Transcript Statics Lecture

Statics
Using 2 index cards:
Create a structure or system of structures that
will elevate two textbooks at least 1.5cm off your
desk
Statics
Kentucky & Indiana Bridge
Chicago
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)
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)
1.
2.
3.
4.
Total degrees in a triangle: 180
Three angles of the triangle below: A, B, and
Three sides of the triangle below: x, y, and r
Pythagorean Theorem:
B
x2 + y2 = r2
r
A
y
x
C
C
Trigonometric functions are ratios of the lengths of the
segments that make up angles.
opp.
sin Q =
=
hyp.
r
Q
x
y
cos Q =
y
r
adj.
x
=
hyp.
r
opp.
tan Q =
=
adj.
y
x
For <A below, calculate Sine, Cosine, and Tangent:
B
2
A
1
3
C
opposite
sin A =
hypotenuse
1
sin A =
2
opposite
tan A =
adjacent
adjacent
cos A =
hypotenuse
tan A = 1
√3
cos A = √3
2
B
c
A
Law of Cosines:
c2 = a2 + b2 – 2ab cos C
a
b
C
Law of Sines:
sin A
sin B
=
a
b
=
sin C
c
1.
Scalar – a variable whose value is expressed only as a
magnitude or quantity
Height, pressure, speed, density, etc.
2.
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.
Properties of Vectors
1. Magnitude
Length implies magnitude of vector
2. Direction
Arrow implies direction of vector
3. Act along the line of their direction
4. No fixed origin
Can be located anywhere in space
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
magnitude
direction
45o
Vectors – Scalar Multiplication
1.
2.
We can multiply any vector by a whole number.
Original direction is maintained, new magnitude.
2
½
Vectors – Addition
1.
2.
We can add two or more vectors together.
2 methods:
1. Graphical Addition/subtraction – redraw vectors head-totail, then draw the resultant vector. (head-to-tail order
does not matter)
Vectors – Rectangular Components
1.
2.
3.
4.
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
Fy
Fx
x
Vectors – Rectangular Components
This means:
vector F
=
vector Fx
+
vector Fy
Remember the addition of vectors:
y
Fy
Fx
x
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
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?
y
Fx = (4 lbs)i
What is Fy?
Fy
Fy = (3 lbs)j
Fx
x
What is F?
F = (4 lbs)i + (3 lbs)j
Vectors – Rectangular Components
If vector
V=ai+bj+ck
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
Fy
sin Q = Fy / F
Fy = F sin Qj
Q
Fx
Vectors – Rectangular Components
When are Fx and Fy Positive/Negative?
Fy +
y
Fy +
Fx +
Fx x
Fx Fy -
Fy -
Fx +
Vectors – Rectangular Components
Complete the following chart in your notebook:
II I
III IV
Vectors
1.
2.
Vectors can be completely represented in two ways:
1. Graphically
2. Sum of vectors in any three independent directions
Vectors can also be added/subtracted in either of those ways:
1.
2. F1 = ai + bj + ck;
F2 = si + tj + uk
F1 + F2 = (a + s)i + (b + t)j + (c + u)k
Vectors
A third way to add, subtract, and otherwise decompose vectors:
R
45o
F1
30o
105o
F2
Use the law of sines or the law of cosines to find R.
Vectors
Brief note about subtraction
1. If F = ai + bj + ck,
then
2. Also, if
F=
Then,
–F=
– F = – ai – bj – ck
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 - Example
A satellite flies without friction in space. Earth’s gravity pulls
downward on the satellite with a force of 200 N. Stray space
junk hits the satellite with a force of 1000 N at 60o to the
horizontal. What is the resultant force acting on the satellite?
1. Sketch and label free-body diagram (all external and reactive
forces acting on the body)
2. Decompose all vectors into rectangular components (x, y, z)
3. Add vectors
Now on to the point…
Statics
Newton’s 3 Laws of Motion:
1. 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:
2. 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:
3. Action/Reaction
Statics
Two conditions for static equilibrium:
1.
Since Force is a vector, this implies
Individually.
Two conditions for static equilibrium:
1.
Two conditions for static equilibrium:
Why isn’t
sufficient?
Two conditions for static equilibrium:
2. 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)
Two conditions for static equilibrium:
MP = F * x
MP = Fy * x
F
F
P
P
x
x
M = (magnitude of force perpendicular to moment arm) * (length
of moment arm) = (magnitude of force) * (perpendicular
distance from point to force)
Moment Examples:
1. Tension test apparatus – unknown and reaction forces?
2.
If a beam supported at its endpoints is given a load F at its
midpoint, what are the supporting forces at the endpoints?
Ra
Rb
Find sum of moments about a or b.
Watch your signs – identify positive
Moment Examples:
3.
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?
What is your method for solving this problem? Remember,
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
Trusses
Joints:
Pin or Hinge (fixed)
Trusses
Supports:
Pin or Hinge (fixed) – 2 unknowns
Reaction in x-direction
Reaction in y-direction
Rax
Ray
Trusses
Supports:
Roller - 1 unknown
Reaction in y-direction only
Ray
Assumptions to analyze simple truss:
1.
2.
3.
4.
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.
1.
2.
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
Truss Analysis Example Problems:
1. A force F is applied to the following equilateral truss.
Determine the force in each member of the truss shown
and state which members are in compression and which
are in tension.
Truss Analysis Example Problems:
2. Using the method of joints, determine the force in each
member of the truss shown. Assume equilateral
triangles.
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.
m = 2j – r (Simple Truss)
Static Stability:
The truss is rigid – it will not collapse.
Conditions of static determinacy and stability of trusses:
Materials Lab Connections:
• Tensile Strength = Force / Area
• Compression is Proportional to 1 / R4
Problem Sheet solutions due Monday