4.BEARING CAPACITY If a footing is subjected to too great a load,some of the soil supporting it will reach a failure state and the footing may experience a bearing capacity failure.The bearing capacity is the limiting pressure that the footing can support. Footing ← 4.1 Shallow Foundations(D/B<1) The solutions presented below have been determined for the situation where a footing sits on the soil surface.The effect of founding the footing below the ground surface,as is common for most buildings,is introduced by assuming that the soil above the footing applies a surcharge as shown below 0 qs=yD 111IF Buried Footing Equivalent Surface Footing In making this assu mption the effects of any restraint from the soil above the base of the footing be neo otion is r vided that the depth of burial isno too great.To There are two methods of theoretical analysis used to investigate the bearing capacity The lower bound approach in which an arrangement of the stresses in the ground is found in equilibrium with the applied loads for which the soil is everywhere in a state of failure.The solution is less than or equal to the true collapse load. .The upper bound approach in which a failure mechanism is postulated and the applied loads in equilibrium with the assumed failure mechanism are determined.The solution is greater than or equal to the true collapse load In the limit,when the mechanism and the stresses are consistent,the two methods give the same solution. ne whenφ≠0 and most of the ch ned so tions are not to to the collapse loads are now avail kely to be in the
4. BEARING CAPACITY If a footing is subjected to too great a load, some of the soil supporting it will reach a failure state and the footing may experience a bearing capacity failure. The bearing capacity is the limiting pressure that the footing can support. 4.1 Shallow Foundations(D/B < 1) The solutions presented below have been determined for the situation where a footing sits on the soil surface. The effect of founding the footing below the ground surface, as is common for most buildings, is introduced by assuming that the soil above the footing applies a surcharge as shown below Buried Footing Equivalent Surface Footing In making this assumption the effects of any restraint from the soil above the base of the footing are ignored. This assumption is reasonable provided that the depth of burial is not too great. To be considered as a shallow foundation D/B should be less than 1. There are two methods of theoretical analysis used to investigate the bearing capacity. • The lower bound approach in which an arrangement of the stresses in the ground is found in equilibrium with the applied loads for which the soil is everywhere in a state of failure. The solution is less than or equal to the true collapse load. • The upper bound approach in which a failure mechanism is postulated and the applied loads in equilibrium with the assumed failure mechanism are determined. The solution is greater than or equal to the true collapse load. • In the limit, when the mechanism and the stresses are consistent, the two methods give the same solution. Accurate solutions to the collapse loads (bearing capacity) are difficult to determine when 0 and most of the charted solutions are not totally reliable. Computerised methods for determining bounds to the collapse loads are now available and are likely to be more widely used in the future. Footing qs = D D B Q Q
4.2 Bearing capacity of a shallow strip footing (plane strain) To illustrate the factors on which the bearing capacity depends a simple (not accurate)lower bound approach is considered below. Footing Surcharge qs ↓ Soil at state of Soil at state of Active Failure Passive with Gv>Gh Failure with Frictionless Gh>Ov Discontinuity When the soil is at a state of failure the Mohr-Coulomb criterion is satisfied,and we have o,=N。o:+2cVNg N。=g,+ccot o ccoto Under the stress is given by Oy =qr +YZ At failure N,=9+yz ccot g.+ccotφ hence N (q:+yz+ccot)-cco In the region away from the footing the horizontal stress will be greater than the vertical stress. At any depthzthe vertical stress is given by Gy qr YZ and at failure N。= Gn+ccot qs +yz ccot and n=N (qs +Yz+ccot)ccot For equilibrium the horizontal forces from the two failure zones must be equal
