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TYPES OF SHALLOW FOOTINGS bc - column width Critical Critical section section for bending for shear 1 2 de 150mm (min) - effective depth determined from 75mm bending and shear (min) cover to B earth face Bmin = bc + 300 mm B is determined by satisfying serviceability and ultimate limit states. (a) Isolated footing (b) Isolated footing and floor slab (c) Combined footing (d) Mat foundation (e) Waffle mat (f) Tendons layout in a post-tensioned mat bw - wall width 0.5 bw (min) bw (min) 2bw (g) Strip footing (h) A strip footing under construction (i) Location of footings near slopes (j) Location of footings on s slope (k) Stepped footings Figure 4.1 Shallow footings and approximate initial dimensions. There are several types of shallow foundations. Some are shown in Figure 4.1. A spread or individual forting, also called isolated footing, transfers individual column loads directly to the soil (Figures 4.1a,b). If a single spread footing interferes with another spread footing, the two can be combined to form a combined footing (Figure 4.1c). When there is a large number of spread footings required for a structure it may he more economical to use a mat or raft foundation. If the overburden pressure of a soil removed to embed a raft foundation is equal to the stress imposed by the structure, the raft is termed fully compensated or floating raft (Figure 4.1d). If the overburden pressure is a fraction of the imposed stress, the raft is partially compensated. Some rafts are thickened in the form of a waffle as shown in Figure 4.1e. Post-tensioned mat or floor slab are often located just below the surface (Figure 4.1f). Raft foundations are also integrated with pile foundations. These are called pile-raft foundations. Walls are normally supported using a strip foundation (Figures 4.1g,h). A strip foundation is one in which the width (B) to length (L) ratio is small (B/L >> 10). On sloping ground, a footing can rest on top of the slope, on the slope or can be stepped as illustrated in Figures 4.1i,j,k. THEORETICAL BEARING CAPACITY BASED ON LIMIT EQUILIBRIUM METHOD The theoretical bearing capacity equations in common use in design practice were obtained using the limit equilibrium method. Prandtl (1920) showed theoretically that a wedge of material is trapped below a rigid plate when it is subjected to concentric loads. Terzaghi (1943) applied Prandtl's theory to a strip footing (length is much longer than its width) with the assumption that the soil is a semi-infinite, homogenous, isotropic, weightless rigid-plastic material. Based on Prantl’s theory, failure of the footing occurs by a wedge of soil below the footing pushing its way downward into the soil. The failure mechanism postulated by Terzaghi (1943) is shown in Figure 4.2a. Df F Considered by Meyerhof Df Df Figure 4.2 Failure mechanisms The zone ABC is a rigid wedge of soil trapped beneath the footing. Zone ABD is fan like with radial slip lines stopping on a logarithm spiral slip line. The other, zone ADE, consists of φ′ φ′ slip lines oriented at angles of 45 + and 45 − to the horizontal and vertical planes, 2 2 respectively. The zone ADE is called the Rankine passive zone. The surfaces, AB and AD, are frictional sliding surfaces and their actions are similar to rough walls being pushed into the soil. The pressure exerted is called passive earth pressure. The shape of the surface BD depends on γB the ratio where γ is the unit weight of the soil and σzo is the overburden pressure. For a σ zo γB weightless soil, → 0, and BD is a logarithmic spiral; for γB = 0, BD is a circle; for γB ≠ 0 , σ zo BD is between a circle and a logarithm spiral. For clean, coarse-grained soils, BD is always a circle (Vesic, 1973). For dense coarse-grained soils or heavily over-consolidated fine-grained soils, the failure planes are expected to reach the ground surface and the failure mode is called general shear failure. For loose coarse-grained soils or lightly over-consolidated fine-grained soils, the failure planes, if they are developed, are expected to lie within the soil layer below the base of the footing and extend laterally. The failure mode is called local shear. Another type of shear failure is possible. For very loose soil, the failure surfaces may be confined to the surfaces of the rigid wedge. This type of