You are making this more complicated than it actually is:
The beam that you are trying to design is static. It is a beam with a fixed support and a simple support. A fixed support can only have force in the vertical and a simple support can have forces in both the vertical and horizontal. Therefore, to figure the moments and forces all you need to do is use the you three equations of equilibrium. These are:
Sum of Moments=0 Sum of Vertical Forces=0 Sum of Horizontal Forces=0
When it comes to moments clockwise reactions are negative and counter clockwise reactions are positive.
There is no horizontal forces in this problem so we can ignore it.
Your vertical load is 250lbs. You need to put in a safety factor of your choosing. I would use 4 to 1 but it is up to you. For this problem I will use a 1000lb load at the end of the beam.
Your beam is 3.75ft long with a fixed support on the left end at 0ft and a simple support somewhere along its length. We will use 1.875ft for the location of the simple support. We will first calculate the reactions about the supports using moment. We will label the fixed support R1 and the simple support R2. Moment is equal to force X distance. So we have:
Sum Moments about R1 1.875R2+3.75(-1000)=0 1.875R2=3750 R2=3750/1.875 R2=2000
Sum Moments about R2 -1.875R1+1.875(-1000)=0 -1.875R1=1875 R1=-1000
Sum All Vertical Forces : -1000(load)+ -1000(R1)+ 2000(R2)=0
All vertical forces are equal to zero as are the moments so the beam is in equilibrium.
Now you know your reactions at R1 and R2.
Now if you are using mild steel the Modulus of Elasticity or E= 30x10^6
E=stress/strain Strain is how much the material will "stretch" or displace under load. Poissons Ratio is also involved here but we will not complicate things with it.
Simply take your load and the cross sectional area of the tube and figure your Stress.
As long as your stress is under the yield stress of 36,000psi you are fine. Since you already have your safety factor of 4:1 all will be well if you are under that number.
You can figure out load at a joint by Using the Method of Joint or Method of Sections.
Maximum Moment will be where the shear stress crosses zero in the shear stress diagram.
If you do not know what these are get any good Statics book and it will explain it. You could also look through Machinerys Handbook.
Also, look into free body diagrams
If you need any more help, let me know. -Steve