Structural Analysis I • Structural Analysis • Trigonometry Concepts • Vectors • Equilibrium • Reactions • Static Determinancy and Stability • Free Body Diagrams • Calculating Bridge Member Forces
Jan 01, 2016
Structural Analysis I
• Structural Analysis• Trigonometry Concepts• Vectors• Equilibrium • Reactions • Static Determinancy and Stability • Free Body Diagrams• Calculating Bridge Member Forces
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
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
How Much Weight Can This Truss Bridge Support?
Pythagorean Theorem
• In a right triangle, the length of the sides are related by the equation:
a2 + b2 = c2
a
b
c
Sine (sin) of an Angle
• In a right triangle, the angles are related to the lengths of the sides by the equations:
sinθ1 = =Opposite a
Hypotenuse c
sinθ2 = =Opposite b
Hypotenuse c
a
b
c
θ1
θ2
Cosine (cos) of an Angle
• In a right triangle, the angles are related to the lengths of the sides by the equations:
cosθ1 = =Adjacent b
Hypotenuse c
cosθ2 = =Adjacent a
Hypotenuse c
a
b
c
θ1
θ2
This Truss Bridge is Built from Right Triangles
a
b
c
θ1
θ2
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
Vectors
• Mathematical quantity that has both magnitude and direction
• Represented by an arrow at an angle θ
• Establish Cartesian Coordinate axis system with horizontal x-axis and vertical y-axis.
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
Θ = 40o
F = 5N
y
x
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.
Vector Component Example
• The billiard ball hit of 5N/40o can be represented by two vector components, Fx and Fy F = 5N
Fx
Fy
θx
y
F = 5N
x
y
Fy Component Example
To calculate Fy, sinθ =
sin40o =
5N * 0.64 = Fy
3.20N = Fy
F = 5N
Fx
Fy
Θ=40o
Opposite Hypotenuse
Fy
5N
Fx Component Example
To calculate Fx, cosθ =
cos40o =
5N * 0.77 = Fx
3.85N = Fx
F = 5N
Fx
Fy
Θ=40o
Adjacent Hypotenuse
Fx
5N
What does this Mean?
Your 5N/40o hit is represented by this vector
F = 5N
Θ=40o
x
yFx = 3.85N
Fy=3.20N
x
y
The exact same force and direction could be achieved if two simultaneous forces are applied directly along the x and y axis
Vector Component Summary
F = 5N
Θ=40o
x
y
Force Name 5N at 40°
Free Body Diagram
x-component 5N * cos 40°
y-component 5N * sin 40°
How do I use these?
• 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?
She pulls with 100 pound force
He pulls with 150 pound force
Boat Pull Solution
• Represent the boat as a point at the (0,0) location
• Represent the pulling forces with vectors
Θm = 50oΘf = 70o
Fm = 150 lb
Ff = 100 lb
x
y
Boat Pull Solution (cont)
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
Separate force Ff into x and y components
Θf = 70o
Ff = 100 lb
-x x
y
Boat Pull Solution (cont)
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
Separate force Fm into x and y components
Θm = 50o
Fm = 150 lb
x
y
Boat Pull Solution (cont)
Force Name Ff FmResultant
(Sum)
Vector Diagram
(See next slide)
x- component-100lb*cos70
= -34.2 lb150lb*cos50
= 96.4 lb62.2 lb
y-component100lb*sin70
= 93.9 lb150lb*sin50= 114.9 lb
208.8 lb
70o
100 lb
x
y
50o
150 lb
x
y
Boat Pull Solution (end)
• 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
Fm
Ff
-x
y
xFTotalX
FTotalY
Equilibrium
• Total forces acting on an object is ‘0’
• Important concept for bridges – they shouldn’t move!
• Σ Fx = 0 means ‘The sum of the forces in the x direction is 0’
• Σ Fy = 0 means ‘The sum of the forces in the y direction is 0’ :
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
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?
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
Determinancy and Stability
• Statically determinate truss: 2j = m + 3
• Statically indeterminate truss: 2j < m + 3
• Unstable truss: 2j > m + 3
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