Slabs on grade

How a ground-bearing slab is designed with the STATIX slab on grade calculator on its Eurocode 2 route. EN 1992-1-1 has no clause for a slab on the ground. So the page takes the materials, punching and cover from EN 1992-1-1, and the load factors from EN 1990. For the slab on the soil it uses published theory: Meyerhof’s yield lines for rack legs and wheels, and Hetenyi’s beam on springs for storage and walls. The lesson ends with the whole course on one page.

1. Learning outcomes

After this lesson you can:

describe the ground by its modulus k and find the radius of relative stiffness l of the slab;
find the flexural tensile strength f_ctd,fl (3.1.8, eq. 3.23) and the moment capacity of a plain slab and of a slab with mesh, check the mesh against A_s,min (9.3.1.1(1), 9.2.1.1(1)), and say what Section 12 changes for a plain slab;
check a rack leg or a wheel at the interior, at a joint and at a joint crossing by yield lines, and punching under it by 6.4;
check uniform storage with aisles and a wall on the slab, with the load factors of EN 1990 6.10;
set out sawn joints, choose the cover for a floor with forklifts (4.4.1.2(13)), and say why a mesh does not replace joints (7.3.2);
say which checks the page does to EN 1992-1-1, which follow other published theory, and which you do by hand.

2. Before you start

RC Design 1 (Eurocode): punching (lesson 7) and detailing (lesson 8); lesson 1 of this course (punching at columns); Foundations 1 (soils and bearing).
The slab on grade calculator (this link opens it on Eurocode 2, EN 1992-1-1), and its guide.
A pocket calculator and about 60 minutes.
The same lesson on the SA annex: RC Design 2, lesson 7 (the page’s SANS 51992-1-1 route).

What the slab on grade page’s Eurocode 2 route uses.
The model. EN 1992-1-1 has no clause for a ground-bearing slab. The page treats the slab as a plate on a bed of springs (Winkler) of modulus k, with E_cm of Table 3.1 and ν 0.2 (3.1.3(4)), and finds the radius of relative stiffness l. It follows no design guide’s text. It uses the published theory that the guides for industrial floors build on. Meyerhof (1962) gives the yield-line collapse loads for a point load at the interior, at an edge and at a corner. Hetenyi (1946) gives the beam on an elastic foundation, for uniform loads with aisles and for walls. It does not use Westergaard’s elastic stresses. A sawn joint counts as a free edge, unless you choose a load transfer under ‘Load across joints’ (15 % for aggregate interlock, 30 % for dowels: practice values, not EN values).
Actions. EN 1990 6.10 with 1.35 G + 1.5 Q (the page does not offer 6.10a and 6.10b). A point load is split into its permanent part (the rack itself, G) and the rest (Q); uniform storage is all Q; a wall has G and Q. The slab’s own weight bends nothing and is not added. A thickening under a wall is checked for soil bearing with the characteristic G + Q, concrete at 24 kN/m³ and an allowable pressure q_a.
Materials. f_ck 16 to 50 MPa; f_ctm and E_cm from Table 3.1; f_ctk,0.05 = 0.7 f_ctm; the depth factor of 3.1.8(1), eq. 3.23 (written for reinforced concrete members), applied to f_ctk,0.05; f_ctd,fl = f_ctk,fl/γ_c with α_ct 1.0 (3.1.6(2)P); γ_c 1.5 and γ_s 1.15 (2.4.2.4). Mesh by SANS 1024 reference (ref 193 = 193 mm²/m each way, five wires per metre), f_yk 485 MPa unless you change it under More inputs. As on the pages of lessons 2 to 5, α_cc and α_ct are 1.0 with no input. A slab with no mesh, or with less steel than A_s,min, is plain concrete and falls in Section 12 (12.1(1)P). For its static loads (racking, storage, walls) 12.3.1 recommends α_ct,pl 0.8, and the page takes 0.8 on a plain face (section 5).
Moments. Plain: M_un = f_ctd,fl h²/6 per metre. With a mesh in the tension face: the larger of M_un and A_s f_yk/γ_s × 0.95d. A top mesh raises M_n (the ring crack on top), a bottom mesh M_p (the radial cracks underneath). For slabs 9.3.1.1(1) applies 9.2.1.1(1): the tension steel should not be less than A_s,min. With less steel the slab is plain concrete under Section 12 (12.1(1)P), so the page counts a mesh only when it meets A_s,min; a plain face gets 0.8 M_un (12.3.1; section 5 and Example B). Uniform loads and walls use the plain moment alone, so that the top does not crack.
Punching. 6.4.3, 6.4.4(1) and 6.4.5(3): at the face of the loaded area (u₀, v_Rd,max = 0.4νf_cd with f_cd = f_ck/1.5) and at 2d (u₁), β 1.0, with the soil pressure inside u₁ left out (on the safe side). With c the side of the square plate, u₀ is 4c at the interior, 3c at an edge and 2c at a corner; 6.4.5(3) caps the last two at c₂ + 3d and 3d, which only matters for plates wider than 1.5d. d and ρ_l come from the tension face: under an interior or edge load the slab sags, so that is the bottom, and by 6.4.4(1) a top mesh is not tension steel; with no bottom mesh the page takes d = 0.75h and ρ_l = 0 (practice). At a corner the tension is at the top, and d and ρ_l come from the top mesh (section 5).
Joints and cover. Sawn contraction joints at the smaller of 25h and 6 m (practice). A bay longer than 1.5 times its width gets an INFO row; the floor is not split again. The page gives the joint plan, the mesh sheets and the concrete volume. It assumes the mesh is cut at every sawn joint, and counts the sheets over the whole floor. The cover of the top mesh by 4.4.1 (Table 4.4N at class S4, without the changes of Table 4.3N; Δc_dev 10 mm); the exposure list has no abrasion classes.
Also on the page: a mesh counts only if it meets A_s,min (9.3.1.1(1), 9.2.1.1(1)), and a plain face takes α_ct,pl 0.8 (12.3.1).
Not checked: crack control (7.3), the allowance for abrasion (4.4.1.2(13)), edge steel for loads near an edge (6.4.2(5), 9.3.1.4), dowels and tied or free-movement joints, curling, settlement and the ground itself (EN 1997), flatness, and steel-fibre concrete. The examples do the first two by hand. Other annexes may use other values.

