## Mxy Moments in concrete floor design

## Mxy Moments in concrete floor design

(OP)

I recently asked the AS3600 committee for clarification on the design of concrete floors based on FEM analysis and specifically how the Mxy moments should be handled. The following is a copy of their response.

As in regard to your technical query we offer the following.

Under no circumstances can torsion be "ignored", it is fundamental to equilibrium!

The generally accepted method of dealing with the twisting moments, Mxy, for the design moments is to use the Wood-Armer equations. In essence, the method involves adding the absolute value of Mxy to the moments Mx and My, using the correct signs of each to give design moments in each direction:

Mx* = Mx + IMxyI

M*y = My + IMxyI <?xml:namespace prefix = o ns = "urn:schemas-microsoft-com:office:office" />

I assume that the context of your question is for the case of slabs. In this case the basic of equation of equilibrium is:

"missing graphic, showing a basic equilibrium equation

d2mx/dx2 + d2my/dy2 - 2 * d2mxy/dxdy + q = 0;

emphasing that equalibrium must be satisfied and members must be ductile"

Provided this equation is satisfied, equilibrium is satisfied. How we treat this will depend on the method of analysis. For example, in most lower bound methods the torsional component of the stress resultants are taken as zero and the load carried, in full, as moments in the x and y directions (or in the case of a one way slab, in one of the x or y directions). Provided that the stress resultants sum to the total load on the slab, equilibrium is satisfied. Note here that torsion has not been "ignored", rather a concisions decision has been taken by the designer to take mxy as zero and, thus, increase one or both the other components. Further, to ensure that the system has sufficient ductility additional compatibility reinforcement (as per AS3600-2009 clause 9.1.3.3(e)) must be placed in the high torsion regions to alleviate any adverse torsional effects and see that the loads can be redistributed to the designers selected load path.

If we were to use a linear-elastic FE package to obtain the design moments, three bending components (mx, my and mxy) and two shear components (vx and vy) will be output. In this case, the torsional component is likely not zero and most certainly cannot be ignored. This is fundamental mechanics. In this case, the usual method of analysis is to determine the yield condition using the "Wood-Armer" equation:

mux = mx + k|mxy|

muy = my + k-1|mxy|

where k is typically taken as k = 1. Here mux and muy include both the normal and torsional components of the moments. To "ignore" the torsional moments in such circumstances violates equilibrium and is dangerous!

On the second question, I think that the code is reasonably clear on the torsional stiffness to be taken:

For equilibrium torsion, the torsional stiffness of the uncracked section should be used.

For compatibility torsion, the torsional stiffness may be taken as zero provided that "the torsion reinforcement requirements of Clause 8.3.7 and the detailing requirements of Clause 8.3.8 are satisfied." (Refer to Clause 8.3.2).

I trust that this answers your questions; am happy to provided further detail if needed.

As in regard to your technical query we offer the following.

Under no circumstances can torsion be "ignored", it is fundamental to equilibrium!

The generally accepted method of dealing with the twisting moments, Mxy, for the design moments is to use the Wood-Armer equations. In essence, the method involves adding the absolute value of Mxy to the moments Mx and My, using the correct signs of each to give design moments in each direction:

Mx* = Mx + IMxyI

M*y = My + IMxyI <?xml:namespace prefix = o ns = "urn:schemas-microsoft-com:office:office" />

I assume that the context of your question is for the case of slabs. In this case the basic of equation of equilibrium is:

"missing graphic, showing a basic equilibrium equation

d2mx/dx2 + d2my/dy2 - 2 * d2mxy/dxdy + q = 0;

emphasing that equalibrium must be satisfied and members must be ductile"

Provided this equation is satisfied, equilibrium is satisfied. How we treat this will depend on the method of analysis. For example, in most lower bound methods the torsional component of the stress resultants are taken as zero and the load carried, in full, as moments in the x and y directions (or in the case of a one way slab, in one of the x or y directions). Provided that the stress resultants sum to the total load on the slab, equilibrium is satisfied. Note here that torsion has not been "ignored", rather a concisions decision has been taken by the designer to take mxy as zero and, thus, increase one or both the other components. Further, to ensure that the system has sufficient ductility additional compatibility reinforcement (as per AS3600-2009 clause 9.1.3.3(e)) must be placed in the high torsion regions to alleviate any adverse torsional effects and see that the loads can be redistributed to the designers selected load path.

If we were to use a linear-elastic FE package to obtain the design moments, three bending components (mx, my and mxy) and two shear components (vx and vy) will be output. In this case, the torsional component is likely not zero and most certainly cannot be ignored. This is fundamental mechanics. In this case, the usual method of analysis is to determine the yield condition using the "Wood-Armer" equation:

mux = mx + k|mxy|

muy = my + k-1|mxy|

where k is typically taken as k = 1. Here mux and muy include both the normal and torsional components of the moments. To "ignore" the torsional moments in such circumstances violates equilibrium and is dangerous!

On the second question, I think that the code is reasonably clear on the torsional stiffness to be taken:

For equilibrium torsion, the torsional stiffness of the uncracked section should be used.

For compatibility torsion, the torsional stiffness may be taken as zero provided that "the torsion reinforcement requirements of Clause 8.3.7 and the detailing requirements of Clause 8.3.8 are satisfied." (Refer to Clause 8.3.2).

I trust that this answers your questions; am happy to provided further detail if needed.

## RE: Mxy Moments in concrete floor design

thanks again RAPT.

Arguing with an engineer is like wrestling with a pig in mud. After a while you realize that they like it

## RE: Mxy Moments in concrete floor design

This essentially answers a query I had a few years back regarding whether it is acceptable to reduce the J value in a frame analysis package to essentially create a "torsion-free element".

It has been a handy tool that I have used in the past. Quickly generating a grillage to get some idea about how the structure behaves. I generally find reducing the torsional stiffness by a few orders of magnitude will be enough to create a torsion-free element.

thread507-247903: Torsional Stiffness (J or Ix) of a Prestressed Conc. Beam

## RE: Mxy Moments in concrete floor design

That rule or a similar one has been in the code for years.

But with plate/slab elements, Mxy is not just due to torsion. To define the moment state on an element other than in the principal direction requires the definition of 3 moments, Mx, My and the twisting moment Mxy.

As we cannot physically reinforce in the principal direction for each element, we use orthogonal reinforcing patterns requiring us to reinforce in the X and Y directions. So we need to determine design moments in these directions. Unfortunately, we have 3 moments, Mx, My and Mxy.

If you reduce the torsional stiffness to 0, you will still get an Mxy moment on most elements as the stress state on each corner is different resulting in a twist on the element.

This cannot be ignored in design as other torsional moments may be based on the rule you have stated, depending on their need to provide equilibrium.