Chapter three

Bearing capacity of shallow foundations

Bearing Capacity: Bearing capacity is the ability of soil to safely carry the pressure from any engineered structure placed upon it without undergoing a shear failure and with the resulting settlements being tolerable for that structure. 3.1A bearing capacity failure is defined as a foundation failure that occurs when the shear stresses in the soil exceed the shear strength of the soil. Bearing capacity failures of foundations can be grouped into three categories, as follows (Vesic, 1963, 1975): 1.General shear failure: As shown in Fig. 3.1 Sudden or catastrophic failure the failure surface in the soil will extend to the ground surface Bulging on the ground surface adjacent to foundation Common failure mode in dense sand When the load is plotted versus settlement of the footing,

there is a distinct load at which the foundation fails and this is designated Qult. The value of Qult divided by the width B and length L of the footing is considered to be the ultimate bearing capacity (qult) of the footing. The ultimate bearing capacity has been defined as the bearing stress that causes a sudden catastrophic failure of the foundation, it occurs when settlement of footing reaches 4 to 10% of B.

2. Local shear failure. As shown in Fig. 3.2, Common in medium soil compaction Significant settlement upon loading Failure surface first develops right below the foundation and then slowly extends outwards with load increments Foundation movement shows sudden jerks first (at qu1) and then after a considerable amount of movement the slip surface may reach the ground. A small amount of bulging may occur next to the foundation.

3. Punching shear. As shown in Fig. 3.3, Common in fairly loose sand or soft clay Failure surface does not extends beyond the zone right beneath the foundation Extensive settlement with a wedge shaped soil zone beneath the

foundation. Vertical shear occurs around the edges of foundation After reaching failure load-settlement curve continues at some

slope and mostly linearly. qu occurs when settlement of footing reaches 15 to 25% of B.

3-2 BEARING-CAPACITY EQUATIONS: I-The Terzaghi Bearing-Capacity Equation The most commonly used bearing capacity equation is that equation developed by Terzaghi (1943). • Terzaghi assumed a general shear failure in order to develop the

following bearing capacity equation:

Nc, Nq and Nγ: are Bearing capacity factors, function of Φ (table3-1)

q : is the effective overburden pressure c : cohesion of soil γ : unit weight of soil q : γ Df ( where Df is the depth of excavation of the footing)

• Terzaghi suggested the following modifications for the previous equations for a local shear failure:

Nc’, Nq’ and Nγ’ are the modified bearing capacity equations obtained by replacing the friction angle Φ by:

II-General Bearing-Capacity Equation: In order to take account of many cases not considered by Terzaghi such as rectangular shape, depth factor, ground and load inclination factors, several authors have presented a more general form of bearing capacity equations as follows:

The recommended bearing capacity factors for use are shown in table 3-2; the shape ,depth and inclination factors recommended for use are given in table 3-3

3-3 Effect of water table

• For undrained loading conditions in clayey soils (Φ=0), (Nc=5.14; Nq=1.0; Nγ = 0), the general bearing capacity equation takes the form :

Where q: total overburden pressure above footing bottom Hence, the net ultimate bearing capacity is:

3-4 Net Ultimate Bearing Capacity The net ultimate bearing capacity is defined as the ultimate pressure per unit area of the foundation that can be supported by the soil in excess of the pressure caused by the surrounding soil at the foundation level. If the difference between the unit weight of concrete used in the foundation and the unit weight of soil surrounding the foundation is assumed to be negligible, then: where qnet,u is the net ultimate bearing capacity.

3-5 the factor of safety: In order to calculate the gross allowable load-bearing capacity of shallow foundations requires the application of a factor of safety (FS) to the gross ultimate bearing capacity:

The net ultimate bearing capacity was defined as :

But it is more correct to use the following expression:

The factor of safety may be at least 3 in all cases, the magnitude of FS should depend on the uncertainties and risks involved for a given condition.

Example 3.1 A square foundation is 1.5 m x 1.5 m in plan. The soil supporting the foundation has a friction angle Φ=20, and c = 15.2 kN/m2. The unit weight of soil γ is 17.8 kN/m3. Determine the allowable gross load on the foundation using Terzaghi bearing capacity equation with a factor of safety (FS) of 4. Assume that the depth of the foundation (Df) is 1 meter. Solution: using equ.(2) :

Referring to table 3.1 for Φ=20 Nc = 17.69, Nq = 7.44 and Nγ =3.64 Thus, qu = (1.3) (15.2) (17.69) +(1 x 17.8) (7.44) + (0.4) (17.8) (1.5) (3.64) 349.55 + 132.43 + 38.87 + 520.85 = 521 kN/m2 So the allowable load per unit area of the foundation is qall =qu/FS = 521/4 =130.25kN/m2 ~130 kN/m2

Example 3.2 A square footing is shown in the following figure, Determine the safe gross load (factor of safety of 3) that the footing can carry using the general bearing

capacity equation. Solution: with c = 0, Fci , Fqi & Fi = 1 (vertical loading), qu = qNqFqsFqd +0.5γBNγs Fγs Fγd For Φ =32, (Table3.2) given, Nq= 23.18 and N γ= 30.22.

