CHAPTER VECTOR MECHANICS FOR ENGINEERS: 4 …ybu.edu.tr/fozturk/contents/files/354Chapter-4.pdf ·...
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1
VECTOR MECHANICS FOR ENGINEERS:
STATICS
Ninth Edition
Ferdinand P. Beer
E. Russell Johnston, Jr.
Lecture Notes:
J. Walt Oler
Texas Tech University
CHAPTER
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4Equilibrium of Rigid
Bodies
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Design of a support
2 - 2
1. Identify the main structure.
2. Where is the load on the
structure?
3. How is the structure fixed to
the ground?
4. How do we decide on the
dimensions of the ground
support?
2
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Contents and Objectives
4 - 3
Draw Free-Body Diagram
Identify Reactions at Supports for a Two-Dimensional Structure
Solve Problems of Equilibrium of a Rigid Body in Two Dimensions
Identify Statically Indeterminate Reactions
Recognize a Two-Force Body
Recognize a Three-Force Body
Solve Problems of Equilibrium of a Rigid Body in Three Dimensions
Recognize Reactions at Supports and Connections for a Three-Dimensional
Structure
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Introduction
4 - 4
• The necessary and sufficient condition for the static equilibrium of a
body are that the resultant force and couple from all external forces
form a system equivalent to zero.
00 FrMF O
000
000
zyx
zyx
MMM
FFF
• Resolving each force and moment into its rectangular components
leads to 6 scalar equations which also express the conditions for static
equilibrium,
• For a rigid body in static equilibrium, the external forces and
moments are balanced and will impart no translational or rotational
motion to the body.
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2 - 5
Free-Body Diagram
The truck ramp has a weight of 1800 N.
The ramp is pinned to the body of the truck and held in the position by the
cable. How can we determine the cable tension and support reactions ?
How are the idealized model and the free body diagram used to do this?
Which diagram above is the idealized model?
Geometry Geometry + Loads
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Free-Body Diagram
4 - 6
First step in static equilibrium analysis of a rigid
body is identification of all forces acting on the
body with a free-body diagram.
• Draw an outlined shape. Cut the body free
form its constraints.
• Label loads and dimensions on FBD necessary
to compute the moments of the forces.
• Indicate point of application and assumed
direction of unknown applied forces. These
usually consist of reactions through which the
ground and other bodies oppose the possible
motion of the rigid body.
• Indicate point of application, magnitude, and
direction of external forces, including the rigid
body weight and couple moments.
4
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Reactions at Supports and Connections for a Two-Dimensional Structure
4 - 7
• Reactions equivalent to a
force with known line of
action.
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Reactions at Supports and Connections for a Two-Dimensional Structure
4 - 8
• Reactions equivalent to a
force of unknown direction
and magnitude.
• Reactions equivalent to a
force of unknown
direction and magnitude
and a couple of unknown
magnitude.
5
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Quiz 1
2 - 9
1. The beam and the cable (with a frictionless pulley at D) support
an 80 kg load at C. In a FBD of only the beam, there are how
many unknowns?
A) 2 forces and 1 couple moment
B) 3 forces and 1 couple moment
C) 3 forces
D) 4 forces
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Quiz 2
2 - 10
2. If the directions of the force and the couple moments are
both reversed, what will happen to the beam?
A) The beam will lift from A.
B) The beam will lift at B.
C) The beam will be restrained.
D) The beam will break.
6
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Quiz 3
2 - 11
Draw a FBD of member ABC, which is supported
by a smooth collar at A, roller at B, and link CD.
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Quiz 4
2 - 12
How many unknown support reactions
are there in this problem?
A) 2 forces and 2 couple moments
B) 1 force and 2 couple moments
C) 3 forces
D) 3 forces and 1 couple moment
7
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Practical Example
2 - 13
A 4000 N of engine is supported by three chains, which are
attached to the spreader bar of a hoist.
