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PPT – Structural Analysis I PowerPoint presentation | free to download - id: 7f5341-OTJjN

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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

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

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

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

This Truss Bridge is Built from Right Triangles

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 ?
- 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

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

Fy Component Example

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

Fx Component Example

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

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

Vector Component Summary

Force Name 5N at 40

Free Body Diagram

x-component 5N cos 40

y-component 5N sin 40

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

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

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

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

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

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

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

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 lt m 3
- Unstable truss 2j gt 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