Why Are My Bending Moment Signs Incorrect in the Direct Stiffness Method?

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The discussion revolves around issues with incorrect bending moment signs in a seven-storey structure analyzed using the Direct Stiffness Method. The user reports accurate force magnitudes but incorrect signs, specifically noting a positive bending moment at the bottom and a negative at the top of each column storey. They suspect the transformation matrix might be the source of the issue and seek validation on its correctness. A suggestion is made to test the code with a simpler structure to troubleshoot the problem. The thread concludes with acknowledgment that a similar query has been resolved on another site.
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Why am I getting incorrect signs
Hi there,

I was wondering if someone might know why I am getting incorrect signs for my structure that I am working on using the Direct Stiffness Method? I am following the procedure that I was taught when I was at University and I can't completely remember everything. I am designing a seven-storey structure that looks like this:

7 storey structure.png


The magnitude of the forces I am getting is accurate, but the signs are incorrect.

Here are the bending moment on the structure:

bm frame.png


You might not be able to see, but for the columns, you get a positive bending moment at the bottom of each storey and a negative bending moment at the top of the storey - this pattern is the same for each storey.

In my code I am getting positive bending moments at the top and the bottom, but like I said the magnitude is quite accurate.

Rather than post the entirity of my code, lets focus on a single column.

The local stiffness matrix for a column has the form:

LSM.png


And I set up the transformation matrix as follows:


Code:
T =

  6x6 table

              U1    V1    theta1    U2    V2    theta2
              __    __    ______    __    __    ______

    U1         0    1       0        0    0       0  
    V1        -1    0       0        0    0       0  
    theta1     0    0       1        0    0       0  
    U2         0    0       0        0    1       0  
    V2         0    0       0       -1    0       0  
    theta2     0    0       0        0    0       1


To compute the global stiffness matrix you use the equation:

1720191801904.png



This will give you the global stiffness matrix! You can then proceed to find the displacements and rotations. Then you can find the internal forces on the structure.

I guess the thing to look at is the transformation matrix. Does it seem correct to you. The local stiffness matrix is definitely correct. I know you guys might not be Structural Engineers, but I am hoping some of you have experience with the Stiffness Method.

Many thanks in advance!
 

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Before I try to remove the dust from what I can recall regarding this topic.
Could you provide a WAY SIMPLER structure to test your code and our knowledge?

Something like this for example:
1723322043165.png
 
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