You need to check to see if the breaking strength of any of the
chains is going to be exceeded. How can you determine the
force acting in each of the chains?
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Equilibrium of a Rigid Body in Two Dimensions
4 - 14
• For all forces and moments acting on a two-
dimensional structure,
Ozyxz MMMMF 00
• Equations of equilibrium become
000 Ayx MFF
where A is any point in the plane of the
structure.
• The 3 equations can be solved for no more
than 3 unknowns.
• The 3 equations can not be augmented with
additional equations, but they can be replaced.
000 BAx MMF
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STEPS FOR SOLVING 2-D EQUILIBRIUM PROBLEMS
2 - 15
1. If not given, establish a suitable x - y coordinate system.
2. Draw a free body diagram (FBD) of the object under analysis.
3. Apply the three equations of equilibrium to solve for the
unknowns.
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Sample Problem 4.1
4 - 16
A fixed crane has a mass of 1000 kg
and is used to lift a 2400 kg crate. It
is held in place by a pin at A and a
rocker at B. The center of gravity of
the crane is located at G.
Determine the components of the
reactions at A and B.
SOLUTION:
• Create a free-body diagram for the crane.
• Determine B by solving the equation for
the sum of the moments of all forces
about A. Note there will be no
contribution from the unknown
reactions at A.
• Determine the reactions at A by
solving the equations for the sum of
all horizontal force components and
all vertical force components.
• Check the values obtained for the
reactions by verifying that the sum of
the moments about B of all forces is
zero.
9
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Sample Problem 4.1
4 - 17
• Create the free-body diagram.
• Check the values obtained.
• Determine B by solving the equation for the
sum of the moments of all forces about A.
0m6kN5.23
m2kN81.9m5.1:0
BM A
107.1kNB
• Determine the reactions at A by solving the
equations for the sum of all horizontal forces
and all vertical forces.
0:0 BAF xx
107.1kNxA
0kN5.23kN81.9:0 yy AF
33.3 kNyA
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Sample Problem 4.3
4 - 18
A loading car is at rest on an inclined
track. The gross weight of the car and
its load is 25 kN, and it is applied at at
G. The cart is held in position by the
cable.
Determine the tension in the cable and
the reaction at each pair of wheels.
SOLUTION:
• Create a free-body diagram for the car
with the coordinate system aligned
with the track.
• Determine the reactions at the wheels
by solving equations for the sum of
moments about points above each axle.
• Determine the cable tension by
solving the equation for the sum of
force components parallel to the track.
• Check the values obtained by verifying
that the sum of force components
perpendicular to the track are zero.
10
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Sample Problem 4.3
4 - 19
• Create a free-body diagram
kN 10.5
25sinkN 25
kN 22.65
25coskN 25
y
x
W
W
• Determine the reactions at the wheels.
0mm 1250
mm 150kN 22.65mm 625kN 10.5:0
2
R
M A
2 8 kNR
0mm 1250
mm 150kN 22.65mm 625kN 10.5:0
1
R
M B
1 2.5 kNR
• Determine the cable tension.
0TkN 22.65:0 xF
22.7 kNT
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Sample Problem 4.4
4 - 20
A 6-m telephone pole of 1600 N is
used to support the wires. Wires T1 =
600 N and T2 = 375 N.
Determine the reaction at the fixed
end A.
SOLUTION:
• Create a free-body diagram for the
telephone cable.
• Solve 3 equilibrium equations for the
reaction force components and
couple at A.
11
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Sample Problem 4.4
4 - 21
• Create a free-body diagram for
the frame and cable.
• Solve 3 equilibrium equations for the
reaction force components and couple.
010cos)N600(20cos)N375(:0 xx AF
N 238.50xA
020sinN 37510sin600NN1600:0 yy AF
N 1832.45yA
:0AM
0 (6m)
20cos375N)6(10cos600N
mM A
N.m00.1431AM
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Notes
2 - 22
1. If there are more unknowns than the number of independent equations,
then we have a statically indeterminate situation. We cannot solve these
problems using just statics.
