fundamentals of engineering supplied-reference handbook - Ventech!

FORCE
A force is a vector quantity. It is defined when its (1)
magnitude, (2) point of application, and (3) direction are
known.
RESULTANT (TWO DIMENSIONS)
The resultant, F, of n forces with components Fx,i and Fy,i
has the magnitude of
⎡⎛
F = ⎢⎜
∑
⎢⎣
⎝
1
2
2 2
1 1 ⎥ ⎥
n ⎞ ⎛ n ⎞ ⎤
Fx,
i ⎟ + ⎜ ∑ Fy,
i ⎟
i=
i=
⎠
⎝
The resultant direction with respect to the x-axis using fourquadrant
angle functions is
θ =
⎛ n
arctan ⎜ ∑ F
⎝ i=
1
y,
i
The vector form of a force is
F = Fx i + Fy j
n
∑ F
i=
1
⎠
x,
i
RESOLUTION OF A FORCE
Fx = F cos θx; Fy = F cos θy; Fz = F cos θz
cos θx = Fx /F; cos θy = Fy /F; cos θz = Fz /F
Separating a force into components (geometry of force is
known R =
2 2 2
x + y + z )
Fx = (x/R)F; Fy = (y/R)F; Fz = (z/R)F
MOMENTS (COUPLES)
A system of two forces that are equal in magnitude, opposite
in direction, and parallel to each other is called a couple.
A moment M is defined as the cross product of the radius
vector r and the force F from a point to the line of action of
the force.
M = r × F; Mx = yFz – zFy,
My = zFx – xFz, and
Mz = xFy – yFx.
SYSTEMS OF FORCES
F = Σ Fn
M = Σ (rn × Fn)
Equilibrium Requirements
Σ Fn = 0
Σ Mn = 0
⎞
⎟
⎠
⎦
STATICS
24
CENTROIDS OF MASSES, AREAS, LENGTHS,
AND VOLUMES
Formulas for centroids, moments of inertia, and first
moment of areas are presented in the MATHEMATICS
section for continuous functions. The following discrete
formulas are for defined regular masses, areas, lengths, and
volumes:
rc = Σ mnrn/Σ mn, where
mn = the mass of each particle making up the system,
rn = the radius vector to each particle from a selected
reference point, and
rc = the radius vector to the center of the total mass
from the selected reference point.
The moment of area (Ma) is defined as
May = Σ xnan
Max = Σ ynan
Maz = Σ znan
The centroid of area is defined as
xac = May /A
yac = Max /A
zac = Maz /A
where A = Σ an
The centroid of a line is defined as
xlc = (Σ xnln)/L, where L = Σ ln
ylc = (Σ ynln)/L
zlc = (Σ znln)/L
The centroid of volume is defined as
xvc = (Σ xnvn)/V, where V = Σ vn
yvc = (Σ ynvn)/V
zvc = (Σ znvn)/V
MOMENT OF INERTIA
The moment of inertia, or the second moment of
area, is defined as
Iy = ∫ x 2 dA
Ix = ∫ y 2 dA
The polar moment of inertia J of an area about a point is
equal to the sum of the moments of inertia of the area about
any two perpendicular axes in the area and passing through
the same point.
Iz = J = Iy + Ix = ∫ (x 2 + y 2 ) dA
= rp 2 A, where
with respect to center of
the coordinate system
rp = the radius of gyration (see page 25).