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Statics
• 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
“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.”
What is Mechanics?
Wikipedia.org/wiki/Isaac_Newton
Chicago
Kentucky & Indiana Bridge
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)
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
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)
Construction is about static
equilibrium (statics)
Statics = no motion (almost).
All forces equal zero.
All torques equal zero.
• A force is a push or pull on an object.
What is a force?
• Gravity is pulling you down.
• The chair is pushing you up.
• Total forces are zero.
kg
N
lb
kglbF 8.9
2.2150
NF 668
How much force does the chair exert?
Are there forces
on you now?
Types of Load
• Concentrated
• Distributed
Let’s pull on a rope.
Sometimes the forces are
not just up and down.
How much tension is in each rope?NNF 3342668
What forces are
on each rope?
Forces are often
at an angle.
In equilibrium, net forces must be zero both
• Right & left
• Up & down
Vectors have magnitude and direction.
Trigonometry Review1. Total degrees in a triangle:
2. Three angles of the triangle below:
3. Three sides of the triangle below:
4. Pythagorean Theorem:
x2 + y2 = r2
A
B
C
y
x
r
Trigonometric functions are ratios of the lengths of the
segments that make up angles.
Q
y
x
r
sin Q = =opp. y hyp. r
cos Q = =adj. x hyp. r
tan Q = =opp. y adj. x
For <A below, calculate Sine, Cosine, and Tangent:
sin A = opposite
hypotenuse
cos A = adjacent
hypotenuse
tan A = opposite adjacent
sin A = 1 2
cos A =
tan A =
√3 2
12
3
B
C
1 √3
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.
Understanding Forces
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
Properties of Vectors
Magnitude, Direction
Vectors - Description
45o
F = 40 lbs 45o
F = 40 lbs @ 45o
magnitude direction
Hat signifies vector quantity
Bold type and an underline F also identify vectors
1. We can multiply any vector by a whole number.
2. Original direction is maintained, new magnitude.
Vectors – Scalar Multiplication
2
½
We can add two or more vectors together - 2 methods:
1. Resolve into rectangular components then add
2. Graphical Addition/subtraction – redraw vectors head-to-tail, then draw the resultant vector. (head-to-tail order does not matter)
Vectors – Addition
y
xFx
Fy
1. It is often useful to break a vector into horizontal and vertical components (rectangular components).
2. Consider the Force vector below.
3. Plot this vector on x-y axis.
4. Project the vector onto x and y axes.
Vectors – Rectangular Components
Vectors – Rectangular Components
y
xFx
Fy
This means:
vector F = vector Fx + vector Fy
Remember the addition of vectors:
Vectors – Rectangular Components
y
xFx
Fy
Fx = Fx i
Vector Fx = Magnitude Fx times vector i
Vector Fy = Magnitude Fy times vector j
Fy = Fy j
F = Fx i + Fy j
i denotes vector in x direction
j denotes vector in y direction
Unit vector
Vectors – Rectangular Components
y
xFx
Fy
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
Vectors – Rectangular Components
If vector
V = a i + b j + c k
then the magnitude of vector V
|V| =
Vectors – Rectangular Components
Fx
Fy
cos Q = Fx / F
Fx = F cos Qi
sin Q = Fy / F
Fy = F sin Qj
What is the relationship between Q, sin Q, and cos Q?
Q
Vectors – Rectangular Components
y
x
Fx +
Fy +
When are Fx and Fy Positive/Negative?
Fx -
Fy +
Fx -Fy -
Fx +Fy -
Vectors – Rectangular Components
III
III IV
1. Vectors can be completely represented in two ways:
1. Graphically
2. Sum of vectors in any three independent directions
2. 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
Use the law of sines or the law of cosines to find R.
Vectors
F1 F2
R45o
105o
30o
Brief note about subtraction
1. If F = ai + bj + ck, then – F = – ai – bj – ck
2. Also, if
F =
Then,
– F =
Vectors
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
Find horizontal and vertical forces
• Simple triangle shows horizontal and
vertical parts.
If the angle at the top is
40o, what are the forces
(A & B)?
NNB
N
B
12.034.09.31
9.3120sin
60N
F2F1
40o
Half of the upward force comes
from each member.