4.2 Bearing capacity of a shallow strip footing (plane strain) To illustrate the factors on which the bearing capacity depends a simple (not accurate) lower bound approach is considered below. When the soil is at a state of failure the Mohr-Coulomb criterion is satisfied, and we have 1 = N 3 + 2 c N or N c c = + + 1 3 cot cot Underneath the footing the vertical stress will be greater than the horizontal stress, that is v = 1 and h = 3, and in the limit these will be related by the Mohr-Coulomb criterion. At any depth z the vertical stress is given by v qf = + z At failure N q z c c f h = + + + cot cot hence h f N = q + z + c − c 1 ( cot ) cot In the region away from the footing the horizontal stress will be greater than the vertical stress. At any depth z the vertical stress is given by v qf = + z and at failure N c q z c h s = + + + cot cot and h = N (qs + z + c cot ) − c cot For equilibrium the horizontal forces from the two failure zones must be equal qf Footing Surcharge qs Soil at state of Active Failure with v > h Soil at state of Passive Failure with Frictionless h > v Discontinuity H
J()dz-J(.)d which gives g+-N. -+ccotH and hence the bearing capacity qr is 9r q.N+HNi-1)ccot(Ni-1) This solution will give a lower bound to the true solution because of the simplified stres distribution assumed in the soil.Nevertheless,similar terms occur in all bearing capacity calculations.These terms relate to the surcharge applied to the soil surface the self weight of the soil 。the effect of cohesion The general bearing capacity formulae are therefore written in the following form qr=q.N。+Yy)N,+cN 2 where the quantities Na.N..and Ne are known as bearing capacity factors.For the shallow end on and can be determined from the 40 50 40 20 20 40 60 N.and N. BEARING CAPACITY FACTORS [After Terzaghi and Peck (1948)]
( ) ( ) h active h passive H H dz = dz 0 0 which gives 1 2 2 2 2 N q H H c H N q H H f s c H + + = + + cot cot and hence the bearing capacity qf is q q N ( ) ( ) H f = s + N − + c N − 2 2 2 2 1 cot 1 This solution will give a lower bound to the true solution because of the simplified stress distribution assumed in the soil. Nevertheless, similar terms occur in all bearing capacity calculations. These terms relate to • the surcharge applied to the soil surface • the self weight of the soil • the effect of cohesion The general bearing capacity formulae are therefore written in the following form q q N B f = s q + N + cNc 2 where the quantities Nq, N, and Nc are known as bearing capacity factors. For the shallow bearing capacity problem values of these factors depend on and can be determined from the chart produced by Terzaghi shown below (see also p 28 in Data Sheets) BEARING CAPACITY FACTORS [After Terzaghi and Peck (1948)] 60 50 40 30 20 10 0 20 40 60 80 N and N 0 10 20 30 40 (degrees) q c N N q N B D a b c d q= D Qf f f Bearing capacity of a shallow foundation ULTIMATE BEARING CAPACITY OF CLAY ( = 0 only) (After A.W. Skempton) 0 1 2 3 4 5 D/B 5 6 7 8 9 N Circle or square Continuous c 5.14 B D N (for rectangle) = (0.84+0.16 ) N (square) L= Length of footing B L c q ult = cNc q = B N + cN + D N continuous footing 1 2 f c f q q = 0.4 BN + 1.3cN + D N square f c f q q = 0.6 RN + 1.3cN + D N circular f c f q qf = cNc + D c Nc BEARING CAPACITY THEORIES OF TERZAGHI AND SKEMPTON
The mechanism analysed by Terzaghi is shown below.Note that it differs from the simple problem analysed above by having a transition region between the regions where the principal stresses are horizontal and vertical. Dr ..k. The shape of the footing also influences the bearing capacity.This can be allowed for by adjusting the terms in the bearing capacity equation. Fora continuous strip footing q=q,N。+YN,+cN Fora square footing qr =q,N 0.4y BN,1.3cN For a circular footing qr =q.N 0.6y BN,1.3cN. bove 4.2.1 Effective Stress Analysis Two situations can be simply analysed. The soil is dry.The total and effective stresses are identical and the analysis is identical to that described above except that the parameters used in the equations are c.Y rather than c.d.v If the water table is m ore than adepth of 1.5B(the footing with)belowthe base of the footing the water can be assumed to have no effect. .The soil below the base of the footing is saturated. IQ=qrB
