Design of Concentric Pad Footings

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Foundation of a structure is the least respected but most important aspect of a structure because a structure without a good foundation is bound to fail (collapse). We must have been acquainted with different types of foundations which include strip footing, pad footing, wide strip footing, combined pad footing, strap foundation, raft foundation, pile foundation and even caisson/well foundation. This is not our interest in this post but to carry out a holistic design of pad footing and to show all we need to know and do to completely carry out design of pad footing.

It is the duty of the geotechnical engineer to carry out the design of foundations. Or he can recommend the foundation type while the structural engineer does the reinforcement design. Two things are of utmost concern to the engineer. These are the likely settlement the soil that would support the foundation would undergo and the bearing capacity of the soil. The settlement of the soil is determined and a foundation type is provided whose settlement would not exceed the allowable settlement. Thus if the settlement expected is not suitable for pad foundation, another suitable foundation would be adopted. The bearing capacity of the soil on the other hand which is a measure of the shear strength of the soil should be such that the soil does not fail in shear. In this case there would be an allowable bearing capacity that should not be exceeded and this bearing capacity should constrain footing width in case of pad footing or determine the type of foundation that would be suitable if the pad footing is not suitable.

These two terms: settlement and bearing capacity are determined from soil exploration tests such as penetration tests (standard penetration tests, cone penetration tests etc) and plate loading tests.

The estimation of bearing capacity of soils from soil exploration is usually based on bearing capacity theories. There are about seven (7) classical bearing capacity theories which are:

1. Rankine theory
2. Prandtl theory
3. Terzaghi theory
4. Skempton’s theory
5. Meyerhof’s theory
6. Hansen’s theory
7. Vesics’ theory

Among these theories, Terzaghi’s theory and Meyerhof’s theory are among the most popular because of many special cases considered in these theories. Particular formulas available in these theories can be used to estimate allowable soil bearing capacity from important parameters and conditions of the soil. The formulas can also be used to size the foundations. One advantage of Meyerhof’s theory is that it can also be used for deep foundations as well as for footings on slopes. We would only consider Terzaghi’s theory in our estimation and sizing. Specialized literatures can be consulted for other theories.

Before we proceed, it would be good to define some bearing capacity terms that are usually considered in foundation design. These are:

1. Ultimate bearing capacity: This is the pressure that would cause shear failure of the supporting soil immediately below and adjacent to the foundation. It is usually denoted by q_u and can be estimated thus,

q_u = P (load)/Area = P/(B x L) where B = breadth of footing and L = length of footing.

2. Safe bearing capacity: This is the maximum value of contact pressure to which the soil can be subjected to without risk of shear failure. It is based on the strength of the soil and invariably, it is the ultimate bearing capacity.

3. Allowable bearing capacity: This is the bearing capacity allowed in other to take care of shortcomings such as errors in design, variations in the strength of materials etc. It is the bearing capacity value normally used in designs. It is usually less than ultimate bearing capacity and based on the factor of safety (F.O.S). Factor of safety is a numeric term used to cater for shortcomings mentioned above. For soils, F.O.S of 3 – 5 is ideal. It is usually denoted by q_a.

q_a = q_u/F.O.S

4. Net bearing capacity: This is described as the increase in pressure at the foundation level. It is the total foundation pressure less the effective weight of the soil per unit area permanently removed. It is denoted by q_n.

q_n = q_u – γz where γ = soil unit weight and z = depth of soil. γz is known as the overburden pressure. It is often denoted as q_o. Thus, q_n = q_u – q_o

According to Terzaghi, the ultimate bearing capacity of soil can be expressed as q_u = c¹N_cS_c + q_o N_qW_q + (1/2)γBN_γS_γW_γ

Where

c¹ = effective cohesion,

B = width of the footing,

q_o = soil overburden pressure = γz

γ = soil unit weight

Other terms in the equation are correction factors

N_c, N_q and N_γ are called bearing capacity factors. These values can be determined when the angle of internal friction of the soil, ϕ is known. As stated elsewhere ϕ can be determined from soil exploration. The table below provides the values of the factors (according to Terzaghi) based on angle of internal friction.