2. The order in which we apply equations may affect the simplicity of the
solution. For example, if we have two unknown vertical forces and one
unknown horizontal force,
then solving FX = 0 first allows us to find the horizontal unknown
quickly.
3. If the answer for an unknown comes out as negative number, then the
sense (direction) of the unknown force is opposite to that assumed when
starting the problem.
12
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Equilibrium of a Two-Force Body
4 - 23
• Consider a plate subjected to two forces F1 and F2 .
• For static equilibrium, the sum of moments about A
must be zero. The moment of F2 must be zero. It
follows that the line of action of F2 must pass
through A.
• Similarly, the line of action of F1 must pass through
B for the sum of moments about B to be zero.
• Requiring that the sum of forces in any direction be
zero leads to the conclusion that F1 and F2 must have
equal magnitude but opposite sense.
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Applications of 2-Force Bodies
2 - 24
In the cases above, members AB can be considered as two-force members,
provided that their weight is neglected.
This fact simplifies the equilibrium analysis of
some rigid bodies since the directions of the
resultant forces at A and B are thus known (along
the line joining points A and B).
13
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Equilibrium of a Three-Force Body
4 - 25
• Consider a rigid body subjected to forces acting at
only 3 points.
• Assuming that their lines of action intersect, the
moment of F1 and F2 about the point of intersection
represented by D is zero.
• Since the rigid body is in equilibrium, the sum of the
moments of F1, F2, and F3 about any axis must be
zero. It follows that the moment of F3 about D must
be zero as well and that the line of action of F3 must
pass through D.
• The lines of action of the three forces must be
concurrent or parallel.
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Sample Problem 4.6
4 - 26
A man raises a 10 kg joist, of
length 4 m, by pulling on a rope.
Find the tension in the rope and
the reaction at A.
SOLUTION:
• Create a free-body diagram of the joist.
Note that the joist is a 3 force body acted
upon by the rope, its weight, and the
reaction at A.
• The three forces must be concurrent for
static equilibrium. Therefore, the reaction
R must pass through the intersection of the
lines of action of the weight and rope
forces. Determine the direction of the
reaction force R.
• Utilize a force triangle to determine the
magnitude of the reaction force R.
14
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Sample Problem 4.6
4 - 27
• Create a free-body diagram of the joist.
• Determine the direction of the reaction
force R.
636.1414.1
313.2tan
m 2.313m 515.0828.2
m 515.020tanm 414.1)2045cot(
m 414.1
m828.245cosm445cos
21
AE
CE
BDBFCE
CDBD
AFAECD
ABAF
6.58
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Sample Problem 4.6
4 - 28
• Determine the magnitude of the reaction
force R.
38.6sin
N 1.98
110sin4.31sin
RT
N 8.147
N9.81
R
T
15
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Statically Indeterminate Reactions
4 - 29
• More unknowns than
equations
Redundant Constraints: When a body has more
supports than necessary to hold it in equilibrium, it
becomes statically indeterminate.
A problem that is statically indeterminate has
more unknowns than equations of equilibrium.
Are statically indeterminate structures used in
practice? Why or why not?
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Improper Constraints
2 - 30
• Equal number unknowns
and equations but
improperly constrained.
Here, we have 6 unknowns but there is nothing
restricting rotation about the AB axis.
16
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Improper Constraints
2 - 31
In some cases, there may be as many unknown reactions as
there are equations of equilibrium.
However, if the supports are not properly constrained, the body
may become unstable for some loading cases.
0AM
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Equilibrium of a Rigid Body in Three Dimensions
4 - 32
• Six scalar equations are required to express the conditions
for the equilibrium of a rigid body in the general three
dimensional case.
