Structural analysis, statically determinate beams.
In this lecture, we are going to answer two questions:
What is a statically determinate beam?
and how do we analyze one?
A beam is a properly supported structural member
capable of carrying loads that are perpendicular to its longitudinal axis.
Such a member is called a beam because the loads cause it to bend, like this.
The analysis of beams involves calculating their support reactions.
In this example, beam analysis entails calculating three support reactions: Ax, Ay, and By.
But what is a statically determinate beam?
A beam is called statically determinate
if its unknown reaction forces can be calculated solely using the equilibrium equations.
And how do we analyze such a beam?
Before we can answer the question, let's review reaction forces in beams.
The three most common beam support types are: pin, roller and fixed supports.
There are two reaction forces associated with a pin support:
A force in the x direction, and a force in the y direction.
A roller support provides one reaction force only: a force that is perpendicular to the support surface.
A fixed support has three reactions:
a force in the x direction, a force in the y direction, and a bending moment about the z-axis.
To clarify, consider the following examples.
Here, we have a simply supported beam.
It rests on a pin and a roller.
Therefore, there are three reaction forces.
Two forces associated with the pin support, and one force associated with the roller support.
Here, the beam is fixed at one end but free at the other end.
So, we end up having three reaction forces.
A force in the x direction, a force in the y direction, and a bending moment.
In order to analyze a beam, it is customary to draw a diagram called the free-body diagram.
It shows the applied and reaction forces, and their locations in the Cartesian coordinate system.
Note how each unknown force has been uniquely labeled.
The free-body diagram helps us correctly formulate the equilibrium equations.
To determine the support reactions, that is, to analyze the beam, we use three equations:
One, sum of the forces in the x direction must be zero.
Two, sum of the forces in the y direction must be zero.
Three, sum of the moments about the z axis at any point, say, at point A must be zero.
The first equation in the expanded form gives us: Ax=0.
The second equation gives us: Ay + By - P = 0.
The third equation gives us: PL/2 – By L = 0.
Solving the third equation for By gives us: By = P/2.
Solving the second equation for Ay gives us: Ay = P/2.
Let's look at two more examples.
Here, we have a cantilever beam subjected to a uniformly distributed load.
To analyze the beam, first we draw its free-body diagram.
We then replace the uniformly distributed load with its equivalent concentrated load.
So now, our free-body diagram looks like this.
Then, we write the equilibrium equations.
Ax = 0
Ay - 20 = 0
Ma - 20 (5) = 0
Solving the last equation for Ma gives: Ma = 100.
From the second equation we get: Ay = 20.
And obviously the first equation gives us: Ax = 0.
In the second example,
we have a beam with a pin support at its left end,
a roller support at its right end,
and another roller support at the middle point.
The beam is subjected to two concentrated loads, like this:
To analyze the beam, we start by drawing its free-body diagram.
Note that the diagram shows four unknown reaction forces:
two forces at the pin support, and one force at each roller support.
The equilibrium equations are:
Ax = 0
Ay + By + Cy - 2 P = 0
5P + 15P - 20Cy - 10By = 0.
Here, we have four unknowns but only three equations.
Therefore, we cannot calculate the unknowns.
The beam is not statically determinate.
This is called a statically indeterminate beam.
We will discuss techniques for analyzing statically indeterminate beams in later lectures.
For now, see if you can analyze the following determinate beams.