Structural Elements for Foundations

Abstract

This chapter presents the design methods of the foundations starting from the basic soil models and following with the verification of the isolated footings and foundation piles. The analysis of continuous foundation beams, grids and rafts together with then problems of structure–foundation interaction are the examined. The calculation of retaining walls is treated with the models of earth pressure and the pertinent verifications of stability. Finally the diaphragm walls, possibly provided with anchoring prestressed tendons, are presented. The final section shows the application of the design procedures to the different foundation elements of the same multi-storey building treated in the preceeding chapters.

The original version of this chapter was revised: For detailed information please see Erratum. The erratum to this chapter is available at 10.1007/978-3-319-52033-9_11

Appendix: Data on Soils and Foundations

9.1.1 Table 9.1: Soil Parameters

The following tables give indicative values for the main soil parameters, necessary for the stability calculations:

k :

subgrade coefficient (expressed in N/mm³)

g :

unit weight (expressed in kg/dm³)

ϕ :

internal friction angle (expressed in degrees)

c :

cohesion (expressed in N/mm²)

a—subgrade coefficient (N/mm³)

Type of soil	k
Coarse gravel	0.150–0.250
Gravel–sand mixtures	0.100–0.150
Dry clay or silt	0.070–0.100
Compact sand	0.050–0.100
Humid clay or silt	0.030–0.060
Fine or soft sand	0.015–0.020
Recent backfills	0.010–0.020
Organic tillage	0.005–0.015

b—unit weight of soils (kg/dm³)

Type of soil	g (soft)	g (compact)
Dry gravela	1.5–1.7	1.8–2.0
Humid gravela	1.7–1.9	1.9–2.1
Saturated gravela	1.9–2.1	2.0–2.2
Dry organicb	1.4–1.6	1.7–1.9
Humid organicb	1.6–1.8	1.8–2.1
Saturated organicb	1.8–2.0	2.0–2.2
Clay, silt	1.7–1.9	2.0–2.2

1. ^aor sand ^btillage

c—internal friction angle of soils

Type of soil	ϕ
Multi-graded coarse compact gravel	45–50
Multi-graded coarse loose gravel	35–40
Multi-graded round compact gravel	40–45
Mono-graded round compact gravel	35–40
Multi-graded round loose gravel	30–35
Mono-graded round loose gravel	25–30
Compact sand	35–40
Loose sand	25–30
Organic (tillage) sand	15–25
Fata organic (tillage)	0–20
Sandy clay	15–25
Fata clay	0–20
Silt	20–25

1. ^aDepending on moisture content, or pore water pressures

d—cohesion of soils (N/mm²)

Type of soil	c
Hard clay	0.100–1.000
Stiff clay	0.050–0.100
Plastic clay	0.020–0.050
Sandy clay	0.010–0.020
Compact silt	0.005–0.010

9.1.2 Chart 9.2: Soil Resistance—Formulas

The following formulas refer to a type of global failure with the formation of a sliding surface in the soil from underneath the foundation up to the ground level. They give the capacity of the foundation in terms of distributed pressure on the horizontal support base of the foundation under the effect of the vertical loads. It is implied that such pressure is constant on the entire resisting support surface, centred on the point O where the resultant of forces is located.

Symbols

r _v :

resisting pressure

ϕ :

internal friction angle of foundation soil

c :

cohesion of foundation soil

g :

unit weight of foundation soil

q :

pressure acting on adjacent peripheral zones

b :

characteristic width of foundation

a :

characteristic length of foundation

A :

area of resisting surface

P _Ed :

resultant of vertical loads on the base of the foundation

Resistance Verification

(see Table 9.6 for partial safety factors)

$$ P_{\text{Rd}} = r_{\text{v}} A \ge \gamma_{\text{R}} P_{\text{Ed}} , $$

where

$$ r_{\text{v}} = s_{\text{q}} N_{\text{q}} q + s_{\text{c}} N_{\text{c}} c + s_{\text{g}} N_{\text{g}} gb/2 $$

with

$$ \begin{aligned} q & = hg^{\prime } \\ h & = {\text{depth of surrounding soil above the base}} \\ g^{\prime} & = {\text{unit weight of surroundings soil}} \\ s_{\text{q}} & = 1 + (b/a)tg\phi \\ s_{\text{c}} & = 1 + (b/a)\left( {N_{\text{q}} /N_{\text{c}} } \right) \\ s_{\text{g}} & = 1 - 0.4(b/a) \\ \gamma_{\text{R}} & = {\text{model coefficient}} \\ \end{aligned} $$

(see Table 9.6)

The characteristic dimensions a, b (a ≥ b) of the foundation are described in the figure.

