Statics and Strength of Materials Formula Sheet

Statics and Strength of Materials Formula Sheet

1. Statics and Strength of Materials Formula Sheet
12/12/94 | A. Ruina
Not given here are the conditions under which the formulae are accurate or useful.
Basic Statics
Free Body Diagram
TheFBD is a picture of any system for which you would like to apply mechanics equations and
of all the external forces and torques which act on the system.
Action & Reaction
If A feels force F FFFFFFFFFand couple M MMMMMMMMMfrom B.
then B feels force ¡FFFFFFFFFFand couple ¡MMMMMMMMMMfrom A.
(With FFFFFFFFFFand¡FFFFFFFFFFacting on the same line of action.)
Force and Moment Balance
These equations apply to every FBD in equilibrium:
F orce Balance
z
}|{
X
All external
forces
FFFFFFFFFF=0000000000
M oment Balance about pt C
z
}|{
X
All external
torques
MMMMMMMMMM
=C
=0000000000
²The torque MMMMMMMMMM
=C
of a force depends on the reference point C. But, for a body in
equilibrium, and for any point C, the sum
of all the torques relative to point C must
add to zero ).
²Dotting the force balance equation with a unit vector gives a scalar equation,
e.g.f
P
FFFFFFFFFFg¢iiiiiiiiii=0 )
P
Fx=0.
²Dotting the moment balance equation with a unit vector gives a scalar equation,
e.g.f
P
MMMMMMMMMM
=C
g¢¸¸¸¸¸¸¸¸¸¸=0 ) net moment about axis in direction ¸¸¸¸¸¸¸¸¸¸throughC=0.
Some Statics Facts and De¯nitions
²The moment of a force is unchanged if the force is slid along its line of action.
²For many purposes the words `moment', `torque', and `couple' have the same meaning.
²Two-force body. If a body in equilibrium has only two forces acting on it then the
two forces must be equal and opposite and have a common line of action.
²Three-force body. If a body in equilibrium has only three forces acting on it then the
three forces must be coplanar and have lines of action that intersect at one point.
²truss:A collection of weightless two-force bodies connected with hinges (2D) or ball
and socket joints (3D).
²Method of joints. Draw free body diagrams of each of the joints in a truss.
²Method of sections. Draw free body diagrams of various regions of a truss. Try to
make the FBD cuts for the sections go through only three bars with unknown forces
(2D).
²Caution: Machine and frame components are often nottwo-force bodies.
²Hydrostatics: p=½gh; F =
R
pdA
Cross Section Geometry
thin-wall
annulus annulus
Definition Composite (circle:c
1
=0) (approx) rectangle
A=
R
dA
P
A
i
¼(c
2
2
¡c
2
1
) 2¼ct bh
J=
R
½
2
dA
¼
2
(c
4
2
¡c
4
1
) 2¼c
3
t
I=
R
y
2
dA
P
(I
i
+d
2
i
A
i
)
¼
4
(c
4
2
¡c
4
1
) ¼c
3
t bh
3
=12
¹y=
R
ydA
R
dA
P
y
i
A
i
P
A
i
center center center
Q=
R
ydA=A
0
¹y
0
P
A
0
i
¹y
0
i
b(
h
2
4
¡y
2
)
2
Stress, strain, and Hooke's Law
Stress Strain Hooke's Law
Normal: ¾=P
?
=A ²=±=L
0
=
L¡L
0
L
0
¾=E²
[²=¾=E+®¢T]
²
tran
=¡º²
long
Shear: ¿=P
k
=A °= change of
formerly right angle
¿=G°
2G=
E
1+º
Stress and deformation of some things
EquilibriumGeometry Results
Tension P=¾A ²=±=L ±=
PL
AE
[±=
PL
AE
+®L¢T]
Torsion T=
R
½¿ dA °=½Á=L Á=
TL
JG
¿=
T½
J
Bending M=¡
R
y¾ dA ²=¡y=½=¡y· u
00
=
M
EI
and
Shear in
dM
dx
=V;
dV
dx
=¡w u
00
=
d
2
dx
2
u=
1
½
=· ¾=
¡My
I
Beams
V=
R
¿dA ¿=
VQ
It
¿t¢x=¢M Q=I
Pressure pAgas=¾A
solid
¾=
pr
2t
(sphere)
Vessels
¾
l
=
pr
2t
(cylinder)
¾c=
pr
t
(cylinder)
Buckling
Critical buckling load = P
crit
=
¼
2
EI
L
2
ef f
.
pinned-pinned clamped-free clamped-clamped clamped-pinned
L
ef f
=L L
ef f
=2L L
ef f
=L=2 L
ef f
=:7L
Mohr's Circle
Rotating the surface of interest an angle µin physical space corresponds to a rotation of 2 µon
the Mohr's circle in the same direction.
C=
¾
1
+¾
2
2
=
¾x+¾y
2
R=
¾
1
¡¾
2
2
=
p
(¾x¡C)
2
+¿
2
xy
=
r
³
¾x¡¾y
2
´
2
+¿
2
xy
tan 2µ=
¿
¾¡C
=
2¿
¾x¡¾y
Miscellaneous
²Power in a shaft: P=T!.
²Saint Venant's Principle: Far from the region of loading, the stresses in a structure
would only change slightly if a load system were replaced with any other load system
having the same net force and moment.