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Dr. D. B. Wallace
Basic Stress Equations
Internal Reactions: 6 Maximum
(3 Force Components & 3 Moment Components) Cut Surface Centroid of Cross Section y Centroid of Cross Section
Shear Forces (τ )
x Cut Surface z
y
Bending Moments (σ)
x
Vx Vy
P
Mx
My
T
z
Normal Force (σ) Force Components Moment Components
Torsional Moment or Torque ( τ )
Normal Force:
Centroid y Cut Surface x z P Axial Force
σ
σ=
P A
l Uniform over the entire cross section. l Axial force must go through c

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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
zxy
PV
y
V
x
Torsional Moment)
ττ
()
σσ
(Bending Moments
zxy
TM
y
M
x
or Torque
Force Components Moment Components
Cut Surface Cut Surface Centroid of Cross SectionCentroid of Cross Section
Normal Force:
Axial Force
zxy
P
Centroid
σσ
Axial Stress
Cut Surface
σ =
PA
l
Uniform over the entire cross section.
l
Axial force must go through centroid.
Shear Forces:
Cross SectionyAa
Point of interestLINE perpendicular to V through point of interest= Length of LINE on the cross section= Area on one side of the LINECentroid of entire cross sectionCentroid of area on one side of the LINE= distance between the two centroids= Area moment of inertia of entire cross sectionabout an axis pependicular to V.
VbAayI
y Shear Force
zxy
V
y
x Shear Force
zxy
V
x
ττττ
τ =⋅ ⋅⋅
VAyIb
a
b g
Note
:
The maximum shear stress for common cross sections are:
Cross Section:Cross Section:
Rectangular:
τ
max
= ⋅
32VA
Solid Circular:
τ
max
= ⋅
43VA
I-Beam orH-Beam:webflange
τ
max
=
VA
web
Thin-walledtube:
τ
max
= ⋅
2VA
Basic Stress EquationsDr. D. B. Wallace
Torque or Torsional Moment:
Solid Circular or Tubular Cross Section:
r = Distance from shaft axis to point of interestR = Shaft RadiusD = Shaft Diameter
JDRJDD
forsolidcircularshaftsforhollowshafts
oi
=⋅=⋅=⋅ −π ππ
4444
32232
e j
Torque
zx
y
T
Cut Surface
ττ
τ =⋅
TrJ
τπτπ
maxmax
=⋅⋅=⋅ ⋅⋅ −
1616
344
TDTDDD
forsolidcircularshaftsforhollowshafts
ooi
e j
Rectangular Cross Section:
Torque
zxy
T
Centroid
ττ
Torsional Stress
Cut Surface
ττ
12
abNote:a > bCross Section:
Method 1:
τ τ
max
.
= = ⋅ ⋅ + ⋅ ⋅
122
318Tabab
b g e j
ONLY
applies to the center of the longest side
Method 2:
τα
12122
, ,
=⋅ ⋅
Tab
a/b
αα αα
1.0.208.2081.5.231.2692.0.246.3093.0.267.3554.0.282.3786.0.299.4028.0.307.41410.0.313.421
∞
.333----Use the appropriate
αα
from the tableon the right to get the shear stress ateither position
1
or
2
.
Other Cross Sections:
Treated in advanced courses.2
Basic Stress EquationsDr. D. B. Wallace
Bending Moment
x Bending Moment
zx
yzxy
M
x
σσσσ
M
y
y Bending Moment
σ σ=⋅=⋅
MyIandMxI
xxyy
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
Moments of Inertia:
hcbDcR
IbhhZIcbh
isperpendiculartoaxis
=⋅= =⋅
32
126IDRZIcDR
=⋅=⋅= =⋅=⋅π ππ π
4433
644324
Parallel Axis Theorem:
I = Moment of inertia about new axis
IIAd
= + ⋅
2
centroiddnew axisArea, A
I
= Moment of inertia about the centroidal axisA = Area of the regiond = perpendicular distance between the two axes.
Maximum Bending Stress Equations:
σπ
max
=⋅⋅
32
3
MD
SolidCircular
b g
σ
max
=⋅⋅
6
2
Mbh
Rectangular
a f
σ
max
=⋅=
McIMZ
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
ρ
.
The beam is assumed to be initially straight. The applied moment,
M
, causes the beam to assume a radius of curvature,
ρρ
.
Before:After:M
M
ρρ
σρ= ⋅
Ey
E
= Modulus of elasticity of the beam material
y
= Perpendicular distance from the centroidal axis to thepoint of interest (same
y
as with bending of astraight beam with
Mx
).
ρρ
= radius of curvature to centroid of cross section3
Basic Stress EquationsDr. D. B. Wallace
Bending Moment in Curved Beam:
Geometry:rrerr
n
σσσσ
oi
centroidcentroidalneutral axisaxis
oi
y
nonlinearstressdistribution
Mcc
io
ρρ
rAdAerr
narean
== −
z
ρ
A = cross sectional area
r
n= radius to neutral axis
r
= radius to centroidal axise = eccentricity
Stresses:
Any Position:Inside
(maximum magnitude)
:Outside:
σ =− ⋅⋅ ⋅ +
MyeAry
n
b g
σ
iii
MceAr
=⋅⋅ ⋅σ
ooo
MceAr
=− ⋅⋅ ⋅
Area Properties for Various Cross Sections:
Cross Section
rdA
area
ρ
z
A
thr
i
r
o
r
Rectangle
ρ
rh
i
+
2trr
oi
⋅
F H GI K J
ln
ht
⋅
thr
i
r
o
r
Trapezoid
ρ
o
t
i
rhtttt
iioio
+⋅ + ⋅⋅ +
23
b gb g
For triangle:
set tior toto
0
ttrtrthrr
oioiiooi
− +⋅ − ⋅⋅
F H GI K J
ln
htt
io
⋅+
2r
Hollow Circle
ρ
abr
2
2222
⋅ − − −
LNMOQP
π
rbra
π⋅ −
ab
22
e j
4

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