Static Equilibrium,
Balance of forces &
Simply Supported Beams
What you need to know to perform basic static analysis on
simply supported beams with point loads and evenly
distributed loads.
Static Equilibrium and
balance of forces
Static equilibrium in a mechanical system is the condition that exists when
there is NO acceleration
•
•
•
•
F=ma must equal 0.
More simply put when considering the analysis of simple beams
There is no motion – everything is static and not moving
Since mass and acceleration are not zero for simple beams that must mean that:
 the sum of the forces acting on the mechanical system must equal zero
AND
 With no rotation (static) the sum of the moments must also equal zero.
Formulas we will be using
𝑀𝑃 = 0
𝐹𝑦 = 0
𝐹𝑥 = 0
For analysis of simple beams we will be using each of the formulas above
How to read the mathematical formula
with a ∑ in it.
𝑀 =0
𝑃
This formula is read as
The sum of
the moments
∑
M
about a point
equals zero
P
An example of taking the sum of would be finding your bowling average:
𝑠𝑐𝑜𝑟𝑒𝑠
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑠𝑐𝑜𝑟𝑒𝑠
Scores
150
125
155
= bowling average
𝑠𝑐𝑜𝑟𝑒𝑠 = 150 + 125 + 155
𝑠𝑐𝑜𝑟𝑒𝑠 = 430
430
3
= bowling average
110 = bowling average
Moment – What is it and what are
Distance - d
its units.
It is a rotational force – a force applied
at a distance from a pivot point
Point of
rotation
Formula for Moment
Force - F
M=F*d
M – moment or a torque
F – force in US customary units is pounds
d – distance from the point of rotation is in inches or feet
What are the units for a moment?
M=F*d
= lbs * ft
ft*lbs or read as foot pounds
Yes you can have a force times a distance
and the result is foot pounds ft*lbs
UNITS - ft*lbs OR in*lbs
Moment caused by A
M=F*d
F = 20 lbs
d = 1 ft
M = 20 lbs(1 ft)
M = 20 ft*lbs
Moment caused by B
M=F*d
F = 10 lbs
d = 2 ft
M = 10 lbs(2 ft)
M = 20 ft*lbs
Static Equilibrium and the see-saw
Static Equilibrium
Forces are balanced
and
moments are balanced
Fr
Fe
Dr
De
𝑀𝑃 = 0
Point p - point of rotation
Moment for the blue guy
Force Fr * distance
Dr
= Fr*Dr
De
= Fe*De
Moment for the red guy
Force Fe * distance
To solve for a mechanical
system in static equilibrium
we START with the equation
that deals with moments
being balanced:
A moment is a Force * distance
The distance is the displacement
of the perpendicular component
of force from the point of
rotation (point p).
For now all forces will be
Perpendicular
Solving static equilibrium problems
moments are positive & negative
BY DEFINITION:
Forces that cause a Counter
Clockwise rotation about
a point are considered
positive moments
Forces that cause a
Clockwise rotation
about a point are considered
negative moments
Fr
+(Fr )(Dr)
Dr
The moment Fr*Dr is a positive moment
about point p because it would cause
a CCW rotation
– (Fe)(De)
Fe
De
The moment Fe*De is a negative moment
about point p because it would cause
a CW rotation
Solving equilibrium problems
first class lever – see-saw
Fr
+(Fr )(Dr)
Dr
– (Fe)(De)
Fe
De
Using the fact that in static equilibrium
the sum of the moments about a point is equal to zero:
use the formula and write an equation with proper signs
𝑀𝑃 = 0
+(Fr )(Dr) – (Fe)(De) = 0
Problem:
If Fe = 10 lbs and De = 8 ft and Fr = 40 lbs. What is the distance Dr need to be
for see-saw to be in static equilibrium?
Solving equation with one unknown
Equation from static analysis and
sum of moments
Write down knowns and unknowns
Fe = 10 lbs.
De = 8 ft.
Fr = 40 lbs.
Dr. = ?