failure is termed punching shear. Each of these collapse mechanism depends on the relative density of the soil and the depth of embedment (Figure 4.3). Figure 4.3 Collapse mechanism based on relative density and embedment depth (Source: Vesic, 1973) In saturated fine-grained soils, the settlement to mobilize general shear failures under a surface footing is 3% to 7% of the foundation width and about 15% for deeply embedded footings. For coarse-grained soils, the corresponding settlements are 5% to 15% for surface footings and 25% for deeply embedded footings. We will consider the theoretical bearing capacity equations for short-term (undrained condition) and long-term (drained condition) separately. A total stress analysis (TSA) utilizing the undrained shear strength, su, is employed for short-term loading of fine-grained soils. An effective stress analysis (ESA) utilizing the effective friction angle, φ′, is employed for long-term and short-term loading of fine-grained and coarse-grained soils. All the equations are based on the following assumptions. 1. The embedment depth is not greater than the width of the footing (Df < B) 2. General shear failure occurs 3. The shear strength of the soil above the footing base is negligible (Terzaghi’s and Hansen’s assumption). 4. The soil above the footing base can be replaced by a surcharge stress (= γDf). 5. The soil is a rigid-plastic, homogenous medium. No displacement occurs prior to collapse. The appropriate friction angle to use for an ESA is the peak friction angle,φ′p. However, φ′p is not a fundamental soil parameter. Its magnitude depends on the ability of the soil to dilate which in turn depends on the relative density and the normal effective stress. We will consider the theoretical net ultimate bearing capacity (qult) of shallow footings based on limit equilibrium proposed by Meyerhof (1963), Vesic (1973) and Hansen (1970). Terzagi’s equation is not given because it has been generally superseded in engineering practice by Meyerhof (1963), Vesic (1973) and Hansen (1970) methods. The author has modified some of these equations to separate total stress analysis from effective stress analysis. Meyerhof’s (1963) equations Vertical load TSA: q ult = 5.14s u s c d c + γD f ESA: q ult = γD f (N q − 1)s q d q + 0.5γB′N γ s γ d γ Inclined load TSA: q ult = 5.14s u d c i c + γD f ESA: q ult = γD f (N q − 1)d q i q w q + 0.5γB′N γ d γ i γ w γ where π tan φ′ φ′ ) ; N γ = (N q − 1) tan(1.4φ′ ) and the other factors are given in Table 4.1. p Nq = e p tan 2 (45o + p 2 Vesic’s (1973) equations Vertical load TSA: q ult = 5.14s u s c d c + γD f ESA: q ult = γD f (N q − 1)s q d q w q + 0.5γB′N γ s γ d γ w γ π tan φ′ φ′ ) ; N γ = 2(N q + 1) tan φ′ and the other factors are given in Table 4.1. p Nq = e p tan 2 (45o + p 2 Hansen’s (1970) equations All loads TSA: qult = 5.14 su (1 + sc + dc - ic - bc - gc) ESA: qult = γ Df (Nq - 1) sq dq iq bq gq wq + 0.5 γ B′ Nγ sγ dγ bγ gγ wγ where π tan φ′ φ′ ) ; N γ = 1.5(N q − 1) tan φ′ and the other factors are given in Table 4.1 p Nq = e p tan 2 (45o + p 2 The factors, Nq and Nγ, are bearing capacity factors that are functions of φ′p; sc, sq and sγ are shape factors, dc, dq, dγ are embedment depth factors, ic, iq, iγ are load inclination factors, bc, bq, bγ are base inclination (base tilt) factors, gc, gq, gγ are ground inclination factors, wq and wγ are ground water factors. Various equations have been proposed for the various factors as given in Table 1. The bearing capacity equations apply for a single resultant load with normal Vn and horizontal components HB parallel to the width, B (the short side) and horizontal components HL parallel to the length, L (the long side). When investigating potential failure along the short side, use HL and for failure along the long side, use HB. Table 4.1 Geometric factors for use in theoretical bearing capacity equations Shape factors Hansen Meyerhof Vesic sc B′ B′ B′ 1 + 0.2 1+ 0.2 ; θ = 0 ; vertical loads only L′ L′ L′ B′ 0.2i cB ; for inclined loads, short side L′ failure L′ 0.2i cB ; for inclined loads, long side B′ failure sq B′ B′ B′ 1 + 0.1K p 1+ tan φ′ 1+ sin φ′ ; θ = 0 ; vertical loads L′ L′ L′ p p φ′ B′ K p = tan 2 (45 + p ); φ′ > 10o only 1 + i qB sin φ′ ; for inclined L′ p p 2 loads, short side failure L′ 1+ i qL sin φ′ ; for inclined loads, B′ p long side failure sγ B′ B′ B′ 1 + 0.1K p = 1 − 0.4 1 + 0.4 φ′ ; θ = 0 ; vertical loads L′ L′ L′ p φ′ only ); φ′ > 10o p K p = tan 2 (45 + p B′i γB 2 1 − 0.4 ; for inclined loads, short L′i γL side failure, sγ > 0.6 should be used L′i γL 1 − 0.4 ; for inclined loads, long B′i γB side failure, sγ > 0.6 should be used Depth factors Hansen Meyerhof Vesic dc Df Df Df 1 + 0.2 1 + 0.4 ; D ≤ B′ ; see 0.4 ; D ≤ B′ ; see note 1 B′ B′ f B′ f note 1 D Df 0.4 tan −1 f ; Df > B′ ; see note 1 1 + 0.4 tan −1 ; B′ B′ Df > B′ ; see note 1 dq Df 2 Df 1 + 0.1 K p ; φ′ > 10o ′ p D f 1+2 tan φ′ (1-sin φ′ ) B′ ; D ≤ B′ p p B 1+2 tan φ′ (1-sin φ′ ) 2 B′ p p D 1+2 tan φ′ (1-sin φ′ ) 2 tan -1 ( f ) B′ p p 1+2 tan φ′ (1-sin φ′ ) 2 tan -1 p p ;D>B ′ ; see note 1 see note 1 dγ Df 1; D f < B′ 1; D f < B′ 1 + 0.1 K p ; φ′ > 10o B′ p Notes: note 1 – set all depth factors to 1 if the shear strength of the soil above the base is small Load inclination factors Hansen Meyerhof ic 2 ⎛ θo ⎞ Hi ⎜1 − ⎟ ; load inclined in direction of 0.5(1 − 1 − ) ; see note 2 ⎝ 90 ⎠ Vn A′s u footing width Q is the applied vertical load with Vn as the normal load, Hi is the horizontal load, the subscript i ⎡ ⎛ α ⎞ ⎤ denotes direction. cos θ ⎢1− ⎜1− a ⎟ sin θ⎥ ; load inclined in Q ⎣ ⎝ π+ 2 ⎠ ⎦ Vn direction of footing length +θ 2 1 αa = adhesion factor that is usually to HB 3 2 for short-term loading HL L B iq 2 Hi 5 ⎛ θo ⎞ (1 − 0.5 ) ; 45o ≤ φ′ ≤ 300 ;i q > 0 ; see note ⎜1 − ⎟ ; load inclined in direction of Vn p ⎝ 90 ⎠ 3 footing width 5 ⎛ ηo ⎞ ⎜ (0.7 − )H i ⎟ ⎛ sin θ ⎞ 450 ⎜1 − ⎟ ; for η = 90 o 0 cos θ ⎜1 − ⎟ ; load inclined in ⎜ sin φ′ ⎟ ⎜ Vn ⎟ ⎝ p ⎠ ⎜ ⎟ direction of footing length ⎝ ⎠ iγ 2 Hi 5 ⎛ θo ⎞ (1 − 0.7 ) ; 45o ≤ φ′ ≤ 300 ;i γ > 0 ; see note ⎜1 − ⎟ ; load inclined in direction of Vn p ⎝ 90 ⎠ 2 footing width 5 ⎛ ηo ⎞ ⎛ sin θ ⎞ ⎜ (0.7 − )H i ⎟ ⎜1 − 450 ⎟ ; for η = 90 o 0 cos θ ⎜1 − ⎟ ; load inclined in ⎜ sin φ′ ⎟ ⎜ Vn ⎟ ⎝ p ⎠ ⎜ ⎟ direction of footing length ⎝ ⎠ Notes: note 2– i c ,i q , i γ = i cB ,i qB , i γB for the horizontal force along the width, Hi = HB, (failure along the long side, HB dominant) or i c , i q , i γ = i cL , i qL , i γL for the horizontal force along the long side, Hi = HL, (failure along short side, HL dominant) Base and ground inclination factors (see Figure ) Hansen bc ηo ; β < φ′ ; ηo + βo < 90o p 147 bq exp( −2η tan φ′ ); η i s in radians p β < φ′ ; ηo + βo < 90o p bγ exp( −2.7 η tan φ′ ); η i s in radians p β < φ′ ; ηo + βo < 90o p gc βo ;; β < φ′ ; ηo + βo < 90o p 147 gq (1 − 0.5 tan β]5 ; β < φ′ ; ηo + βo < 90o p gγ (1 − 0.5 tan β]5 ; β < φ′ ; ηo + βo < 90o p Vn Df +β B/2 Hi B/2 +η Figure 4.4 Footing on a slope: definition of terms in Hansen’s equation The effective width B′ and effective length, L′, giving an effective area, A′ = B′ L′ must be used in the theoretical bearing capacity equations. When the location of the resultant load (load center) is not coincident with the centroid (center of area) of the footing, the footing dimension is theoretically adjusted to align the load center with the centroid. Some possible cases are shown in Figure 4.5. Vn Vn Vn eB eL My Mx L L L B B B B′ = B, L′ = L B′ = B - 2eB, L′ = L - 2eL Mx M A′ = A = BL A′ = B′L′ eB = ;e L = Y Vn Vn B′ = B - 2eB, L′ = L - 2eL A′ = B′L′ Q Vn +θ eB HB eL HL L L B B B′ = B - 2eB, L′ = L - 2eL A′ = B′L′ Use both the inclination factors and the effective width in the equations Figure 4.5 Some possible cases of eccentric loads You have to make some adjustment to the theoretical bearing capacity equations for ground water condition. The term γDf in the bearing capacity equations for an ESA refers to the vertical stress of the soil above the base of the foundation. The last term γB refers to the vertical stress of a soil mass of thickness, B, below the base of the footing. You need to check which one of three ground water situations is applicable to your project. Situation 1: Ground water level at a level B below the base of the footing. No correction is required, i.e., wq = wγ = 1 Situation 2: Ground water level within a depth B below the base of the footing Df < z < (B + Df ) (Figure 4.3a). wq = 1, w γ = ( z-Df ) + γ′ D z (1 + f − ) B γ sat B B z Df Df z (Df –z) B B (a) Ground water within a depth B below base (b) Ground water within embedment depth Figure 4.3 Ground water effects below base of footing. Situation 3: Ground water level within the embedment depth 0 ≤ z ≤ Df (Figure 4.3b) z γ′ z γ′ wq = + (1 − ) and w γ = D f γ sat Df γ sat DESIGN OF SHALLOW FOOTINGS It is customary for practicing engineers to make an estimate of the footing size and then check that it satisfies serviceability and ultimate limit states. Typical initial dimensions for isolated and strip footings are shown in Figure Remember that these dimensions are only initial guidelines. The procedures to design a shallow rectangle footing using ASD are as follows: 1. Select a footing size 2. Determine settlement using a load factor of 1. 