The values of N _q, N _c ed N _g are given in Table 9.3 as a function of ϕ.

9.1.3 Table 9.3: Parameters of Resistance Formulas

The following table gives, as a function of the internal friction angle ϕ of soil, the values of the three parameters N _q, N _c and N _g of the formula of Chart 9.2. These values are derived from

$$ \begin{aligned} N_{\text{q}} & = e^{\pi tg\phi } tg^{2} \left( {\frac{\pi }{4} + \frac{\phi }{2}} \right) \\ N_{\text{c}} & = (N_{\text{q}} - 1)/tg\phi \\ N_{\text{g}} & = 2(N_{\text{q}} + 1)tg\phi \\ \\ \end{aligned} $$

tgϕ	Nq	Nc	Ng	Nq/Nc	ϕ
0.00	1.00	5.14	0.00	0.195	0.0
0.01	1.05	5.28	0.04	0.200	0.6
0.02	1.11	5.42	0.08	0.205	1.1
0.03	1.17	5.56	0.13	0.210	1.7
0.04	1.23	5.71	0.18	0.215	2.3
0.05	1.29	5.86	0.23	0.221	2.9
0.06	1.36	6.02	0.28	0.226	3.4
0.07	1.43	6.19	0.34	0.232	4.0
0.08	1.51	6.36	0.40	0.237	4.6
0.09	1.59	6.53	0.47	0.243	5.1
0.10	1.67	6.72	0.53	0.249	5.7
0.11	1.76	6.91	0.61	0.255	6.3
0.12	1.85	7.10	0.68	0.261	6.8
0.13	1.95	7.31	0.77	0.267	7.4
0.14	2.05	7.52	0.85	0.273	8.0
0.15	2.16	7.73	0.95	0.279	8.5
0.16	2.27	7.96	1.05	0.286	9.1
0.17	2.39	8.19	1.15	0.292	9.6
0.18	2.52	8.43	1.27	0.299	10.2
0.19	2.65	8.69	1.39	0.305	10.8
0.20	2.79	8.95	1.52	0.312	11.3
0.21	2.93	9.21	1.65	0.319	11.9
0.22	3.09	9.49	1.80	0.325	12.4
0.23	3.25	9.78	1.95	0.332	13.0
0.24	3.42	10.08	2.12	0.339	13.5
0.25	3.60	10.39	2.30	0.346	14.0
0.26	3.79	10.71	2.49	0.353	14.6
0.27	3.98	11.05	2.69	0.361	15.1
0.28	4.19	11.39	2.91	0.368	15.6
0.29	4.41	11.75	3.14	0.375	16.2
0.30	4.64	12.12	3.38	0.383	16.7
0.31	4.88	12.50	3.64	0.390	17.2
0.32	5.13	12.90	3.92	0.398	17.7
0.33	5.39	13.32	4.22	0.405	18.3
0.34	5.67	13.74	4.54	0.413	18.8
0.35	5.97	14.19	4.88	0.420	19.3
0.36	6.27	14.65	5.24	0.428	19.8
0.37	6.60	15.12	5.62	0.436	20.3
0.38	6.94	15.62	6.03	0.444	20.8
0.39	7.29	16.13	6.47	0.452	21.3
0.40	7.67	16.66	6.93	0.460	21.8
0.41	8.06	17.21	7.43	0.468	22.3
0.42	8.47	17.79	7.95	0.476	22.8
0.43	8.90	18.38	8.52	0.484	23.3
0.44	9.36	18.99	9.11	0.493	23.7
0.45	9.83	19.63	9.75	0.501	24.2
0.46	10.33	20.29	10.43	0.509	24.7
0.47	10.86	20.98	11.15	0.518	25.2
0.48	11.41	21.69	11.91	0.526	25.6
0.49	11.99	22.42	12.73	0.535	26.1
0.50	12.59	23.19	13.59	0.543	26.6
0.51	13.23	23.98	14.51	0.552	27.0
0.52	13.90	24.80	15.49	0.560	27.5
0.53	14.60	25.65	16.53	0.569	27.9
0.54	15.33	26.54	17.64	0.578	28.4
0.55	16.10	27.46	18.81	0.586	28.8
0.56	16.91	28.41	20.06	0.595	29.2
0.57	17.75	29.39	21.38	0.604	29.7