The mechanism analysed by Terzaghi is shown below. Note that it differs from the simple problem analysed above by having a transition region between the regions where the principal stresses are horizontal and vertical. BEARING CAPACITY FACTORS [After Terzaghi and Peck (1948)] 60 50 40 30 20 10 0 20 40 60 80 N and N 0 10 20 30 40 ( d e g r e e s ) q c N Nq N B D a b c d q= D Qf f f Bearing capacity of a shallow foundation ULTIMATE BEARING CAPACITY OF CLAY ( = 0 only) (After A.W. Skempton) 0 1 2 3 4 5 D/B 5 6 7 8 9 N Circle or square Continuous c 5.14 B D N (for rectangle) = (0.84+0.16 ) N (square) L= Length of footing B L c q ult = cNc q = B N + cN + D N continuous footing 1 2 f c f q q = 0.4 BN + 1.3cN + D N square f c f q q = 0.6 RN + 1.3cN + D N circular f c f q qf = cNc + D c Nc BEARING CAPACITY THEORIES OF TERZAGHI AND SKEMPTON The shape of the footing also influences the bearing capacity. This can be allowed for by adjusting the terms in the bearing capacity equation. For a continuous strip footing q q N B f = s q + N + cNc 2 For a square footing qf = qs Nq + 0 BN + 13cNc .4 . For a circular footing qf = qs Nq + 0 6 BN + 13cNc . . The analysis has been presented in terms of total stress. This can be used to evaluate the short term undrained bearing capacity. To evaluate the long term bearing capacity an effective stress analysis is required. This is very similar to the total stress analysis considered above. 4.2.1 Effective Stress Analysis Two situations can be simply analysed. • The soil is dry. The total and effective stresses are identical and the analysis is identical to that described above except that the parameters used in the equations are c´, ´, dry rather than cu, u, sat. If the water table is more than a depth of 1.5 B (the footing width) below the base of the footing the water can be assumed to have no effect. • The soil below the base of the footing is saturated. Df qs = D Q = qf B u = uo
Notation The effective bearing capacity qf=qf·o The effective surcharge q's=qs·0 The effective(submerged)unit weighty' Ysat Yw These quantities are usd because they give the variation of stresswh depth.Fo under the footing the total vertical stress,pore pressure and effective vertical stress at any depthzare =qr +Yz u=u。+YwZ o:G,-u=qi y'z Following through the same analysis as before but using the effective stress failure criterion which is given by N c'cot o;+c'cot中 gives qi =q:N +H(Ni-1)+e'cote(Ni-1) which can be more generally writtenas qf=qN。+YN,+cN 2 ersgmgapacftosactaitaotocegnombdoceNacthitctoacanme qr q'f uo The be The simple analysis considered so far has soil strength pa ·rate of loading(d ndranedl g0 wa (a turated) foundation shape(strip footing,square or circle) Additional factors which also influence the capacity are
Notation The effective bearing capacity q’f = qf - uo The effective surcharge q’s = qs - uo The effective (submerged) unit weight’ = sat - w These quantities are used because they give the variation of effective stress with depth. For example under the footing the total vertical stress, pore pressure and effective vertical stress at any depth z are v qf = + z u u z = o + w = − = + v v u qf z Following through the same analysis as before but using the effective stress failure criterion which is given by N c c = + + 1 3 cot cot gives = + ( ) ( ) q q N − + − H f s N c N 2 2 2 2 1 cot 1 which can be more generally written as = + q q N + B f s q N c Nc 2 where the bearing capacity factors are identical to those given before. Note that the total bearing capacity is qf and qf = q’f + uo The bearing capacity is a function of many factors. The simple analysis considered so far has accounted for • soil strength parameters • rate of loading (drained or undrained) • groundwater conditions (dry or saturated) • foundation shape (strip footing, square or circle) Additional factors which also influence the capacity are