3. Check that settlement is less than or equal to the serviceability limit state. If settlement exceeds serviceability limit state, increase the footing size and recalculate. If settlement is much less (> 10% ) than the serviceability limit state, then reduce footing size and recalculate. 4. Calculate ultimate bearing capacity (qult) using the bearing capacity equations if lab tests results are available or from empirical equations using field test data. For fine-grained soils, calculate the ultimate bearing capacity for both short-term (TSA) and long-term (ESA) conditions. For free draining coarse-grained soils, calculate the ultimate bearing capacity for long-term (ESA) conditions, which is also applicable to short-term conditions. q ult 5. Calculate the allowable bearing capacity using q a = + γDf FS My Mx 6. If the eccentricities, eB = ; eL = , where x and y are the axes parallel to the width P P 6e B 6e and length respectively, are not zero then calculate Γ = or Γ = L B L 7. Calculate B′ = B − 2eB and L′ = L − 2eL 8. Calculate ( Pa )max = P(1 + Γ) where Pa is the applied load and Γ is the higher value calculated in item 6. 9. Check that q a B′L′ ≥ ( Pa )max The performance or resistance factors to determine the bearing capacity of shallow footings based on LFRD are shown in Table 7X.A Table 7 X. A Performance factors for bearing capacity calculations using LRFD Resistance Factor, ωR ESA: Coarse-grained and fine –grained soils • φ′cs from lab tests 0.95* • φ′p from lab tests 0.80 • SPT 0.45 • CPT 0.55 • Plate load test 0.55 TSA: Fine-grained soils • su from lab tests 0.60 • Shear vane 0.60 • CPT 0.50 • Plate load test 0.55 Rock 0.6 * Author’s recommendation The procedures to design a shallow rectangle footing using LRFD are as follows: 1. Select a footing size. 2. Determine settlement using a load factor of 1. 3. Check that settlement is less than or equal to the serviceability limit state. If settlement exceeds serviceability limit state, increase the footing size and recalculate. If settlement is much less (> 10% ) than the serviceability limit state, then reduce footing size and recalculate. 4. Determine the maximum factored load (Pu) by combining loads according to IBC (2000) or other codes. 5. Calculate ultimate bearing capacity (qult) using the bearing capacity equations if lab tests results are available or from empirical equations using field test data. For fine-grained soils, calculate the ultimate bearing capacity for both short-term (TSA) and long-term (ESA) conditions. For free draining coarse-grained soils, calculate the ultimate bearing capacity for long-term (ESA) conditions, which is also applicable to short-term conditions. My Mx 6. If the eccentricities, eB = ; eL = , where x and y are the axes parallel to the width P P and length respectively, are not zero then calculate B′ = B − 2eB and L′ = L − 2eL Pu 7. Calculate q u = ( if eB = 0, then B = B′; if eL = 0, then L = L′) B′L′ 8. Calculate the gross resistance R = ψR qult + ρi γ Df where γ Df is the effective soil pressure above the footing base, ρi is the dead load factor for earth pressure (usually 0.9). 9. Check that R ≥ q u Suggested changes if serviceability and ultimate limit states are not satisfied after first trial. Case 1: Settlement too large Actions: (1) Increase size of footing. This will reduce the applied stress. (2) Consider a mat foundation if isolated footing becomes too large with the potential for stress overlaps. The mat foundation will spread the load over a larger area than isolated footings thereby reducing the applied stress. (3) Consider soil improvement such as dynamic compaction (coarse-grained soils) and soil reinforcement by, for example, geotextiles (see Chapter 11). (4) Consider deep foundations. Case 2: Ultimate bearing capacity too low. Actions: Same as above