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Concrete Slab and Footing Design

Strength design of reinforced-concrete slabs and footings with the ACI 318 equations as mathematics and every code factor entered (φ, load factors, moment coefficients, strip percentages, the 0.0018 minimum): a one-way slab by the simple-span forms or entered moment coefficients with flexure, bar spacing, one-way shear (the basic and the size-effect Vc), temperature steel, crack control and the minimum thickness; a two-way flat plate by the direct design method with M0, the negative / positive split, the column-strip and middle-strip moments, punching shear at the column with the three vc expressions and the unbalanced-moment transfer γv Msc c/Jc, and a drop-panel option; a spread footing sized from the service load on the net allowable pressure with one-way shear at d, two-way shear at d/2, flexure at the column face, bars with the minimum, development length, bearing at the column base and dowels; a combined or strap footing with the resultant, the plan size for uniform pressure, the shear and moment diagrams by beam analysis, longitudinal and transverse flexure and punching at each column; and a wall footing per unit length. Every check prints the nominal and φ-factored strength, the demand-to-capacity ratio and the LRFD margin. The spread footing reproduces the FEMA P-751 Chapter 5 example; the default is an illustrative 14 × 14 in column with PD = 150 and PL = 100 kips on 4 ksf, f′c = 4,000 p​si, fy = 60 ksi.

  • One-way slab: wu from entered load factors, M and V by the simple-span forms or entered coefficients, As required by the Ku quadratic, bar spacing, Vc basic or with the ACI 318-19 size effect, temperature steel, crack-control spacing, minimum thickness

  • Two-way flat plate, direct design method: M0 = qu l2 ln²/8, entered negative / positive fractions and column-strip percentages, strip steel, punching shear with vc = min[4, 2 + 4/β, 2 + αs d/bo] λ√f′c and the transfer fraction γv, drop panels

  • Spread, combined / strap and wall footings: plan size on the net allowable pressure (soil and surcharge deducted), one-way shear at d, two-way shear at d/2 with the perimeter truncated at a near edge, flexure at the face, bars with the 0.0018 minimum, development length, bearing and dowels, beam diagrams of the combined footing

  • φ, load factors, coefficients and limits are entered inputs with the ACI 318-19 values as defaults; acceptance against FEMA P-751 Chapter 5 and an independent numpy re-implementation

Concrete Slab and Footing Design

Unit System: /

Analysis


Materials, Strength-Reduction and Load Factors (ACI 318-19 values as defaults, entered)















Soil and Bearing (footings; the net allowable pressure deducts the footing weight, the soil above it and the surcharge)







Spread Footing (plan dimensions in ft; blank B and L = square footing sized from the bearing check)
























Combined or Strap Footing (two columns along L; column 1 nearest the left end; plan lengths in ft)
































Wall Footing (per unit length of wall; loads per ft or per m)















One-Way Slab (12 in strip; loads in psf or kN/m²)


















Two-Way Flat Plate, Direct Design Method (interior panel of a regular layout; spans centre to centre)


























Method, Conventions, Sources and Limits | Method page
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Concrete Slab and Footing Design

Concrete Slab and Footing Design: Method, Conventions, Sources and Limits

What the tool does

The tool carries out the strength design (ACI 318 load and resistance factor method, "LRFD" in the margin boxes) of five reinforced-concrete elements, one at a time: a one-way slab strip, a two-way flat plate by the direct design method, a spread footing, a combined or strap footing, and a wall footing. The ACI 318 provisions are used as equations; every code factor is an entered input with the ACI 318-19 value as its default (the strength-reduction factors φ, the load factors on dead and live load, the minimum steel ratio 0.0018, the tension-controlled strain limit, the moment coefficients and strip percentages of the slab methods, the development-length factors, the spacing limit). The numerics are in U.S. customary units inside (inches, pounds, psi, because the √f′c expressions are written for psi) and converted for display; the SI form is therefore the exact conversion of the psi form, not the rounded SI coefficients printed in the code (0.17 for 2, 0.33 for 4, and so on).