soil compressibility embed ment (D/B>1) 。inclined loading eccer load 。 non-homogeneous soil These are usually taken into account by introducing correction factors to the conventional hearing capacity factors.Pages 74 and 75 of the Data Sheets give formulae for some of these correction factors. It should also be noted that pages 69 to 71 of the Data Sheets give more(theoretically)accurate bearing capacity factors However,in practice the Terzaghi factors are still widely used.There are two important points to note about these sets of factors The bearing capacity equation assumes that the effects of c',Y,and can be superimposed. However,this is not correct as there is an interaction between the three effects.This is a consequence of the plastic nature of the soil response. .The formulae give the ultimate bearing capacity.In practice the soil will deform significantly s m Local (yed 30 e aep ing at a oad le then sprea as the loa s increase d.Only when the echanism exis adopted to keep settlements avoid prob ms with local failure Example A 5 m wide strip footing is constructed on saturated clay with properties cu=25 kN/m2,=0, c'=2 kN/m2 =25 and ysat=15 kN/m2 Determine the short term and long term bearing capacities if the water table is at the soil surface and the footing is founded 2 m below the surface.The figure below shows the actual and idealised problem to be analysed. Q=qrB D=2m 1.Short term-Undrained analysis The position of the water table is not important,but the soil must be saturated for an undrained analysis to be app qs=Ysat D=15×2=30kPa
• soil compressibility • embedment (D/B > 1) • inclined loading • eccentric loading • non-homogeneous soil These are usually taken into account by introducing correction factors to the conventional bearing capacity factors. Pages 74 and 75 of the Data Sheets give formulae for some of these correction factors. It should also be noted that pages 69 to 71 of the Data Sheets give more (theoretically) accurate bearing capacity factors. However, in practice the Terzaghi factors are still widely used. There are two important points to note about these sets of factors • The bearing capacity equation assumes that the effects of c', , and ' can be superimposed. However, this is not correct as there is an interaction between the three effects. This is a consequence of the plastic nature of the soil response. • The formulae give the ultimate bearing capacity. In practice the soil will deform significantly before general bearing failure occurs and large settlements may occur. Local failure (yield) will occur at some depth beneath the footing at a load less than the ultimate collapse load. The zone of plastic (yielding) soil will then spread as the load is increased. Only when the failure zone extends to the surface will a failure mechanism exist. A minimum load factor of 3 against ultimate failure is usually adopted to keep settlements within acceptable bounds, and to avoid problems with local failure. Example A 5 m wide strip footing is constructed on saturated clay with properties cu = 25 kN/m2 , u = 0, c' = 2 kN/m2 , ' = 25o , and sat = 15 kN/m2 . Determine the short term and long term bearing capacities if the water table is at the soil surface and the footing is founded 2 m below the surface. The figure below shows the actual and idealised problem to be analysed. 1. Short term - Undrained analysis The position of the water table is not important, but the soil must be saturated for an undrained analysis to be appropriate. qs = sat D = 15 2 = 30 kPa qs D = 2m B=5m Q Q=qf B
