Statics

x2 + y2 = r2 Total degrees in a triangle: 180 Three angles of the triangle below: Three sides of the triangle below: Pythagorean Theorem: x2 + y2 = r2 180 A, B, and C x, y, and r B r y HYPOTENUSE A C x

Trigonometric functions are ratios of the lengths of the segments that make up angles. sin Q = = opp. y hyp. r r cos Q = = adj. x hyp. r y Q x tan Q = = opp. y adj. x

For <A below, calculate Sine, Cosine, and Tangent: sin A = opposite hypotenuse B 1 2 1 2 opposite adjacent sin A = tan A = A C 1 √3 cos A = adjacent hypotenuse tan A = √3 2 cos A =

Law of Cosines: c2 = a2 + b2 – 2ab cos C Law of Sines: sin A sin B sin C a b c B c a A C b = =

Statics Using 2 index cards and a piece of tape: Create the tallest structure you can. Scoring: 1 pt for each cm higher than 5 1.5 pts for each 5 cm2 of material (cards and tape) saved

Structures must not break, twist, deform, collapse Static structure Static structure with some moving parts Movable structure Moving structure

What is a structure? What’s its purpose? A body that can resist applied forces without changing shape or size (apart from elastic deformations) What’s its purpose? Transmit forces from one place to another Provide shelter Art

Types of Structures Mass Framed Shells

Types of Load Concentrated Distributed

What is Mechanics? “Branch of science concerned with the behavior of physical bodies when subjected to forces or displacements, and the subsequent effects of the bodies on their environment.”

Kentucky & Indiana Bridge Statics What is Statics? Branch of Mechanics that deals with objects/materials that are stationary or in uniform motion. Forces are balanced. Examples: 1. A book lying on a table (statics) 2. Water being held behind a dam (hydrostatics) Kentucky & Indiana Bridge Chicago http://www.brianmicklethwait.com/images/uploads/ChicagoBridges.jpg http://images.google.com/imgres?imgurl=http://www.sunnysideoflouisville.org/landmarks/images/k%26i.jpg&imgrefurl=http://sunnysideoflouisville.org/landmarks/landmarks.htm&usg=__Ap_TH9tJ7I97o1uM_sVOL8RzKSs=&h=303&w=438&sz=106&hl=en&start=53&um=1&tbnid=yktXJOXmXMIlSM:&tbnh=88&tbnw=127&prev=/images%3Fq%3Dtruss%2Bbridges%26ndsp%3D21%26hl%3Den%26client%3Dfirefox-a%26rls%3Dorg.mozilla:en-US:official%26sa%3DN%26start%3D42%26um%3D1

Dynamics Dynamics is the branch of Mechanics that deals with objects/materials that are accelerating due to an imbalance of forces. Examples: 1. A rollercoaster executing a loop (dynamics) 2. Flow of water from a hose (hydrodynamics) http://www.cc.gatech.edu/cpl/projects/graphcuttextures/data/interaction/LittleRiver.jpg

Height, pressure, speed, density, etc. Scalar – a variable whose value is expressed only as a magnitude or quantity Height, pressure, speed, density, etc. Vector – a variable whose value is expressed both as a magnitude and direction Displacement, force, velocity, momentum, etc. 3. Tensor – a variable whose values are collections of vectors, such as stress on a material, the curvature of space-time (General Theory of Relativity), gyroscopic motion, etc. http://www.treehugger.com/fusion-pool-dining-table-pool.jpg

Length implies magnitude of vector Direction Properties of Vectors Magnitude Length implies magnitude of vector Direction Arrow implies direction of vector Act along the line of their direction No fixed origin Can be located anywhere in space

Vectors - Description F = 40 lbs 45o F = 40 lbs @ 45o Bold type and an underline F also identify vectors Vectors - Description Hat signifies vector quantity Magnitude, Direction F = 40 lbs 45o F = 40 lbs @ 45o http://www.christmastraditions.com/Merchand/Bethany/Hallown/2008/EB7758.jpg magnitude direction 40 lbs 45o

Free-body diagrams #1 on Statics problem sheet

Vectors – Scalar Multiplication We can multiply any vector by a whole number. Original direction is maintained, new magnitude. 2 ½

Vectors – Addition We can add two or more vectors together. 2 methods: Graphical Addition/subtraction – redraw vectors head-to-tail, then draw the resultant vector. (head-to-tail order does not matter)

Vectors – Rectangular Components It is often useful to break a vector into horizontal and vertical components (rectangular components). Consider the Force vector below. Plot this vector on x-y axis. Project the vector onto x and y axes. y F Fy Fx x

Vectors – Rectangular Components This means: vector F = vector Fx + vector Fy Remember the addition of vectors: y F Fy Fx x

F = Fx i + Fy j Vectors – Rectangular Components y F Fy x Fx Unit vector Vectors – Rectangular Components Vector Fx = Magnitude Fx times vector i F = Fx i + Fy j Fx = Fx i i denotes vector in x direction y Vector Fy = Magnitude Fy times vector j F Fy = Fy j Fy j denotes vector in y direction Fx x

