Direct Stiffness Method - Why am I getting incorrect signs

[Tygra_1983]

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:

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

Here are the bending moment on the structure:

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:

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:

Kglobal = T(transpose)*KLocal*T


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.

Many thanks in advance

 

[Celt83]

To say it another way the direct stiffness method (DSM) gives you the results such that if you pulled that member out in isolation it's isolated free body diagram would be in static equilibrium with the resulting joint loads determined by the DSM solution.

 

[rb1957]

yeah, that's what I'm trying to highlight. The joint shears are what happens at the joints due to constant shear in the beam. The joint shears are a load/reaction pair.

The member shear creates different signed shears at the joints, because the +ve sign convention changes at the joints (or with which end of the beam (which side of the section cut) you're looking at. Celt has a nice view of a section cut, looking at the LH side of the cut, but draw the RH side of the cut ... shears and moments have the opposite sign convention.

This is super important to understand !

"Hoffen wir mal, dass alles gut geht !"
General Paulus, Nov 1942, outside Stalingrad after the launch of Operation Uranus.