Structural Analysis I

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Structural Analysis I

• Structural Analysis
• Trigonometry Concepts
• Vectors
• Equilibrium
• Reactions
• Static Determinancy and Stability
• Free Body Diagrams
• Calculating Bridge Member Forces

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Learning Objectives

• Define structural analysis
• Calculate using the Pythagoreon Theorem, sin, and
  cos
• Calculate the components of a force vector
• Add two force vectors together
• Understand the concept of equilibrium
• Calculate reactions
• Determine if a truss is stable

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Structural Analysis

• Structural analysis is a mathematical examination
  of a complex structure
• Analysis breaks a complex system down to
  individual component parts
• Uses geometry, trigonometry, algebra, and basic
  physics

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How Much Weight Can This Truss Bridge Support?
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Pythagorean Theorem

• In a right triangle, the length of the sides are
  related by the equation
• a2 b2 c2

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Sine (sin) of an Angle

• In a right triangle, the angles are related to
  the lengths of the sides by the equations
• sin?1

Opposite b Hypotenuse c
sin?2
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Cosine (cos) of an Angle

• In a right triangle, the angles are related to
  the lengths of the sides by the equations
• cos?1

Adjacent a Hypotenuse c
cos?2
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This Truss Bridge is Built from Right Triangles
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Trigonometry Tips for Structural Analysis

• A truss bridge is constructed from members
  arranged in right triangles
• Sin and cos relate both lengths AND magnitude of
  internal forces
• Sin and cos are ratios

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Vectors

• Mathematical quantity that has both magnitude and
  direction
• Represented by an arrow at an angle ?
• Establish Cartesian Coordinate axis system with
  horizontal x-axis and vertical y-axis.

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Vector Example

• Suppose you hit a billiard ball with a force of 5
  newtons at a 40o angle
• This is represented by a force vector

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Vector Components

• Every vector can be broken into two parts, one
  vector with magnitude in the x-direction and one
  with magnitude in the y-direction.
• Determine these two components for structural
  analysis.

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Vector Component Example

• The billiard ball hit of 5N/40o can be
  represented by two vector components, Fx and Fy

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Fy Component Example

• To calculate Fy, sin?
• sin40o
• 5N 0.64 Fy
• 3.20N Fy

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Fx Component Example

• To calculate Fx, cos?
• cos40o
• 5N 0.77 Fx
• 3.85N Fx

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What does this Mean?

Your 5N/40o hit is represented by this vector
The exact same force and direction could be
achieved if two simultaneous forces are applied
directly along the x and y axis
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Vector Component Summary
Force Name 5N at 40
Free Body Diagram
x-component 5N cos 40
y-component 5N sin 40
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How do I use these?
She pulls with 100 pound force

• Calculate net forces on an object
• Example Two people each pull a rope connected
  to a boat. What is the net force on the boat?

He pulls with 150 pound force
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Boat Pull Solution
y

• Represent the boat as a point at the (0,0)
  location
• Represent the pulling forces with vectors

Fm 150 lb
Ff 100 lb
Tm 50o
Tf 70o
x
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Boat Pull Solution (cont)
Separate force Ff into x and y components

• First analyse the force Ff
• x-component -100 lb cos70
• x-component -34.2 lb
• y-component 100 lb sin70
• y-component 93.9 lb

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Boat Pull Solution (cont)
Separate force Fm into x and y components

• Next analyse the force Fm
• x-component 150 lb cos50
• x-component 96.4 lb
• y-component 150 lb sin50
• y-component 114.9 lb

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Boat Pull Solution (cont)
Force Name Ff Fm Resultant (Sum)
Vector Diagram (See next slide)
x- component -100lbcos70 -34.2 lb 150lbcos50 96.4 lb 62.2 lb
y-component 100lbsin70 93.9 lb 150lbsin50 114.9 lb 208.8 lb
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Boat Pull Solution (end)
y

• White represents forces applied directly to the
  boat
• Gray represents the sum of the x and y components
  of Ff and Fm
• Yellow represents the resultant vector

FTotalY
Fm
Ff
-x
x
FTotalX
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Equilibrium

• Total forces acting on an object is 0
• Important concept for bridges they shouldnt
  move!
• S Fx 0 means The sum of the forces in the x
  direction is 0
• S Fy 0 means The sum of the forces in the y
  direction is 0

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Reactions

• Forces developed at structure supports to
  maintain equilibrium.
• Ex If a 3kg jug of water rests on the ground,
  there is a 3kg reaction (Ra) keeping the bottle
  from going to the center of the earth.

3kg
Ra 3kg
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Reactions

• A bridge across a river has a 200 lb man in the
  center. What are the reactions at each end,
  assuming the bridge has no weight?

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Determinancy and Stability

• Statically determinant trusses can be analyzed by
  the Method of Joints
• Statically indeterminant bridges require more
  complex analysis techniques
• Unstable truss does not have enough members to
  form a rigid structure

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Determinancy and Stability

• Statically determinate truss 2j m 3
• Statically indeterminate truss 2j lt m 3
• Unstable truss 2j gt m 3

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Acknowledgements

• This presentation is based on Learning Activity
  3, Analyze and Evaluate a Truss from the book by
  Colonel Stephen J. Ressler, P.E., Ph.D.,
  Designing and Building File-Folder Bridges