000
000
zyx
zyx
MMM
FFF
• These equations can be solved for no more than 6 unknowns
which generally represent reactions at supports or connections.
• The scalar equations are conveniently obtained by applying the
vector forms of the conditions for equilibrium,
00 FrMF O
17
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Applications
2 - 33
Ball-and-socket joints and journal bearings are often used in
mechanical systems. To design the joints or bearings, the support
reactions at these joints and the loads must be determined.
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Applications
2 - 34
If A is moved to a lower position D, will
the force in the rod change or remain the
same? By making such a change without
understanding if there is a change in
forces, failure might occur.
The tie rod from point A is used to support
the overhang at the entrance of a building.
It is pin connected to the wall at A and to
the center of the overhang B.
18
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Application
2 - 35
The crane, which weighs 1550 N, is supporting a oil drum.
How do you determine the largest oil drum weight that the
crane can support without overturning ?
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Reactions at Supports and Connections for a Three-Dimensional Structure
4 - 36
19
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Reactions at Supports and Connections for a Three-
Dimensional Structure
4 - 37
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2 - 38
Exchange arm for a suspension
Universal joint in transmission shaft of
a truck
20
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2 - 39
Bearing for trolley wheels
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Sample Problem 4.8
4 - 40
A sign of uniform density weighs
1200 N and is supported by a ball-
and-socket joint at A and by two
cables.
Determine the tension in each cable
and the reaction at A.
SOLUTION:
• Create a free-body diagram for the sign.
• Apply the conditions for static
equilibrium to develop equations for
the unknown reactions.
21
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Sample Problem 4.8
4 - 41
• Create a free-body diagram for the
sign.
Since there are only 5 unknowns,
the sign is partially constrain. It is
free to rotate about the x axis. It is,
however, in equilibrium for the
given loading.
kjiT
kjiT
rr
rrTT
kjiT
kjiT
rr
rrTT
EC
EC
EC
ECECEC
BD
BD
BD
BDBDBD
72
73
76
32
31
32
7
236
6.3
4.22.14.2
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Sample Problem 4.8
4 - 42
• Apply the conditions for
static equilibrium to
develop equations for the
unknown reactions.
0m . N 1440771.08.0:
0514.06.1:
0N 1200m 1.2
0:
0N 1200:
0:
0N 1200
72
32
73
31
76
32
ECBD
ECBD
ECEBDBA
ECBDz
ECBDy
ECBDx
ECBD
TTk
TTj
jiTrTrM
TTAk
TTAj
TTAi
jTTAF
kjiA
TT ECBD
N 100.1N 419N 1502
N 1402N 451
Solve the 5 equations for the 5 unknowns,
22
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Quiz 5
2 - 43
1. The rod AB is supported using
two cables at B and a ball-and-
socket joint at A. How many
unknown support reactions exist
in this problem?
A) 5 force and 1 moment reaction
B) 5 force reactions
C) 3 force and 3 moment
reactions
D) 4 force and 2 moment
reactions
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2 - 44
2. If an additional couple moment in the
vertical direction is applied to rod AB at
point C, then what will happen to the
rod?
A) The rod remains in equilibrium as the
cables provide the necessary support
reactions.
B) The rod remains in equilibrium as the
ball-and-socket joint will provide the
necessary resistive reactions.
C) The rod becomes unstable as the cables
cannot support compressive forces.
D) The rod becomes unstable since a
moment about AB cannot be restricted.
23
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Quiz 6
2 - 45
1. A plate is supported by a ball-and-
socket joint at A, a roller joint at B,
and a cable at C. How many
unknown support reactions are
there in this problem?
A) 4 forces and 2 moments
B) 6 forces
C) 5 forces
D) 4 forces and 1 moment
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2 - 46
2. What will be the easiest way to determine the force
reaction BZ ?
A) Scalar equation FZ = 0
B) Vector equation MA = 0
C) Scalar equation MZ = 0
D) Scalar equation MY = 0