3. The slab and the ground

A slab on grade. A warehouse or factory floor sits on a compacted granular sub-base over the natural ground, the subgrade, often with a plastic membrane between. The membrane cuts the friction that holds back the slab as it shrinks, and stops the base drawing water out of the fresh concrete. Racking, stored goods and forklifts bear on the slab, and the slab spreads them into the ground. EN 1992-1-1 has no clause for such a slab. The page takes the concrete’s strength, the punching check and the cover from EN 1992-1-1, the load factors from EN 1990, and the slab on the soil from published theory.

The ground as springs. The page models the ground as a bed of separate springs (Winkler): k is the pressure per unit settlement. k is not a soil constant: it depends on the size of the loaded area, so take it from plate-bearing tests or from the geotechnical engineer, not from a table. A good sub-base raises the k the slab feels. The page takes one k for the sub-base and the subgrade together. The ground itself (bearing, settlement) is designed to EN 1997, and the page does not check it.

The radius of relative stiffness. How far a load spreads depends on the slab’s stiffness against the ground’s: the radius of relative stiffness l, a length. A thicker slab or softer ground gives a larger l, and the load is shared by a wider area of slab. ν = 0.2 is the Poisson’s ratio of uncracked concrete (3.1.3(4)). For a long strip, such as the slab beside an aisle or under a wall, the page uses the beam-on-springs factor λ; 1/λ is also a length.

First crack or collapse. Westergaard worked out the elastic stress under a load on a slab on springs: it tells you when the first crack forms underneath. Meyerhof showed that the slab carries much more before it fails, because the cracks turn into hinges (yield lines) and the ground keeps pushing back. The page uses Meyerhof’s collapse loads as the ultimate capacity, against the factored load of EN 1990 6.10. The guides for industrial floors use the same yield-line results; the page takes them from the published theory, not from any one guide.

