ch.04 force system resultants
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2/13/2013
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04. Force System Resultants
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 4.01 Force System Resultants
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 an 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
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 4.02 Force System Resultants
1. Moment of a Force Scalar Formulation
- The moment of a force about a point provides a measure of
the tendency for rotation (sometimes called a torque)
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
= = = = = 0
Engineering Mechanics Statics 4.03 Force System Resultants
1. Moment of a Force Scalar Formulation
- Magnitude
=
- Direction
- Resultant Moment
+ ()= : ()= 11 22 + 33
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 4.04 Force System Resultants
1. Moment of a Force Scalar Formulation
- Example 4.1 For each case illustrated in the figure,
determine the moment of the force about point
Solution
= 100 2 = 200
= 50 0.75 = 37.5
= 40 (4 +2300) = 229
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 4.05 Force System Resultants
1. Moment of a Force Scalar Formulation
- Example 4.1 For each case illustrated in the figure,
determine the moment of the force about point
Solution
= 60 1450 = 42.4
= 7 (4 1) = 21.0
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 4.06 Force System Resultants
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2. Cross Product
The cross product of two vectors
=
- Magnitude
= , =
- Direction
Specified by the right-hand rule
curling the fingers of the right hand from vector (cross) to
vector , the thumb points in the direction of
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 4.07 Force System Resultants
2. Cross Product
- Laws of Operation
Commutative law
=
Multiplication by a scalar
= = =
Distributive law
+ = ( ) + ( )
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 4.08 Force System Resultants
2. Cross Product
- Cartesian Vector Formulation
=
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 4.09 Force System Resultants
3. Moment of a Force-Vector Formulation
- The moment of a force about point
=
- Magnitude
= = =
- Direction
Determined by the right-hand rule
as it applies to the cross product
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 4.11 Force System Resultants
3. Moment of a Force-Vector Formulation
- Principle of Transmissibility
= 1 = 2 = 3
- Cartesian Vector Formulation
= =
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 4.11 Force System Resultants
3. Moment of a Force-Vector Formulation
- Resultant Moment of a System of Forces
= ( )
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 4.12 Force System Resultants
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3. Moment of a Force-Vector Formulation
- Example 4.3 Determine the moment produced by the force about point . Express the result as a Cartesian vector
= 12
= 4 + 12
=
= 2 (4 + 12 ) (12 )
42 + 122 + (12)2
= 0.4588 + 1.376 1.376
=
= 0 0 12
0.4588 1.376 1.376
= 16.5 + 5.5
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 4.13 Force System Resultants
Solution =
3. Moment of a Force-Vector Formulation
- Example 4.4 Two forces act on the rod, determine the
resultant moment they create about the flange at . Express the result as a Cartesian vector
Solution
= 5
= 4 + 5 2
=
= 1 + 2
= 0 5 0
60 40 20
+ 4 5 280 40 30
= 30 40 + 60
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 4.14 Force System Resultants
4. Principle of Moments (Varignons Theorem)
- The moment of a force about a point is equal to the sum of the moments of the components of the force about the point
= = 1 + 2 = 1 + 2
For two-dimensional problems
= =
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 4.15 Force System Resultants
4. Principle of Moments (Varignons Theorem)
- Example 4.6 Determine the moment of the force about point
Solution 1
= 3 750
= 2.898
=
= 5 2.89
= 14.5
Solution 2
+ =
= 5450 3300
5450 3300
= 14.5
= 14.5
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 4.16 Force System Resultants
4. Principle of Moments (Varignons Theorem)
Solution 3
+ =
= 5750
= 14.5
= 14.5
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 4.17 Force System Resultants
4. Principle of Moments (Varignons Theorem)
- Example 4.6 Force acts at the end of the angle bracket, determine the moment of the force about point
Solution 1
+ = 400300 0.2 400300 0.4
= 98.6
= 98.6
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 4.18 Force System Resultants
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4. Principle of Moments (Varignons Theorem)
- Example 4.6 Force acts at the end of the angle bracket, determine the moment of the force about point
Solution 2
= 0.4 0.2
= 400300 400300 = 200 346.4
= =
0.4 0.2 0200.0 346.4 0