3-6 Eccentrically Loaded Foundations When foundations are subjected to moments in addition to the vertical load, the distribution of pressure by the foundation on the soil is not uniform. The distribution of nominal pressure is:

where Q : total vertical load M: moment on the foundation

Meyerhof suggested “the effective area method” for a good estimate of pressure distribution :it is a step by step procedure for determining qu: 1)- with the eccentricity : Thus we obtain:

and

• For e/B = 1/6 , qmin =0 • For e/B < 1/6 , qmin >0 fig.3.4(a)

• For e/B > 1/6 , qmin <0 fig.3.4(b):

development of soil tension, or a separation between soil and foundation, in this case :

2) Determine the effective dimensions of the foundation as B’ : effective width = B - 2e L’ :effective length = L Note that: if the eccentricity were in the direction of the length of the foundation, then : L’ =L - 2e. B’ =B; The smaller of the two dimensions (that is, L’ and B’) is the effective width of the foundation

(a)

(b)

Fig.3.4

3) Use the general bearing capacity equation to determine the ultimate bearing capacity as:

Note that: • replace B and L by B’ and L’ to compute Fcs, Fqs, and Fγs, (from

Table 3-3) • Use B and L to determine Fcd, Fqd, and Fγd 4) The total ultimate load that the foundation can sustain is where A’ : effective area=B’.L’

5) FS = Qall /Q ; Qall = Qult/FS or Qall,net= Qult/FS – Q soil = qu’.A’/FS –q.A

3-7Foundations with Two-Way Eccentricity:

When we determine the effective area (A’), effective width (B’), and effective length (L’),four possible cases may arise (Highter and Anders, 1985). The effective area is such that its centroid coincides with the load:

Fig.3.5

Fig.3.6

Fig.3.7

Fig3.8

Example 3.3

Example 3.4 A square foundation is shown in next Figure ,with eL = 0.3 m and eB =0.15 m. Assume two-way eccentricity and determine the ultimate load, Qult.

Solution:

Implies to case II Refer to fig.3.6

3.7BEARING CAPACITY FOR FOOTINGS ON LAYERED SOILS

It may be necessary to place footings on stratified deposits where the rupture zone will extend into the lower layer(s) depending on their thickness and require some modification of qult

There are three general cases of the footing on a layered soil as follows: I. Footing on layered clays (all Φ=0) a. Top layer stronger than lower layer (cu1 > cu2) b. Top layer weaker than lower layer (cu1 < cu2) Case 1: cu1 > cu2

i-) H/B is relatively small, thus failure in the soil under the footing will take place by punching in the top layer followed by a general shear failure in the bottom soil layer (fig.3.9(a)), the ultimate bearing capacity can be expressed as:

Where: B= width of foundation L= length of foundation ca= adhesion along the interface (fig.3.10) Nc = 5.14 (for Φ =0)

ii-) H/B is relatively large: the failure surface in the soil at ultimate load will be fully contained in the top soil layer (fig.3.9(b)) , We should consider the smaller value between qu(1) and qu(2) as the ultimate bearing capacity qu. Case 2: cu1 < cu2

the ultimate bearing capacity can be given as follows:

(a)

Fig3.9

fig.3.10 fig.3.11

II. Footing on dense sand overlying soft clay:

i-) H/B is relatively small For continuous footing:

For rectangular footing:

Where Φ and γ relates to the top sand layer Ks : punching shear resistance coefficient which variation is given in fig(3.11) ii-) H/B is relatively large For continuous footing:

For rectangular footing:

Qu is the smaller of qu(1) and qu(2)

3.8-Bearing capacity from SPT The SPT is widely used to obtain the bearing capacity of soils directly, on the basis of tolerable settlement considerations, Meyerhof (1956) proposed a correlation for the net allowable bearing pressure with the SPT-N values, and corresponding to 1 in (25.4 mm) of estimated total settlement:

Where N: the corrected standard penetration number from B/2 above footing to 2B below footing B: the footing width in meters

In order to account for depth effect, different tolerable settlement (other than 1 in.)and better control of SPT procedure, a modified form of the previous equations has been proposed as follows:

Where S = tolerable total settlement (in mm or in.) Fd= depth factor = 1+0.33(Df/B) ≤1.33

3.9-Bearing capacity from CPT Some empirical equations have also been proposed to correlate between qult and the cone penetration resistance (qc) as follows: A- sand:

B- Clay: