Corbels and nibs

How a reinforced concrete corbel and a continuous nib are designed with the STATIX corbel calculator on its Eurocode 2 route. It covers the strut-and-tie model of Annex J.3 and 6.5, the nodes and the tie, the horizontal force, the anchorage of the tie, the links, the bearing of 10.9.5, and nibs and half joints.

1. Learning outcomes

After this lesson you can:

tell a corbel from a short cantilever (J.3(1): a_c < z₀) and draw its strut-and-tie model (6.5, Figure J.5);
find the lever arm z₀, the strut angle θ and the tie force F_td with the horizontal force H_Ed, and check the strut and both nodes (6.5.2, 6.5.4);
anchor the main tie at the front and in the column (J.3(4), 8.4, 8.6) and choose the links by J.3(2) or J.3(3);
check the bearing: its stress, the net and the nominal bearing length (10.9.5) and the partially loaded area (6.7);
design a continuous nib as a corbel per metre, hang its load up into the beam (6.2.1(9)) and know the half-joint models (10.9.4.6);
say which checks on the page follow EN 1992-1-1, which are the page’s own choices, and which you do by hand.

2. Before you start

RC Design 1 (Eurocode), lesson 3 (shear and links) and lesson 8 (anchorage and detailing).
The corbel calculator, set to Eurocode 2 (EN 1992-1-1).
A pocket calculator and about 60 minutes.
The same lesson on the SA code: RC Design 2, lesson 4 (SANS 10100-1).

What the corbel page’s Eurocode 2 route uses.
One corbel on the face of a column, with up to four load points, or a corbel on each face (double-sided). You give the design load F_Ed (the page calls it F_v, ULS): work 1.35 G_k + 1.5 Q_k by EN 1990 6.10 first. The page then uses H_Ed = the larger of your F_h and 0.2 F_Ed. The 0.2 F_Ed is the page’s own floor, common practice: EN 1992-1-1 gives no minimum, and 10.9.5.1(2)P asks for the restraint forces to be taken into account.
Materials. f_ck (cylinder, 12 to 50 MPa) and f_yk, γ_c 1.5, γ_s 1.15. α_cc and α_ct are 1.0, the recommended values of 3.1.6, as on the pages of lessons 2 and 3; this page has no input for them. f_ctm from Table 3.1.
Strut and tie. J.3(1) with Figure J.5 and 6.5: 1.0 ≤ tan θ ≤ 2.5 (a steeper strut is laid at tan θ = 2.5 by moving the load arm out to z₀/2.5). The strut and the node at the foot of the corbel work at 0.6ν′f_cd (eq. 6.56, ν′ = 1 − f_ck/250), the node under the plate at k₂ν′f_cd with k₂ 0.85 (eq. 6.61), the tie at f_yd (6.5.3). The page takes moments about the lower node inside the column, as Figure J.5 draws it: arm a_c + w/2, w = F_Ed/(b 0.6ν′f_cd) (section 3 and Example A).
Links and shear. J.3(2) with k₁ 0.25 and J.3(3) with k₂ 0.5; V_Rd,c by 6.2.2(1) with σ_cp = 0; F_Ed ≤ 0.5 b d ν f_cd (eq. 6.5) without the β of 6.2.2(6). A_s,min of 9.2.1.1(1) and 4 % at most on the main tie: the page’s choice, a corbel taken as a short beam (J.3 gives no minimum).
Anchorage. J.3(4): in the corbel from the inner face of the plate, in the column from the near-face column bars to the far face. 8.4 with η₁ 0.7 when h > 250 mm and α₁ only (the other α at 1, on the safe side); a welded cross bar by 8.6(2) (F_wd = 0.5 A_sf_yd) or a loop (Table 8.1N, eq. 8.1); links by 8.5.
Bearing. 10.9.5.2(2): f_Rd = 0.4 f_cd dry, f_bed ≤ 0.85 f_cd bedded (give f_bed under More inputs, or the row reads NOT CHECKED), 0.85 f_cd on a steel plate; b₁ ≤ 600 mm when dry (10.9.5.2(3)). The net length a₁ by the row of Table 10.2 for a concentrated support (a beam), always with the 20 mm of an isolated member (10.9.5.3); the front edge a₂ by Table 10.3; 6.7 with A_c1 = A_c0.
Cover and fire. 4.4.1 (Table 4.4N for your exposure class, class S4, Δc_dev 10 mm); EN 1992-1-2 Table 5.5 with the corbel taken as a simply supported beam.
Not checked: the distance a₃ and the deviations Δa₂, Δa₃ of eq. 10.6, the edge distance of a bearing over loops (10.9.4.7), crack control (J.3(5)), the column under the corbel’s moment, continuous nibs (on this route the page says NOT CHECKED: EN 1992-1-1 has no nib clause) and half joints. Other annexes may use other values.