Flexure of a strip

Every flexural check uses the equivalent rectangular stress block with a single layer of tension steel. For a factored moment Mu on a strip of width b and effective depth d, the steel required is found from Rn = Mu/(φ b d²) and ρ = (0.85 f′c/fy)[1 − √(1 − 2Rn/(0.85 f′c))], the closed solution of φ As fy(d − a/2) = Mu with a = As fy/(0.85 f′c b) (the Ku form of USACE EM 1110-2-2104 Appendix D), with φ the tension-controlled value. The strength of the bars provided is then computed exactly: a, c = a/β1, the net tensile strain εt = 0.003(d − c)/c, φ by εt (0.65 below εty = fy/Es, the entered φf at or above the tension-controlled limit, linear between), and φMn = φ As fy(d − a/2). If the steel does not yield the neutral axis is solved from strain compatibility and the steel stress Esεs is used. The minimum reinforcement of a slab or footing is ρmin b h on the gross section (ACI 318-19 Tables 7.6.1.1 and 8.6.1.1, 0.0018 for Grade 60), and the required area is the larger of the two. Bar spacing: the largest spacing (rounded down to ½ in) that provides the required area, capped at 3h and the entered absolute limit (18 in) for one-way slabs and footings, 2h for two-way slabs, 5h for temperature steel (7.7.2.3, 8.7.2.2, 24.4.3.3), and for the one-way slab also at the crack-control spacing s ≤ 15(40,000/fs) − 2.5cc ≤ 12(40,000/fs) with fs = ⅔fy (24.3.2).

Shear

One-way shear is checked at d from the face of the support on the full width: Vc = 2λ√f′c b d (ACI 318-19 Table 22.5.5.1(a)) or, as an option, the 2019 size-effect form for members without shear reinforcement, Vc = 8λsλρw1/3√f′c b d with λs = √(2/(1 + d/10)) ≤ 1 and ρw the ratio of the bars provided (form (c)); both are capped at 5λ√f′c b d. Two-way (punching) shear is checked on the perimeter bo at d/2 from the column faces against vc = the least of 4λ√f′c, (2 + 4/β)λ√f′c and (2 + αsd/bo)λ√f′c (Table 22.6.5.2; β the long-to-short side of the column, αs 40, 30 or 20 for interior, edge and corner columns). Where a slab or footing edge lies within d/2 of the column the perimeter is truncated at the edge and the side along that edge is omitted; the number of sides printed says which case applies. The factored shear is Pu − qu × (area inside the perimeter) for a footing and qu × (tributary area − area inside the perimeter) for a slab.

Moment transfer at a slab-column connection: the fraction γv = 1 − 1/(1 + ⅔√(b1/b2)) of the unbalanced moment Msc is transferred by eccentric shear, adding γv Msc cAB/Jc to Vu/(bod) (8.4.4.2). Jc is the polar-type property of the critical section about its centroidal axis, computed as the sum over the perimeter sides of d l³/12 + l d³/12 + d l x² for the sides perpendicular to the axis and d l x² for the sides parallel to it (R8.4.4.2.3); the test script compares it with the closed forms for interior, edge and corner columns. For an interior support of a slab designed by the direct design method with equal adjacent spans the unbalanced moment is Msc = 0.07(0.5 qLu) l2 ln² (8.10.7.3); any other value can be entered.

One-way slab

A 12 in strip. The factored load is wu = γD(γch + superimposed dead) + γL live. Moments and shears are either the simple-span forms wul²/8 and wul/2 or the approximate method of ACI 318-19 6.5 with the coefficients entered (M = C wu ln², V = Cv wu ln/2; the defaults 1/10, 1/14 and 1.15 are the interior-support / end-span values and the tooltips list the others). The design moment is the larger of the negative and positive values and the shear is taken at d from the support face. The output gives the steel required and its spacing, the strength of the bars entered, the shear check, the shrinkage and temperature steel, and the deflection-control minimum thickness l divided by the entered ratio (Table 7.3.1.1: 20, 24, 28, 10).

Two-way flat plate, direct design method

For the panel l1 (direction of analysis) by l2, the clear span is ln = l1 − c1 (not less than 0.65 l1) and the total static moment is M0 = qu l2 ln²/8 (8.10.3.2). An interior span takes 0.65 M0 negative and 0.35 M0 positive (8.10.4.1); an end span of a flat plate without edge beams takes 0.70 / 0.52 / 0.26 M0 at the interior negative, positive and exterior negative sections (Table 8.10.4.2, column 3); the fractions are entered and the span-type select fills the defaults. The column strip is 0.25 min(l1, l2) each side of the column centreline and takes the entered percentage of each moment (75 % negative, 60 % positive and 100 % exterior negative for a flat plate, αf1 = 0 and βt = 0, Tables 8.10.5.1 to 8.10.5.5); the middle strip takes the rest. Each strip is designed as a strip of its width with the average d = h − cover − db; bars at 2h or 18 in at most. Punching is checked at the column chosen (interior, edge or corner, the edge flush with the column face), with the one-way shear across the full width at d and the minimum thickness ln divided by the entered ratio (Table 8.3.1.1: 30 exterior / 33 interior without drops, 33 / 36 with drops). A drop panel adds its projection to d for the column-strip negative steel and the punching check at the column, and a second perimeter at d/2 outside the drop is checked with the slab depth; the tool reports whether the drop meets the l/3 plan and h/4 projection of 8.2.4. The method's own limits (0.5 ≤ l2/l1 ≤ 2, L ≤ 2D, three spans each way, regular columns) are reported, not enforced.

