What is this probable shear?
First, you need to remember that we already defined our seismic loads either by static definition or by response spectrum. In fact, this seismic load in combination to gravity loads is what will be used to derive the initial flexural reinforcement before we can perform check number 3 in my previous article.
But that is not the probable shear.
In as far as our discussion is concerned, we can consider the seismic load we just described as the “actual seismic load” which already served its purpose in design.
This probable shear stipulated in the code, is a scenario where shear forces larger than the ones derived from the analysis are likely to develop in the moment frame as the structure gyrates to the seismic rhythm. Hence the term ‘probable’ shear. And to reiterate, this is not the shear that can be derived from the structural analysis so don’t go looking for this anywhere in the ETABS model.
With this check, the bars were assumed to incur stresses that are 25% higher than the yield strength, which will develop higher flexural stresses and consequently, higher shear stresses in the beams. The beam should then be capable of resisting this shear for the beam to be ductile.
This is done manually which I’m going to expound below. This requires some imagination by the way, which is very important because you will need to be able to visualize the deformed shape and the resulting strains so that you can understand the equations with confidence.
What we will do is to freeze one moment in time while our building sways to the right and left and analyze what’s happening to the beam.
Sway to the Right…
When the frame sways to the right, the deformed shape will be as shown above. Notice the direction of the induced probable moments due to the sway, including which side of the beam (top or bottom) at the support is in tension. Both probable moments have the same direction sense which is counterclockwise.
1. Probable moments
Looking at the figure above, for the MPL-R, the bottom of the beam is in tension, i.e., the bottom bars are also in tension.
And for MPR-R, the top of the beam is in tension, hence, the top bars will also be in tension.
2. Deriving the ultimate distributed load wu
If we’re going to get the free body diagram of the deformed shape, this is what we will arrive at. Remember that we still have gravity loads in addition to the probable shear, and the load factors to be considered is as per above. S corresponds to the snow load effects, which will be ignored in this example. SDS is “the design spectral response acceleration parameter at short periods” which we will assume to be 1.374. So keying it in the formula for ‘wu’, we have (1.20+0.2*1.374)D + 1.0L or
wu = 1.4748D + 1.0L
You might be wondering how in the world did the ‘wu’ have both positive and negative signs. Well, that is by taking the summation of moments on the left and on the right of the beam. By visual inspection, the + sign will govern the value of Ve. But both negative and positive are presented nonetheless, both in the code and in here for completeness and clarity.
3. The probable shear Ve
Probable shear will counteract the probable moments as shown in the above illustration. This is the Ve that we need to design the shear reinforcement of our beam with.
Now let’s key in the actual values so you can appreciate this better.
4. Calculating wu
Say our beam has a tributary width of 6.0m, carries a 250mm thick slab, and super-imposed dead and live loads of 3 and 5 KPa, respectively. So wu yields the value shown below.
5. Calculating the probable moments
The calculation is pretty much the same with calculating the nominal flexural strength, with the exemption of fy which should be increased to 1.25fy to account for higher tensile stresses beyond the designated fy. In our example, it would be steel stresses of up to 1.25*420 = 525 MPa. The rest is standard calculations.
You can find the required flexural reinforcement which we checked in the previous article.
6. Calculating the probable shear Ve corresponding to the sway to the right.
With all the parameters calculated, we can now proceed with calculating the Ve.
How was it so far? All good?
We were able to calculate the probable shear when the frame sways to the right. And if you’re asking ‘how about when the frame sways to the left?’, well you are asking the right question and we should also consider that. Remember that we have to get the largest probable shear values on both situations that the structure sways both to the left and right. And the largest value of Ve will be used to design the shear reinforcement of both the left and right ends of the beam.
Actually, since the top and bottom bars on both beam ends are identical, we will arrive at the same probable shear values. But nevertheless, I will still go through the entire process. Remember, it is not always that we have the same reinforcement on both ends of the beam, hence it is imperative that we always check this one.
Sway to the Left…
As the frame sways to the left, the deformed shape will be as shown above. Try to notice the different direction of the induced probable moments due to the sway. Previously it was counterclockwise. Now both probable moments have the same clockwise direction.
7. Probable moments
For MPL-L, the top of the beam is in tension, that is, the top bars will also be in tension.
And for MPR-L, the bottom of the beam is in tension, which implies that the bottom bars will also be in tension.
8. Deriving the ultimate distributed load wu
Regardless if the frame sways to the left or right, the gravity loads will still be the same. So we will use the previously calculated uniformly distributed wu in item 4 above.
9. Calculating the probable moments
Again, be extra careful which particular bars you are going to use. Always remember that the two probable moments have the same direction. And in this case, both are clockwise.
As you can see, because of symmetry of the beam end reinforcement, the sum of probable moments is the same as that in item 5.
10. Calculating the probable shear Ve induced by the sway to the left
And finally, we will be able to get the probable shear which is the same probable shear that we got considering the moment frame swaying to the right like what I’ve said earlier.
Probable to be used for shear design
The maximum probable shear, considering the moment frame swaying to both left and right, is 729 KN. This is what we will consider for the design of shear links for both beam ends.
This shear value, is usually a lot higher than the factored shear (the ultimate load combination of gravity and seismic loads) derived from analysis. So the beam geometry and shear reinforcement that initially worked using the forces from ETABS analysis, may no longer work when checked against probable shear.
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