Deflection of a Reinforced Concrete Beam at Safe Service Load
An Experiment in Reinforced Concrete Design
Background
In the previous lesson (here) we determined the `safe' load for the beam, presuming similar beams will be cast, and put to future use (say, in structures). Not only must our beams be safe for future use - they must also be `serviceable' - meaning they must not have excessive deflection (sag or bounce) at these future loads. We evaluate deflection at the `safe' load level, not the failure level, when we looked at breaking the beam, or `ultimate', when we looked at safety.
Since we will consider the deflection of the beam at a `load level' way less than ultimate, it will be less cracked. And we have a tool for handling this ... the Effective Moment of Inertia, Ie.
Effective Moment of Inertia
From before (here) ...
I e = Ig (Mcr/Ma)3 + Icr [1 - (Mcr/Ma)3] ...
where ...
Ig = 393 in.4 ... and ... Icr = 117 in.4
and ...
Mcr = 3266 lb-ft (also unchanged).
Ma is now the moment at safe load ...
M = ωL2 /8 + P L/ 4 ... note that this is WITHOUT the load factors ...
Ma= 54.4 plf (6 ft)2 / 8 + 5480 lb (6 ft) / 4 = 245 + 8220 = 8465 lb-ft.
So,
Ie = 393 (3266 / 8465)3 + 117 [ 1 - (3266 / 8465)3 ] = 393 (0.386)3 + 117 [ 1 - (0.386)3 ] =
... Ie= 393 (0.057) + 117 [ 1- 0.057 ] = 22.6 + 110.3 = 133 in.4.
Well, a little less cracked.
Dumping this into our deflection equation ...
Wait! ... there are several checks we do for deflection in reinforced concrete ...
Let's just do one ... Immediate Deflection due to Live Load ...
The Code requires, say for floors*, that this not exceed L/360.
So, let's do it ...
Δ P, safe= P safe L3 / 48 E Ie = 5480 lb (72 in.)3 / (48 x 3,600,000 psi x 133 in.4) = 0.089 in.
( ... assuming 4000 psi concrete ... )
Let's see if this is satisfactory ...
L / 360 = 72 in. / 360 = 0.20 in.
Since Δ P, safe = 0.09 in. ≤ L/360 = 0.20 in. ... GOOD!
Note that the deflection under the safe service load is about half that at `failure'. We would expect that, since the load is about half that at failure, and up until failure, or at least close, the beam is `elastic'.
We'll do some (elastic spring) stiffness calcs in the next lesson.
* Floors not supporting or attached to nonstructural elements likely to be damaged by large deflections
There are other `kinds' of deflection that we look at with reinforced concrete, but we won't get into those here.
References
Safe Load on an Experimental Reinforced Concrete Beam, Jeff Filler, Associated Content.
Strength Calculations for an Experimental Reinforced Concrete Beam, Jeff Filler, Associated Content.
Building Code Requirements for Structural Concrete, ACI 318, American Concrete Institute, P.O. Box 9094, Farmington hills, Michigan, 48333.
Published by Jeff Filler
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- Reinforced Concrete WallsLesson describing the Code requirements of reinforced concrete walls with link to calculation example.
Flexure of Reinforced Concrete BeamsLesson gives design equations with example and discussion for flexural strength of a reinforced concrete beam and also presents Code minimum and maximum reinforcement limits- Beam Shear in Reinforced Concrete - RevisedLesson provides basic equations for shear strength in reinforced concrete with example calculations.
- Introduction to Deflections of Reinforced Concrete BeamsLesson discusses the prescriptive limitations in the Code allowing us to `not' have to do detailed calculations with regard to deflections. Lesson also discusses creep in reinforced concrete.
Deflections of an Uncracked Reinforced Concrete BeamArticle goes through equations and example calculations for deflection of an uncracked reinforced concrete beam.
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