Vectors – Rectangular Components Each grid space represents 1 lb force. What is Fx? Fx = (4 lbs)i What is Fy? Fy = (3 lbs)j What is F? F = (4 lbs)i + (3 lbs)j y F Fy Fx x

Vectors – Rectangular Components If vector V = a i + b j + c k then the magnitude of vector V |V| =

Vectors – Rectangular Components What is the relationship between Q, sin Q, and cos Q? cos Q = Fx / F Fx = F cos Qi sin Q = Fy / F Fy = F sin Qj F Fy Q Fx

Vectors – Rectangular Components When are Fx and Fy Positive/Negative? Fy + Fy + y F Fx + Fx - F x F F Fx - Fx + Fy - Fy -

Vectors – Rectangular Components II I III IV

Vectors Vectors can be completely represented in two ways: Graphically Sum of vectors in any three independent directions Vectors can also be added/subtracted in either of those ways: F1 = ai + bj + ck; F2 = si + tj + uk F1 + F2 = (a + s)i + (b + t)j + (c + u)k

Use the law of sines or the law of cosines to find R. Vectors Use the law of sines or the law of cosines to find R. R 45o 30o F1 105o F2

Vectors Brief note about subtraction If F = ai + bj + ck, then – F = – ai – bj – ck Also, if F = Then, – F =

R = SFxi + SFyj + SFzk Resultant Forces Resultant forces are the overall combination of all forces acting on a body. 1) find sum of forces in x-direction 2) find sum of forces in y-direction 3) find sum of forces in z-direction 3) Write as single vector in rectangular components R = SFxi + SFyj + SFzk

Resultant Forces Only! Resultant forces #2 and #3 Statics Problem Sheet Resultant Forces Only!

Statics Newton’s 3 Laws of Motion: Now on to the point… Newton’s 3 Laws of Motion: A body at rest will stay at rest, a body in motion will stay in motion, unless acted upon by an external force This is the condition for static equilibrium In other words…the net force acting upon a body is Zero

Newton’s 3 Laws of Motion: Force is proportional to mass times acceleration: F = ma If in static equilibrium, the net force acting upon a body is Zero What does this tell us about the acceleration of the body? It is Zero

Newton’s 3 Laws of Motion: Action/Reaction

Statics Two conditions for static equilibrium: Since Force is a vector, this implies Individually.

Two conditions for static equilibrium: 1.

Resultant and Reaction forces #2 and #3 Statics Problem Sheet Reaction Forces Only!

Two conditions for static equilibrium: Why isn’t sufficient?

Two conditions for static equilibrium: About any point on an object, Moment M (or torque t) is a scalar quantity that describes the amount of “twist” at a point. M = (magnitude of force perpendicular to moment arm) * (length of moment arm) = (magnitude of force) * (perpendicular distance from point to force)

MP = F * x MP = Fy * x Two conditions for static equilibrium: M = (magnitude of force perpendicular to moment arm) * (length of moment arm) = (magnitude of force) * (perpendicular distance from point to force) F F P P x x

Tension test apparatus – added load of lever? Moment Examples: An “L” lever is pinned at the center P and holds load F at the end of its shorter leg. What force is required at Q to hold the load? What is the force on the pin at P holding the lever? Tension test apparatus – added load of lever?

Let’s Return to Saving Pablo…

Trusses Trusses: A practical and economic solution to many structural engineering challenges Simple truss – consists of tension and compression members held together by hinge or pin joints Rigid truss – will not collapse http://www.aetn.org/__data/assets/image/0016/20824/highbridge.jpg

Trusses Joints: Pin or Hinge (fixed) http://hsc.csu.edu.au/engineering_studies/civil_structures/1-1/truss.jpg

Trusses Rax Ray Supports: Pin or Hinge (fixed) – 2 unknowns Reaction in x-direction Reaction in y-direction Rax http://hsc.csu.edu.au/engineering_studies/civil_structures/1-1/truss.jpg Ray

Trusses Ray Supports: Roller - 1 unknown Reaction in y-direction only http://hsc.csu.edu.au/engineering_studies/civil_structures/1-1/truss.jpg Ray

2 conditions for static equilibrium: Assumptions to analyze simple truss: Joints are assumed to be frictionless, so forces can only be transmitted in the direction of the members. Members are assumed to be massless. Loads can be applied only at joints (or nodes). Members are assumed to be perfectly rigid. 2 conditions for static equilibrium: Sum of forces at each joint (or node) = 0 Moment about any joint (or node) = 0 Start with Entire Truss Equilibrium Equations http://www.aetn.org/__data/assets/image/0016/20824/highbridge.jpg

Method of Joints Problem #4 from Statics Problem Sheet

Method of Joints

Method of Joints

Method of Joints

Method of Joints

Truss Analysis Example Problems: Using the method of joints, determine the force in each member of the truss shown and identify whether each is in compression or tension.

Static determinacy and stability: Statically Determinant: All unknown reactions and forces in members can be determined by the methods of statics – all equilibrium equations can be satisfied. Static Stability: The truss is rigid – it will not collapse.

Conditions of static determinacy and stability of trusses:

Homework: Method of Joints