Calculating the Length of Bent Bars
By Dick Birley, President of Condor Rebar Consultants, Inc.
First published in Concrete International Magazine, March 2006
Tight tolerances lead to some interesting observations
When reinforcing bars are fabricated with bends, the straight bar is initially cut to a length that is less than the sum of the specified dimensions of the bent bar. The difference between the detailed length and the cut length is the "bend curvature deduction" and may be known in the trade by other names such as gain, creep, and gyp. Generally, fabricators and programmers obtain the bend curvature deduction from a bend deduction table, such as the one shown in Table 1, that lists the deductions for 45- and 90-degree bends of common bar sizes.
Figure 1 shows a No. 8 (No. 25) bar with a standard hook and sides measuring 1 ft 4 in. and 4 ft (400 and 1220 mm). The detailed length of this bar would be the sum of the two sides, or 5 ft 4 in. (1620 mm). From Table 1, the deduction for a 90-degree bend on a No. 8 (No. 25) bar is 2-1/2 in. (65 mm). Thus, in this case, the cut length of the bar would be 5 ft 1-1/2 in. (1555 mm).
The usual standard for measuring the cut length of a bar is along the actual centerline of the bar, which corresponds to the neutral axis of the bar cross section prior to bending. The cut length of a bent bar is shorter than the sum of the finished dimensions for two reasons. The first reason is obvious—the fillets created at the bend points have an arc length that is shorter than the sum of the intersecting tangents. This component of the bend deduction can be found with a simple mathematical calculation. The second reason isn't so obvious. While the outer fibers of the bar are free to elongate, the inner fibers of the bar are restrained by friction against the bending mandrel—the neutral axis therefore shifts inward toward the mandrel. This component cannot be easily calculated. Fortunately, however, the discrepancy resulting from calculating lengths based on the actual centerline is rarely a concern.
A few years ago, some interesting surprises were found while detailing the reinforcing for a large precast segmental bridge. There were more than 200 different bar shapes, of which most were multi-sided, closed stirrups with varying angles. The client insisted that the cut length of the bars had to be within a tolerance of 10 mm (3/8 in.), regardless of the number of bends on the bar or the angles. Obviously, this could not be accomplished by using a chart. The ability to calculate the precise bend deduction for each bend at any angle on each bar size had to be introduced into the detailing software.
To meet this challenge, the calculations had to include a term that would allow the effective centerline of the bend to shift inward from the actual centerline. One way to include this effect in the arc length calculations is to use an effective centerline located a distance ƒ(db/2) from the inside of the bend, where ƒ is a variable labeled the friction factor. With zero friction, ƒ = 1 and the effective centerline is located at the actual centerline. With zero slip, ƒ = 0 and the effective centerline is located at the inside face of the bend.
For a given bar size and mandrel, a calibration for the ƒ value can be made by carefully measuring the lengths of straight bars, bending them to the same angle, measuring and summing the dimensions of the two resulting sides, deducting the original straight length from this sum to find the total bend curvature deduction, and solving the equations defined in Fig. 2.
Many factors, including bar size, steel grade, angle of the rib to the mandrel, the mandrel material, the amount of wear on the mandrel, the bending speed, and the bar temperature (the shop operated through winter and summer), were found to affect the value of ƒ. There were four bending machines in the shop, and each one was assigned a value for ƒ for each bar size. Bending speed was set at a constant prescribed rate for each bender, and the whole process was rechecked every couple of weeks. The 10 mm (3/8 in.) tolerance could be successfully met, provided the client could successfully control the various factors affecting ƒ.
Surprisingly, the value of ƒ was usually about 0.2 to 0.25 and rarely approached 0.3. If the bending speed was increased, ƒ would drop to as low as 0.1. There was a small amount of Grade 75 (520 MPa) bar on the project for which ƒ had to be set to zero. This seemed to indicate that friction with the mandrel was so high (due to the force required to bend the bar) that there was no slip along the inner curve and that all of the elongation was along the outside curve of the bar.
Considering the values for ƒ, it became apparent that, when a bar was bent, there was much more elongation along the outside of the bend than anticipated. If the mandrel was new and very smooth, there seemed to be less friction with the mandrel, which increased the value of ƒ. Increasing the bending speed seemed to increase friction against the mandrel, which decreased the value of ƒ. In the case of Grade 75 (520 MPa) bars with an ƒof 0, the difference between elongations along the outside and inside of the bend was extreme.
As an aside, the client found the task of monitoring the factors affecting the bending so onerous that the requirement for the 10 mm (3/8 in.) cutting tolerance was quietly dropped, and the shop gradually returned to normal fabricating practices.
Thanks to M. Lount, FACI, for contributing the calculations for the bend curvature deduction and identifying the friction factor.
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