“Where there’s friction, there’s fun” our professor in Physics 11 way back in college told us.
We all gave him knowing smiles. Sensing something else, he proceeded to explain the mathematical explanation that relates ‘fun’ to friction.
Combining Newton’s 1st and 3rd law of motion, the block won’t move given that force F does not exceed μn. So putting both forces in equilibrium, F = µN.
Fun right? It is indeed. Brace yourself though, this is going to be weird stuff.
In the Middle East where there are significant temperature fluctuations all year round, a structure should be provided with the necessary reinforcement to inhibit the formation of cracks during casting. And that means you’ve got to have information on the construction methodology and the sequence of pouring before you can do your calculations. That is the short term effect of temperature.
On the other hand, the implication of fluctuating temperatures on the structure during its lifetime is called the long term effect.
I will not discuss about the short term effects, CIRIA already took care of that. I will discuss however how we considered the long term effects in ETABS. But first, let us have some flashback on temperature, cracking and its effects on the structure. Time to set the imagination mode on.
When the temperature rises, a body tends to expand. But for the rigid bar, the expansion is prevented by the fixed support and thus, the internal force that resulted due to the rise in temperature tends to “compress” our rigid bar. Are you able to follow?
And when the temperature drops, a body tends to shrink. And so for our rigid bar, this shrinking or shortening is prevented by the fixed support and thus the resulting internal force is tension.
When a beam, slab, column, or wall or any part of the structure bends and exceeds the modulus of rupture, it cracks. Depending on the magnitude of the bending moment, it can generate cracks ranging from microscopic to visible cracks. Although there are no researches yet that link cracks to reinforcement corrosion, cracks can be very unsightly. And though there are no significant reduction in the strength of structural members due to “normal” cracks, non-technical people can get overly concerned and the said cracks might cause undue panic in extreme instances.
Now, pulling a cable for example, beyond its allowable pulling force will produce tears or cracks in its strands before rupturing.
For our rigid bar bending under its own weight and other imposed loads while being simultaneously subjected to tensile stresses can produce significant cracks due to the combination of pulling and bending. And because concrete has a very low tensile strength, additional bars must be provided to limit the formation of the said cracks.
Temperature gains that induce compressive stresses tend to “seal” the cracks and thus the compression – bending action is not as critical as the tension-bending action.
Confused yet? It can get confusing alright. The rigid bar is simple. However, for a building with varying stiffness and complex interaction between structural members, calculation of combined tensile and bending stresses is next to impossible, except with the use of computers.
Ok, hold on tight. We’re about to include friction in the chaos.
The Fun Friction
For our rigid bar model, let’s replace the fixed supports by a friction mechanism. If we apply a drop in temperature, the bar tends to shorten and shrink. But depending on the end reaction and coefficient of friction, the support may not yield, and so there is tension in the bar. But if the end support moves, the bar is relieved from tension.
For structures founded on soil which serves as the point of contact between the structure and friction, the interaction is much more complex. Normally, one would expect that there will be no movement under the presence of heavy loads such as columns and walls supporting multi-storeys compared to columns supporting just one storey. Again this depends on the force normal (perpendicular) to friction forces and the coefficient of friction. And again, throw in all the geometrical complexities, variety in loading and differences in temperature exposure, and it will render the structure very complicated. Where there are large restraints such as walls, there tends to exist large amounts of tensile stresses due to the change in temperature.
And once you were able to model all of these, all you need to do is look for hotspots: large tensile stresses including the bending moments (recall that cracks are produced by bending and tensile stresses), key in the forces on spreadsheets that check crack widths and you will be able to find the adequate reinforcement.
Modelling Friction in ETABS
Springs are not the correct means of modelling friction. It is only applicable for piles, since we can derive the horizontal springs with ease. But not friction, because once the area of concern moved, the restraint is now released and the horizontal spring already has a constant value.
Credit goes to Dale, my design manager who dug deep into ETABS to explore the friction isolator link.
First, we need to model a link. That is something like a “column” support between the raft and the soil. Height of which does not matter so long as we set the parameters correctly. But first, the point support below the link should be fixed.
Now to define these links, we go to the Link Property Data. For Link type, we chose “Friction Isolator”, tick Directions U1, U2, and U3 and tick all boxes under Non Linear.
Under Properties, click “Modify/Show for U1…” U1 is the direction of Z axis – translation (up/down). Stiffness under the Linear and Nonlinear Properties is the tributary soil subgrade modulus, hence we have Edge, Corner, and Interior. Interior has a full value of the subgrade modulus, Edge has only half, and Corner is a quarter of the total value of subgrade modulus.
For both U2 and U3 that denote the X and Y translation (horizontal), set everything else to zero except for friction which should be as per ground investigation report. And the Effective stiffness should be as big as possible (will discuss that in the next paragraphs). According to Dale, he hasn’t yet discovered why there should be an effective lateral spring for friction but based on preliminary analysis, the larger the value, the more accurate the results will be.
Having set all of those, we must now define nonlinear cases for the load cases. The aim of which is, to load the gravity loads first (that is, normal forces should engage first before friction can take place). And so for this example, we started our nonlinear case with both self-weight and superimposed dead loads. And we continue from this nonlinear state in applying the temperature change.
Now we have to verify the results. Friction force is a function of the coefficient of friction and the force normal to the surface. The shear forces of the links in the X and Y direction must be combined in vector form (that is SRSS or square root of sum of squares) and the result should be equal to the product of friction coefficient and the axial load on the link. And the greater the horizontal effective stiffness is, the greater is the accuracy of the results. Why? We cannot answer it just yet. We still have a lot to read to answer that.
We haven’t used this method yet extensively as this is still relatively new to all of us. This may be crude but in my judgment, this is the best way to model friction compared to how we did it previously. It’s just a lot more tedious especially in the modelling process but this is the nearest exact mathematical solution that we can find. And comparing it to the crude methods we did previously, this is the next best invaluable tool for thermal analysis.