Annex D

(informative)

# A sample analytical method for bearing resistance calculation

## D.1 Symbols used in Annex D

(1) The following symbols are used in Annex D.

*A*' = *B*' × *L*' the design effective foundation area

*b* the design values of the factors for the inclination of the base, with subscripts *c*, *q* and γ

*B* the foundation width

*B*' the effective foundation width

*D* the embedment depth

*e* the eccentricity of the resultant action, with subscripts *B* and *L*

*i* the inclination factors of the load, with subscripts cohesion *c*, surcharge *q* and weight density γ

*i* hydraulic gradient

*L* the foundation length

*L*' the effective foundation length

*m* exponent in formulas for the inclination factor *I*

*N* the bearing capacity factors, with subscripts for *c*, *q* and γ

*q* overburden or surcharge pressure at the level of the foundation base

*q*' the design effective overburden pressure at the level of the foundation base

*s* the shape factors of the foundation base, with subscripts for *c*, *q* and γ

*V* the vertical load

α the inclination of the foundation base to the horizontal

γ' the design effective weight density of the soil below the foundation level

θ direction angle of *H*

(2) The notations used in this method are given in Figure D.1.

## D.2 General

(1) Approximate equations for the design vertical bearing resistance, derived from plasticity theory and experimental results, may be used. Allowance should be made for the effects of the following:

- the strength of the ground, generally represented by the design values of
*c*_{u},*c*' and φ'; - eccentricity and inclination of design loads;
- the shape, depth and inclination of the foundation;
- the inclination of the ground surface;
- ground-water pressures and hydraulic gradients;
- the variability of the ground, especially layering.

## D.3 Undrained conditions

(1) The design bearing resistance may be calculated from:

*R*/

*A*' = (π + 2)

*c*

_{u}

*b*

_{c}

*s*

_{c}

*i*

_{c}+

*q*

with the dimensionless factors for:

- the inclination of the foundation base:
*b*_{c}= 1 – 2α/(π + 2); - the shape of the foundation:

*s*_{c} = 1 + 0,2(*B*'/*L*'), for a rectangular shape;

*s*_{c} = 1,2, for a square or circular shape.

- the inclination of the load, caused by a horizontal load
*H*:

with *H* ≤ *A'**c*_{u}.

## D.4 Drained conditions

(1) The design bearing resistance may be calculated from:

*R*/

*A*' =

*c*'

*N*

_{c}

*b*

_{c}

*s*

_{c}

*i*

_{c}+

*q*'

*N*

_{q}

*b*

_{q}

*s*

_{q}

*i*

_{q}+ 0,5γ'

*B*'

*N*

_{γ}

*b*

_{γ}

*s*

_{γ}

*i*

_{γ}

with the design values of dimensionless factors for:

- the bearing resistance:

*N*_{q} = e^{πtanφ'}tan^{2}(45° + φ'/2)

*N*_{c} = (*N*_{q} – 1)cot φ'

*N*_{γ} = 2(*N*q – 1)tan φ', where δ ≥ φ'/2 (rough base)

- the inclination of the foundation base:

*b*_{c} = *b*_{q} – (1 – *b*_{q})/*N*_{c} × tan φ'

*b*_{q} = *b*_{γ} = (1 – α · tan φ’)^{2}

- the shape of foundation:

*s*_{q} = 1 + (*B*'/L')sin φ', for a rectangular shape;

*s*_{q} = 1 + sin φ', for a square or circular shape;

*s*_{γ}= 1 – 0,3(*B*'/*L*'), for a rectangular shape;

*s*_{γ} = 0,7, for a square or circular shape

*s*_{c}= (*s*_{q}·*N*_{q}– 1)/(*N*_{q}– 1) for rectangular, square or circular shape;- the inclination of the load, caused by a horizontal load
*H*:

*i*_{c} = *i*_{q} – (1 – *i*_{q})/*N*_{c} · tan φ';

*i*_{q} = [1 – *H*/(*V* + *A*'*c*'cot φ')]^{m};

*i*_{γ} = [1 – *H*/(*V* + *A*'*c*'cot φ')]^{m+1}.

where:

*m* = *m*_{B} = [2 + (*B*'/*L*')]/[1 + (*B*'/*L*')] when *H* acts in the direction of *B*';

*m* = *m*_{L} = [2 + (*L*'/*B*')]/[1 + (*L*'/*B*')] when *H* acts in the direction of *L*'.

In cases where the horizontal load component acts in a direction forming an angle θ with the direction of *L*', *m* may be calculated by:

*m* = *m*_{θ} = *m*_{L}cos^{2}θ + *m*_{B}sin^{2}θ.