0.58	18.64	30.41	22.78	0.613	30.1
0.59	19.57	31.47	24.27	0.622	30.5
0.60	20.54	32.57	25.85	0.631	31.0
0.61	21.57	33.72	27.53	0.640	31.4
0.62	22.64	34.90	29.31	0.649	31.8
0.63	23.76	36.13	31.20	0.658	32.2
0.64	24.93	37.40	33.20	0.667	32.6
0.65	26.17	38.72	35.32	0.676	33.0
0.66	27.46	40.09	37.56	0.685	33.4
0.67	28.81	41.51	39.94	0.694	33.8
0.68	30.23	42.98	42.47	0.703	34.2
0.69	31.71	44.51	45.14	0.712	34.6
0.70	33.26	46.09	47.97	0.722	35.0
0.71	34.89	47.73	50.96	0.731	35.4
0.72	36.59	49.44	54.14	0.740	35.8
0.73	38.38	51.20	57.49	0.750	36.1
0.74	40.25	53.04	61.05	0.759	36.5
0.75	42.20	54.94	64.80	0.768	36.9
0.76	44.25	56.91	68.78	0.778	37.2
0.77	46.39	58.95	72.99	0.787	37.6
0.78	48.64	61.07	77.43	0.796	38.0
0.79	50.99	63.27	82.14	0.806	38.3
0.80	53.44	65.55	87.11	0.815	38.7
0.81	56.01	67.92	92.36	0.825	39.0
0.82	58.71	70.37	97.92	0.834	39.4
0.83	61.52	72.92	103.78	0.844	39.7
0.84	64.47	75.55	109.98	0.853	40.0
0.85	67.55	78.29	116.53	0.863	40.4
0.86	70.77	81.13	123.45	0.872	40.7
0.87	74.14	84.07	130.75	0.882	41.0
0.88	77.67	87.12	138.46	0.891	41.3
0.89	81.36	90.29	146.60	0.901	41.7
0.90	85.21	93.57	155.19	0.911	42.0
0.91	89.25	96.98	164.25	0.920	42.3
0.92	93.46	100.51	173.81	0.930	42.6
0.93	97.87	104.17	183.91	0.940	42.9
0.94	102.48	107.96	194.55	0.949	43.2
0.95	107.30	111.90	205.78	0.959	43.5
0.96	112.34	115.98	217.61	0.969	43.8
0.97	117.61	120.21	230.10	0.978	44.1
0.98	123.11	124.60	243.26	0.988	44.4
0.99	128.86	129.15	257.13	0.998	44.7
1.00	134.87	133.87	271.75	1.007	45.0
1.01	141.16	138.77	287.15	1.017	45.3
1.02	147.72	143.84	303.39	1.027	45.6
1.03	154.58	149.10	320.49	1.037	45.8
1.04	161.74	154.56	338.51	1.046	46.1
1.05	169.23	160.22	357.48	1.056	46.4
1.06	177.05	166.08	377.46	1.066	46.7
1.07	185.22	172.17	398.51	1.076	46.9
1.08	193.75	178.47	420.66	1.086	47.2
1.09	202.66	185.01	443.98	1.095	47.5
1.10	211.97	191.79	468.53	1.105	47.7
1.11	221.69	198.82	494.37	1.115	48.0
1.12	231.84	206.11	521.56	1.125	48.2
1.13	242.44	213.66	550.17	1.135	48.5
1.14	253.51	221.50	580.28	1.145	48.7
1.15	265.06	229.62	611.94	1.154	49.0
1.16	277.13	238.04	645.25	1.164	49.2
1.17	289.72	246.77	680.28	1.174	49.5
1.18	302.87	255.82	717.12	1.184	49.7
1.19	316.59	265.20	755.86	1.194	50.0
1.20	330.91	274.93	796.59	1.204	50.2

9.1.4 Chart 9.4: Lateral Earth Pressure

The following formulas refer to the pressures applied on the vertical face of a retaining wall by a horizontal embankment.