4. Point loads by yield lines

The yield lines under a point load. A heavy rack leg or wheel first cracks the slab radially underneath (sagging, tension at the bottom). As the load grows, a ring crack opens on top further out (hogging, tension at the top), and the slab fails when the pattern becomes a mechanism. The collapse load depends on the moment capacity along both sets of lines: M_p for the bottom face and M_n for the top face, per metre.

Meyerhof’s collapse loads, as the page uses them. The contact area of a base plate or a tyre is turned into a circle of the same area, radius a, and a/l measures how big it is against the slab. For a/l = 0 and for a/l of 0.2 or more Meyerhof gives the formulas in the table; between 0 and 0.2 the page takes a straight line from one to the other. An edge load has fewer yield lines than an interior one, and a corner load only one line across the corner, which needs M_n alone.

Where the load stands. A leg beside a sawn joint has only half a slab around it; a leg where two joints cross has a quarter. The page checks the interior, an edge and a corner, as you choose under ‘Where’. It treats a joint as a free edge unless you choose a share of the load to cross it: 15 % for the aggregate interlock of a sawn joint, 30 % for a dowelled joint. These are practice values that you choose; EN 1992-1-1 gives none. A sawn joint opens as the slab shrinks, and the interlock falls away as it opens.

5. Strength, mesh and punching

The plain moment. A plain slab cracks when the stress at its face reaches the flexural tensile strength. 3.1.8(1) relates the mean flexural strength of ‘reinforced concrete members’ to f_ctm and the depth: a thin slab breaks at a higher stress than a thick one. It says the same relation may be used with the characteristic value. The page applies it to f_ctk,0.05 and divides by γ_c, with α_ct 1.0 of 3.1.6(2)P. But a slab with less steel than A_s,min (ref 193 here) is plain concrete: 12.1(1)P puts plain members, and members with less than the minimum reinforcement, in Section 12. The list in 12.1(2) gives examples only. It leaves out the effects of rotating machines and traffic loads, so a forklift wheel, but not static racking, storage or a wall. For those static loads 12.3.1 recommends α_ct,pl 0.8, and the page takes it on a plain face (it takes it for a wheel too, on the safe side). Eq. (12.1) as written also has no depth factor: f_ctd,pl = 0.8 × 2.0275/1.5 = 1.081 MPa for C30/37, against the page’s 0.8 × 1.926 = 1.541 MPa at 175 mm. The depth factor on a plain ground slab is published practice (3.1.2(7)P points to 3.1.8(1) for the flexural strength), not Section 12 as written. The examples give the page’s figures with α_ct,pl 0.8, and say what dropping the depth factor as well would do.

Plain, mesh or fibre. A plain slab has M_un on both faces. A mesh adds strength once that face has cracked: a top mesh raises M_n, a bottom mesh M_p, and where the mesh counts the page takes the larger of M_un and the mesh moment. For slabs 9.3.1.1(1) applies 9.2.1.1(1): the tension steel should not be less than A_s,min. A mesh below A_s,min adds nothing once the section cracks, so the section is taken as plain concrete, in Section 12 (12.1(1)P). The Note to 9.3.1.1(1) also allows 1.2 times the steel the ULS needs where brittle failure is unlikely. For plain concrete 12.5(1) allows a plastic analysis only where it can be justified. Yield lines on a plain slab on the ground rest on Meyerhof’s tests and the support of the subgrade (section 3), which is the justification 12.5(1) asks for; it comes from the published theory, not from EN 1992-1-1. The page counts a mesh only when it meets A_s,min. Steel-fibre concrete is not offered: EN 1992-1-1 has no rules for it, and its design needs the residual strengths that the fibre supplier gets from beam tests.

Punching under a leg. A small, heavy load can punch through the slab. 6.4 checks the shear stress at the face of the base plate (u₀ = 4c, 3c or 2c, with c the side of the square plate) and on the basic control perimeter at 2d (u₁), which runs round the plate and stops at an edge or joint (Figure 6.15). A rack leg carries no moment, so β = 1.0. The ground pushes up inside the perimeter and helps; 6.4.4(2) lets a column base count that, but the page leaves it out, on the safe side. Without bonded tension steel, v_Rd,c falls back to v_min. For a load nearer an edge than d, 6.4.2(5) asks for edge steel (9.3.1.4), which a sawn joint does not have.

