as an introduction to structural
analysis
we must first explain some principles of statics.
we will discuss stability. What is stability? how can we discover if a system is unstable?
and what are the differences between a stable and unstable system?
Next, we will explain what are supports? and why are they important?
what are the different types of supports?
and why do reactions develop in supports?
finally we will explain what is
equilibrium
and how can we find equilibrium equations?
Stability.
So what is a stable system ?
when a stable body is pushed by a force
it will move proportional to the strength of the force.
A small force will cause a small movement.
A stable body returns to its original position when the force is removed
An unstable body will  experience a large movement
in response to a very small force
the body will not return to its original position when the force is removed
What is a support? why is it important?
When a support is added to a point  it restricts the movement of that point
in one or more directions
when a beam is left in the air, it falls freely under its weight.
It's unstable
when a support is added at the left end, it
will prevent the movement of that end
but the beam will continue to rotate and
the right end will continue to fall
until an additional support is added to
the right end.
At this point in time the beam will become stable.
Supports are needed to maintain stability.
Support reactions. What is a reaction?
When a car is pushed on a smooth level ground
it will roll freely with no resistance.
To stop the car you need to push back with an equal and opposite force
the force used to stop the car is called a reaction
because the amount of effort you
need to exert to stop the car
doesn't depend on your decision
it only depends on the magnitude of the original force pushing the car
For any support when movement is
allowed in  a certain direction
no reaction develops in that direction.
A reaction will develop in any direction when motion is restricted in that direction
support types
in 2D statics, a point can experience three
movement types
or 3 degrees of freedom
2 translations in 2 perpendicular directions and one rotation.
A point with three motions
allowed is a free point with no support
A roller support restricts only one
translation at a point
the point is allowed to move in the perpendicular direction
and is allowed to rotate
therefore the roller support has one
reaction force
A hinged is support
restricts two perpendicular translations
but the point is allowed to rotate
The hinged support has two reactions
A Fixed support restricts 3 motion
types.
The fixed support has 3 reactions
2 reaction forces and 1 reaction moment
A Fixed roller restricts a points from rotation and translation in one direction
but the point is allowed to move in the perpendicular direction
therefore the fixed roller has one reaction force and one reaction moment
What is the equilibrium?
Equilibrium is a state related to the
forces and reactions acting on the body
it means the net sum of forces and reaction acting on the body is zero and
therefore the body is stationary
stability on the other hand is a
property that depends on the geometry
and kinematics of the body
if the body has enough supports to generate the
needed reactions to resist any kind of
force configuration equilibrium can be
attained and the body becomes stable
So can an unstable body be in equilibrium?
The answer to this question yes
an unstable body can reach equilibrium for certain force configurations
only for certain force configurations
but cannot reach equilibrium for all the possible force configurations
what are the equilibrium equations?
to reach a state
of equilibrium three equations of
equilibrium have to be fulfilled
first the force in the horizontal or X direction must be resisted
by the reaction in the X Direction.
∑X =0
Second the force in the
vertical or Y direction must be resisted
by reactions in the Y Direction
∑Y =0
Finally, the moment about any point caused by the forces must be resisted
by the moment about the same exact point caused by the reactions
∑M =0
thank you for watching
