Development Length for an Experimental Beam

Earlier (here) we found that our experiment beam would support a maximum concentrated load P = P_y = 9600 lb before the steel in the beam yields, and then be loaded further to a value of P = P_ult = 10,100 lb at which the steel has yielded and the concrete also crushes. This condition requires that the full (yield) strength of the steel is `developed' ... that the load required of the steel is effectively transferred to the steel via the bond between the steel and the concrete. Current practice of ensuring this `transfer' is to require a minimum length of contact between bar and concrete ... the so-called `development length'. Development length varies per bars size, strength, concrete strength, bar cover and spacing with regard to other bars, how much of the strength of the steel is being called upon, and other factors. We will turn to the Code equation for development length for our experimental beam; tables have also been developed for certain bar sizes, concrete and steel strengths, and other specific conditions, such as Table 13.8 of the Ambrose text.

From an earlier lesson (here), ...

... Chapter 12 in the ACI Code gives us our basic equation for development length, ℓ_d , ...

ℓ_d = { (3/40) (f_y /√f '_c) (α β γ λ ) / [ (c + k_{t r}) / d_b ] } d_b

where,

f_y = specified yield strength of the reinforcement (psi) ... in our case 60,000 psi.

f '_c = specified 28-day compressive strength of the concrete (psi) ... (and where √f '_c carries the units of psi), ... in our case 4000 psi.

α = bar location factor, which is 1.0 except for horizontal bars with more than 12 in. of fresh concrete cast below them, ... so, 1.0.

β = bar coating factor, which is 1.0 for uncoated bars, and 1.2 or so for epoxy coating ... the epoxy somewhat smoothing out the valleys and ridges (?) ... So ... 1.0.

γ = bar size factor, 0.8 for # 6 and smaller bars; 1.0 otherwise ... So ... 0.8.

λ = lightweight concrete factor, 1.0 for normalweight concrete ... 1.0.

c = spacing / cover dimension, being the smaller of the distance from the center of the reinforcement bar(s) to the nearest face of concrete, or half the center-to-center spacing of the reinforcement, ... in our case ... the distance to the nearest concrete face is 2.0 in. ... and there are no other bars so there really isn't a spacing number.

k_{t r} is a transverse reinforcement index ... (see below, but) which may be taken as 0, ... so let's take it to be 0 ...

... and ...

d _b = is the diameter of the bar being developed ... in our case # 6 is 0.75 in.

The Code places an upper limit on the quantity (c + k_{t r}) / d_b of 2.5.

So,

α = 1.0 ... β = 1.0 ... γ = 0.8 ... λ = 1.0 ...

c = 2.0, so ...

... thus ... (c + k_{t r}) / d_b = (2.0 + 0) / 0.75 = 2.67 ...

... but not greater than 2.5 ... so ... 2.5.

Thus,

ℓ_d = { (3/40) (f_y /√f '_c) (α β γ λ ) / [ (c + k_{t r}) / d_b ] } d_b =

... [(3/40) (60,000 / √4000 ) (1.0) (1.0)(0.8)(1.0) / 2.5] d_b =

ℓ_d = ... (22.8) d_b ... = 22.8 (0.75 in.) = 17.1 in.

So, ... our beam is 7 ft long. I spans supports 6.0 ft apart. The maximum moment is at mid-span, where we need the reinforcement fully developed. We have not discussed the exact rebar length, but let's say that it is 6ft - 4 in. ... so that there is 2 in. lapping past each support. In our experiment the supports will provide a bearing length of essentially zero inches (you'll see). This means that from where the steel needs to be developed, going both directions, we have 3 ft - 2 in. of bar. We need 17.1 in. of bar to develop the steel. GOOD!

Summarizing ... available length of rebar for developing bond = 88 in. ≥ 17 in. = development length required ... good!

And we'll need the 17.1 in. later, also.

Notes:

1. If the design load for the beam is less than the safe load (incorporating appropriate factors of safety), then the demand on the development of reinforcement is less. This is handled in the Code by allowing the development length to be reduced by the ratio As _needed / As _provided.

References

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

Simplified Engineering for Architects and Builders, Ambrose, J. and P. Tripeny, 10^th edition, John Wiley & Sons, Hoboken, New Jersey.

Reinforcement Development Length or Bond, Jeff Filler, Associated Content.

Building Code Requirements for Structural Concrete, ACI 318, American Concrete Institute, P.O. Box 9094, Farmington hills, Michigan, 48333.