= 98.6
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 4.19 Force System Resultants
Fundamental Problems
- F4.1 Determine the moment of the force about point
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 4.20 Force System Resultants
Fundamental Problems
- F4.2 Determine the moment of the force about point
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 4.21 Force System Resultants
Fundamental Problems
- F4.3 Determine the moment of the force about point
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 4.22 Force System Resultants
Fundamental Problems
- F4.4 Determine the moment of the force about point
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 4.23 Force System Resultants
Fundamental Problems
- F4.5 Determine the moment of the force about point . Neglect the thickness of the member
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 4.24 Force System Resultants
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Fundamental Problems
- F4.6 Determine the moment of the force about point
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 4.25 Force System Resultants
5. Moment of a Force about a Specified Axis
- Scalar Analysis
= =
- Vector Analysis
= 0 = ( )
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 4.26 Force System Resultants
5. Moment of a Force about a Specified Axis
- The moment of about a axis
= ( + + )
+ ( )
= ( ) =
,, : ,, components of the unit vector of axis
,, : , , components of the position vector
extended from any point on the axis to any
point on the line of action of the force
,, : ,, components of the force vector
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 4.27 Force System Resultants
= ( )
=
=
5. Moment of a Force about a Specified Axis
- Example 4.7 Determine the resultant moment of the three
forces 1 60 , 2 50 , 3 40 about the ,, axes
Solution
= 1 1 + 2
2 + 3
3
= 60 2 + 50 2 + 0 = 220
and
= 0 50 3 40 2 = 230
= 0 + 0 40 2 = 80
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 4.28 Force System Resultants
5. Moment of a Force about a Specified Axis
- Example 4.8 Determine the moment produced by the force
, which tends to rotate the rod about the axis
Solution
=
where
=
=0.4 + 0.2
0.42 + 0.22
= 0.8944 + 0.4472
is directed from any point on the axis to any point on the line of action of
, such as: , , , , choose
= = 0.6
= 300
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 4.29 Force System Resultants
5. Moment of a Force about a Specified Axis
= 0.8944 + 0.4472
= 0.6
= 300
=
=0.8944 0.4472 0
0.6 0 00 0 300
= 80.5
Expressing as a Cartesian vector
=
= 80.50 0.8944 + 0.4472
= 72.0 + 36.0 ()
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 4.30 Force System Resultants
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5. Moment of a Force about a Specified Axis
- Example 4.9 Determine the magnitude of the moment of force
about segment of the pipe assembly
Solution
=
where
=
=0.3 + 0.4
0.32 + 0.42= 0.6 + 0.8
= = 0.5 + 0.5
=
= 3000.4 0.4 + 0.2
0.42 + (0.4)2+0.22
= 200 200 + 100 HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 4.31 Force System Resultants
5. Moment of a Force about a Specified Axis
= 0.6 + 0.8
= 0.5 + 0.5
= 200 200 + 100
=
=0.6 0.8 00.5 0 0.5200 200 100
= 100
Expressing as a Cartesian vector
=
= 100 0.6 + 0.8
= 60.0 + 80.0 ()
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 4.32 Force System Resultants
Fundamental Problems
- F4.13 Determine the magnitude of the moment of the force
= 300 200 + 150 () about the axis. Express the result as a Cartesian vector
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 4.33 Force System Resultants
Fundamental Problems
- F4.14 Determine the magnitude of the moment of the force
= 300 200 + 150 () about the axis. Express the result as a Cartesian vector
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 4.34 Force System Resultants
Fundamental Problems
- F4.15 Determine the magnitude of the moment of the 200 force about the axis
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 4.35 Force System Resultants
Fundamental Problems
- F4.16 Determine the magnitude of the moment of the force
about the axis
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 4.36 Force System Resultants
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Fundamental Problems
- F4.17 Determine the magnitude of the moment of the force
= 50 40 + 20 () about the axis. Express the result as a Catersian vector
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 4.37 Force System Resultants
Fundamental Problems
- F4.18 Determine the magnitude of the moment of force about the , , axes. Use a scalar analysis
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 4.38 Force System Resultants
6. Moment of a Couple
- A couple: two parallel forces that have the same magnitude but
opposite in direction separated by a perpendicular distance