Symbols

ξ :

depth of a stratum from the surface of the retained soil

q :

superimposed surface load applied on the retained soil

σ _v :

vertical pressure applied at a depth ξ

p _h :

horizontal pressure on the wall due to active pressure

r _h :

horizontal pressure on the wall due to passive resistance

see also Chart 9.2.

Lateral Earth Pressures

$$ \begin{aligned} & p_{\text{h}} = - 2c\sqrt {\lambda_{\text{a}} } + \lambda_{\text{a}} \sigma_{\text{v}} \quad ( \ge 0)\quad {\text{active pressure}} \\ & r_{\text{h}} = + 2c\sqrt {\lambda_{\text{p}} } + \lambda_{\text{p}} \sigma_{\text{v}} \quad {\text{passive resistance}} \\ \end{aligned} $$

with

$$ \begin{aligned} & \sigma_{\text{v}} = q + g\xi \\ & \lambda_{\text{a}} = tg^{2} \left( {\frac{\pi }{4} - \frac{\phi }{2}} \right)\quad {\text{active pressure coefficient}} \\ & \lambda_{\text{p}} = tg^{2} \left( {\frac{\pi }{4} + \frac{\phi }{2}} \right) = \frac{1}{{\lambda_{\text{a}} }}\quad {\text{passive resistance coefficient}} \\ \end{aligned} $$

The values of λ _a and λ _p are shown in Table 9.5.

9.1.5 Table 9.5: Active and Passive Pressure Coefficients

For the meaning of symbols see Charts 9.2 and 9.4.

tgϕ	λ a	λ p	ϕ	tgϕ	λ a	λ p	ϕ
0.00	1.000	1.000	0.0
0.01	0.980	1.020	0.6	0.42	0.442	2.264	22.8
0.02	0.961	1.041	1.2	0.44	0.426	2.349	23.8
0.03	0.942	1.062	1.7	0.46	0.411	2.436	24.7
0.04	0.923	1.083	2.3	0.48	0.396	2.526	25.6
0.05	0.905	1.105	2.9	0.50	0.382	2.618	26.6
0.06	0.887	1.127	3.4	0.52	0.369	2.713	27.5
0.07	0.869	1.150	4.0	0.54	0.356	2.811	28.4
0.08	0.852	1.173	4.6	0.56	0.344	2.911	29.3
0.09	0.835	1.197	5.1	0.58	0.332	3.014	30.1
0.10	0.819	1.221	5.7	0.60	0.321	3.119	31.0
0.11	0.803	1.246	6.3	0.62	0.310	3.228	31.8
0.12	0.787	1.271	6.8	0.64	0.299	3.339	32.6
0.13	0.772	1.296	7.4	0.66	0.290	3.453	33.4
0.14	0.756	1.322	8.0	0.68	0.280	3.569	34.2
0.15	0.742	1.348	8.5	0.70	0.271	3.689	35.0
0.16	0.727	1.375	9.1	0.72	0.262	3.811	35.8
0.17	0.713	1.403	9.7	0.74	0.254	3.936	36.5
0.18	0.699	1.431	10.2	0.76	0.246	4.064	37.2
0.19	0.685	1.459	10.8	0.78	0.238	4.195	38.0
0.20	0.672	1.488	11.3	0.80	0.231	4.329	38.7
0.21	0.659	1.517	11.9	0.82	0.224	4.466	39.4
0.22	0.646	1.547	12.4	0.84	0.217	4.605	40.0
0.23	0.634	1.578	13.0	0.86	0.211	4.748	40.7
0.24	0.622	1.609	13.5	0.88	0.204	4.893	41.4
0.25	0.610	1.640	14.0	0.90	0.198	5.042	42.0
0.26	0.598	1.672	14.6	0.92	0.193	5.193	42.6
0.27	0.586	1.705	15.1	0.94	0.187	5.347	43.2
0.28	0.575	1.738	15.6	0.96	0.182	5.505	43.8
0.29	0.564	1.772	16.2	0.98	0.177	5.665	44.4
0.30	0.554	1.806	16.7	1.00	0.172	5.828	45.0
0.31	0.543	1.841	17.2	1.02	0.167	5.995	45.6
0.32	0.533	1.877	17.7	1.04	0.162	6.164	46.1
0.33	0.523	1.913	18.3	1.06	0.158	6.337	46.7
0.34	0.513	1.949	18.8	1.08	0.154	6.512	47.2
0.35	0.503	1.987	19.3	1.10	0.149	6.691	47.7
0.36	0.494	2.024	19.8	1.12	0.146	6.872	48.2
0.37	0.485	2.063	20.3	1.14	0.142	7.057	48.7
0.38	0.476	2.102	20.8	1.16	0.138	7.244	49.2
0.39	0.467	2.141	21.3	1.18	0.134	7.435	49.7
0.40	0.458	2.182	21.8	1.20	0.131	7.629	50.2

