Dr. D. B. Wallace
Basic Stress Equations
Centroid of
Cross Section
Internal Reactions:
Centroid of
Cross Section
Shear Forces
(τ )
y
6 Maximum
(3 Force Components
& 3 Moment Components)
x
"Cut Surface"
Vx
Bending Moments
(σ)
x
"Cut Surface"
Mx
z
P
Vy
y
My
Normal Force
(σ)
T
z
Torsional Moment
or Torque ( τ )
Force Components
Moment Components
Normal Force:
Centroid y
σ
x
"Cut Surface"
P
A
σ=
z
P
Axial Force
l Uniform over the entire cross section.
l Axial force must go through centroid.
Axial Stress
Shear Forces:
Cross Section
y
"y" Shear Force
Point of interest
LINE perpendicular to V through point of interest
V
x
Vy
y
b = Length of LINE on the cross section
z
τ
Aa = Area on one side of the LINE
Centroid of entire cross section
y
"x" Shear Force
Vx
Centroid of area on one side of the LINE
Aa
τ
x
I = Area moment of inertia of entire cross section
about an axis pependicular to V.
z
y = distance between the two centroids
τ=
b
V ⋅ Aa ⋅ y
I⋅b
g
Note: The maximum shear stress for common cross sections are:
Cross Section:
Cross Section:
τ max = 3 2 ⋅ V A
Rectangular:
I-Beam or
H-Beam:
flange
web τ max = V A web
Solid Circular:
Thin-walled
tube:
τ max = 4 3 ⋅ V A
τ max = 2 ⋅ V A
Basic Stress Equations
Dr. D. B. Wallace
Torque or Torsional Moment:
Solid Circular or Tubular Cross Section:
y
τ=
x
"Cut Surface"
z
T
Torque
τ max =
τ max =
T⋅r
J
τ
16 ⋅ T
π ⋅ D3
16 ⋅ T ⋅ Do
e
j
J=
J=
for solid circular shafts
π ⋅ Do 4 − D i 4
r = Distance from shaft axis to point of interest
R = Shaft Radius
D = Shaft Diameter
π ⋅ D4 π ⋅ R 4
=
32
2
e
π ⋅ Do4 − Di4
for solid circular shafts
j
32
for hollow shafts
for hollow shafts
Rectangular Cross Section:
τ2
y
Cross Section:
b
Centroid
τ1
x
"Cut Surface"
z
T
Torque
Note:
a>b
a
Torsional Stress
Method 1:
b
τ max = τ1 = T ⋅ 3 ⋅ a + 1.8 ⋅ b
g ea
2
⋅ b2
j
ONLY applies to the center of the longest side
Method 2:
τ1,2 =
T
α1,2 ⋅ a ⋅ b
2
Use the appropriate α from the table
on the right to get the shear stress at
either position 1 or 2.
a/b
1.0
1.5
2.0
3.0
4.0
6.0
8.0
10.0
∞
Other Cross Sections:
Treated in advanced courses.
2
α1
.208
.231
.246
.267
.282
.299
.307
.313
.333
α2
.208
.269
.309
.355
.378
.402
.414
.421
----
Basic Stress Equations
Dr. D. B. Wallace
Bending Moment
"x" Bending Moment
y
x
Mx
σ=
σ
z
Mx ⋅ y
Ix
and
σ=
My ⋅ x
Iy
"y" Bending Moment
y
where: Mx and My are moments about indicated axes
y and x are perpendicular from indicated axes
Ix and Iy are moments of inertia about indicated axes
x
z
My
σ
Moments of Inertia:
b
b ⋅ h3
I=
12
c
h
D
h is perpendicular to axis
c
R
b ⋅ h2
I
=
c
6
Z=
I =
π ⋅ D4
π ⋅ R4
=
64
4
Z =
I
π ⋅ D3
π ⋅ R3
=
=
c
32
4
Parallel Axis Theorem:
I = Moment of inertia about new axis
I = Moment of inertia about the centroidal axis
A = Area of the region
d = perpendicular distance between the two axes.
new axis
Area, A
d
I = I + A ⋅d2
centroid
Maximum Bending Stress Equations:
σ max =
M⋅c
M
=
I
Z
σ max =
32 ⋅ M
π⋅D
3
bSolid Circular g
σ max =
6⋅ M
b ⋅ h2
a Rectangular f
The section modulus, Z, can be found in many tables of properties of common cross sections (i.e., I-beams,
channels, angle iron, etc.).
Bending Stress Equation Based on Known Radius of Curvature of Bend, ρ.
The beam is assumed to be initially straight. The applied moment, M, causes the beam to assume a radius of
curvature, ρ.
Before:
σ = E⋅
After:
y
ρ
E = Modulus of elasticity of the beam material
M
ρ
y = Perpendicular distance from the centroidal axis to the
point of interest (same y as with bending of a
straight beam with Mx).
M
ρ = radius of curvature to centroid of cross section
3
Basic Stress Equations
Dr. D. B. Wallace
Bending Moment in Curved Beam:
Geometry:
σo
nonlinear
stress
distribution
centroid
centroidal
axis
y
co
e
ci
σi
rn
neutral axis
ρ r
ri
A
dA
area ρ
e = r − rn
rn =
ro
z
A = cross sectional area
rn = radius to neutral axis
r = radius to centroidal axis
e = eccentricity
M
Stresses:
Any Position:
σ=
Inside (maximum magnitude):
−M ⋅ y
e ⋅ A ⋅ rn + y
b
σi =
g
Outside:
M ⋅ ci
e ⋅ A ⋅ ri
σo =
− M ⋅ co
e ⋅ A ⋅ ro
Area Properties for Various Cross Sections:
Cross Section
Rectangle
r
ρ
z
r
ri +
t
dA
ρ
A
FG r IJ
Hr K
h⋅t
area
h
2
t ⋅ ln
o
i
ri
h
ro
Trapezoid
r
ρ
ti
ri
ri +
to
h
ro
Hollow Circle
r
a
ρ
b
h ⋅ ti + 2 ⋅ to
3⋅ t i + t o
b
g
g
to − ti +
For triangle:
set ti or to to 0
r
2⋅π
b
4
LM
N
FG IJ
H K
ro ⋅ t i − ri ⋅ t o
r
⋅ ln o
h
ri
r 2 − b2 − r 2 − a 2
OP
Q
h⋅
e
ti + t o
2
π ⋅ a 2 − b2
j
Basic Stress Equations
Dr. D. B. Wallace
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.
Centroidal
Axis
c
r
M⋅c
I
Inside Fiber:
σ i = Ki ⋅
Outside Fiber:
σ o = Ko ⋅
M
M⋅c
I
B
A
b/8
4.0
B
A
b
b/4
c
Values of Ki for inside fiber as at A
3.5
B
B
A
A
c
U or T
3.0
b/2
Curvature
Factor
Round or Elliptical
b
A
B
c
2.5
Trapezoidal
b/3
b/6
2.0
B
b
B
A
A
c
I or hollow rectangular
Ki
M
r
1.5
1.0
Ko
I or hollow rectangular
U or T
0.5
Values of Ko for outside fiber as at B
Round, Elliptical or Trapezoidal
0
1
2
3
4
5
6
7
Amount of curvature, r/c
5
8
9
10
11