Jonathan Ochshorn

© 2012 Jonathan Ochshorn. This discussion of web steel (stirrups) in reinforced concrete beams, based on ACI-318 (*Building Code Requirements for Structural Concrete*) supplements a free online stirrup spacing calculator that I developed. The following symbols or notations are used: *Vu* = design shear force (includes load factors); *Vc* = concrete capacity to resist shear in lb units, taken as 2 x *b* x *d* x *f'c ^{0.5}* (where

Reinforced concrete beams with uniformly-distributed loads have a shear diagram with maximum positive shear at the left support, maximum negative shear at the right support, and a linear variation between those two extremes. This creates two triangular shear diagrams on either side of the beam's centerline, each with the same absolute values of shear force. For this reason, the required web steel can be determined for a single triangle, and the stirrups deployed there can be reproduced as a mirror image on the other half of the beam. As can be seen in Figure 11, the maximum shear force is actually taken at a distance *d* from the face of the beam support.

In the shear force diagrams used here, *Vu* is divided by the strength-reduction factor φ so that its relationship to the steel and concrete capacities to resist shear becomes clearer: *Vu* / φ = *Vc* + *Vs*. Within the shear force triangle, up to four zones may be present: Zone I, where the shear force *Vu* (divided by the strength reduction safety factor, φ), is less than or equal to 0.5*Vc*, requires no web steel at all; Zone II is where *Vu* / φ is less than or equal to *Vc* (but excluding Zone I) and only requires minimum web steel with spacing, *s* = 2*Asfy*/(0.75*b* × *f'c*^{0.5}) __<__ 2*Asfy*/(50*b*); Zone III is where any calculated spacing cannot exceed the "regular" maximum spacing determined by the smaller of *d*/2, 24 in., or the spacing corresponding to minimum web steel; and Zone IV is where any calculated spacing cannot exceed the "reduced" maximum spacing determined by the smaller of *d*/4, 12 in., or the spacing corresponding to minimum web steel. As can be seen in the diagram, the boundary between Zone III and Zone IV — i.e., between "regular" and "reduced" maximum spacing — is defined as the point where *Vu* / φ equals 3*Vc*.

Because the maximum shear force may occur in any of these four zones (shown in Figures 1-9: click on any image to enlarge), and because the computed stirrup spacing at any point along the length of the beam, *s* = 2*Asfyd*/*Vs*, may or may not fall within the limits governed by the two maximum spacing criteria (*d*/2, 24 in., or min. web steel vs. *d*/4, 12 in., or min. web steel), there are nine possible patterns of stirrup spacing that can develop, as follows:

*s* = 2*Asfyd*/*Vs*, is needed up until the point where regular maximum spacing governs, with no web steel needed in Zone I (see Figure 4).

*s* = 2*Asfyd*/*Vs*, is needed in Zone III up until the point where regular maximum spacing governs, with no web steel needed in Zone I (see Figure 6).

*s* = 2*Asfyd*/*Vs*, is needed until reduced maximum spacing begins in Zone IV, regular maximum spacing is needed in Zones III and II, with no web steel needed in Zone I (see Figure 7).

*s* = 2*Asfyd*/*Vs*, is needed until reduced maximum spacing begins in Zone IV, then the spacing, *s* = 2*Asfyd*/*Vs*, is needed beginning in Zone III until regular maximum spacing governs, with no web steel needed in Zone I (see Figure 8).

*s* = 2*Asfyd*/*Vs*, is needed until regular maximum spacing begins in Zone III, with no web steel needed in Zone I (see Figure 9).

These nine possible patterns, and the stirrup spacing that results, are summarized in Figure 10.

The stirrup spacing options described here — based on simplified calculations permitted in ACI-318 (*Building Code Requirements for Structural Concrete*) — are nevertheless somewhat arbitrary. For example, it is possible to find a single spacing corresponding to the greatest shear force, and to use that spacing throughout the beam. At the other extreme, it is also possible to continuously change the stirrup spacing as the shear force changes along the length of a beam. Between these two extremes, a wide range of choices are possible, including those outlined here. The criterion I have used is to change the spacing at key points where maximum spacing limits become viable. It is, of course, possible to determine additional points governed by intermediate spacings.

Figures 1-9 show triangular shear diagrams extending from a value of zero at the beam centerline to a maximum value of 5*Vc*, above which reinforced concrete beams are not permitted. But the "strength-reduced" shear force in an actual beam (i.e., *Vu* / φ) may never reach that maximum value. As can be seen in Figure 11, these diagrams are not intended to represent the shear forces present in any particular beam, but only the four zones within one of which the maximum shear force, *Vu*/φ, must occur. Figure 11 shows a uniformly-loaded beam that happens to fall into category "D" shown in Figure 4; that is, the maximum shear force occurs in Zone III, and this maximum shear force is greater than the shear force corresponding to "regular" maximum spacing. In such a case, calculated minimum spacing can be extended from the face of the beam's support (actually, the first stirrup typically starts at a distance *s*/2 from the face of the beam's support, where *s* is the initial calculated minimum spacing) up until "regular" maximum spacing controls the design. This "regular" maximum spacing then extends through part of Zone III and all of Zone II up until the point where stirrups are no longer needed (i.e., in Zone I).

First posted 5 June 2012. Last updated: 27 May 2015