9.1.6 Table 9.6: Partial Safety Factors

The following table gives the values of the partial safety factors, the ones to be used to amplify forces:

$$ \left\{ \begin{aligned} & G_{{1{\text{d}}}} = \gamma_{{{\text{G}}1}} G1\quad \left( {\text{structural self-weight}} \right) \\ & G_{{2{\text{d}}}} = \gamma_{{{\text{G}}2 \, }} G2\quad \left( {\text{superimposed dead load}} \right) \\ & Q_{\text{d}} = \gamma_{\text{G}} Q_{\text{k}} \quad \left( {{\text{live loads{-}variable actions}}} \right) \\ \end{aligned} \right\} $$

and the ones to be used to reduce the soil characteristics

$$ \begin{aligned} & \left( {tg\phi } \right)_{\text{d}} = \left( {tg\phi } \right)_{\text{k}} /\gamma_{\phi } \quad \left( {\text{internal friction}} \right) \\ & c_{\text{d}} = c_{\text{k}} /\gamma_{\text{c}} \quad \left( {\text{cohesion}} \right) \\ & c_{\text{ud}} = c_{\text{uk}} /\gamma_{\text{cu}} \quad \left( {\text{undrained cohesion}} \right) \\ & g_{\text{d}} = g/\gamma_{{{\gamma }}} \quad \left( {\text{weight of soil}} \right) \\ \end{aligned} $$

There are three types of verifications, referred to three different ultimate limit states of the resisting system

EQU:

stability verifications against the possible loss of equilibrium of the structure as rigid body (with irrelevant mechanical properties of the soil);

STR:

verifications of resistance of the foundation element against the possible failure of its critical zones (with elastic reaction of the soil);

GEO:

verifications of stability of the soil against its possible global failure (see Chart 9.2) or any other type of failure (including the verifications of overturning and sliding)

In particular, the undrained cohesion c _u is used, together with ϕ = 0, in place of the cohesion c in the formulas of resistance and lateral pressures (see Charts 9.2 and 9.4) for the short-term verification of soft clays.

It is implied that for each couple of values, the lesser or greater is used depending on whether the action is favourable or unfavourable,

The coefficients referred to the soil always reduce its characteristics, with respect to both the possible lower resistance and the possible greater active pressure,

The coefficients shown here are related to the ones of Charts 3.1 and 3.2 for the forces and the ones of Charts 2.2 and 2.3 for the resistance of materials (verifications of the type STR).

