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Dr. D. B. Wallace
Basic Stress Equations
Internal Reactions: 6 Maximum (3 Force Components & 3 Moment Components)
Normal Force
(ττ)
(σσ)
Shear Forces
z
x
y
VxVy P Torsional Moment (ττ)
(σσ)
Bending Moments
z
x
y
MxMy T
or Torque Force Components (^) Moment Components
"Cut Surface" "Cut Surface"
Centroid ofCross Section Centroid of Cross Section
Normal Force:
Axial Force
z
x
y
P
Centroid σ σ
Axial Stress
"Cut Surface" (^) σ = P A
l Uniform over the entire cross section. l Axial force must go through centroid.
Shear Forces:
Cross Section
y Aa
Point of interest LINE perpendicular to V through point of interest = Length of LINE on the cross section = Area on one side of the LINE Centroid of entire cross section Centroid of area on one side of the LINE
= distance between the two centroids
= Area moment of inertia of entire cross section about an axis pependicular to V.
V b Aa
y
I
"y" Shear Force
z
x
y
Vy "x" Shear Force
z
x
y
Vx
τ τ
ττ
τ =
V A y I b
b (^) a g
Note : The maximum shear stress for common cross sections are: Cross Section: Cross Section:
Rectangular: τ max = 3 2 ⋅V A Solid Circular: τ max = 4 3 V A⋅
I-Beam or H-Beam: (^) flange (^) web τ max = V Aweb
Thin-walled tube: τ max = 2 ⋅V A
Torque or Torsional Moment:
Solid Circular or Tubular Cross Section: r = Distance from shaft axis to point of interest R = Shaft Radius D = Shaft Diameter J D^ R
J
D D
for solid circular shafts
for hollow shafts
o i
= ⋅^ = ⋅
π π
π
4 4
4 4
e j
Torque
z
x
y
T
"Cut Surface"
τ = T r⋅ J
τ π τ π
max
max
= ⋅ ⋅ = ⋅^ ⋅ ⋅ −
16
16
3
4 4
T D T D D D
for solid circular shafts
o for hollow shafts e (^) o i j
Rectangular Cross Section:
Torque
z
x
y
T
Centroid
τ τ
Torsional Stress
"Cut Surface" (^) ττ 1
2
a
b Note: a > b
Cross Section:
Method 1: τ (^) max = τ 1 = T ⋅ b 3 ⋅ a + 1 8_._ ⋅ b g ea 2 ⋅b^2 j (^) ONLY applies to the center of the longest side
Method 2:
τ 1 2 , α1 2 , 2
T
a b
a/b (^) αα 1 αα 2 1.0 .208. 1.5 .231. 2.0 .246. 3.0 .267. 4.0 .282. 6.0 .299. 8.0 .307. 10.0 .313. ∞ .333^ ----
Use the appropriate αα from the table on the right to get the shear stress at either position 1 or 2.
Other Cross Sections: Treated in advanced courses.
Bending Moment in Curved Beam:
Geometry:
r r
e
σσ rn r
σσ o
i
centroidal^ centroid
neutral axis
axis
o i
y
nonlinearstress distribution
M
c ci
o
r AdA
e r r
n area n
z ρ
A = cross sectional area rn = radius to neutral axis r = radius to centroidal axis e = eccentricity
Stresses:
Any Position: Inside (maximum magnitude) : Outside: σ = −^ ⋅ ⋅ ⋅ +
M y
e A b rn yg
σi i i
M c e A r
σo o o
M c e A r
= −^ ⋅
Area Properties for Various Cross Sections:
Cross Section r
dA
z area ρ A
t ri h ro
r
Rectangle
ρ (^) r (^) i + h 2 t^
r r
o i
⋅ F
HG
I
KJ
ln (^) h ⋅t
t ri h ro
r
Trapezoid
ρ ti o
r
h t t i (^) t t i o i o
+ ⋅^ +^ ⋅
b g
b g
For triangle: set ti or to to 0
t t r^ t^ r^ t h
r o i (^) r o i i o o i
− + ⋅^ −^ ⋅^ ⋅
F
HG
I
KJ
ln h ⋅ t^ i^ +to 2
r
Hollow Circle
ρ
a
b
r 2
⋅ L^2 − 2 − 2 −^2
NM^
O
π r b r a QP
π ⋅ ea 2 −b^2 j
Bending Moment in Curved Beam (Inside/Outside Stresses):
Stresses for the inside and outside fibers of a curved beam in pure bending can be approximated from the straight beam equation as modified by an appropriate curvature factor as determined from the graph below [ i refers to the inside, and o refers to the outside]. The curvature factor magnitude depends on the amount of curvature (determined by the ratio r/c ) and the cross section shape. r is the radius of curvature of the beam centroidal axis, and c is the distance from the centroidal axis to the inside fiber.
M
M
Centroidal Axis
r
c
Inside Fiber: σi Ki^
M c I
Outside Fiber: σo Ko^
M c I
K
i
o
Curvature Factor
Amount of curvature, r/c
K (^) r
b c
b c
b c
c b/
b/
b/8 b/
b/ B A B^ A
B A
B A
B (^) A
B (^) A (^) B (^) A
Values of K (^) i for inside fiber as at A
Values of K (^) o for outside fiber as at B
U or T
Round or Elliptical
Trapezoidal
I or hollow rectangular
Round, Elliptical or Trapezoidal
U or T
I or hollow rectangular
