Stiffness Calculations of an Experimental Reinforced Concrete Beam

Background

One final set of calcs ... stiffness calcs. These are `academic' in a sense, but interesting (at least I think so). (But) they are also very `physical'; in other words, the idea of `stiffness' describes (physical) reality. As long as a structural member hasn't failed, we should be able to load it some amount, and it will deform (deflect, sag, whatever we want to call it). We unload it, and it `springs' back. We load it more, and it deforms more. We unload that ... and it (still) springs back. We load it too much, and it doesn't spring back ... we failed the beam ... yield steel ... it is permanently bent, deformed. So, structural members act like springs. We load them; they deform; we unload them; they spring back to original shape; we load them too much; ugh ... they don't spring back; they look ugly; we throw them away.

We measure this `springy-ness' with what we call `stiffness' (or `spring constant). The `stiffness' is defined as `the amount of load required to produce a unit amount of deflection'. Usually we consider the unit amount of deflection as oneinch (though it could be anything). So, ... stiffness is the amount of load to produce one inch of deflection.

In terms of an equation ...

... k = P / Δ ... (F / x for you physics and engineering people) ...

Now, structural members tend to be very stiff. In other words, it will take A LOT of force (say, pounds) to produce this `one inch' of deflection.

And, in fact, many of our structrural members will BREAK FIRST.

But we still use the `measure' of k ... to describe the behavior (at smaller loads).

And, further, in reinforced concrete, as the concrete begins to crack, even before we consider the beam `failed' ... this stiffness changes.

Stiffness of the Un-cracked Beam

So, let's calculate the stiffness of our experimental concrete beam at several stages.

Before the beam cracks (at all) it will have an `un-cracked stiffness' ... we can get this (easily) from our P_cr and Δ_cr values ...

... k_cr = k _uncracked = P_cr / Δ_cr = 2014 lb / 0.011 in. = 183,000 lb/in.

What this `says' is that the beam will deflect 1 in. under a P of 183,000 lb. Obviously it will crack, break, be totally destroyed at a P much less than 183,000 lb. It's a measure, a rate, a ratio. The beam will deflect 1.0 in. per every 183,000 lb. Now, we won't ever load it to that, ... but what if we load it to, say, 600 lb. This value is less than Pcr, so such a question is valid.

We can answer the question several ways:

1) ... if for every 183,000 lb we would get 1.0 in. ... then for 600 lb we would expect 600/183,000 of 1 inch ... or 0.003 in.

2) ... we can take the above equation and cast it into a form ... Δ = P / k ...

... Δ = 600 lb / (183,000 lb/in.) = 0.003 in.

3) ... we could go back to our deflection equation ... Δ = P L³ / 48 EI.

Stiffness at (Before) Failure

We can use the same approach to determine the stiffness just before steel yield ...

... k_y = P_y / Δ _y ...

Going back to our earlier lessons ( ... here ... and ...) ... P_y = 9600 lb and Δ _y = 0.172 in.

So,

... k_y = 9600 lb / 0.172 in. = 56,000 lb/in.

Note how the beam is way less stiff. Well, yeah, it is way more cracked!

We might notice that it's 56/183 = about one third as stiff as the un-cracked beam. Yeah, our cracked Moment of Inertia is about one third as much.

Stiffness at Safe Service Load

We could also calculate the stiffness at the safe service load.

...k _{@ safe load}= P _safe / Δ _safe = 5480 lb / 0.089 in. = 62,000 lb/in.

This number is probably mostly academic ... but note that it's a bit stiffer at that stage than the `fully cracked' beam.

Concluding Remarks

A beam of `pure' steel will have essentially a single stiffness up until yield. A wood beam will have a single stiffness up until extreme fiber rupture, unless the load is sustained, in which case wood tends to creep (deflect some more). A single stiffness value is a characteristic of behavior that we also call `linear', or `linear elastic'. Doubling the load doubles the deflection (... linear) ... and when we unload the member, it springs back to original configuration ( ... elastic). Before a reinforced concrete beam cracks we (might) consider it linear elastic. But once concrete cracks, it doesn't un-crack. And the more it cracks, the more stiffness we lose. In a supposed `continuum' between the load that cracks the beam and the load that yields the steel we could say we have a changing stiffness (I_e varying from I_g to I_cr). But if we unload the beam, say, from just below yield, down to ... nothing, it stays cracked; we don't go back to Ig, though we might go back to pretty close to an undeformed shape (unless the loading was sustained, and the concrete `crept'). If we `bend' the beam beyond steel yield, as we have earlier discussed, the stiffness becomes undefined, as we will have more and more deflection (and, we could say, permanent deflection), with little or no more actual load (value).

Yes, concrete does creep. We won't go there now.

References

Strength Calculations of an Experimental Beam, Jeff Filler, Associated Content.

Safe Load for an Experimental Reinforced Concrete Beam, Jeff Filler, Associated Content.

Deflection of an Experimental Beam at Safe Load, Jeff Filler, Associated Content.