		EQU	STR	GEO
Forces–loads
Structural self-weight	γ G1	0.9 ÷ 1.1	1.0 ÷ 1.3	1.0
Other permanent loads	γ G2	0.0 ÷ 1.5	0.0 ÷ 1.5	0.0 ÷ 1.3
Live loads–variable actions	γ Q	0.0 ÷ 1.5	0.0 ÷ 1.5	0.0 ÷ 1.3
Soil parameters
Friction angle	γ φ	1.25	1.25	1.25
Cohesion	γ c	1.25	1.25	1.25
Undrained cohesion	γ cu	1.4	1.4	1.4
Weight of soil	γ γ	1.0	1.0	1.0
Model coefficients
Overturning	$\upgamma_{\text{R}}^{{\prime }}$	1.0	–	1.8
Sliding	$\upgamma_{\text{R}}^{{\prime }}$	1.1	–	1.1
Soil resistance	γ R	–	–	1.8
Driven piles (tip bearing)	γ b	–	–	1.45
Driven piles (skin friction)	γ s	–	–	1.45
Bored piles (end bearing)	$\upgamma_{\text{b}}^{{\prime }}$	–	–	1.7
bored piles (skin friction)	$\upgamma_{\text{s}}^{{\prime }}$	–	–	1.45

9.1.7 Chart 9.7: Construction Requirements of Foundations

With reference mainly to the durability requirements, the following minimum measures are recommended.

Reinforcement Cover

In order to take into account the lower construction precision of the foundation works, the reinforcement covers should be appropriately increased with respect to the ones given in Table 2.17.

Bored piles	75 mm
Surface cast against the excavation	75 mm
Surface cast against levelled ground	50 mm
Surface cast against blinding	35 mm
Footings (except the base)	40 mm
Beams (except the base)	40 mm
Walls: surface against retained soil	30 mm

These values should be taken as minimum design values and include the following tolerances:

Footings	± 15 mm
Beams	± 10 mm
Walls	± 5 mm
Bored piles	± 50 mm

For aggressive soils, the minimum values of covers shown above should be increased by 25 mm.

Minimum Reinforcement

If their size is relatively big and the possible cracking due to shrinkage does not compromise the resistance significantly, the foundations can be made of unreinforced or lightly reinforced concrete. In this case, the following prescriptions on minimum reinforcement do not apply.

For continuous tie beams and other slender tying elements that resist axial tension forces, when the significant length can lead to early cracking due to shrinkage, a minimum longitudinal reinforcement should be provided equal to

$$ A_{\text{s}} \ge A_{\text{c}} f_{\text{ctm}} /f_{{y{\text{k}}}} $$

similar to reinforced concrete ties (see Chart 2.14).

For element predominantly in bending, such as foundation beams, a minimum reinforcement on the edge in tension should be provided

$$ A_{\text{s}} \ge \frac{1}{2}y_{\text{c}}^{\prime } bf_{\text{ctm}} /f_{\text{yk}} $$

similar to uncracked reinforced concrete beams (see Chart 3.19), where $y_{\text{c}}^{{\prime }}$ indicates the depth of the portion in tension and b indicates its width.

9.1.8 Chart 9.8: Verifications Against Overturning and Sliding

The following formulas refer to the equilibrium of the isolated foundation as a rigid body, whose stability relies only on the support base.

Symbols

P _Ed :

vertical action on the support base of the foundation

e :

eccentricity of the vertical action with respect to the centre

a :

length of the foundation (orthogonal to e)

b :

width of the foundation (parallel to e)

x :

width of the loaded limit strip

H _Ed :

horizontal force on the support base of the foundation

μ :

soil-foundation friction coefficient

see also Charts 9.2 and 9.4.

Overturning Verification

We refer to combined compression and uniaxial bending on a rectangular foundation, whose support base is not entirely in compression, with the dimensions of the resisting footprint a·x (Charts 9.2 and 9.3). For a given vertical action P _Ed, the limit equilibrium situation is characterized by the value x defined as follows.

The overturning verification is set with

$$ e_{\lim } > e, $$

where

$$ e_{\lim } = (b - x)/2 $$

Cohesive soil—general case

The width x of the loaded strip is obtained solving the equation:

$$ r_{\text{l}} (x)x^{2} + r_{\text{o}}^{\prime } (x)x - \gamma_{\text{R}} p_{\text{Ed}} = 0 $$

with

$$ \begin{aligned} & r_{\text{l}} = s_{\text{g}} N_{\text{g}} g/2 \\ & r_{\text{o}} = s_{\text{q}} N_{\text{q}} q + s_{\text{c}} N_{\text{c}} c \\ & p_{\text{Ed}} = P_{\text{Ed}} /a \\ & s_{\text{g}} = 1 - 0.4\beta_{\text{x}} \\ & s_{\text{q}} = 1 + \beta_{\text{x}} tg\phi \\ & s_{\text{c}} = 1 + \beta_{\text{x}} N_{\text{q}} /N_{\text{c}} \\ & \beta_{\text{x}} = x/a\quad se\quad x \le a \\ & \beta_{\text{x}} = a/x\quad se\quad x > a \\ \end{aligned} $$

