Rebar Development and Splices: Method

Rebar Development and Splices: Method, Equations and Limits

What the tool does

The tool gives the lengths over which a deformed reinforcing bar must be embedded or lapped to develop its yield strength: the tension development length l_d, the compression development length l_dc, the development length of a standard hook l_dh and of a headed bar l_dt, the tension lap l_st and the compression lap l_sc. The relations are those of ACI 318-19 Chapter 25, written out as mathematics with the section numbers. The code is cited and none of its tables is reproduced: every leading coefficient is an input in the group "Code Coefficients" and every modification factor ψ is an input, each with the code value as its default and that value named in its tooltip. The examples the tool reproduces are those printed in two public FEMA documents (below).

Units

The code equations are dimensional. In customary units f_y and f′_c are in psi and lengths in inches; √f′_c is the square root of the number of psi and is limited to 100 (Section 25.4.1.4). The tool computes in those units. When SI is selected the inputs are converted exactly and the results are shown in mm and MPa, with the coefficients converted too: with √(145.038 psi/MPa) = 12.043, the coefficient 3/40 becomes 1/1.107, the hook divisor 55 becomes 23.02 (the d_b^1.5 term brings in √25.4), the head divisor 75 becomes 31.39, 1/50 becomes 0.2409, 0.0003 becomes 0.0435 and 0.0005 becomes 0.0725. The SI edition of the code prints these rounded (1/1.1, 23, 31, 0.24, 0.043, 0.071) and rounds the minimum lengths to 300, 200 and 150 mm, so a hand calculation with the SI edition differs from the tool by up to about 2 percent.

Tension development of a straight bar (25.4.2)

General equation: l_d = C_t · (f_y / (λ √f′_c)) · (ψ_t ψ_e ψ_s ψ_g / ((c_b + K_tr)/d_b)) · d_b, with C_t = 3/40, never less than 12 in.

c_b is the lesser of the distance from the centre of the bar to the nearest concrete surface and one-half of the centre-to-centre spacing of the bars being developed. The tool takes the first as cover + stirrup diameter + d_b/2 from the inputs. K_tr = 40 A_tr / (s n), where A_tr is the area of all transverse reinforcement within the spacing s that crosses the potential plane of splitting and n is the number of bars developed along that plane; the code permits K_tr = 0. The confinement term (c_b + K_tr)/d_b is limited to 2.5: below that value the concrete splits along the bar, above it the bar pulls out and more cover or more stirrups no longer help. The equation descends from the regression of beam-splice tests by Orangun, Jirsa and Breen (1977).

Factors (code values, Table 25.4.2.5): λ = 0.75 for lightweight concrete, 1.0 for normalweight; ψ_t = 1.3 for horizontal bars with more than 12 in of fresh concrete cast below, else 1.0; ψ_e = 1.5 for epoxy-coated bars with clear cover under 3 d_b or clear spacing under 6 d_b, 1.2 for other epoxy-coated bars, 1.0 uncoated; the product ψ_t ψ_e is limited to 1.7; ψ_s = 0.8 for No. 6 and smaller bars, 1.0 for No. 7 and larger; ψ_g = 1.0 for Grade 40 and 60, 1.15 for Grade 80, 1.3 for Grade 100 (new in the 2019 edition). The tool takes a grade between two of these at the higher value and puts the size break at 0.8125 in, so that a 20M bar (19.5 mm) counts with No. 6.

Simplified table (Table 25.4.2.3). The table is the same equation with the confinement term fixed. Its first row (clear spacing and clear cover of at least d_b with at least the code-minimum stirrups or ties along l_d, or clear spacing of at least 2 d_b and clear cover of at least d_b) uses 1.5, which gives the divisors 20 (No. 7 and larger) and 25 (ψ_s = 0.8); "other cases" uses 1.0, which gives 40/3 and 50/3. The tool finds the row from the clear spacing, the clear cover and the "Minimum Stirrups" select and prints the table value beside the general one. When the second condition of the first row holds, the actual term is at least 1.5 and the general equation can never be longer; under the first condition the table relies on the stirrups, and with small K_tr the general equation can come out longer, which the page then says.

