1. 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.

2. 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
𝑀 𝑃 = 0
𝐹 𝑦 = 0
𝐹 𝑥 = 0
For analysis of simple beams we will be using each of the formulas above

3. How to read the mathematical formula with a ∑ in it.
𝑀 𝑃 = 0
This formula is read as
The sum of
the moments
about a point
equals zero ∑ M P
An example of taking the sum of would be finding your bowling average:
𝑠𝑐𝑜𝑟𝑒𝑠 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑠𝑐𝑜𝑟𝑒𝑠 = bowling average
Scores
150
125
155
𝑠𝑐𝑜𝑟𝑒𝑠=
= bowling average
𝑠𝑐𝑜𝑟𝑒𝑠=430
110 = bowling average

4. Moment – What is it and what are its units.
Distance - d
It is a rotational force – a force applied
at a distance from a pivot point
Point of
rotation
Formula for Moment
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
Force - F
Moment caused by A
M = F * d
F = 20 lbs
d = 1 ft
M = 20 lbs(1 ft)
M = 20 ft*lbs
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
Moment caused by B
M = F * d
F = 10 lbs
d = 2 ft
M = 10 lbs(2 ft)
M = 20 ft*lbs
UNITS - ft*lbs OR in*lbs

5. Static Equilibrium and the see-saw
Forces are balanced
and
moments are balanced
To solve for a mechanical
system in static equilibrium
we START with the equation
that deals with moments
being balanced: Fr Fe Dr De
𝑀 𝑃 = 0
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
Point p - point of rotation
Moment for the blue guy
Force Fr *
distance Dr
= Fr*Dr
Moment for the red guy
Force Fe
distance De
= Fe*De *

6. 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 De Dr Fe
+(Fr )(Dr)
– (Fe)(De)
The moment Fr*Dr is a positive moment
about point p because it would cause
a CCW rotation
The moment Fe*De is a negative moment
about point p because it would cause
a CW rotation

7. Solving equilibrium problems first class lever – see-sawFr De Dr Fe
+(Fr )(Dr)
– (Fe)(De)
Using the fact that in static equilibrium
the sum of the moments about a point is equal to zero:
𝑀 𝑃 = 0
use the formula and write an equation with proper signs
+(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?

8. 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
+ (10 lbs.)(8ft) (10 lbs.)(8ft)
Get rid of negative sign by adding
same thing to both sides of equals sign
Isolate unknown Dr or get Dr by itself
Cancel units because 𝑠𝑜𝑚𝑒 𝑡ℎ𝑖𝑛𝑔 𝑠𝑎𝑚𝑒 𝑡ℎ𝑖𝑛𝑔 = 1
(40 lbs.) (Dr) = (10 lbs.)(8ft)
(40 lbs.) (40 lbs.)
Dr = (10)(8ft) 40
Parentheses not needed around Dr
and simplify fraction
Dr = 2 ft
Check that solution is correct

9. 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.
X axis
- neg
Y axis
+ pos
Formulas
𝐹 𝑦 = 0
𝐹 𝑥 = 0
Lets first consider the forces in the y – plane.
Forces pointing down are negative and forces pointing up are positive
The see-saw example below will need a REACTION force to be in static equilibrium.
Therefore using this formula: Fr De Dr Fe
𝐹 𝑦 = 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
F Reaction
𝐹 𝑦 = + F Reaction – Fr – Fe = 0

10. Solving equation with one unknown
Equation from static analysis and
sum of forces in y direction
Write down knowns and unknowns
Fe = 10 lbs. F Reaction = ?
Fr = 40 lbs.
+ F Reaction – Fr – Fe = 0
+ F Reaction – 40 lbs. – 10 lbs. = 0
Substitute knowns into equation
With units
F Reaction – 40 lbs lbs = 0
F reaction lbs = 0
+ 50 lbs lbs
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.
𝐹 𝑥 = 0

11. Types of reaction forces for simply supported beams
Pin connection
Roller
Need to understand the reaction forces for the different types of supports.

12. Free Body Diagram Simply supported beam
P1 Load = 100 lbs.
What would you
estimate the end
reactions to be?
Fax A B
Y reaction force at B
roller
Y reaction force at A
pin
Fay
Fby

13. Moment calculations for simply supported beam with point load
P1 Load = 100 lbs.
10’ 5’
Fax A B
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
𝑀 𝐴 = 0
Fax(0ft) + Fay(0ft) – P1(5ft) + Fby(10ft) = 0
Fby = 10lbs(5)
substitute
Fax(0ft) + Fay(0ft) – 100lbs(5ft) + Fby(10ft) = 0
Fby = 50lbs
– 100lbs(5ft) + Fby(10ft) = 0

14. Next equation to use in solution
𝐹 𝑦 = 0
P1 Load = 100 lbs.
Fax A B
Fay
Fby = 50 lbs
𝐹 𝑦 = 0
Write equation
+ Fay + Fby -100 lbs = 0
Substitute known values `
+ Fay + 50 lbs -100 lbs = 0
Collect like terms
+ Fay lbs = 0
Solve for unknown
Fay = 50 lbs
Fax = 0 because there are no other forces in the x-direction therefore must be zero