+(Fr )(Dr) – (Fe)(De) = 0
(40 lbs.) (Dr) – (10 lbs.)(8ft.) = 0
Get rid of negative sign by adding
+ (10 lbs.)(8ft) + (10 lbs.)(8ft) same thing to both sides of equals sign
(40 lbs.) (Dr) = (10 lbs.)(8ft)
(40 lbs.)
(40 lbs.)
Dr = (10)(8ft)
40
Dr = 2 ft
Isolate unknown Dr or get Dr by itself
𝑠𝑜𝑚𝑒 𝑡ℎ𝑖𝑛𝑔
Cancel units because 𝑠𝑎𝑚𝑒 𝑡ℎ𝑖𝑛𝑔 = 1
Parentheses not needed around Dr
and simplify fraction
Check that solution is correct
What about translational forces
those in the X and Y directions?
The forces in any system in static equilibrium need to balance.
Another way of saying this is the forces need to sum up to equal zero.
Y axis
+ pos
Formulas
𝐹𝑦 = 0
𝐹𝑥 = 0
- neg
+ pos
Lets first consider the forces in the y – plane.
X axis
Forces pointing down are negative and forces pointing up are positive
- neg
The see-saw example below will need a REACTION force to be in static equilibrium.
Therefore using this formula:
Fr
Fe
Dr
De
F Reaction
𝐹𝑦 = 0
F Reaction is up positive
+ F Reaction
Fr is down negative
- Fr
Fe is down negative
- Fe
Summing the forces in the y direction yields
𝐹𝑦 = + F Reaction – Fr – Fe = 0
Solving equation with one unknown
Equation from static analysis and
sum of forces in y direction
+ F Reaction – Fr – Fe = 0
+ F Reaction – 40 lbs. – 10 lbs. = 0
F Reaction – 40 lbs. - 10 lbs = 0
F reaction
- 50 lbs.
= 0
+ 50 lbs
+ 50 lbs
Write down knowns and unknowns
Fe = 10 lbs.
F Reaction = ?
Fr = 40 lbs.
Substitute knowns into equation
With units
Collect like terms
Add 50 lbs each side of equals
F Reaction = 50 lbs.
What about the forces in the X- Direction?
NO FORCES in the X-direction for this simplification of a see-saw.
𝐹𝑥 = 0
Types of reaction forces for
simply supported beams
Simply supported beam
Pin connection
Roller
Need to understand the reaction forces for the different types of supports.
Free Body Diagram Simply
supported beam
What would you
estimate the end
reactions to be?
P1 Load = 100 lbs.
Fax
Y reaction force at A
pin
A
B
Fay
Y reaction force at B
roller
Fby
Moment calculations for simply
supported beam with point load
P1 Load = 100 lbs.
5’
Fax
10’
B
A
Fay
Fby
Which point is best use to calculate the sum of moments?
We can choose any point to calculate the moments about.
If we choose Point A we will have be able to eliminate 2 unknowns because
the distance from the force Fax and Fay is zero to the point of rotation.
Note: M=F*d if d=0 then product of F*0= 0.
Fby(10ft) = 100lbs(5ft)
10ft 10ft
Fax(0ft) + Fay(0ft) – P1(5ft) + Fby(10ft) = 0
𝑀𝐴 = 0
Fby = 10lbs(5)
Fax(0ft) + Fay(0ft) – 100lbs(5ft) + Fby(10ft) = 0
substitute
Fby = 50lbs
– 100lbs(5ft) + Fby(10ft) = 0
Next equation to use in solution
𝐹𝑦 = 0
P1 Load = 100 lbs.
Fax
A
B
Fby = 50 lbs
Fay
𝐹𝑦 = 0
Write equation
Substitute known values
Collect like terms
Solve for unknown
+ Fay + Fby -100 lbs = 0
+ Fay + 50` lbs -100 lbs = 0
+ Fay - 50 lbs = 0
Fay = 50 lbs
Fax = 0 because there are no other forces in the x-direction therefore must be zero