Spread footing

The plan size comes from the service load on the net allowable pressure qnet = qa − γch − γs(Df − h) − qs (the footing weight, the soil above it and the surcharge are deducted from the gross allowable value), or from the factored load on an entered factored soil capacity as in the FEMA P-751 strength check; a blank B and L give a square footing rounded up to 3 in. The factored pressure for the concrete design is qu = Pu/(BL): the footing and soil weights are carried directly by the soil and produce no shear or moment in the footing. The effective depth is d = h − cover − 1.5db for the upper layer of the mat (the convention of FEMA P-751 5.1.2.3); the lower layer, placed along the longer side, is one bar deeper, and the tool gives the longer direction the larger d. The checks are one-way shear at d from the column face in each direction, punching at d/2, flexure at the column face in each direction (Mu = qu b x²/2, x the cantilever), the bars with the minimum and the spacing limit, the tension development length of the bars from the face against the straight length available (face to the end less the cover), bearing at the column base on both the footing side (0.85f′cA1√(A2/A1) ≤ 2, A2 the 2:1 frustum within the footing depth, 22.8.3.2) and the column side, and the dowels (0.005 A1 plus any excess of Pu over φBn divided by φfy, 16.3.4; compression development ldc = max[fyψr/(50λ√f′c), 0.0003fyψr]db ≥ 8 in, 25.4.9). The effective column dimension is entered: the concrete column side, halfway between the centre and the face of a masonry column, or halfway between the face and the base-plate edge of a steel column (13.2.7.2).

Development length: the simplified form of Table 25.4.2.3 for the case of clear spacing and cover at least db (fyψtψe/(25λ√f′c) db for No. 6 and smaller, divisor 20 for No. 7 and larger), or the general form 3/40 · fy/(λ√f′c) · ψtψeψs/((cb + Ktr)/db) · db with the confinement term entered (not more than 2.5), both at least 12 in. The hook length used where a straight bar does not fit is the ACI 318-14 form ldh = 0.02ψefy/(λ√f′c) db as USACE EM 1110-2-2104 (2016) applies it; the tool reports a straight-bar shortfall and leaves the hook to the designer.

Combined and strap footings

Two columns along the footing length, column 1 nearest the left end (a property line when x1 = c1/2). For the rectangular combined footing the length is 2x̄, twice the distance from the left end to the resultant of the service loads, so that the service pressure is uniform; an entered length gives a linear service pressure q = P/(BL)(1 ± 6e/L) and the width comes from the service load on qnet. The factored pressure is linear with the factored eccentricity. The footing is then analysed as a beam on the upward load B qu(x) with the factored column loads as point loads: V(x) and M(x) are sampled along the length (closure at the far end is printed), the maximum positive moment sizes the bottom steel on the full width, the maximum negative moment between the columns the top steel, one-way shear is taken as the largest |V| at d from either face of either column, punching at each column with the perimeter truncated at a near edge, and a transverse beam c1 + 2kd wide under each column (k = 0.75 the usual practice; ACI 318 does not define this width, Bowles is cited) carries the column load spread across B as a cantilever (B − c2)/2 each side. For the strap footing a footing of entered length sits under each column, footing 1 offset inward by e1, and a rigid weightless strap balances the eccentricity: R1 = P1S/(S − e1), R2 = P − R1, uniform pressure under each footing, widths from the reactions; the same beam engine gives the diagrams, the strap carrying the constant shear R1u − P1u and the negative moment (top steel on the strap section with the beam minimum max(3√f′c, 200) b d/fy).

Wall footing

Per unit length of wall: width from the service load on qnet (rounded up to 3 in) unless entered, one-way shear at d from the wall face, the moment at the face of a concrete wall or halfway between the centre and the face of a masonry wall (13.2.7.1), transverse bars with the minimum and the spacing limit (entered spacing or the largest that provides the required area), development from the critical section, and longitudinal temperature steel ρminBh.

