1. Statics

2. 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

3. 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

4. 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 =

5. 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
= =

6. 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

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

8. 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

9. Types of Structures
Mass
Framed
Shells

10. Types of Load
Concentrated
Distributed

11. 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.”

12. 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

13. 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)

14. 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.

15. 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

16. 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 o
F = 40 45o
magnitude
direction
40 lbs
45o

17. Free-body diagrams
#1 on Statics problem sheet

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

19. 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)

20. 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

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

22. 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

23. 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

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

25. 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

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

27. Vectors – Rectangular ComponentsII I
III IV

28. 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

29. 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

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

31. 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

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

33. 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

34. 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

35. Newton’s 3 Laws of Motion:
Action/Reaction

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

37. Two conditions for static equilibrium:1.

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

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

40. 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)

41. 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

42. 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?

43. Let’s Return to Saving Pablo…

44. 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

45. Trusses Joints: Pin or Hinge (fixed)

46. Trusses Rax Ray Supports: Pin or Hinge (fixed) – 2 unknowns
Reaction in x-direction
Reaction in y-direction
Rax
Ray

47. Trusses Ray Supports: Roller - 1 unknown Reaction in y-direction only
Ray

48. 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

49. Method of Joints
Problem #4 from Statics Problem Sheet

50. Method of Joints

51. Method of Joints

52. Method of Joints

53. Method of Joints

54. 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.

55. 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.

56. Conditions of static determinacy and stability of trusses:

57. Homework: Method of Joints