- A couple moment: the moment produced by a couple
= +
=
=
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 4.39 Force System Resultants
6. Moment of a Couple
- Scalar Formulation
=
- Vector Formulation
=
- Equivalent Couples
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 4.40 Force System Resultants
6. Moment of a Couple
- Resultant Couple Moment
=
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 4.41 Force System Resultants
6. Moment of a Couple
- Example 4.10 Determine the resultant couple moment of the
three couples acting on the plate
Solution
+ = : = 11 + 22 33
= 200 4 + 450 3 300 5
= 950
= 950
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 4.42 Force System Resultants
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6. Moment of a Couple
- Example 4.11 Determine the magnitude and direction of the
couple moment acting on the gear
Solution
+ = : = 600300 0.2 600300 0.2
= 43.9
+ = : = 600300 0.2 600300 0.2
= 43.9 HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 4.43 Force System Resultants
or
6. Moment of a Couple
Note
The same result can also be obtained using
=
: the perpendicular distance between the lines of action of the couple forces
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 4.44 Force System Resultants
6. Moment of a Couple
- Example 4.12 Determine the couple moment acting on the
pipe. Segment is directed 300 below the plane
Solution 1
= 25 + 25
= 8 25 + (6300 + 8 6300) 25
= 200 129.9 + 200
= 130 () HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 4.45 Force System Resultants
6. Moment of a Couple
It is easier to take moments of the couple forces about a point
lying on the line of action of one of the forces, e.g., point A
In this case the moment of the force at is zero, so that
= 25
= (6300 6300) 25
= 130 () HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 4.46 Force System Resultants
6. Moment of a Couple
Solution 2
Taking moments of the forces about either point or point
= = 25 6300 = 25 5.196 = 129.9
Applying the right-hand rule, acts in the direction, thus
= 130 () HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 4.47 Force System Resultants
6. Moment of a Couple
- Example 4.13 Replace the two couples acting on the pipe
column by a resultant couple moment
Solution
1 = = 150 0.4 = 60 1 = 60
2 = = 0.3 1254
5 125
3
5 = 22.5 +30
= 1 + 2 = 60 + 22.5 + 30 ()
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 4.48 Force System Resultants
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Fundamental Problems
- F4.19 Determine the resultant couple moment acting on the
beam
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 4.49 Force System Resultants
Fundamental Problems
- F4.20 Determine the resultant couple moment acting on the
triangular plate
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 4.50 Force System Resultants
Fundamental Problems
- F4.21 Determine the magnitude of so that the resultant couple moment acting on the beam is 1.5 clockwise
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 4.51 Force System Resultants
Fundamental Problems
- F4.22 Determine the couple moment acting on the beam
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 4.52 Force System Resultants
Fundamental Problems
- F4.23 Determine the resultant couple moment acting on the
pipe assembly
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 4.53 Force System Resultants
Fundamental Problems
- F4.24 Determine the couple moment acting on the pipe
assembly and express the result as a Cartesian vector
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 4.54 Force System Resultants
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7. Simplification of a Force and Couple System
- System of Forces and Couple Moments
=
= +
If the force system lies in the plane and any couple moments are perpendicular to this plane, then the above
equations reduce to the following three scalar equations
= ,
= , = +
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 4.55 Force System Resultants
7. Simplification of a Force and Couple System
- Example 4.14 Replace the force and couple system by an
equivalent resultant force and couple
moment acting at point
Solution
Force summation
+ = 3300 +
3
55 = 5.598
+
= 3300 4
55 4
= 6.50 = 6.50
Moment summation
+ = 3300 0.23300 0.1
+3
550.1
4
550.540.2
= 2.46 = 2.46
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 4.56 Force System Resultants
7. Simplification of a Force and Couple System
Summation
+ = 3300 +
3
55 = 5.598
+
= 6.50 = 6.50
+ = 2.46
Using the Pythagorean theorem
= 2 +
2
= 5.5982 + 6.502
= 8.58
Its direction
= 1
=
16.50
5.598= 49.30
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 4.57 Force System Resultants
7. Simplification of a Force and Couple System
- Example 4.15 Replace the force and couple system acting on
the member by an equivalent resultant