Foru=0 Ny=0,N=1,Ne=5.14 (From Terzaghi's chart) For=0 values can be more accurately read from Skempton's chart also on p28 in Data Sheets,and this is discussed further below. 4r g.N,T,+eN. qr =30x1 +0+25x5.14 =158.5kPa (Bearingcapacity) Q qrx B 158.5x5 792.5 kN/m (Bearing Force) 2.Long term-Effective stress analysis qs=30kPa =2×9.8=19.6kPa q's=10.4 kPa y=15-9.8=5.2kPa For'=250 Ng=13,Ne=24.5,N=10 (From Terzaghi's chart) iNN,e'N, qr=10.4×13+0.5×5.2x5×10+2×24.5=314.2kPa qf=314.2+19.6=333.8kPa Q =1669kN/m 4.2.2 Total stress analysis for=0 The analysis is more straightforward for=0soil.The bearing capacity formula reduces to gr Ne cu qs Values of Ne can be obtained from the chart produced by Skempton shown below(p28 in Data Sheets).It can be seen that Ne varies with the depth to width ratio D/B and with the shape of the footing.This chart is also applicable to deep foundations,that is with D/B>1. It is often assumed that the stress due to the weight of the footing.and any soil used to bury the footing,is equivalent to the soil stress qs and thus the net bearing capacity can be written qult Cu Ne
For u = 0 N = q = 1, Nc = 5.14 (From Terzaghi’s chart) For u = 0 values can be more accurately read from Skempton’s chart also on p28 in Data Sheets, and this is discussed further below. q q N B f = s q + N + cNc 2 qf = 30 1 + 0 + 25 5.14 = 158.5 kPa (Bearing capacity) Q = qf B = 158.5 5 = 792.5 kN/m (Bearing Force) 2. Long term - Effective stress analysis qs = 30 kPa uo = 2 9.8 = 19.6 kPa q’s = 10.4 kPa ’ = 15 - 9.8 = 5.2 kPa For ’ = 25o Nq = 13, Nc = 24.5, N = 10 (From Terzaghi’s chart) = + q q N + B f s q N c Nc 2 q’f = 10.4 13 + 0.5 5.2 5 10 + 2 24.5 = 314.2 kPa qf = 314.2 + 19.6 = 333.8 kPa Q = 1669 kN/m 4.2.2 Total stress analysis for u = 0 The analysis is more straightforward for u = 0 soil. The bearing capacity formula reduces to Values of Nc can be obtained from the chart produced by Skempton shown below (p28 in Data Sheets). It can be seen that Nc varies with the depth to width ratio D/B and with the shape of the footing. This chart is also applicable to deep foundations, that is with D/B > 1. It is often assumed that the stress due to the weight of the footing, and any soil used to bury the footing, is equivalent to the soil stress qs and thus the net bearing capacity can be written qult = cu Nc f c u s q = N c + q
-5.14 N (or rectangle =0.84+0.163)Ne(sqre） D/B Length of footing ULTIMATE BEARING CAPACITY OF CLAY(-0ony)(ARer A.W.Skempton) 4r-cN.+rD 4.3 Bottom heave of excavations in clay During the excavation of well supported trenches and larger holes in the ground it is p ossible for kind of bearine can acity failure underlying clay into the excavation. D heave For0,and constant undrained strength cu and the Factor of Safety ssg。-8 = Values for Ne can be determined from Skempton's chart given above.The restraining effects of the soil around the xcavation have been ignored.For shallow excavations this has the effect of slightly increasing the factor of safety
BEARING CAPACITY FACTORS [After Terzaghi and Peck (1948)] 60 50 40 30 20 10 0 20 40 60 80 N and N 0 10 20 30 40 (degrees) q c N N q N B D a b c d q= D Qf f f Bearing capacity of a shallow foundation ULTIMATE BEARING CAPACITY OF CLAY ( = 0 only) (After A.W. Skempton) 0 1 2 3 4 5 D/B 5 6 7 8 9 N Circle or square Continuous c 5.14 B D N (for rectangle) = (0.84+0.16 ) N (square) L= Length of footing B L c q ult = cNc q = B N + cN + D N continuous footing 1 f 2 c f q q = 0.4 BN + 1.3cN + D N square f c f q q = 0.6 RN + 1.3cN + D N circular f c f q qf = cNc + D c Nc BEARING CAPACITY THEORIES OF TERZAGHI AND SKEMPTON 4.3 Bottom heave of excavations in clay During the excavation of well supported trenches and larger holes in the ground it is possible for failure to occur as the soil heaves up into the base of the excavation. This mode of failure is a kind of bearing capacity failure. The weight of clay besides the excavation tends to push the underlying clay into the excavation. For = 0, and constant undrained strength cu The bearing capacity (pressure) = cu Nc The driving pressure causing failure = D and the Factor of Safety = Bearing capacity Stress cau g failure c N D u c sin = Values for Nc can be determined from Skempton’s chart given above. The restraining effects of the soil around the excavation have been ignored. For shallow excavations this has the effect of slightly increasing the factor of safety. D B heave
