Structural Analysis I Structural Analysis Trigonometry Concepts Vectors

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: a 2 + b 2 = c 2 c a b

Sine (sin) of an Angle • In a right triangle, the angles are related to the lengths of the sides by the equations: c Opposite sinθ 1 = = Hypotenuse a c θ 1 Opposite sinθ 2 = = Hypotenuse b c θ 2 a b

Cosine (cos) of an Angle • In a right triangle, the angles are related to the lengths of the sides by the equations: c Adjacent b cosθ 1 = = Hypotenuse c θ 1 Adjacent a cosθ 2 = = Hypotenuse c θ 2 a b

This Truss Bridge is Built from Right Triangles θ 2 c a θ 1 b

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 y • Suppose you hit a billiard ball with a force of 5 newtons at a 40 o angle • This is represented by a force vector F = 5 N Θ = 40 o 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 y F = 5 N • The billiard ball hit of 5 N/40 o can be represented by two vector components, Fx and Fy x y F = 5 N Fy θ Fx x

Fy Component Example To calculate Fy, sinθ = sin 40 o F = 5 N Fy Θ=40 o Fx = Opposite Hypotenuse Fy 5 N 5 N * 0. 64 = Fy 3. 20 N = Fy

Fx Component Example To calculate Fx, cosθ = Adjacent Hypotenuse cos 40 o = F = 5 N Fy Θ=40 o Fx Fx 5 N 5 N * 0. 77 = Fx 3. 85 N = Fx

What does this Mean? y y Fx = 3. 85 N F = 5 N Θ=40 o Your 5 N/40 o hit is represented by this vector x Fy=3. 20 N x 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 5 N at 40° y Free Body Diagram F = 5 N Θ=40 o x-component 5 N * cos 40° y-component 5 N * sin 40° x

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 y • Represent the boat as a point at the (0, 0) location • Represent the pulling forces with vectors Fm = 150 lb Ff = 100 lb Θf = 70 o Θm = 50 o x

Boat Pull Solution (cont) Separate force Ff into x and y components y First analyse the force Ff • • x-component = -100 lb * cos 70° x-component = -34. 2 lb y-component = 100 lb * sin 70° y-component = 93. 9 lb Ff = 100 lb Θf = 70 o -x x

Boat Pull Solution (cont) Separate force Fm into x and y components Next analyse the force Fm • • x-component = 150 lb * cos 50° x-component = 96. 4 lb • • y-component = 150 lb * sin 50° y-component = 114. 9 lb y Fm = 150 lb Θm = 50 o x

Boat Pull Solution (cont) Force Name Vector Diagram x- component y-component Ff y Fm y 150 lb (See next slide) 100 lb 70 o x -100 lb*cos 70 = -34. 2 lb 100 lb*sin 70 = 93. 9 lb Resultant (Sum) 50 o x 150 lb*cos 50 = 96. 4 lb 150 lb*sin 50 = 114. 9 lb 62. 2 lb 208. 8 lb

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 y FTotal. Y Fm Ff -x x FTotal. X

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 3 kg jug of water rests on the ground, there is a 3 kg reaction (Ra) keeping the bottle from going to the center of the earth. 3 kg Ra = 3 kg

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: 2 j = m + 3 • Statically indeterminate truss: 2 j < m + 3 • Unstable truss: 2 j > 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