Table 1; bearing capacity factors based on ϕ

N_c represents the contribution due to constant component of shear strength. N_q represent the contribution due to surcharge pressure while N_γ represent the contribution resulting from weight of the soil.

S_c and S_γ can also be found in the equation. These are called shape factors that is the shape of the footing.

For strip footing, S_c = S_γ = 1

For square/rectangular footings, S_c = 1 + 0.3 (B/L) and _Sγ = 1 – 0.2 (B/L) where B and L are breadth and length of footing respectively.

For circular footings, S_c = 1.3 and S_γ = 0.6

W_q and W_γ are water table correction factors.

There are three conditions applicable in this situation which include:

1. When the water table is above the bottom of the footing as shown in Figure 1 below that is Z_g ≤ Z

W_q = 0.5 (1 + (Z_g/Z)); W_γ = 0.5

2. When the water table is below the bottom of the footing as shown in Figure 2 below

In this case, W_q =1; W_γ = 0.5 (1 + (Z_b/Z)) where Z_b = Z_g – Z

3. When the water table is exactly at the bottom of the footing. In this case, W_q = 1; W_γ = 0.5

Structural design of the footing

The structural design of footing begins with sizing of the footing which requires knowledge of the bearing capacity of the soil (usually allowable bearing capacity) and the load coming on the footing from the column. The estimation of load from column (both axial loads and moments) can be done manually or gotten from software analysis. As stated elsewhere, the bearing capacity of soil can be calculated based on results from soil exploration.

Note: Some of the parameters in shape factor are functions of B and L. But this is more applicable in rectangular footings because in square footings, B/L = 1.

Example

A proposed square concentric pad footing is to carry a dead load of 500 kN and live load of 300 kN from a column of 300 x 300 mm dimensions at a depth of 0.75 m in a sandy clay deposit. The shear strength parameters of the soil are c = 10 kN/m² and ϕ = 40^o. Assuming that the unit weight of the soil is 18 kN/m³ and a factor of safety against shear of 3.0, determine the allowable bearing capacity of the soil if the water table is at the base of the foundation. Also carry out the structural design of the footing based BS 8110.

Ultimate bearing capacity, q_u = C¹N_cS_c + q_o N_qW_q + (1/2)γBN_γS_γW_γ

Solution

c¹ = 10 kN/m²;

From the table 1 above, based on ϕ = 40^o, N_c = 95.7; N_q = 81.3; N_γ = 100.4

Since it is a square footing, S_c = 1+0.3 (1) = 1.3 and S_γ = 1-0.2(1) = 0.8

For water table correction, since the water table is at the base of the footing (third condition), W_q = 1; W_γ = 0.5

q_u = 10 x 95.7 x 1.3 + 18 x 0.75 x 81.3 x 1 + 0.5 x 18 x B x 100.4 x 0.8 x 0.5 =1244.1 + 1097.55 + 361.44 B

q_u = 2341.65 + 361.44 B. Using F.O.S of 3, allowable bearing pressure, q_a = q_u/FOS = (2341.64 + 361.44B)/3 = 780.55 + 120.48B

q_a = 780.55 + 120.48B  (1)                                               

The loads on the column are 500 kN dead load and 300 kN live load,

Total unfactored load, S_k = G_k + Q_k = 500 + 300 = 800 kN

Total factored load, N = 1.4 G_k + 1.6 Q_k = (1.4 x 500) + (1.6 x 300) = 700 + 480 = 1180 kN

To size the footing, we need to first determine the area of the footing

Area of footing = (G_k + Q_k + W)/q_a = (S_k + W)/qa, where S_k is the unfactored load with value of 800 kN. W is the self -weight of the footing and it is given as a percentage of the unfactored load. The table 2 below provides the percentage range for different kinds of footings.

Table 2; Selfweight, W of footing as percentage of unfactored column load, S_k

1. Assess W

Since our question is about pad footing, W = 6% of 800 = 48 kN

Thus, total load = S_k + W = 800 + 48 = 848 kN

Area of footing = B x L = B² since B = L (square footing).

