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RE: x brace question

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In response to Bill Sherman- The radii can't just be added, but by adding them, you get the same result as a bracing stiffness analysis

In the classic example of a pinned-rotationally fixed column with a horizontal spring at the fixed rotation end, 

P*delta = Q*L  and Q = k*delta 
where Q= the force in the spring , k=stiffness the of spring, and L= Length of the column

The required stiffness for effective length=1.0 is k=Pcr/L

Pcr for elastic buckling is pi^2*E*I/L^2

Therefore, considering the stiffness of the tension member to be k=48EI/L^3, for the tension member to brace the compression member, pi^2>48     -of course this is never the case. It does show that the non-compression diagonal has some contribution to reducing the effective length of the compression member even without including the effect of tension restoring force.

Thus far, the analysis has ignored the effect of tension in the tension member.  P*delta= Q*L for the tension member as well as the compression member- this time the spring force is in the opposite direction than for the compression member.  Since Q tension = Q compression , wouldn't the tension brace always provide enough restraint to provide for N=2 buckling?  N=2 buckling corresponds to K=0.5- the same result obtained by adding radii of gyration.

There is the issue of initial out-of-straightness from fabrication and erection tolerance that will cause the real K to be greater than 0.5.

I guess I'm an optimist and think K=0.67 is o.k.

Another note: I never referred to the contribution of the out of plane gusset plates to the end restraint of the members in the system.  Of course, there is some small restraint that varies depending on the configuration of the connections and member types