force and couple moment acting at point
Solution
Force summation
+ =
3
5500 = 300 = 300
+
=4
5500 750 = 350 = 350
Moment summation
+ =4
5500 2.5
3
5500 1
750 1.25 + 200
= 37.5 = 37.5
and = 3002 + 3502 = 461, = 1 350
300= 49.40
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 4.58 Force System Resultants
7. Simplification of a Force and Couple System
- Example 4.16 The structural member is subjected to a couple
moment and forces 1 and 2. Replace this system by an equivalent resultant force and couple moment acting at its
base, point
Solution
1 = 800
2 = 300
= 300
=0.15 + 0.1
(0.15)2+0.12
= 249.6 + 166.4
= 4
5500 +
3
5500 = 400 + 300
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 4.59 Force System Resultants
7. Simplification of a Force and Couple System
Force summation
= = 1 + 2 = 800 249.6 + 166
= 250 + 166 800 ()
Moment summation
= +
= + 1 + 2
= 400 + 300 + 800
+
0.15 0.1 1249.6 166.4 0
= 166 650 + 300 ()
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 4.60 Force System Resultants
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Fundamental Problems
- F4.25 Replace the loading system by an equivalent resultant
force and couple moment acting at point
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 4.61 Force System Resultants
Fundamental Problems
- F4.26 Replace the loading system by an equivalent resultant
force and couple moment acting at point
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 4.62 Force System Resultants
Fundamental Problems
- F4.27 Replace the loading system by an equivalent resultant
force and couple moment acting at point
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 4.63 Force System Resultants
Fundamental Problems
- F4.28 Replace the loading system by an equivalent resultant
force and couple moment acting at point
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 4.64 Force System Resultants
Fundamental Problems
- F4.29 Replace the loading system by an equivalent resultant
force and couple moment acting at point
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 4.65 Force System Resultants
Fundamental Problems
- F4.30 Replace the loading system by an equivalent resultant
force and couple moment acting at point
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 4.66 Force System Resultants
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8. Further Simplification of a Force and Couple System
- Concurrent force system: the system in which the lines of
action of all the forces intersect at a common point
=
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 4.67 Force System Resultants
8. Further Simplification of a Force and Couple System
- Coplanar force system
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 4.68 Force System Resultants
=
=
=
=
= /
8. Further Simplification of a Force and Couple System
- Parallel force system
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 4.68 Force System Resultants
=
=
= = /
8. Further Simplification of a Force and Couple System
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 4.69 Force System Resultants
8. Further Simplification of a Force and Couple System
- Reduction to a wrench
= +
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 4.71 Force System Resultants
, = /
8. Further Simplification of a Force and Couple System
- Example 4.17 Replace the force and couple moment system
acting on the beam by an equivalent resultant force, and find where
its line of action intersects the beam, measured from point
Solution
Force summation
+ =
3
58 = 4.8 = 4.8
+
= 4 +4
58 = 2.4 = 2.4
and = 4.82 + 2.42 = 5.37, = 1
2.4
4.8= 26.60
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 4.72 Force System Resultants
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8. Further Simplification of a Force and Couple System
= 4.8
= 2.4
Moment summation
+ =
= 4 1.5 15 3
58 0. 5
4
58 4.5
= 2.25
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 4.73 Force System Resultants
8. Further Simplification of a Force and Couple System
- Example 4.19 The jib crane is
subjected to three coplanar forces.
Replace this loading by an equivalent
resultant force and specify where the
resultants line of action intersects the column and boom
Solution
Force summation
+ =
3
5250 175 = 325
+
= 4
5250 60 = 260
and = 3252 + 2602 = 416
= 1260
325= 38.70
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 4.74 Force System Resultants
8. Further Simplification of a Force and Couple System
Moment summation
+ = 325 260 0
= 175 5 60 3
+3
5250 11
4
5250 8
= 2.29
By the principle of transmissibility,
can be placed at a distance where it intersects
+ = 325 11 260
= 175 5 60 3
+3
5250 11
4
5250 8
= 10.9
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 4.75 Force System Resultants
8. Further Simplification of a Force and Couple System
- Example 4.18 The slab is subjected to four parallel forces.
