
B
A A B B A A B B
B
Six rules for constructing shear-force and
bending-moment diagrams
Rule 1: ∆ V = P 0
2
Rule 2: ∆ V = V − V = w( x)
dx
Rule 3: dV
dx
=
2 1
wx ( )
2
Rule 4: ∆ M = M − M = V dx
Rule 5: dM
dx
= V
Rule 6: ∆ M = −M 0
Flexure
2 1
∫
x
x1
Flexural strain and stress
1
E
ε x =− y σ x =− y
ρ
ρ
Flexure Formula
My
Mc M
σ x =− or σ max = = where S =
Iz
I S
Transformed-section method for beams of two materials
[where material (2) is transformed into an equivalent
amount of material (1)]
E2
My
My
n = σx1
=− σx2
=−n
E
I
I
1
∫
B
x
x1
transformed
Bending due to eccentric axial load
F My
σ x = −
A I
z
Unsymmetric bending of arbitrary cross sections
⎡ Iz z − Iyz
y⎤
⎡ Iyy Iyzz
σ x = ⎢ M y
M
2 ⎥ + − + ⎤
⎢
2 ⎥ z
⎣ II y z−
I yz ⎦ ⎣ II y z−
I yz ⎦
or
( MI z y + MI y yz ) y ( MI y z+
MI z yz ) z
σ x =−
+
2 2
II y z−
I yz II y z−
I yz
MI y z+
MI z yz
tan β =
MI + MI
z y y yz
Unsymmetric bending of symmetric cross sections
Mz y My z
σ x = −
I I
y
z
tan β =
MI
y z
MI
z
y
transformed
I
c
Bending of curved bars
σ
x
M r
=−
rA r
( n − r)
( − r )
c
n
where r
n
=
∫
A
A
dA
r
Horizontal shear stress associated with bending
VQ
τ H = where Q = Σ yiAi
It
Shear flow formula
VQ
q =
I
Shear flow, fastener spacing, and fastener shear relationship
qs ≤ nV = n τ A
f f f f f
For circular cross sections,
2
3
1
3
Q = r = d (solid sections)
3 12
Q = 2 [ 3 3
1
R − r ] = [ 3 3
D − d ] (hollow sections)
3
12
Beam deflections
Elastic curve relations between w, V, M, θ, and v for constant EI
Deflection = v
dv
Slope = = θ (forsmall deflections)
dx
M EI d 2
Moment =
v
2
dx
dM
V EI dv 3
Shear = =
3
dx dx
dV
Load w EI d 4
= =
v
4
dx dx
Plane stress transformations
t
σt
σn
τtn
θ
τnt
y
τnt
τtn
θ
σt
σn
Stresses on an arbitrary plane
2 2
σ = σ cos θ + σ sin θ + 2τ sinθcosθ
n x y xy
2 2
σ = σ sin θ + σ cos θ − 2τ sinθcosθ
t x y xy
2 2
τ =−( σ − σ )sinθcos θ + τ (cos θ − sin θ)
nt x y xy
n
x
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