Designing beams in braced bays for axial load

Question

How do I design a beam for axial load if it is in a diaphragm?

Background

In a typical steel building modeled in Tekla Structural Designer, the lateral loading in the structure would be expected to be distributed by a floor diaphragm. The diaphragm transfers the lateral load to the elements of the lateral system - typically braced bays or shear walls.

In the case of braced bays - formed of columns, beams and braces as illustrated in the examples below - and when the diaphragm is set to be rigid (which is the default) all of the load in the diaphragm will be transferred directly to the braces.  The beam in the braced bay between the columns in the floor (and hence in the diaphragm) will have zero axial load.  This is because a 'rigid' diaphgram is exactly that; it is infinitely stiff (in plane).  The rigid diaphgram works analytically by constraining together all the nodes within its area/ along its edge to behave as a rigid body (in plane); i.e. while the diaphragm and hence nodes can displace/ rotate en-masse, their in-plane relative distances remain unchanged. 

Next, axial load develops in members due to their elastic extension/ compression (internal member forces result from displacement, not vice-versa).  Ergo, if there is no extension/ compression then no force can develop.  It should be clear then that this is the reason no axial force will develop in members that are within a rigid diaphgram* - because the relative positions of their end nodes remains (perfectly) unchanged hence extension or compression is impossible.

• (*Note that the single exception to this rule is force resulting from Temperature loads.  This is because the mechanism for such loads is entirely different to that of externally applied loads discussed in this article. For temperature loads, the member itself tries to intrinsically lengthen or contract (due to an increase or decrease in temperature respectively), and force only results from restraining this displacement)

It could be considered that this may lead to an unconservative design of these beams if the engineer assumes they take all of the brace force (as axial load) that is in fact transfered (analytically) entirely by the diaphgram.  If this is indeed the assumption, it is a) conservative and b) cannot be what is occuring in reality.  Consider;

• For a diaphgram to actually work as a diaphram it must have a finite (in fact a considerable) in plane stiffness.
• For the diaphgram to transfer force to the elements of the lateral system, these must be tied into the diaphragm - if they were not then they would simply behave independently of it and the system of bracing would not work.
• If the elements are tied into the diaphgram AND the diaphgram has in-plane stiffness, then, in the case of the beam in question, it must share load with the diaphragm.  Assuming all the bracing force is in the beam is contradictory in that it makes incompatible assumptions that cannot co-exist; that the diaphgram has in-plane stiffness and is behaving as a diaphgram and that it has zero in-plane stiffness and does not restrain the elastic extension of the beam to some extent.  This is an analytical impossiblity as well as a real-world one.

Answer

To address this issue, the engineer may be tempted to try to release the beam nodes from the diaphragm (which is a modeling option), however this will not work successfully if at all.  Consider - if you remove the beam nodes from the diaphragm (which are common with the braces and columns where they connect), how can it also simultaneously remain in the diaphram to have load transferred to it?  Some thought should lead to the conclusion that this also is impossible (just as the assumption discussed above is - that the diaphgram can simultaneously have considerable and zero stiffness).  For this reason, this is not our recommendation.

Another option - which is both analytically possible and which models what must actually occur in reality - is to set the diaphragm as semi rigid.  In this case, as they must in reality, the beam and diaphragm will share load.  This is because a semi rigid diaphgram (SRD) has a finite stiffness, so nodes within it can change their relative distances (in-plane) and so axial extension/ compression can occur in beams within it, hence axial force can develop. How much force develops in the beam depends on the relative (axial) stiffness of it vs that of the SRD.  This is something the engineer can control and which is a matter of their judgement.

Example

Consider the example below in which the diaphragm is set to rigid.  Examining the distribution of forces after running analysis (any of the static types) we will see that there is no axial force in the beams within the braced bay.

 

Image

If we now change the diaphragm to semi rigid, we will see that the axial force is now present in these beams.  Moreover, they will now be designed for this force.  Other beams in the frame (not within the braced bay) may also develop some small axial force.  While this may not be large enough to require an increase in section, you could if you wish change the ignored forces limit so this is not considered by design.  Importantly, the brace forces are essentially unchanged so this method will not result in under-design of these.

Image

You may also wish to adjust the stiffness of the semi-rigid diaphragm to increase or decrease the magnitude of the forces in the elements within the diaphragm.  This can be done via the properties window for the slabs that form the diaphragm.  Please refer to this article for more information on adjusting diaphgram stiffness.

Finally, another approach  - and the only option if you wish to use the assumption of the beam taking all the brace force - would be to calculate this from the brace forces and perform your own additional design checks externally including this axial load.

See also

Design Options - Design Forces
Effect of diaphragm stiffness