Excess reinforcement (25.4.10). Where more steel is provided than required, l_d and l_dc may be multiplied by A_s,required / A_s,provided, but not below the minimum, and not at noncontinuous supports, where development of f_y is required, for bars required to be continuous, for hooked, headed and mechanically anchored bars, in seismic-force-resisting systems of Seismic Design Categories C to F, or for the anchorage of piles to pile caps in those categories. The tool applies the ratio only when the select says so. The lap lengths never use it.

High-strength bars (25.4.2.2). For f_y of 80,000 psi or more with bars closer than 6 in on centre the code requires transverse reinforcement with K_tr of at least 0.5 d_b; the tool prints a note when that is not met.

Compression development (25.4.9)

l_dc = the greater of f_y ψ_r / (50 λ √f′_c) · d_b and 0.0003 f_y ψ_r d_b, never less than 8 in. ψ_r = 0.75 for bars enclosed by a spiral, by closely spaced ties (No. 4 at 4 in or less) or hoops, else 1.0. The two expressions cross at f′_c = 4,444 psi. Hooks and heads do not help in compression.

Standard hooks (25.4.3 and 25.3)

ACI 318-19: l_dh = f_y ψ_e ψ_r ψ_o ψ_c / (55 λ √f′_c) · d_b^1.5, never less than 8 d_b and 6 in. ψ_e = 1.2 for epoxy-coated bars; ψ_r = 1.0 for No. 11 and smaller bars with confining ties A_th of at least 0.4 A_hs or with hooks spaced at 6 d_b or more, else 1.6; ψ_o = 1.0 for No. 11 and smaller bars ending inside a column core with side cover of at least 2.5 in or with side cover of at least 6 d_b, else 1.25; ψ_c = f′_c/15,000 + 0.6 below 6,000 psi, else 1.0. The defaults of the form are the unfavourable values 1.6 and 1.25, which fit closely spaced beam bars hooked outside a column core; set both to 1.0 for a beam bar hooked into a confined column. For a bar larger than No. 11 the tool uses at least 1.6 and 1.25 whatever is entered, in the result and in the bar schedule.

ACI 318-14 (selectable): l_dh = f_y ψ_e ψ_c ψ_r / (50 λ √f′_c) · d_b, with the same minimums; there ψ_c = 0.7 is a cover factor (side cover of at least 2.5 in and, for a 90 degree hook, 2 in on the extension) and ψ_r = 0.8 a tie factor. That edition allowed the excess-reinforcement ratio on hooks, and the tool applies it only with this form. The 2019 equation is longer for large bars and for unconfined, closely spaced hooks.

Geometry (Table 25.3.1). Inside bend diameter 6 d_b for No. 3 to No. 8, 8 d_b for No. 9 to No. 11, 10 d_b for No. 14 and No. 18. Straight extension 12 d_b for a 90 degree hook; the greater of 4 d_b and 2.5 in for a 180 degree hook. The tool also prints the out-to-out dimension and the bar length beyond the start of the bend (arc along the bar centre line plus the extension). Stirrup and tie hooks (Table 25.3.2): bend diameter 4 d_b for No. 3 to No. 5 and 6 d_b for No. 6 to No. 8; extension the greater of 6 d_b and 3 in for 90 degree hooks of No. 3 to No. 5 and for 135 degree hooks, 12 d_b for 90 degree hooks of No. 6 to No. 8, the greater of 4 d_b and 2.5 in for 180 degree hooks. l_dh is measured from the critical section to the outside of the hook.

Headed bars (25.4.4)

ACI 318-19: l_dt = f_y ψ_e ψ_p ψ_o ψ_c / (75 √f′_c) · d_b^1.5, never less than 8 d_b and 6 in, with ψ_p = 1.0 for No. 11 and smaller bars with parallel ties A_tt of at least 0.3 A_hs or spaced at 6 d_b or more, else 1.6, and ψ_e, ψ_o, ψ_c as for hooks. ACI 318-14: l_dt = 0.016 f_y ψ_e / √f′_c · d_b with f′_c taken as at most 6,000 psi. A head may be used only for bars not larger than No. 11 in normalweight concrete, with a net bearing area of at least 4 A_b, clear cover of at least 2 d_b and centre-to-centre spacing of at least 3 d_b (2014: clear spacing of at least 4 d_b and f_y of at most 60,000 psi); the tool checks these and says which fail.