Figures and margins

Each element has a drawn figure (plan and section with the critical sections; the combined footing in elevation with the pressure blocks; the slab panel with its strips and the critical perimeter) and a chart of the computed case (demand-to-capacity ratios against the thickness, the moment strength against the bar spacing, the combined-footing shear and moment diagrams). Every check prints the nominal strength, the φ-factored strength, the demand-to-capacity ratio and a margin box M.S. = φRn/U − 1 labelled LRFD; the bearing check is a service comparison qnet/q − 1 (or a factored one on the entered capacity).

Sources and the copyright rule

ACI 318-19 (and 318-14 for the hook length and the 0.005 strain limit) is cited for its provisions and used as mathematics; no ACI table, chart or example is reproduced, and every table value the design needs is an entered input with the code's value as its default. The published acceptance cases are public-domain government documents: the spread footings of FEMA P-751 Chapter 5 (interior, perimeter and corner footings of a seven-story building: plan size, one-way and two-way shear, flexure, bars) and the same example in FEMA 451 Chapter 4, the development lengths printed in FEMA P-751 Chapters 7 and 8, and the stem flexure and the hook length of USACE EM 1110-2-2104 (2016) Appendix D. No public-domain printed example of a one-way slab, a two-way flat plate by the direct design method, a combined footing or a wall footing was found; those elements are checked against the closed forms and an independent numpy re-implementation, including the punching Jc closed forms, the combined-footing diagrams by dense integration and twenty seeded footings. The default case (a 14 × 14 in column with PD = 150 and PL = 100 kips on 4 ksf, 4-ft embedment, 24 in thick, No. 7 bars) is illustrative.

Limits

Not covered: moment on a spread footing (eccentric soil pressure), footings on piles and pile caps, shear reinforcement in slabs or footings, the equivalent frame method, slab deflections, pattern live loading, strut-and-tie for deep footings, seismic detailing (ACI 318 Chapter 18) and tie beams, the serviceability of the soil (settlement), and sloped or stepped footings. The direct design method's applicability conditions are reported, not enforced. The FEMA P-751 corner footing's printed φMn = 75.2 ft-kips is not reproduced by its own data (6 No. 5 at d = 9.5 in give 77.6 ft-kips); the EM 1110-2-2104 (2016) page D-17 hook length prints the formula without the bar diameter (18.97 in for 0.02fy/√f′c; with db = 1.41 in it is 26.75 in); both are recorded in the test script and the References README.



References

American Concrete Institute (2019). Building Code Requirements for Structural Concrete and Commentary. ACI 318-19, Farmington Hills, MI.

American Concrete Institute (2014). Building Code Requirements for Structural Concrete and Commentary. ACI 318-14, Farmington Hills, MI.

Building Seismic Safety Council (2012). NEHRP Recommended Seismic Provisions: Design Examples. FEMA P-751, Federal Emergency Management Agency, Washington, DC.

Building Seismic Safety Council (2006). NEHRP Recommended Provisions: Design Examples. FEMA 451, Federal Emergency Management Agency, Washington, DC.

U.S. Army Corps of Engineers (2016). Strength Design for Reinforced Concrete Hydraulic Structures. EM 1110-2-2104, Washington, DC.

U.S. Army Corps of Engineers (1992). Strength Design for Reinforced-Concrete Hydraulic Structures. EM 1110-2-2104, Washington, DC.

Naval Facilities Engineering Command (1986). Foundations and Earth Structures. NAVFAC DM-7.02, Alexandria, VA.

American Society of Civil Engineers (2017). Minimum Design Loads and Associated Criteria for Buildings and Other Structures. ASCE/SEI 7-16, Reston, VA.

Whitney, C. S. (1937). "Design of Reinforced Concrete Members under Flexure or Combined Flexure and Direct Compression." Journal of the American Concrete Institute, Vol. 33, No. 3, pp. 483–498.

Moe, J. (1961). Shearing Strength of Reinforced Concrete Slabs and Footings under Concentrated Loads. Development Department Bulletin D47, Portland Cement Association, Skokie, IL.

Hanson, N. W., and Hanson, J. M. (1968). "Shear and Moment Transfer between Concrete Slabs and Columns." Journal of the PCA Research and Development Laboratories, Vol. 10, No. 1, pp. 2–16.

Bowles, J. E. (1996). Foundation Analysis and Design, 5th ed. McGraw-Hill, New York.

Wight, J. K., and MacGregor, J. G. (2012). Reinforced Concrete: Mechanics and Design, 6th ed. Pearson, Upper Saddle River, NJ.

Worked Examples

Published examples this tool reproduces; each page lists the inputs to enter and the printed answers.

All worked examples

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