NN
A
A
N
9.3194.0
30
3020cos
Look at point F1 for horizontal
member.
A
B
Practice Problem #1
The traction device is applied to a broken leg as
shown. What weight is needed if the traction force
pulling the leg straight out (right) is 165 N? (The
tension in the rope equals the weight.)
Practice Problem #2
Students want to hang a 1200 N cannon from ropes on
the football goalpost as shown. If the goalposts are 5
meters apart and the ropes are 3 meters long, would a
rope which breaks at 1000 N be good enough?
Practice Problem #3
A stop light is held by two cables as shown. If the stop
light weighs 120 N, what are the tensions in the two
cables?
Sometimes forces don't
move things, they rotate
them. How do you open a door?
Where is the best place to push on it?
Torque causes rotation.
Torque requires a force and a lever arm.
Give examples of
things you rotate?
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.
Individually.
Since Force is a vector, this implies
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
M = (magnitude of force perpendicular to moment arm) * (length of moment arm) = (magnitude of force) * (perpendicular distance from point to force)
P
F
x
P
F
x
Moment Examples:
1. 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?
2. Tension test apparatus – added load of lever?
Objects are in static
equilibrium if:
No net forces
No net torques
(moments)
Place a meter stick across
two scales. Put a 200 g
weight on it.
200 g
scale
200 g
scale
100 g 100g
150 g 50g
No net forces
Equate forces up & down.
Equate forces right and left.
No net torques
Pick any point.
Equate CW and CCW torques.
What should the
scales read?
1. Equate up and down forces.
2. Equate right and left forces.
3. Pick a pivot point.
4. Equate CW and CCW torques.
gFF 20021
x
cmFcmg 10030200 2
What should the scales
read?
1. Equate up and down forces.
2. Equate right and left forces.
3. Pick a pivot point.
4. Equate CW and CCW torques.
gggFF 50020030021
cmFcmgcmg 1006030030200 2
x
gFgF 260and240 12
Practice Problem #1
Consider a 12 m beam supported at each end by
two forces F1 and F2. A 10 N force is held by the
beam 4 m from the left side as shown. Find the
support forces F1 and F2.
Practice Problem #2
We have a board with a weight of 8 N that is
20 m long. A 5 N force is held 8 m from the left
side. Find the forces F1 and F2 that are
located at the ends of the board.
Practice Problem #3
Find F1 and F2. Find the forces on each
side of the triangle labels a, b and c.
Indicate whether these are tension or
compression forces.
Statics in bridges.
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
1. Joints are assumed to be frictionless, so forces can only be transmitted in the direction of the members.
2. Members are assumed to be massless. 3. Loads can be applied only at joints (or nodes). 4. Members are assumed to be perfectly rigid.
2 conditions for static equilibrium:1. Sum of forces at each joint (or node) = 02. Moment about any joint (or node) = 0
Assumptions to analyze simple truss:
Consider this popsicle stick
triangle
What parts are under compression? Tension?
fishing line
Examine triangle
members.
Look at any point on bridge.
Forces = 0 and torques = 0.
Determine which members are under
tension (like a string)?
compression (like a rod)?
compression
tensioncompression compression
load
load
Trusses Joints:
Pin or Hinge (fixed)
TrussesSupports:
Pin or Hinge (fixed) – 2 unknowns
Reaction in x-direction
Reaction in y-direction
RAx
RAyRoller - 1 unknown
RDy
Reaction only in y-direction
Method of Joints Problem:
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.
Now examine truss
members.
compression
tension
compression tension
compression
Use symmetry
to examine
members. Which members are
under
tension (like a string)?
compression (like a rod)?
compression
tension
compressiontension
compression
Load
Conditions of static determinacy and stability of trusses:
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.
Use Bridge Designer
http://engineering.jhu.edu/ei/about-ei/course-materials/
Statics, Structures, and Bridge Project Section
Virtual Lab Bridge Designer
Calculate the forces on a triangle.
Homework
• Use Bridge Designer for an initial design of a spaghetti bridge.
– 50 cm span (must be > 50 cm)
– <25 cm high
– Weight hung from bottom, center of bridge.
• For your design, determine the weakest and strongest members.
– Use knowledge of spaghetti properties from lab tests.
