# Utilizing symmetry in statics problem

I am given the structure below, and supposed to find the forces occuring in $$A, F, E, D, C$$ and $$B$$. The task hints of using symmetry, as several forces are equal in magnitude.

Supposedly, $$A=B$$, and $$G=H$$, where the image below depicts the top beam, and these forces.

My question is therefore why $$A=B$$ and $$H=G$$. I have asked other students/T.As., and they say it is obvious or intuitive, but can not really explain. Can anyone help?

• The pairs A & B and H & G (as well as C & D and E & F for that matter) are symmetric. The load is evenly distributed and symmetric about both horizontal axes through the center of mass of the system. If you create a free body diagram and do a sum of forces and moments you should see that.
– jko
Sep 2 at 11:39
• Imagine that another student was given the same problem, but the diagram given to them have the labels A and G interchanged with B and H. 1 would they arrive at the same answer ? 2 After seeing their diagram would you be able to tell if the labels were interchanged or just the viewing angle used to draw the diagram ?
– AJN
Sep 2 at 12:05

For the system shown, intuitively we can cut the beam at halfway between the support points G & H, so the left half is a mirror image of the right half regarding geometry, loading, support reactions, and member internal forces, so we can analyze half of the structure to obtain the same results. However, the trick is what type of support needs to be assigned to the joint at the cut.

From the full structure, it is expected that there is a positive (smiley) moment at the mid-point of the interior span, so we need to keep the joint fixed to draw moment to the joint, however, by doing so the resulting moment won't be correct (which will be negative), so we realize, in order to get the correct result, we need to modify the joint support specification.

Now, by examining the expected deflection of the full structure at the same location of the cut, we know the joint must be able to move freely in the vertical direction to allow deflection to occur, so we need to remove the restrain in the Y direction. Note, by releasing the vertical support at the cut, the moment curve is corrected as well.

The graph below shows the analyses results of the reduced model, which has the rightmost support fixed with Y fixity released, and the full structure.

General Procedures:

1. Cut the structure at a point that will result in one side structure is the mirror image of the other.

2. Inspect the moment and deflection curves on the whole structure at the cut location to determine the support type to be specified to the joint of the cut. Consider partial release of support fixity if necessary.