Which face is in tension. Right under an interior or edge load the slab bends down into the ground: tension at the bottom. The hogging that opens the ring crack on top lies further out. 6.4.4(1) takes ρ_l from the bonded tension steel, and d is measured to it, so at the punching perimeter a top mesh counts for neither. Without bottom steel there is no tension steel: v_Rd,c = v_min, and practice takes d = 0.75h. The page uses this rule under interior and edge loads (Examples A and B). At a corner the critical bending is hogging across the corner, so there the top mesh is in the tension face.

6. Loads, joints and cover

Loads. Racking legs and posts are point loads on their base plates: the rack’s own weight is permanent (G) and the stored pallets are imposed (Q, EN 1991-1-1 category E). EN 1990 6.10 with the recommended factors of Table A1.2(B) gives 1.35 G + 1.5 Q. A forklift wheel is a point load on its tyre contact area; EN 1991-1-1 6.3.2.3 gives the axle loads of forklift classes and a dynamic factor to multiply them by. Bulk storage is a uniform load. A wall built on the slab is a line load, with a thickened strip under it if the slab cannot carry it.

Uniform loads and walls. A load spread evenly over the whole floor bends nothing. Storage with aisles does: the loaded areas either side bend the slab upwards in the empty aisle and can crack its top. The page finds the worst aisle width from Hetenyi’s beam on springs, so you need not know the layout. It uses the plain moment M_un: a crack in the floor surface is not wanted, so the top mesh is not counted. A wall is a line load on the same beam, inside the slab or at a free edge. If the slab cannot carry it, give a thickening (Thickening B × h_t) and the page checks that instead, for soil bearing and size.

Joints. Concrete shrinks as it dries, and the top dries first, so a slab shortens and its edges curl up. EN 1992-1-1 asks that these movements be considered and restraint kept down, for example by joints (2.3.3(1)P, (2)); its 30 m joint distance of 2.3.3(3) is about frames, not floors. Sawn contraction joints decide where the slab cracks: the saw cut weakens it and the crack forms below the cut. A free-movement joint is formed, with dowels that slide, so it opens freely and still carries shear. A tied joint has tie bars that keep it shut. The page sets out sawn joints only, at the smaller of 25h and 6 m, and flags (INFO) a bay longer than 1.5 times its width (practice). It assumes the mesh is cut at every sawn joint, so the joint can open; its sheet count runs over the whole floor.

Shrinkage and steel. A bay between sawn joints shortens by about ε_cs times its length, so each joint opens by that much over the years, less what the sub-base friction holds back. A floor without joints would need enough steel to spread the shrinkage into many fine cracks. By 7.3.2 that steel is set by the cracking force of the whole section in tension: k_c = 1.0, and k = 1.0 for slabs up to 300 mm thick. A light mesh is a fraction of that. It does not stop cracks; it only holds the faces of a crack together. Joints decide where the cracks go.

Cover on a working floor. A warehouse floor is inside and dry, XC1, but forklifts wear its surface away. The recommended structural class is S4; Table 4.3N lowers it by one class for C30/37 or stronger in XC1, so c_min,dur is 10 mm (Table 4.4N). The page keeps S4 without the Table 4.3N changes (15 mm), which is on the safe side. 4.4.1.2(13) lets you allow for wear with a sacrificial layer added to c_min. The recommended values are 5 mm for XM1 (air tyres), 10 mm for XM2 (forklifts with air or solid rubber tyres) and 15 mm for XM3 (elastomer or steel tyres, tracks). The aggregate matters as much as the cover. The page’s exposure list has no XM classes, so type the larger cover in yourself.

What the page covers on its Eurocode 2 route, and what you add. The sub-base and the membrane are in the k you type; their thickness and compaction come from the geotechnical design. Dowelled joints, curling, the settlement of the ground and the flatness needed for very narrow aisles are the designer’s to add.