Cohesive soil—β _x assigned

For a rectangular foundation, if an assumption is made for the ratio β _x , consequently giving a constant value to the three coefficients s _g, s _q and s _c, the following solution is obtained

$$ x \cong \left\{ { - r_{\text{o}} + \sqrt {r_{\text{o}}^{2} + 4r_{\text{l}} \gamma_{\text{R}} p_{\text{Ed}} } } \right\}/(2r_{\text{l}} ) $$

The solution can be refined re-evaluating the three coefficients s _g, s _q and s _c on the basis of the calculated x.

Cohesive soil—strip footing

For a strip footing with a ≫ b, being x/a ≅ 0 one has s _g = s _q = s _c = 1, therefore one obtains:

$$ x = \left\{ { - r_{\text{o}} + \sqrt {r_{\text{o}}^{2} + 4r_{\text{l}} \gamma_{\text{R}} p_{\text{Ed}} } } \right\}/(2r_{\text{l}} ) $$

Non-cohesive soil—without surrounding pressure

With c = q = 0 and with an assumption on β _x for the evaluation of s _g = 1–0.4β _x , one obtains

$$ x \cong \sqrt {2\gamma_{\text{R}} p_{\text{Ed}} /s_{\text{g}} N_{\text{g}} g} $$

The solution can be refined re-evaluating s _g on the basis of the calculated x.

For a strip footing (s _g = 1) the revised solution is obtained directly with

$$ x = \sqrt {2\gamma_{\text{R}} p_{\text{Ed}} /N_{\text{g}} g} $$

Sliding Verification

When the horizontal translational equilibrium relies on the friction of the support base, it is verified when

$$ H_{\text{Ed}} < \mu P_{\text{Ed}} $$

with μ ≤ tgϕ.

9.1.9 Chart 9.9: Reinforced Concrete Footings: Resistance Verifications

We refer to a stocky footing, with a parallelepiped shape, to support a centred column. It is implied that such footing is reinforced with an orthogonal grid of bars at the bottom.

Symbols

P _Ed :

vertical action transferred from the column to the footing

M _Ed :

bending moment from the column (along a′)

H _Ed :

shear force from the column (along a′)

a′, b′:

sides of the column

h :

footing depth

a, b :

sides of the footing (parallel to a′, b′)

G :

footing self-weight

A _sa, A _sb :

footing reinforcement along a and b

d _a, d _b :

footing effective depths along a and b

see also Charts 2.2 and 2.3.

Verifications of Resistance

The resistance of each part of the footing shall be related to the pressure received back from the soil, distributed on the support base as deduced on the basis of the applied loads, with the assumptions of elastic behaviour of the soil and infinitely rigid footing.

Centred load (e = 0)

$$ \sigma_{\text{g}} = \frac{{P_{\text{Ed}} }}{ab}\quad {\text{constant pressure }}\left( {{\text{without}}\,G} \right) $$

• reinforcement along a

  $$ \begin{aligned} & s_{\text{a}} = (a - a^{\prime} )/2\quad {\text{footing protrusion}} \\ & P_{\text{ad}} = s_{\text{a}} b\sigma_{\text{g}} = \frac{a - a^{\prime} }{2a}P_{\text{d}} \\ & A_{\text{sa}} \ge \lambda_{\text{a}} P_{\text{Ed}} /f_{\text{yd}} \\ \end{aligned} $$

with

$$ \begin{aligned} & \lambda_{\text{a}} = l_{\text{a}} /d_{\text{a}} \quad l_{\text{a}} = c_{\text{a}} + s_{\text{a}} /2 \\ & c_{\text{a}} = \hbox{min} (0.2d_{\text{a}} ,a^{\prime} /4) \\ \end{aligned} $$