3. How a corbel works

When a bracket is a corbel. EN 1992-1-1 J.3(1) calls a bracket a corbel when the load line is closer to the column face than the lever arm z₀ of its tie, a_c < z₀, and lets it be designed with a strut-and-tie model by 6.5. A longer bracket bends like a short cantilever: design it by 6.1 and 6.2. The load comes onto a bearing plate a_c from the face; a horizontal force H_Ed acts on the top surface. h_c is the depth at the column face.

The strut-and-tie model of Figure J.5. The load goes down a concrete strut from the plate to a node at the foot of the corbel, in the column, and a steel tie along the top holds the top of the strut. Moments about the lower node give the tie force: F_td z₀ = F_Ed a + H_Ed (z₀ + a_H). Here a is the arm of F_Ed about the node and a_H the height of the bearing surface above the tie. The tie carries H_Ed as well. The strut slope must lie between tan θ = 1.0 and 2.5 (J.3(1)), with θ measured from the tie: 45° to 68°, steeper than the 21.8° to 45° that 6.2.3 allows in a beam.

Finding z₀. The lower node is a compression node (6.5.4(4) a)). The page takes it hydrostatic: the same stress on every face, equal to the strut limit 0.6ν′f_cd. The face that takes the horizontal force (a vertical face) is then x = (F_td − H_Ed)/(bσ_Rd,max) high, and its centre lies z₀ = d − x/2 below the tie. Put that in the moment equation and z₀ follows from a quadratic. If z₀/a comes out above 2.5, the strut is too steep for J.3(1): the page then moves the load arm out to z₀/2.5, which raises the tie force a little.

Where the lower node sits. The node carries F_Ed down into the column, so its bottom face, which takes F_Ed, is w = F_Ed/(bσ) wide and must lie in the column: under the corbel’s soffit nothing pushes up. Figure J.5 draws the node inside the column. The arm of F_Ed about the node centre is then a_c + w/2. The page does this: arm a_c + w/2, or z₀/2.5 when the strut would be steeper than tan θ = 2.5. Moments about a point on the column face, as some older texts take them, give an F_td low by roughly F_Ed(w/2)/z₀, and tie and anchorage results that are not on the safe side. Example A shows both.

4. The nodes, the tie, the links and the horizontal force

Stress limits. A strut that crosses cracks, with tension across it, is held to 0.6ν′f_cd (6.5.2(2)), with the recommended ν′ = 1 − f_ck/250. A node is checked on its faces: up to k₁ν′f_cd where only struts meet, k₂ν′f_cd where one tie is anchored and k₃ν′f_cd where ties run two ways (6.5.4(4), recommended k values). 6.5.4(5) allows up to 10 % more in some cases, such as triaxial compression; the page does not take it. 6.5.2(3) also points to 6.2.2 and 6.2.3 when a strut runs straight from a load to a support, as in a corbel.

The node under the plate. The tie is anchored here, so this is a compression-tension node (6.5.4(4) b), Figure 6.27). Two stresses are checked against k₂ν′f_cd: the bearing under the plate, F_Ed over the plate area, and the strut where it leaves the node. The strut is a₁ sinθ + u cosθ wide, where a₁ is the plate length and u = 2a_H the height of the node round the tie. The page also keeps the strut end within the strut limit 0.6ν′f_cd, on the safe side.

Anchoring the main tie. J.3(4) asks for the tie to be anchored at both ends. At the front the length starts at the plate’s inner edge, so the bar has little room: a welded transverse bar (8.6) or a loop (Figure 8.1, mandrel by Table 8.1N and eq. 8.1) does the work. In the column the length counts from the vertical bars at the near face, and the bar runs to the far face, usually bent down. The basic length is l_b,rqd = (φ/4)(σ_sd/f_bd) (8.4.3), with f_bd = 2.25η₁η₂f_ctd (8.4.2) and η₁ = 0.7 for a top bar in a member over 250 mm deep. A welded cross bar of 14 to 32 mm takes F_btd (eq. 8.8N, at most the weld strength F_wd), and σ_sd drops by F_btd/A_s (8.6(2)). The anchorage is never less than l_b,min (eq. 8.6).

Links. A short corbel, a_c ≤ 0.5h_c, gets closed horizontal or inclined links of at least k₁A_s,main in addition to the main tie (J.3(2), recommended k₁ = 0.25). They hold the strut together against splitting. A longer one, a_c > 0.5h_c, needs closed vertical links of at least k₂F_Ed/f_yd only where F_Ed exceeds V_Rd,c of 6.2.2 (J.3(3), recommended k₂ = 0.5). A note: Figure J.6 a) carries two labels: ΣA_s,lnk ≥ A_s,main beside the links and A_s,lnk ≥ k₁A_s,main below; the text gives only k₁. The page checks the text, summing every leg of every link. SANS 10100-1 asks for half the main tie. Check what your National Annex says.

