Vector Analysis The Basics

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Vector AnalysisThe Basics

• Vectors are used to analyze combinations of
  forces.
• The Parallelogram Rule is used to add two
  vectors.
• Vectors may be broken into components.

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Vector Basics

• Forces may be represented as vectors. A vector
  consists of a scalar value (magnitude) and a
  direction. The direction is usually measured in
  degrees, counterclockwise from the positive
  portion of the x-axis (East).

Magnitude
r
?
direction
Line of Action
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The Parallelogram Rule - 1

• Draw lines from the heads of the vectors that are
  parallel to the other vector.

Force Diagram
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The Parallelogram Rule - 1

• Draw lines from the heads of the vectors that are
  parallel to the other vector.

• Draw a vector from the intersection of the two
  vectors to the intersection of the two new lines.

Force Diagram

• This vector is known as the Resultant, and is
  equivalent to the action of the original two.

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The Parallelogram Rule - 2

• Apply the rule multiple times for more than two
  vectors

Force Diagram
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The Parallelogram Rule - 3
The rule may be used in reverse to find vector
components.
r1
rlt?
r2
r1 r sin(?) r2 r cos(?)
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The Parallelogram Rule - 3
A very useful special case of the Parallelogram
Rule.
r1y
rlt?
r2x
r² r1² r2² y² x² tan? (r1 r2) (y
x)
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The Parallelogram Rule - 4
Relation of components to the Parallelogram Rule.
Vector addition with the Parallelogram Rule.
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The Parallelogram Rule - 4
Relation of components to the Parallelogram Rule.
Components of first vector.
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The Parallelogram Rule - 4
Components of second vector.
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The Parallelogram Rule - 4
Components of both vectors.
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The Parallelogram Rule - 4
Both methods yield the same results.
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The Parallelogram Rule - 4
Geometries of both methods superimposed.
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Static Equilibrium
The Equilibrant
Resultant of addition of two vectors.
Vector that causes equilibrium is the same
magnitude but opposite direction -rlt?
rlt(? 180) (rlt? resultant)
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Graphic Vector AnalysisThe Polygon Rule

• The Polygon Rule is the simplest, most direct
  vector analysis method.
• Graphic vector analysis requires precise drawing
  and measurement for accurate results.
• When properly executed, vector magnitude,
  direction and sense may be determined for two
  unknown vector values

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Application of the Polygon Rule - 1
Reference tags (optional)

• Unknown force vector

2

• 3

• Unknown force vector

1

• These Unknown force vectors have unknown
  magnitudes but have known directions. The
  polygon method will work only if the number of
  unknown values is limited to two.

Force Diagram

• Known force vector

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Application of the Polygon Rule - 2
2
1. Copy vectors to form a polygon.

• 3

1
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Application of the Polygon Rule - 3
2. Draw vector 3 from head of vector 1 to
intersection
2

• 3

1
3
1
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Application of the Polygon Rule - 4

• Note when drawn correctly, each vector will be
  drawn from the head of one vector to the tail of
  the next. In this case, the vectors trace a
  clockwise path.

2

• 3

1
3
3. Draw vector 2 from head of vector 3 to Tail of
1.
1
2
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Application of the Polygon Rule - 5
4. Draw vector 2 3 on force diagram.
2

• 3

1
The analysis is complete.
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Alternate Polygon Construction - 1
Copy vectors in a different arrangement
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Alternate Polygon Construction - 2
Draw new vectors starting from the head of the
known vector.

• Note the vectors are drawn in a counterclockwise
  manner, but each is drawn from the head of one
  vector to the tail of the next one.

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Alternate Polygon Construction - 3
Copy vectors back to force diagram.

• The results are the same, no matter what the
  drawing order or clock sense of the path.

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Complex Force Polygons - 1
Complex vector systems may be analyzed in the
same way.

• As long as the number of unknowns is limited to
  two, the analysis may be completed.

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Complex Force Polygons - 2
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Complex Force Polygons - 3
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Complex Force Polygons - 4
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Complex Force Polygons - 5

• Any of the arrangements shown are valid and yield
  the same results

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Graphic Vector AnalysisTrusses

• Calculate reactions.
• Apply Polygon Rule to each joint in a truss.

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Trusses and Framework
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Trusses
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The Truss and Applied Loads

• Forces in the truss above may be solved by
  applying the Polygon Rule to each joint.
• The simplest first step is to solve for the
  reaction forces.

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Static Equilibrium of the Truss

• In this symmetrical arrangement, the two
  reactions will be equal but opposite in direction
  to the three applied forces.
• The Polygon Rule may be used to determine the
  static equilibrium of the truss as a whole.

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Reactions

• Since all the vectors are parallel, the two
  unknown vectors cannot be determined by
  intersection, but each will be equal to half the
  distance of the sum of the applied forces.

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Reactions

• Since all the vectors are parallel, the two
  unknown vectors cannot be determined by
  intersection, but each will be equal to half the
  distance of the sum of the applied forces.

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External Forces
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Solving for Each Joint

• Break down the problem into individual joints.

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Solving for Each Joint
3
2
4
1

• The order of solution is shown above.
• In each case a joint with only two unknowns will
  be found.

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Solving for Each Joint
3
2
4
1

• The order of solution is shown above.
• In each case a joint with only two unknowns will
  be found.

• The order of solution could also be 1,2,4,3.
• The symmetry of the truss will be considered
  later.

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Solving for Each Joint

• Find a joint with only two unknowns and apply the
  Polygon Rule.
• The results from this joint will allow you to
  solve others.

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Joint 1
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Joint 1
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Joint 1
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Joint 1

• Note when drawn correctly, each vector will be
  drawn from the head of one vector to the tail of
  the next. In this case, the vectors trace a
  clockwise path.

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Joint 1
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Compression and Tension
A vector pushing into a joint indicates
compression in that member
C
A vector pulling away from a joint indicates
tension in that member
T
The analysis of this joint is complete.
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Joint 2
C
T

• The value of the last vector may be used to
  determine one of the vectors in the next joint.
  It will be equal in magnitude, but opposite in
  direction.

• The second joint now has only two unknowns.

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Joint 2
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Joint 2
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Joint 2
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Joint 2
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Joint 3
C
C
C
T

• The results of the last joint may be used to
  calculate the next joint.

• Since the truss is symmetrical, the right hand
  element will be equal to the left hand element.

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Joint 3
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Joint 3
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Joint 3
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Joint 4
C
C
T
C
C
C
C
T
T

• The fourth joint does not need any analysis since
  all elements on the left hand side have been
  determined.

• Elements on the right hand side are symmetrical
  with the left.

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END