Lap splices (25.5)

Tension. Class A, 1.0 l_d, where the steel provided is at least twice that required over the whole lap and at most 50 percent of it is spliced within the lap length; Class B, 1.3 l_d, in every other case; never less than 12 in. Here l_d is the equation value without its 12 in minimum and without the excess-reinforcement ratio (25.5.1.4), which is why a Class B lap of a small bar can be shorter than 1.3 × 12 in. Bars larger than No. 11 are not lap spliced. In a non-contact lap of a flexural member the spliced bars may be no farther apart, centre to centre, than the lesser of one-fifth of the lap length and 6 in. For bars of different size the lap is the greater of l_d of the larger bar and l_st of the smaller.

Compression. l_sc = 0.0005 f_y d_b for f_y up to 60,000 psi and (0.0009 f_y − 24) d_b above; above 80,000 psi not less than the tension lap; the lap so found is one-third longer where f′_c is below 3,000 psi; never less than 12 in. In columns the lap may be multiplied by 0.83 where ties through the lap have an effective area of at least 0.0015 h s in both directions and by 0.75 within a spiral (10.7.5.2.1), entered as the column factor; the tool applies it to the compression expression and not to the tension-lap floor of bars above 80,000 psi. Column bars in tension follow the tension classes: Class A only where the bar stress is at most 0.5 f_y, at most half the bars are spliced at a section and the laps are staggered by l_d.

Bundled bars (25.6)

A bundle of n bars is treated as one bar of the same total area, d_eq = d_b √n, for cover, spacing, the row of the simplified table and the confinement term. The development or lap length is that of the individual bar, increased 20 percent for a three-bar bundle and 33 percent for a four-bar bundle. With the bundle select the cover and spacing entered are those of the bundle, and the tool computes c_b from the bundle's equivalent diameter.

Embedment check and the stress developed

The available length is compared with the required length of the chosen anchorage and printed as a margin of safety, M.S. = l_available / l_required − 1, with the method named ACI. For a straight bar the stress that a shorter embedment develops is taken in proportion, f_s = f_y l / l_d (at most f_y), with l_d the equation value; this is the usual assumption of uniform average bond. It is not printed for hooks, heads and laps: their anchorage strength does not grow in proportion to the length, which is the reason the 2019 code withdrew the excess-reinforcement ratio for hooked and headed bars.

Bar schedule and chart

The schedule repeats all lengths for No. 3 to No. 18 (ASTM A615 nominal diameters and areas) and, on request, for the metric sizes 10M to 55M, with the cover, spacing, transverse steel and factors of the form; ψ_s and ψ_g follow the size and the grade and c_b follows each diameter. Figure 2 plots the columns against the bar diameter. The hook and head lengths grow faster than the straight length because of the 1.5 power.

Checks against published values

FEMA P-2192-V1 (2021), which uses ACI 318-19: p. 161, No. 7 bar, f′_c = 8,000 psi, ψ_t = 1.3, table value 38.2 in; pp. 167–168, No. 8 coupling-beam bars at 1.25 times the table value, about 55 in. FEMA P-751 (2012): p. 7-47, No. 8 top bar, f′_c = 5,000 psi, confinement term 2.5, l_d = 33.1 in, Class B lap 43 in and 34 in for the bottom bars; p. 8-14, No. 3 bar, 14.2 in; p. 8-38, No. 9 hook by the earlier hook form with the 0.7 cover factor, 22.5 in. USACE EM 1110-2-2104 (2016) p. D-17 prints the earlier hook form for a No. 11 bar. The tool reproduces each printed value; the test script lists them with 40 seeded cases checked against an independent implementation.

Limits

Not covered: welded deformed and plain wire reinforcement and its laps, mechanical and welded splices (which must develop at least 1.25 f_y), mechanically anchored bars qualified by test, prestressing strand, the seismic development lengths of Chapter 18 (hooks in special moment frame joints), the rules for cutting off and extending flexural reinforcement, hooks or heads on bundled bars, and the AASHTO LRFD bridge specifications, whose development equations have a different form and are not offered. The designer chooses the ψ factors: the tool does not verify that the ties, covers and spacings that justify a favourable factor are actually present.