I said I would explain the difference in static capacity and energy to crush, one more time. I don't know why. I believe it will be greeted with "TALK,TALK,TALK,TALK,TALK,TALK,TALK,TALK,TALK,TALK" but, knowing that, it's no one's fault but my own that time is being committed to ferreting out truth from BS. A lot of people subscribe to the BS, and I don't owe them a damn thing, but here goes.
The graphs below illustrate the difference between a support's static capacity and the energy required to crush it, and the relation between the two. The graphs are load-displacement graphs for three different contrived cases of abstract supports, the same type of graph Bazant supplied in his paper to describe the load response of a typical steel column in axial compression (posted twice several pages back).
The horizontal axis is displacement of an imposed load through distance to fully compact a crushing support. The vertical axis is the resisting force provided by the support. Units are fraction of the total support crushing distance
h and multiples of a hypothetical nominal design load
mg for horizontal and vertical, respectively. These are natural and convenient units for the problem, which is expressed independent of any particular mass or support height.
Each graph has areas shaded in a color and a gray texture. The latter will be explained in a moment. The height of the color portion represents the resistive force given by the support at each position as it's crushed, a simple graph of force versus position. There are only two force values, high/low (or OK/fail), to eliminate irrelevant detail. The force is higher than the imposed load in the first segment of travel then drops at some point to less than the load.
Static Capacity:
The support's maximum static capacity is the high value at the beginning. Units of force.
Crush Energy:
The work (mechanical energy) done in fully crushing the support is given by W = ∫F(h)dh which is the shaded area under the force curve. Units of energy.
The static capacity says what load can be supported by an intact member. The crushing energy is PART of what dictates motion over time of a collapse in progress. Two different things that cannot be rolled together into the single psikeyhackr criteria of "as weak as possible". Turning to the graphs:
Each of these cases represent a support which, while intact, has the capacity to hold a static load of mg with some amount of reserve. None of these supports will fail with a static load of
mg, but all will fail and even crush completely if the load is impacting at sufficient (as yet undetermined) velocity.
Case 1 at the top depicts a support which will yield under a static load 1.5 times the design load, so I'll loosely refer to it as having a factor of safety (FOS) of 1.5. Case 2 has a greater static capacity with an FOS of 2.0, Case 3 greater still at FOS of 2.5.
Clearly, the first of the three is weakest in terms of static capacity, having only a 50% margin of safety, but we'll see that it is the strongest in terms of energy to crush - which is the quantity of interest in a progressive collapse. Case 1 provides its peak resistive force over the first half of travel (even though it's being crushed) whereas Case 2 has peak capacity for 10% of the travel; Case 3 is a blip at only 2%.
Recall the shaded areas are the work done in crushing - force times distance - and it's obvious that Case 1 consumes much more energy in crushing because it maintains its somewhat lower initial capacity for a much longer period of travel, then drops to a lower capacity which is still 50% of the load. By contrast, supports in Cases 2 and 3 fail abruptly and more decisively, dropping quickly to very low capacity (zero in Case 3) for most of the travel.
Case 1 could be thought of as an approximation to stiff gel eventually going into shear fracture. Case 2, a steel column. Case 3, a glass rod. Just to put faces on the abstract properties.
Now for the rectangular gray bands. Each of these have the same area as the regions shaded in color, therefore represent the same amount of energy. The height of this rectangular gray band is the
equivalent average resistive force over the entire crushing distance.
If a support is crushed to full compaction, the dynamics at the beginning and end of this interval are the same as if there were this average force acting on the impacting load the entire time.Note in Case 1 that the average force is equal to the load
mg, meaning there is (over the entire displacement) no net average force acting on the impacting load. Gravity pulls downward, and the support pushes back by exactly the same amount. What would actually happen is the load would decelerate in the first half of travel, then accelerate in the last half. On the whole these alternating regions would balance and the load neither gain nor lose velocity as the collapse progresses were it not for losses due to inelastic collision. If the load impacts with insufficient momentum to crush a Case 1 support past half compaction, the collapse will arrest without the support being fully crushed.
In summary, these three cases show how static capacity has no direct relation to energy of crushing. The support which holds the greatest load crushes most easily, and vice-versa.