Chapter -4- Force System Resultant … · To present methods for determining the resultants of...
Transcript of Chapter -4- Force System Resultant … · To present methods for determining the resultants of...
1
Ishik University / Sulaimani
Civil Engineering Department
Chapter -4-
Force System Resultant 1
Ishik
Univ
ersity-S
ula
imani
Assista
nt L
ectu
rer - A
smaa A
bdulm
aje
ed
Ishik
Univ
ersity
-Sula
imani
Assista
nt L
ectu
rer - A
smaa A
bdulm
aje
ed
2
2
CHAPTER OBJECTIVES
To discuss the concept of the moment of a force and show
how to calculate it in two and three dimensions.
To provide a method for finding the moment of a force about
a specified axis.
To define the moment of a couple.
To present methods for determining the resultants of non-
concurrent force systems.
To indicate how to reduce a simple distributed loading to a
resultant force having a specified location.
Ishik
Univ
ersity-S
ula
imani
Assistan
t Lectu
rer - Asm
aa Abdulm
ajeed
3
CHAPTER OUTLINE
Moment of a Force – Scalar Formation
Principle of Moments
Moment of a Force about a Specified Axis
Moment of a Couple
Equivalent System
Resultants of a Force and Couple System
Reduction of a Simple Distributed Loading
Ishik
Univ
ersity-S
ula
imani
Assistan
t Lectu
rer - Asm
aa Abdulm
ajeed
4
3
4.1 MOMENT OF A FORCE – SCALAR FORMATION
Moment of a force about a point or axis – a measure of the
tendency of the force to cause a body to rotate about the
point or axis.
Case 1
Consider horizontal force Fx,
which acts perpendicular to
the handle of the wrench and is
located dy from the point O.
Ishik
Univ
ersity-S
ula
imani
Assistan
t Lectu
rer - Asm
aa Abdulm
ajeed
5
Fx tends to turn the pipe about the z axis.
The larger the force or the distance dy, the greater the
turning effect.
Torque – tendency of
rotation caused by Fx
or simple moment (Mo) z
Moment axis (z) is
perpendicular to shaded
plane (x-y)
Fx and dy lies on the
shaded plane (x-y)
Moment axis (z) intersects
the plane at point O
Ishik
Univ
ersity-S
ula
imani
Assistan
t Lectu
rer - Asm
aa Abdulm
ajeed
6
4
Case 2
Apply force Fz to the wrench.
Pipe does not rotate about z axis.
Tendency to rotate about x axis.
The pipe may not actually rotate
Fz creates tendency for rotation
so moment (Mo) x is produced.
Moment axis (x) is perpendicular to shaded plane (y-z).
Fz and dy lies on the shaded plane (y-z).
Ishik
Univ
ersity-S
ula
imani
Assistan
t Lectu
rer - Asm
aa Abdulm
ajeed
7
Case 3
Apply force Fy to the wrench.
No moment is produced about point O.
Lack of tendency to rotate as line of action passes through O.
Ishik
Univ
ersity-S
ula
imani
Assistan
t Lectu
rer - Asm
aa Abdulm
ajeed
8
5
In General
Consider the force F and the point O which lies in the shaded
plane.
The moment MO about point O, or about an axis passing
through O and perpendicular to the plane, is a vector quantity.
Moment MO has its specified magnitude and direction.
Ishik
Univ
ersity-S
ula
imani
Assistan
t Lectu
rer - Asm
aa Abdulm
ajeed
9
Magnitude
For magnitude of MO, MO = Fd where
d = moment arm or perpendicular distance from the axis at
point O to its line of action of the force.
Units for moment is N.m.
Direction
Direction of MO is specified by using
“right hand rule”.
- fingers of the right hand are curled
to follow the sense of rotation when
force rotates about point O.
- Thumb points along the moment axis
to give the direction and sense of the moment
vector.
- Moment vector is upwards and perpendicular
to the shaded plane.
Ishik
Univ
ersity-S
ula
imani
Assistan
t Lectu
rer - Asm
aa Abdulm
ajeed
10
6
Direction
MO is shown by a vector arrow with a curl to
distinguish it from force vector.
Example (Fig b)
MO is represented by the
counterclockwise curl, which
indicates the action of F.
Ishik
Univ
ersity-S
ula
imani
Assistan
t Lectu
rer - Asm
aa Abdulm
ajeed
11
Direction
Arrowhead shows the sense of
rotation caused by F Using the
right hand rule, the direction and
sense of the moment vector points
out of the page.
In 2D problems, moment of the force
is found about a point O.
Moment acts about an axis perpendicular to the plane
containing F and d .
Moment axis intersects the plane at point O.
Ishik
Univ
ersity-S
ula
imani
Assistan
t Lectu
rer - Asm
aa Abdulm
ajeed
12
7
Resultant Moment of a System of Coplanar Forces
Resultant moment, MRo = addition of the moments of all the
forces algebraically since all moment forces are collinear.
MRo = ∑Fd
taking clockwise to be negative.
o A clockwise curl is written along the equation to
indicate that a positive moment if directed along the +
z axis and negative along the – z axis
Ishik
Univ
ersity-S
ula
imani
Assistan
t Lectu
rer - Asm
aa Abdulm
ajeed
13
14
Ishik
Univ
ersity-S
ula
imani
Assistan
t Lectu
rer - Asm
aa Abdulm
ajeed
8
Moment of a force does not always cause rotation.
