Safe Load on an Experimental Reinforced Concrete Beam
An Experiment in Reinforced Concrete Design
Background
In the previous lesson (here) we determined `how' the beam would fail. We first determined Pcr, the load that would (first) crack the beam, and then what would happen next ... would it shear, concrete crush, steel break, etc.??? To do that, we calculated Pc, Py, PV and also Pult, namely, the theoretical) load P that would first crush concrete in the compression zone, yield the steel in the tension zone, fail the beam in shear (diagonal tension), and also the load corresponding to yielding the steel, and yielding it some more to the point the concrete (also) crushes. And so the lesser of these ( Pc, Py, PV) would be expected to (eventually) happen next, and would be what we could call `failure'. And in some sense, whatever happens first ... is be `failure' of the beam, with the other numbers not really being applicable (except, perhaps, Pult).
We found, in terms of numbers, ...
Pcr = 2010 lb
Pc = 10,100 lb
Py = 9600 lb
PV = 10,100 lb
Pult = 10,100 lb
In terms of what we would expect to happen, then, would be ...
The beam would first crack at 2010 lb ...
Cracking (and sagging) will continue ... up to ...
Steel starting to yield at 9600 lb ... (since Py< Pc and PV)
And since steel yields before concrete crushes ... we can also expect the steel to keep yielding until we also crush concrete ... at Pult = 10,100 lb.
But, we are also pretty `near' the shear strength of the beam ... it would not be surprising to see the beam beginning to, or completely, shear, also.
We also calculated the expected deflections along the way ...
A deflection of 0.0011 in. under its own weight ...
A deflection of 0.011 in. (more) at Pcr ...
A (net) deflection of 0.175 in. just as the steel yields.
Deflection calculations become meaningless after that, since the beam will no longer be elastic. But, the beam will very much continue to deflect (deform).
Safe Load
Now we will calculate the `safe load' (P) for the beam (Psafe). The `safe' load is the allowable future use load for the beam, assuming, say, we want to market this beam (for future use in a structure or structures). (Actually, not this beam, because this beam will be very much ruined when we are done with it, but, say, future cast similar beams.) Determining some future safe load will incorporate factors of safety. We will use the Strength Design approach (SD), and let's assume that the future use will involve a concentrated Occupancy Live Load, P, coming down at mid-span, kinda like we modeled in the lab.
Interesting with this `experiment' is that the load values associated with the different `kinds' of failure were (are) all pretty close. Even more interesting will be the calculation of the `safe' load, since different kinds of failure involve (by Code) different factors of safety. In generic form,
U = 1.2 D + 1.6 L,
where,
U is the `ultimate' load condition or load combination,
D stands for Dead load,
L for Live (Occupancy) load,
... and 1.2 and 1.6 are the corresponding factors of safety associated with `loads' (Load Factors).
BUT,
... since we are now in the realm of the `Code' ...
We will only look at Pult and PV ...
We are NOT ALLOWED to use a beam in which the steel doesn't yield before the concrete crushes, and, if that is indeed the case (steel yielding first), we then take it on up to concrete also crushing (after steel yield, safely). So,
... setting up equations for PV, safe and Pult, safe ...
SHEAR ...
Starting with ... V = ω L /2 + P/2 ...
where the first term is the self-weight ... which we will consider Dead (D) ...
and the second term is our Live (future) load ...
and bringing in our factors of safety ...
Vu = 1.2 (ω L /2) + 1.6 P/2 ...
... 1.2 (54.4 plf x 6 ft / 2) + 1.6 P / 2 = ...
Vu = 196 lb + 0.8 P ...
We now can `set' this equal to our shear strength ... except in the Strength Design approach the strengths must also be adjusted with a factors of safety, in this case the strength reduction factor for shear ... 0.75.
So,
Our factored shear strength is ... φ Vc = 0.75 (5218 lb, from before) ... equals ... 3914 lb.
Now we will set factored shear load equal to factored shear strength, and solve for the safe load dealing with shear ...
Vu = 196 lb + 0.8 P = φ Vc = 3914 lb.
P = (3914 lb - 196 lb ) / 0.8 = ...
P V, safe = 4649 lb.
FLEXURE
We'll do essentially the same thing with our moment equation and the ultimate flexural failure in the beam ... (steel yield and concrete crushing) ...
In terms of load ...
... generic ... M = ωL2 /8 + P L/ 4.
for our beam where the uniform load is the self weight, and is `Dead' ... and the P is the (future, concentrated) `Live' load ... (not implying at all that all future loads are necessarily concentrated or Live)
Mu = 1.2 ωL2 /8 + 1. 6 P L/ 4 ...
= ... 1.2 (245 lb-ft) + 1.6 P (6 ft) / 4 = ...
= ... 294 lb-ft + 2.4 P ft.
We must also apply a strength reduction factor to the ultimate bending strength. The φ for flexure where steel yielding controls is 0.90.
So, ... φ Mn = φ Mult = 0.90 (14,950 lb-ft from before) ... 13,455 lb-ft.
Setting Mu = φ Mn ...
... 294 lb-ft + 2.4 P = φ Mn = 13,455 lb-ft ... and solving for P ...
... P = (13,455 lb-ft - 294 lb-ft ) / 2.4 = ...
... Pult, safe = 5484 lb.
WHOA! ...
This is interesting. Note that SHEAR controls. In terms of how the beam will fail ... we get steel yielding. In terms of `design' ... shear controls.
The design `Safe' or `Allowable' Concentrated Occupancy Live load for the beam is ...
... Psafe = 5480 lb.
Note that the safe load is approximately half of the load at failure, which is overall typical for members in structural design.
Next will look at the deflection at `safe' load (here).
References
Strength Calculations of an Experimental Beam, Jeff Filler, Associated Content.
Building Code Requirements for Structural Concrete, ACI 318, American Concrete Institute, P.O. Box 9094, Farmington hills, Michigan, 48333.
Load Combinations in the Strength Design of Structural Concrete, Jeff Filler, Associated Content.
Load Factors in the Strength Design of Structural Concrete, Jeff Filler, Associated Content.
Deflection of a Reinforced Concrete Beam at Safe Service Load, Jeff Filler, Associated Content.
Published by Jeff Filler
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