Recall that, q_a = load/area

Thus, q_a = 800/B²  (2)                                                            

2. Assess the breadth of the footing, B

Equation (1) = (2)

780.55 + 120.48 B = 848/B²

B² (780.55 + 120.48 B) = 848; 780.55 B² + 120.48 B³ = 848

120.48 B³ +780.55 B² – 848 = 0. From the solution, B = 971 mm, say 1000 mm

Area provided = B² = 1² = 1 m²

3. Estimate the thickness of the footing

h = W/24A_prov = 48/ (24 x 1) = 2 m

The depth is too much. Let’s increase the width B in other to reduce the depth

Assuming B = 2000 mm = 2 m

A_prov = 4 m²

h = 48/ (24 x 4) = 0.5 m = 500 mm

Let’s try h = 600 mm

Concrete cover, cc = 50 mm

Bar diameter, θ = 16 mm

Effective depth, d = h – cc – θ/2 = 600 – 50 – 8 = 542 mm

4. Check for punching shear at column face

ν_punch = N/ 2(a₁ + a₂)d = [(1.4 x 500) + (1.6 x 300)]/2(300 + 300) 542 = 1180 x 10³/650400 = 1.814 N/mm²

ν_punch ˂ 0.8√fcu or 5 N/mm²

1.814 ˂ 4 N/mm² or 5 N/mm²                         Punching shear is satisfied!!!

5. Determination of bending moment and design of section

B = 2000 mm = 2 m; L = B; h = 600 mm = 0.6 m; a1 = 300 mm = 0.3 m

Moment arm, j = (B-a)/2 = (2 – 0.3)/2 = 0.85m

q = N/A_prov. = 1180/4 = 295 kN/m²; m = (qj²)/2 = (295 x 0.85²)/2 = 106.57 kNm/m run

k = M/f_cubd² = (106.57 x 10⁶)/30 x 1000 x 542² = 0.0121 ˂ 0.156 Ok

Lever arm, la = 0.5 + √0.25 – (k/0.9) = 0.986, say la = 0.95

Z = lad = 0.95 x 542 = 515 mm

As_req = M/0.95f_yZ = (106.57 x 10⁶)/0.95 x 410 x 515 = 533 mm²/m

As_min = 0.13% bh = 0.0013 x 1000 x 600 = 780 mm²/m

Adopt As_min as As_req = 780 mm²/m

Provide Y16@250 mm C/C btm EW (As_prov = 804 mm²/m)

6. Determine Concrete Shear strength (critical shear strength)

ν_c = 0.632 ((f_cu/25) x p)^1/3 x (400/d)^1/4

N.B. (400/d)^1/4 ≥ 1; If d ˃ 400, ignore the expression. Thus is our case

ν_c = 0.632 ((f_cu/25) x p)^1/3 ; but p = (100As_prov)/bd = (100 x 780)/(1000 x 542) = 0.144

ν_c = 0.632 (1.2 x 0.144)^1/3 = 0.632 x 0.557 = 0.352 N/mm²

7. Check for shear at critical section distance, d, measured from column face

j – d = 850 – 542 = 308 mm = 0.308 m

Shear at critical section, V = q (j – d) = 295 x 0.308 = 90.86 kN/m

Shear stress, ν = V/bd = (90.86 x 103)/1000 x 542 = 0.168 ˂ 0.352 N/mm² Ok

8. Check for punching shear at distance 1.5d from column face

Area under punching, A_p = BL – (a₁ + 3d) (a₂ + 3d) = B² – (a₁ + 3d)² = 2² – (0.3 + 3(0.542))²

A_p = 4 – 3.7095 = 0.2905 m² (NB: if 1.5d is outside the footing width or if A_p is zero (0) or negative (-ve), there is no need to check for punching shear).

Perimeter of punching, P_p = 2 (a₁ + a₂ + 6d) = 2 (0.3 + 0.3 + 6(0.542)) = 7.704m

Punching shear strength, ν_p = (q x A_p)/P_p x d = (295 x 0.2905)/ (7.704 x 0.542) = 20.524 kN/m² = 0.0205 N/mm²

ν_p ˂ v_c    Ok

Find attached PDF copy of solved example with Tekla Tedds

Manual design seems more economical than the output from Tekla Tedds