RC walls and slender walls

How a reinforced concrete wall is designed with the STATIX RC wall calculator on its Eurocode 2 route. It covers walls as members (9.6), the effective length and slenderness (5.8.3), imperfections and the minimum eccentricity, second-order moments by nominal curvature (5.8.8), in-plane bending and the end strip, in-plane shear (6.2), and the steel and cover rules.

1. Learning outcomes

After this lesson you can:

tell a wall from a column (9.6.1) and a braced wall from an unbraced one (5.8.1);
work the effective length l₀ from the end restraints (5.8.3.2, eq. 5.15 and 5.16, Table 12.1) and judge the slenderness λ against λ_lim (5.8.3.1);
add the imperfection (5.2), the minimum eccentricity (6.1(4)) and the second-order moment by nominal curvature (5.8.8);
share N and M along the wall, and check the end strip and the whole wall for N with M (6.1);
check in-plane shear by 6.2 with the wall as a deep member, and the steel and cover rules of 9.6 and 4.4.1;
say which checks on the page follow EN 1992-1-1, which method comes from SANS 10100-1, and which you do by hand.

2. Before you start

RC Design 1 (Eurocode), lesson 5 (columns: effective length, slenderness, nominal curvature, N-M interaction) and lesson 3 (shear and links).
The RC wall 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 3 (SANS 10100-1).

What the RC wall page’s Eurocode 2 route uses.
One wall, one storey. A solid wall, l_w long and h thick, at one storey, braced or unbraced, with l₀ = β × the storey height you give (the page asks for the clear height). The actions are one ULS case, or characteristic G, Q and W. The page combines them by EN 1990 6.10: 1.35 G + 1.5 Q and 1.0 G + 1.5 Q. With wind it adds 1.35 G + 1.5 Q + 0.9 W, 1.35 G + 1.5 ψ₀Q + 1.5 W, 1.35 G + 1.5 W and 1.0 G + 1.5 W, with the wind both ways.
Materials. f_ck (cylinder) and f_yk 450 or 500 MPa, γ_c 1.5, γ_s 1.15. α_cc is 1.0, the recommended value of 3.1.6(1), as on the PT slab page of lesson 2; the course’s other worked values use 0.85, and this page has no input for it. The rectangular block of 3.1.7(3); steel with the horizontal top branch of 3.2.7(2) b).
Along the wall. EN 1992-1-1 gives no rule for sharing N and M along a wall. The page uses the method of SANS 10100-1 4.8.4.4 a): elastic, with no tension in the concrete. Once part of the wall is in tension it solves the cracked transformed section with E_s/E_cm. The end strip takes the force per metre at the very end.
Slenderness. λ = l₀/i, and λ_lim of 5.8.3.1 with A 0.7, C 0.7 and B from ω, worked on the end strip. e_i = l₀/400 when braced, θ_il₀/2 when unbraced (θ₀ 1/200, α_h from the storey height). Nominal curvature by 5.8.8 with φ_ef 2 and n_bal 0.4, and c = 8 when there is an out-of-plane moment (10 without). e₀ = max(h/30, 20 mm) out of plane and max(l_w/30, 20 mm) in plane (6.1(4)).
Shear. 6.2 with b_w = h, d = 0.8 l_w, z = 0.9d, ρ_l from the vertical bars in the tension half and σ_cp = N/A_c. The horizontal bars are the shear reinforcement, with 1 ≤ cot θ ≤ 2.5, ν₁ = 0.6(1 − f_ck/250) and α_cw 1. The extra tension of 6.2.3(7), ΔF_td = 0.5 V_Ed cot θ with the cot θ of the shear row, is a row of its own: M_Ed + ΔF_tdz against the whole-wall M_Rd. Section checked says whether the section is the point of maximum moment, where it adds nothing.
Detailing, cover and fire. 9.6.2 to 9.6.4 with the recommended values, the horizontal bars outside the vertical bars. Cover by 4.4.1 (Table 4.4N for the exposure class you choose, class S4, Δc_dev 10). Fire by EN 1992-1-2 Table 5.4 with μ_fi 0.7, exposed on both sides unless you choose one, for braced walls only.
Not checked: crack control (7.3), the global second-order effects of the bracing system (5.8.3.3), the forces in the floors that hold the wall, openings and seismic detailing (EN 1998-1). Other annexes may use other values.

3. Walls, bracing and slenderness

What a wall is. EN 1992-1-1 9.6.1(1) applies the wall rules to a reinforced member whose length is at least four times its thickness. Below that it is a column (9.5), which RC Design 1 (Eurocode) lesson 5 covers; the calculator refuses it and says so. A wall that mostly bends out of its plane is designed by the slab rules (9.3), as 9.6.1(1) says.

Braced and unbraced. Out of its plane a wall is braced when the floors at its top and bottom cannot sway that way, because cores or walls in the other direction hold them. 5.8.1 calls those the bracing members; in its own plane a shear wall is one of them. The effective length l₀ follows from the restraint at the ends (5.8.3.2(3)). Eq. 5.15 gives between 0.5 l and l for a braced member, eq. 5.16 at least l for an unbraced one; l is the clear height.

The effective length from the restraints. The page asks for β, with l₀ = β × the clear height. Work it by eq. 5.15 from k at each end: the stiffness of the walls above and below over the stiffness of the slabs that hold the joint, the slabs taken as cracked. The Note to 5.8.3.2(3) recommends k of at least 0.1, as fully rigid restraint is rare. 5.8.3.2(7) lets returns (walls at right angles) cut l₀ further by the factor of 12.6.5.1, Table 12.1 (its own β), applied to the l₀ of eq. 5.15, not to the clear height. Held at the top and bottom only, Table 12.1 gives 1.0: no further cut. Example A works β = l₀/l by hand.

Short or slender. 5.8.3.2 defines the slenderness λ = l₀/i, with i the radius of gyration of the concrete section: h/√12 for a wall. 5.8.3.1 lets second-order effects be ignored when λ is below λ_lim = 20·A·B·C/√n (eq. 5.13N, recommended). The limit falls as the relative axial force n rises, so a wall end heavily loaded by in-plane bending turns slender long before the average does. The page takes A = 0.7 and C = 0.7, the values the Note allows when φ_ef and r_m are not known, and works B from ω = A_sf_yd/(A_cf_cd). EN 1992-1-1 has no upper slenderness limit for walls; SANS 10100-1 has one.

Imperfection and nominal curvature. 5.2(7) a) adds an eccentricity e_i for a wall that is not quite straight; in a braced wall the simple value e_i = l₀/400 is allowed. A slender wall then bows a further e₂ under its load. 5.8.8 estimates it from a nominal curvature 1/r with the steel at yield, corrected by K_r for high axial load and K_φ for creep. Then e₂ = (1/r) l₀²/c, with c = 10, or as low as 8 where the first-order moment is the same over the whole height (5.8.8.2(4)). The design moment is M_0Ed + M₂, and never less than N e₀ (6.1(4)). With two layers d is the depth to the far layer; with one central layer d = h/2 (5.8.8.3(2)).

4. Bending along the wall, and shear

The force along the wall. The axial load and the in-plane moment from wind share out along the wall, more at one end. EN 1992-1-1 has no rule for this; the page uses the method of SANS 10100-1 4.8.4.4 a): an elastic distribution with no tension in the concrete. While the eccentricity e = M/N is at most l_w/6 the whole wall is in compression and the distribution is a straight line. Past that the page solves the cracked transformed section: concrete in compression only, every bar row elastic in tension and compression, with the modular ratio E_s/E_cm (3.2.7(4), Table 3.1).

The end strip. The most heavily loaded metre at the compressed end is checked like a column strip 1 000 mm wide and h deep. It carries its force per metre n_w with the out-of-plane moment, the imperfection and, when slender, the second-order moment, never less than n_we₀. λ_lim and M₂ use the n of this strip. The page takes n_w at the very end, the peak, on the safe side. At the other end the vertical bars alone must carry any tension.

The whole wall in plane. The wall is also one section of depth l_w, checked by 6.1 with plane sections and no concrete tension. The concrete takes the rectangular block of 3.1.7(3) (ηf_cd over λx = 0.8x up to C50/60), with ε_cu3 3.5 ‰ at the compressed end. Every bar row works at its own strain, up to f_yd at yield (3.2.7(2) b)). The moment is at least N e₀ with e₀ = max(l_w/30, 20 mm) in the plane. A wall can pass as a whole and still crush at its end: the end strip is the check that finds that.

In-plane shear. EN 1992-1-1 has no in-plane shear clause for walls. The page applies 6.2 to the wall as a member whose depth is its length: b_w = h, d = 0.8 l_w, z = 0.9d. V_Rd,c of 6.2.2(1) takes ρ_l from the vertical bars in the tension half and σ_cp = N/A_c. Where V_Ed is larger, the horizontal bars carry it as links at 90° (6.2.3). The page chooses cot θ between 1 and 2.5 for the largest resistance, up to V_Rd,max, and does not add V_Rd,c to V_Rd,s (6.2.1(3), (5)). The upper limit V_Rd,max at cot θ = 1 is a row of its own.

The wall as a deep member. In a squat wall, not much taller than it is long, plane sections do not stay plane (6.1(1)P) and the load runs in struts to the base. 9.6.1(1) lets you size the steel with a strut-and-tie model (6.5). The page uses 6.2 for every wall; for a squat wall add a strut-and-tie check by hand. The truss of 6.2.3 also puts extra tension in the vertical bars at the tension end, ΔF_td = 0.5 V_Ed cot θ (6.2.1(7), 6.2.3(7)), with M_Ed/z + ΔF_td capped at M_Ed,max/z. The page adds it as its own row, with the cot θ of its shear row, at every section above the point of maximum moment. Example A shows how.

5. Steel, cover and what is left to you

The steel limits. 9.6.2 sets the vertical steel between A_s,vmin and A_s,vmax (recommended 0.002 A_c and 0.04 A_c, the maximum doubled at laps) and the bar spacing. 9.6.3 asks for horizontal bars at each surface, at least a quarter of the vertical steel and 0.1 % of A_c. Where the vertical steel passes 2 % of A_c, 9.6.4(1) asks for links as in a column (9.5.3). With the vertical bars nearest the faces, 9.6.4(2) asks for 4 links per square metre, unless its Note lets them off; the page always puts the horizontal bars outside, so that rule does not arise. Below A_s,vmin the wall is a plain or lightly reinforced wall (Section 12); the page does not cover those, and fails its minimum-steel row.

One layer or two. Two layers, one near each face, give the wall bending strength both ways and the depth d = h − a for the curvature. Here a is the axis distance of the vertical bars: cover, horizontal bar and half the vertical bar. One central layer is cheaper, but d drops to h/2: in a 200 mm wall about 100 mm against 160 mm, so the nominal curvature and M₂ grow by 1.6. 9.6.2(2) asks for half of A_s,vmin at each face where the minimum governs, and 9.6.3(1) for horizontal bars at each surface; with one layer the page warns about both.

Cover and fire. The nominal cover is c_min + Δc_dev (4.4.1.1, 4.4.1.3), with c_min the largest of the bond value (Table 4.2: the bar size), the durability value for the exposure class and structural class (Table 4.4N) and 10 mm. The page takes class S4 and Δc_dev 10 mm, the recommended values, and checks the cover to the outer (horizontal) bars and to the vertical bars behind them. For fire it checks the wall thickness and the axis distance a of the vertical bars in EN 1992-1-2 Table 5.4, for braced walls only.

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. One method comes from SANS 10100-1: the elastic, no-tension distribution of N and M along the wall (4.8.4.4 a)), as EN 1992-1-1 has no rule for it. The bending schedule uses SANS 282 bar shapes and the SANS 10100-1 table 22 deduction on every route, with laps by 8.7.3, and its note on links cites SANS 10100-1 4.11.4.5.2 (on this route 9.6.4(1)). The page designs one solid wall at one storey. Openings (design each pier as a wall), crack control (lesson 6), the bracing system as a whole and the forces in the floors that hold the wall are yours.