• reinforcement along b

  $$ \begin{aligned} & s_{\text{b}} = (b - b^{\prime} )/2\quad {\text{footing protrusion}} \\ & P_{\text{bd}} = s_{\text{b}} a\sigma_{\text{v}} = \frac{b - b^{\prime} }{2b}P_{\text{Ed}} \\ & A_{\text{sd}} \ge \lambda_{\text{b}} P_{\text{bd}} /f_{\text{yd}} \\ \end{aligned} $$

with

$$ \begin{aligned} & \lambda_{\text{b}} = l_{\text{b}} /d_{\text{b}} \quad l_{\text{b}} = c_{\text{b}} + s_{\text{b}} /2 \\ & c_{\text{b}} = \hbox{min} (0.2d_{\text{b}} ,b^{\prime} /4) \\ \end{aligned} $$

• concrete resistance

  $$ P_{\text{Ed}}^{\prime } = \left( {1 - \frac{a^{\prime} b^{\prime} }{ab}} \right)P_{\text{Ed}} < P_{\text{rc}} $$

with

$$ P_{\text{rc}} = 0.8f_{\text{cd}} \left( {\frac{{b^\prime d_{\text{a}} }}{{1 + \lambda_{\text{a}}^{2} }} + \frac{{a^\prime d_{\text{b}} }}{{1 + \lambda_{\text{b}}^{2} }}} \right) $$

Eccentric load (e > 0)

$$ \begin{aligned} & N = P_{\text{Ed}} + G \\ & M = M_{\text{Ed}} + H_{\text{Ed}} h \\ & e = M/N \\ \end{aligned} $$

• Base entirely in compression (e ≤ a/6)

  $$ \begin{aligned} & \sigma_{\text{o}} = P_{\text{Ed}} /ab\quad {\text{centroidal }}\left( {{\text{without}}\,G} \right) \\ & \sigma = 6M/a^{2} b\quad {\text{due to bending moment}} \\ & \sigma_{\text{g}}^{\prime } = \sigma_{\text{o}} + (a^{\prime}/a)\sigma \quad {\text{at column edge}} \\ & \sigma_{\text{g}}^{\prime \prime } = (1 - a^{\prime} /a)\sigma \quad {\text{increment at footing edge}} \\ \end{aligned} $$

pressures resultant

$$ P_{\text{ad}} = s_{\text{a}} b\sigma_{\text{g}}^{\prime } + s_{\text{a}} b\sigma_{\text{g}}^{\prime \prime } /2 $$

resultant position

$$ \begin{aligned} & u = \left( {s_{\text{a}}^{2} b\sigma_{\text{g}}^{\prime } /2 + s_{\text{a}}^{2} b\sigma_{\text{g}}^{\prime \prime } /6} \right)/P_{\text{ad}} \\ & A_{\text{sa}} \ge \lambda_{\text{a}} P_{\text{ad}} /f_{\text{sd}} \\ \end{aligned} $$

with

$$ \begin{aligned} \lambda_{\text{a}} & = l_{\text{a}} /d_{\text{a}} \quad l_{\text{a}} = c_{\text{a}} + s_{\text{a}} - u \\ c_{\text{a}} & = \hbox{min} (0.2d,a^{\prime } /4) \\ \end{aligned} $$

• Base not entirely in compression (e > a/6)

  $$ \begin{aligned} & x = 3(a/2 - e)\quad {\text{zone in compression}} \\ & \sigma_{\text{g}} = 2N/bx\quad {\text{maximum at the edge}} \\ & \sigma_{\text{o}} = G/ab\quad {\text{pad self-weight}} \\ \end{aligned} $$

with x ^≥ s _a one has

$$ \begin{aligned} & \sigma_{\text{g}}^{\prime } = \left( {1 - s_{\text{a}} /x} \right)\sigma_{\text{g}} - \sigma_{\text{o}} \quad {\text{column edge }}\left( {{\text{without}}\,G} \right) \\ & \sigma_{\text{g}}^{\prime \prime } = \left( {s_{\text{a}} /x} \right)\sigma_{\text{g}} - \sigma_{\text{o}} \quad {\text{increment at pad edge}} \\ \end{aligned} $$

the verification A _sa follows as for the previous case.