Force F tends to rotate the beam clockwise about A
with moment.
MA = FdA
Force F tends to rotate the beam counterclockwise
about B with moment.
MB = FdB
Hence support at A prevents the rotation.
15
Ishik
Univ
ersity-S
ula
imani
Assistan
t Lectu
rer - Asm
aa Abdulm
ajeed
16
Ishik
Univ
ersity-S
ula
imani
Assistan
t Lectu
rer - Asm
aa Abdulm
ajeed
9
17
Ishik
Univ
ersity-S
ula
imani
Assistan
t Lectu
rer - Asm
aa Abdulm
ajeed
Example 4.1
For each case, determine the moment of the force about point O.
18
Ishik
Univ
ersity-S
ula
imani
Assistan
t Lectu
rer - Asm
aa Abdulm
ajeed
10
Solution;
Line of action is extended as a dashed line to establish moment
arm d.
Tendency to rotate is indicated and the orbit is shown as a
colored curl.
)(.5.37)75.0)(50()(
)(.200)2)(100()(
CWmNmNMb
CWmNmNMa
o
o
Ishik
Univ
ersity-S
ula
imani
Assistan
t Lectu
rer - Asm
aa Abdulm
ajeed
19
Solution;
)(.0.21)14)(7()(
)(.4.42)45sin1)(60()(
)(.229)30cos24)(40()(
CCWmkNmmkNMe
CCWmNmNMd
CWmNmmNMc
o
o
o
Ishik
Univ
ersity-S
ula
imani
Assistan
t Lectu
rer - Asm
aa Abdulm
ajeed
20
11
Example 4.2
Determine the moments of the 800N force acting on the frame
about points A, B, C and D.
Ishik
Univ
ersity-S
ula
imani
Assistan
t Lectu
rer - Asm
aa Abdulm
ajeed
21
Solution;
Scalar Analysis
Line of action of F passes through C
)CCW(m.N400)m5.0)(N800(M
m.kN0)m0)(N800(M
)CW(m.N1200)m5.1)(N800(M
)CW(m.N2000)m5.2)(N800(M
D
C
B
A
22
Ishik
Univ
ersity-S
ula
imani
Assistan
t Lectu
rer - Asm
aa Abdulm
ajeed
12
Ishik
Univ
ersity-S
ula
imani
Assistan
t Lectu
rer - Asm
aa Abdulm
ajeed
23
Example 4.3
Determine the resultant moment of the four forces
acting on the rod shown in figure about point O. Ish
ik U
niv
ersity-S
ula
imani
Assistan
t Lectu
rer - Asm
aa Abdulm
ajeed
24
13
4.2 PRINCIPLES OF MOMENTS
The guy cable exerts a force F on the
pole and creates a moment about the
base at A,
MA = Fd
If the force is replaced by Fx and Fy
at point B where the cable acts on
the pole, the sum of moment about
point A yields the same resultant
moment. 25
Ishik
Univ
ersity-S
ula
imani
Assistan
t Lectu
rer - Asm
aa Abdulm
ajeed
Fy create zero moment about A
MA = Fxh
Apply principle of
transmissibility and slide the
force where line of action
intersects the ground at C, Fx
create zero moment about A
MA = Fyb
26
Ishik
Univ
ersity-S
ula
imani
Assistan
t Lectu
rer - Asm
aa Abdulm
ajeed
14
Example 4.3
The force F acts at the end of the angle bracket. Determine the
moment of the force about point O.
Ishik
Univ
ersity-S
ula
imani
Assistan
t Lectu
rer - Asm
aa Abdulm
ajeed
27
Solution
Method 1:
MO = 400sin30°N(0.2m)-400cos30°N(0.4m)
= -98.6N.m
= 98.6N.m (CCW)
As a Cartesian vector,
MO = {-98.6k}N.m
Ishik
Univ
ersity-S
ula
imani
Assistan
t Lectu
rer - Asm
aa Abdulm
ajeed
28
15
Solution
Method 2:
Express as Cartesian vector
r = {0.4i – 0.2j}N
F = {400sin30°i – 400cos30°j}N
= {200.0i – 346.4j}N
For moment,
mNk
kji
FXrMO
.6.98
04.3460.200
02.04.0
29
Ishik
Univ
ersity-S
ula
imani
Assistan
t Lectu
rer - Asm
aa Abdulm
ajeed
30
Ishik
Univ
ersity-S
ula
imani
Assistan
t Lectu
rer - Asm
aa Abdulm
ajeed
Example 4.4
The member is subjected to a force of F=6 kN. If θ= 45º,
determine the moment produced by F about point A.
Ans;
16
31
Ishik
Univ
ersity-S
ula
imani
Assistan
t Lectu
rer - Asm
aa Abdulm
ajeed
Example 4.5
The two boys push on the gate with forces of FA=30 Ib and FB=50
Ib as shown. Determine the moment of each force about C. Which
way will the gate rotate, clockwise or counterclockwise? Neglect the
thickness of the gate.
Ans;
The gate will
rotate C.C.W.