Shear near the support. 6.2.2(6) lets the share of a load within 0.5d to 2d of a support be cut by β = a_v/2d when V_Rd,c is checked, provided the tie is fully anchored; its Figure 6.4 b) shows a corbel. The load without β must still pass eq. 6.5, the crushing limit of the concrete. 6.2.2(7) lets corbels be designed by strut and tie instead, which is the J.3 route. The page checks eq. 6.5 without β, and in J.3(3) compares F_Ed itself with V_Rd,c, on the safe side.

The horizontal force. A precast beam on a corbel shrinks, creeps and moves with temperature. If its bearing cannot slide freely, it pulls on the corbel. 10.9.5.1(2)P asks for these restraint actions in the design of the members next to such a bearing, but EN 1992-1-1 gives no number. The page uses H_Ed = the larger of your F_h and 0.2 F_Ed, its own floor. H_Ed acts at the bearing surface, a_H above the tie, so the tie takes H_Ed(z₀ + a_H)/z₀. Separate bars welded to the bearing plate (F_h bars on the page, an option it takes from SANS 10100-1 6.2.4.2.4) can carry H_Ed itself (A_s,h ≥ H_Ed/f_yd, anchored in the column); the main tie then keeps only F_Eda/z₀ + H_Eda_H/z₀.

5. Bearings, nibs and half joints

The bearing length (10.9.5.2(1), Figure 10.6). A simple bearing needs a nominal length a. It is the net length a₁ plus a₂, lost at the front of the supporting member (Table 10.3), plus a₃, lost at the end of the supported member (Table 10.4). Add the deviations: Δa₂ for the distance between supports (Table 10.5) and Δa₃ = l_n/2500 for the member length. An isolated member, one not tied to its neighbours, needs 20 mm more (10.9.5.3(1)P). The page checks a₁ (with the 20 mm) against the plate length and a₂ against the distance to the front; a₃ and the deviations are yours.

Bearing strength. Where nothing else is specified, 10.9.5.2(2) gives f_Rd = 0.4 f_cd for a dry bearing and f_bed ≤ 0.85 f_cd otherwise, with f_cd the lower of the two members. Without mortar or a pad to spread the load, the bearing width b₁ counts up to 600 mm only (10.9.5.2(3)). The net length a₁ = F_Ed/(b₁f_Rd) is never less than the row of Table 10.2 for the support: line supports of floors and roofs, or concentrated supports such as beams. 10.9.4.3(3) asks for good workmanship at dry joints and keeps the mean stress between plane dry surfaces to 0.3 f_cd; a soft pad that spreads sideways needs bars near the surfaces by eq. 10.5 (10.9.4.3(4), (5)). The page uses the beam row for every corbel, and 6.7 with no spread for the partially loaded area.

Nibs. A continuous nib along a beam carries floor units, such as hollowcore, on a line bearing. EN 1992-1-1 has no nib clause; the page’s Eurocode 2 route says NOT CHECKED for a nib. Design it by hand as a corbel per metre run: the strut and tie of J.3 and 6.5, the tie anchored from the inner edge of the bearing, links by J.3(2) or (3), the bearing as a line support of Table 10.2. The load enters the beam near its bottom, so the beam’s links at that face must hang it up to the top, in addition to the links for shear (6.2.1(9)).

Half joints. A beam end cut down to sit on a support, a dapped end, is designed by strut and tie (10.9.4.6). Figure 10.4 gives two models, which may be combined. In one, vertical links close to the re-entrant corner hang the reaction up into the full beam, and a horizontal tie in the nib holds the strut. In the other, inclined bars run from the top of the nib across the corner into the bottom of the beam. Each tie is anchored past its node. The corbel page does not design half joints; the nib of Example B uses the same ideas.

Which code, and what is left to you. Every check row on the Eurocode 2 route names its EN 1992-1-1 (or EN 1992-1-2) clause. Some choices are the page’s own and are stated in its rows or warnings: the floor H_Ed ≥ 0.2 F_Ed, the tie minimum of a beam, the beam row of Table 10.2 and the 20 mm of an isolated member on every corbel. Two more come from SANS 10100-1: the link zone over the upper 2d/3 (6.2.4.2.5) and the F_h bars welded to the plate (6.2.4.2.4), an option that EN 1992-1-1 does not require. The page puts the lower node inside the column, as Figure J.5 draws it (the strut row and calc sheet section 2: arm a = a_c + w/2, or z₀/2.5; section 3). Nibs, half joints, the full eq. 10.6, the hanging steel in a beam that carries a nib, crack control and the column itself are yours. The bending schedule uses SANS 282 bar shapes on every route.