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Structural Analysis I

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Title: Test Slide Author: john hull Last modified by: John R. Hull Created Date: 12/24/2000 12:31:30 AM Document presentation format: On-screen Show – PowerPoint PPT presentation

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Title: Structural Analysis I


1
Structural Analysis I
  • Structural Analysis
  • Trigonometry Concepts
  • Vectors
  • Equilibrium
  • Reactions
  • Static Determinancy and Stability
  • Free Body Diagrams
  • Calculating Bridge Member Forces

2
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

3
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

4
How Much Weight Can This Truss Bridge Support?
5
Pythagorean Theorem
  • In a right triangle, the length of the sides are
    related by the equation
  • a2 b2 c2

6
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
7
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
8
This Truss Bridge is Built from Right Triangles
9
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

10
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.

11
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

12
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.

13
Vector Component Example
  • The billiard ball hit of 5N/40o can be
    represented by two vector components, Fx and Fy

14
Fy Component Example
  • To calculate Fy, sin?
  • sin40o
  • 5N 0.64 Fy
  • 3.20N Fy

15
Fx Component Example
  • To calculate Fx, cos?
  • cos40o
  • 5N 0.77 Fx
  • 3.85N Fx

16
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
17
Vector Component Summary
Force Name 5N at 40
Free Body Diagram
x-component 5N cos 40
y-component 5N sin 40
18
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
19
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
20
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

21
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

22
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
23
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
24
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

25
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
26
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?

27
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

28
Determinancy and Stability
  • Statically determinate truss 2j m 3
  • Statically indeterminate truss 2j lt m 3
  • Unstable truss 2j gt m 3

29
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
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