Determine the magnitude and direction of a resultant force
equivalent to the given force system and locate its point of
application on the slab
Force summation
+ = 600+100400500
= 1400 = 1400
Moment summation
+ = 1400
= 100 5 400 10
= 2.50
+
= 1400
= 600 8 100 6
= 3 HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 4.76 Force System Resultants
Solution
Fundamental Problems
- F4.31 Replace the loading system by an equivalent resultant
force and specify where the resultants line of action intersects the beam measured from
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 4.78 Force System Resultants
Fundamental Problems
- F4.32 Replace the loading system by an equivalent resultant
force and specify where the resultants line of action intersects the beam measured from
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 4.79 Force System Resultants
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Fundamental Problems
- F4.33 Replace the loading system by an equivalent resultant
force and specify where the resultants line of action intersects the beam measured from
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 4.80 Force System Resultants
Fundamental Problems
- F4.34 Replace the loading system by an equivalent resultant
force and specify where the resultants line of action intersects the member measured from
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 4.81 Force System Resultants
Fundamental Problems
- F4.35 Replace the loading system by an equivalent single
resultant force and specify the and coordinates of its line of action
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 4.82 Force System Resultants
Fundamental Problems
- F4.36 Replace the loading system by an equivalent single
resultant force and specify the and coordinates of its line of action
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 4.83 Force System Resultants
9. Reduction of a Simple Distribute Loading
- Surface forces are caused by direct contact
- Concentrated forces are applied to a point
- Linear Distributed forces () (force/length) are idealizations
The resultant force is equivalent to the area under distributed loading curve and acts through centroid or geometric center of
this area
- Body force is developed when one body exerts a force on
another body without direct physical contact HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 4.83 Force System Resultants
9. Reduction of a Simple Distribute Loading
- Uniform Loading Along a Single Axis
= /2
= = ()
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 4.84 Force System Resultants
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9. Reduction of a Simple Distribute Loading
- Magnitude of Resultant Force
+ =
=
=
- Location of Resultant Force
= =
=
=
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 4.85 Force System Resultants
9. Reduction of a Simple Distribute Loading
- Example 4.21 Determine the magnitude and location of the
equivalent resultant force acting on the shaft
Solution
+ =
= 6022
0
= 603
3 0
2
= 160
The location of measured from
=
= 602
2
0
6022
0
=
604
4 0
2
160
= 1.5
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 4.86 Force System Resultants
9. Reduction of a Simple Distribute Loading
- Example 4.22 A distributed loading of Pa acts over the top
surface of the beam. Determine the magnitude and location of
the equivalent resultant force
Solution
= 800 0.2 = 160/
=1
29 1440 = 6480 = 6.48
= 9 1
39 = 6
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 4.87 Force System Resultants
9. Reduction of a Simple Distribute Loading
- Example 4.23 The granular material exerts the distributed
loading on the beam. Determine the magnitude and location of
the equivalent resultant of this load
Solution
Divide the trapezoidal loading into a rectangular and triangular
loading
a trapezoidal loading = a rectangular + a triangular loading
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 4.88 Force System Resultants
+ =
+ =
9. Reduction of a Simple Distribute Loading
The magnitude of the force represented by each of these
loadings is equal to its associated area
1 =1
29 50 = 225/, 1 =
1
39 = 3
2 = 9 50 = 450/, 2 =1
29 = 4.5
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 4.89 Force System Resultants
+ =
9. Reduction of a Simple Distribute Loading
The two parallel forces 1 and 2 can be reduced to a single
resultant
+ = = 225 + 450 = 675
+ = 675 = 3 225 + 4.5 450 = 4
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 4.90 Force System Resultants
-
2/13/2013
16
Fundamental Problems
- F4.37 Determine the resultant force and specify where it acts
on the beam measured from
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 4.91 Force System Resultants
Fundamental Problems
- F4.38 Determine the resultant force and specify where it acts
on the beam measured from
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 4.92 Force System Resultants
Fundamental Problems
- F4.39 Determine the resultant force and specify where it acts
on the beam measured from
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 4.93 Force System Resultants
Fundamental Problems
- F4.40 Determine the resultant force and specify where it acts
on the beam measured from
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 4.94 Force System Resultants
Fundamental Problems
- F4.41 Determine the resultant force and specify where it acts
on the beam measured from
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 4.95 Force System Resultants
Fundamental Problems
- F4.42 Determine the resultant force and specify where it acts
on the beam measured from
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 4.96 Force System Resultants
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