The truss problem for this semester is the problem below. Find the maximum compression and the maximum tension member.
Here is a little help on the solution. The first thing you must solve is the forces in the Y direction at A and J. We know that Jy + Ay = 2000 + 3000 + 5000 by simply adding all external forces in the Y direction.
Another easy way to solve for Jy and Ay is to do moments around any of the elements. Assuming each member has a length of 1 then these equations would be;
Moment around A: -(1)(2000) – (2)(3000) – 3(4000) + 4(Jy) = 0
Moment around J: -(1)(5000) – (2)(3000) – 3(2000) + 4(Ay) = 0
You will find that all of these equations are consistent and will give you the same Force in Y at A and J
More complex is to do a Force balance at each of the members, I will give 2 examples here to help you out. I assume all members are in compression. Under compression the force vectors will always point towards the node. Positive and Negative signs for the vectors are simply being in the positive X or Y as assumed.
Forces at D
The forces in the X direction are;
F(BD) – F(DF) + cos(60) F(CD) – cos(60) F(DE) = 0
F(BD) – F(DF) + 0.5 F(CD) – 0.5 (F(DE) = 0
The forces in the Y direction are;
cos(30) F(CD) + cos(30) F(DE) = 0
0.866 F(CD) + 0.866 F(DE)
Forces at E
Forces in the X direction
F(CE) – F(EG) + cos(60) F(DE) – cos(60) F(EF) = 0
F(CE) – F(EG) + 0.5 F(DE) – 0.5 F(EF) = 0
Forces in the Y direction
– cos(30) F(DE) – cos(30) F(EF) – 3000 = 0
-0.866 F(DE) – 0.866 F(EF) – 3000 = 0
Setting up Matrix
I recommend setting up the matrix in Excel as it makes it easier to organize the truss members. here is a spreadsheet with these (without the equations). In setting up the equations you must include the Y equations anywhere where there is an external force (only Y as the equations are only in the Y directions). These nodes are A, C, E, G, J – I recommend simply doing both X and Y at these nodes. You will have 18 possible equations, but you only need 15.
Once you do this and select the equations it is simply a matter of filling in the blanks for the equations.
I made one change to this (add Dx and removed Bx) and filled in some of the blanks. So you can see how I fill in the blanks on the spreadsheet.
Here is the solution (for max and min) I got
Maximum Tension: 7072.75 at member: EG
Maximum Compression: 6639.72 at member: HJ
A hand solution of the problem (Method of Joints) is at Solving the Warren Truss by Hand