4.4 Deep(Pile)foundations Piles are relatively long and slender members used to transmit foundation loads through soil strata of low bearing capacity to deeper soil or rock having a higher bearing capacity.The method by which this occurs is the basis of the simplest pile type classification.We have two main pile types: 1. End-bearing piles PILES SOFT SOIL ROCK 2 Friction(or floating)piles PILES SOFT SOIL Strength with depth For both pile types a further distinction is required based on their method ofinstallation a. Driven(or displacement)piles: These piles. re-formed before beir driven acked gene ground. b Bored piles: For these piles a hole is first bored in the ground, and the pile is then usually formed in the hole
4.4 Deep (Pile) foundations Piles are relatively long and slender members used to transmit foundation loads through soil strata of low bearing capacity to deeper soil or rock having a higher bearing capacity. The method by which this occurs is the basis of the simplest pile type classification. We have two main pile types: 1. End-bearing piles 2. Friction (or floating) piles For both pile types a further distinction is required based on their method of installation. a. Driven (or displacement) piles: These piles are generally pre-formed before being driven, jacked, screwed or hammered into the ground. b. Bored piles: For these piles a hole is first bored in the ground, and the pile is then usually formed in the hole. ROCK PILES SOFT SOIL PILES SOFT SOIL Strength increases with depth
These categories may be further subdivided into Large Displacement .Preformed-driven into the ground and left in position -Solid-Timber/Concrete -Hollow with aclosed end-Steel or concrete tubes .Formed in-situ-closed-ended tubular driven then withdrawn filling void withconcrete Small displacement ·Screw piles .Steel tube and H-sections-(Tube sections may plug and become large displacement) No displacemen then filled with concrete.During construction the hole steel casing 4.4.I Loads applied to Piles Combinations of vertical,horizontal and moment loading may be applied at the soil surface from the overlying structure. For the majority of foundations the loads applied to the piles are primarily vertical.Horizna loads arising from wind loads on structures are usually relatively small and are ignored. sin jetties,foundations for Only the a nalysis of piles subiected to ical loads is nsidered here.The analysis of pile and moment loading is more comple because of the nature of the soil- structure interaction. Apart from their ability to transmit foundation loads to underlying strata piles are also widely used as a means of controlling settlement and differential settlement.In these notes only the ultimate axial capacity is considered
These categories may be further subdivided into Large Displacement • Preformed - driven into the ground and left in position - Solid - Timber/Concrete - Hollow with a closed end - Steel or concrete tubes • Formed in-situ - closed-ended tubular driven then withdrawn filling void with concrete Small displacement • Screw piles • Steel tube and H-sections - (Tube sections may plug and become large displacement) No displacement • Void formed by boring or excavation then filled with concrete. During construction the hole may need to be supported for which there are two main options - steel casing - drilling mud 4.4.1 Loads applied to Piles Combinations of vertical, horizontal and moment loading may be applied at the soil surface from the overlying structure. For the majority of foundations the loads applied to the piles are primarily vertical. Horizontal loads arising from wind loads on structures are usually relatively small and are ignored. However, for piles in jetties, foundations for bridge piers, tall chimneys, and offshore piled foundations the lateral resistance is an important consideration. Only the analysis of piles subjected to vertical loads is considered here. The analysis of piles subjected to lateral and moment loading is more complex because of the nature of the soilstructure interaction. Apart from their ability to transmit foundation loads to underlying strata piles are also widely used as a means of controlling settlement and differential settlement. In these notes only the ultimate axial capacity is considered
