Home

User Guide for ABC - Analysis of Bearing Capacity v1.0

1. Xp T X B T Xc 2 20 Rp Re sin Op 0 us Xp Clearly the computational burden in terms of floating point operations is significantly higher in axial symmetry than it is in plane strain This is another reason why it invariably takes ABC somewhat longer to solve a circular footing problem than a comparable strip footing problem 2 5 Stress field construction 2 5 1 Smooth footings solution type 1 Figure 2 6a shows a completed mesh of characteristics for a smooth strip footing problem For clarity very coarse subdivision counts are used for both d and the fan zone 5 and 10 respectively This type of mesh in which all of the characteristics proceed to the footing is referred to in ABC as solution type 1 Note that it is only necessary to calculate half of the stress field which is symmetric about the z axis Suppose that the mesh has been partially constructed to the stage shown in Figure 2 6b and that a new a characteristic is to be added The calculation of solution points proceeds in a clockwise direction using the finite difference procedures CalcAB and CalcA described in the 2004 C M Martin O NX 4 HERS cS Dad Tractian Major Princ b Partway through construction Traction Major Princ c CalcAB Traction Major Princ 2004 C M Martin ABC v1 0 d CalcAB Figure 2 6 Solution type 1 showing a completed mesh b f cons
2. a ET Figure 2 9 Rough strip footing with ABC Hough stip phi F 2100 4 0 0703 d B 0 0014 ABC v1 0 18 to preserve the accuracy and efficiency of the method for large values of F as outlined in the previous section When is less than about 1 and in particular when 0 problems of a quite different nature are encountered when F becomes large especially if the footing 1s rough The source of these problems is apparent from Figure 2 9 which shows for a rough strip footing on purely cohesive soil with 20 how the mesh of characteristics changes as F increases Note that when 0 F reduces to kB c and the form of the mesh is not affected by the unit weight y or the surcharge q These analyses could be reproduced by for example fixing k 21kPa m 0 y 0 1 and 4 0 then choosing suitable values of c 0 05 0 02 0 01 kPa to give the desired values of F 20 50 100 In all cases solution type 3 is applicable Figure 2 9 is designed to show that as F increases e the overall size of the mesh decreases when F it degenerates altogether e size of the false head region decreases again when F it degenerates altogether In terms of the mesh size parameters d and d although both distances tend to zero as F oo they do so in such a way that the ratio d d also tends to zero The structure of the mesh near the axis of symmetry then
3. and Intersect chord AC with chord Solve cf equation 2 4 but avoiding the singularity prone tan e functions xc x Joos ate 25 E E ue 2 12 xc xp oos 959 26 2 2 for and Update field variables Calculate Ro cos sing where c kze and solve cf equation 2 5 R k Oc O 8 8 ytan k xc x4 Y zc z4 coso 2 13 Icd 55 6 Xg Y zc zy for and Check convergence If on the first iteration set x x 25 2z oc 6 0 and repeat from Step 1 Otherwise test each of the four components for convergence x2 lt Tol lzc ze lt Tol iius loc oe lt Tol o o 6 lt Tol old old old old If any component fails set Ze 2 and repeat from Step 1 Otherwise return the converged solution Back up strategy If convergence has not been achieved within 1 iterations solve the four nonlinear equations 2 40 B and 2 5a 8 using the MINPACK subroutine HYBRD Mor et al 1980 specifying a convergence tolerance and an iteration limit Max t2 and taking the latest fixed point iterate x Zc oc as the initial approximation to the solution If HYBRD fails to converge report a fatal error Otherwise return the converged solution 2004 C M M
4. Initial mesh of characteristics for kB c 6 ABC Validation 4 3 10 KB cO 6 Adjusted Mesh qu 9 692 kPa Traction Major Princ 2004 C M Martin ABC v1 0 71 4 3 11 Smooth circular footing on general cohesive frictional soil Cox 1962 considered a series of problems involving a smooth circular footing on a general soil He defined a relative cohesion c which in the present notation is equivalent to c qtang the term that appears in the denominator of F in equation 4 1 characterise the relative importance of self weight and cohesion surcharge he used a dimensionless ratio g 18 2 Since Cox was only concerned with homogeneous soil k 0 there 15 a simple relationship between his the dimensionless ratio F used here namely F 2G tano In Cox 1962 the results are presented as average bearing pressures normalised with respect to c TABLE 1 VALUES OF THE MEAN YIELD POINT PRESSURE pj4 c FOR PUNCH INDENTATION Since the results in the first column had already been obtained by Cox et al 1961 and the bearing capacity for 0 is independent of only 16 new calculations were required Cox 1962 To reproduce these analyses in ABC it is convenient to fix 1kPa Kk 20 B 2m and g 0 the unit weight y then corresponds directly to G and the average bearing pressure q corresponds directly to the normalised pressure in Cox s table Converged results from A
5. Shield R T 1955 On the plastic flow of metals under conditions of axial symmetry Proc R Soc London Ser A Vol 233 pp 267 287 Sokolovskii V V 1965 Statics of granular media New York Pergamon Tani K amp Craig W H 1995 Bearing capacity of circular foundations on soft clay of strength increasing with depth Soils and Foundations Vol 35 No 4 pp 21 35 Ukritchon B Whittle A J amp Klangvijit C 2003 Calculations of bearing capacity factor using numerical limit analyses J Geotech and Geoenv Eng ASCE Vol 129 No 6 pp 468 474 This is basically a translation of the preceding reference but it includes additional worked examples 2004 C M Martin
6. M ARR EN OOK HAN AN ANE OA SS ASA ARAN AK VP Tennent BARA CULLEN ID RN NANI RS MNT NC PARA ACERO RO ol MAMANS MN MN n THREE PEF n n ane A E P Ps o hat n V dam A AJ nt o TO A E M ES Pony AN SAN OR AMAR Qus ERA inne x ANI e 99 au Major Princ Rough strip footing on sand with no surcharge 30 note adaptively added characteristics Initial mesh of characteristics for T z cr wa n T n ot wa uu T 1 4 P 2 ot m E gt 1 a a n Ty be hea Traction Major Princ 2004 C M Martin ABC v1 0 229 4 2 9 Smooth strip footing on non homogeneous undrained clay 0 The bearing capacity 1s independent of y and is given by qa N q where f kB c This problem has been studied previously by Davis amp Booker 1973 Houlsby amp Wroth 1983 and Tani amp Craig 1995 It is convenient to fix c 0 0 or any other value B 1m and 4 0 then choose k to give the desired kB c the average bearing pressure q then corresponds directly
7. along the a segment AC These are supplemented by the geometrical condition ze 0 and the fact that Oc 0 where is evaluated from one of equations 2 11 Details of the calculation procedure are given below footing Figure 2 5 Calculation of new solution point on underside of footing CalcA CalcA In Out 20 0 Preliminaries Calculate R c sinb where c Set 0 0 using equation 2 115 2 11r as appropriate footing 2004 C M Martin ABC v1 0 9 Intersect chord with footing Solve cf equation 2 40 a eos te ze z SAE 2 16 Zo 0 2 17 for and 2 Evaluate Noting that Ro 1 where is a function of solve cf equation 2 50 Rc a 9 ytanp k xe x y ze z4 2 18 for Return the solution CalcA requires no iteration so its code is simpler and faster than that of CalcAB It is however called far less frequently 2 4 2 Axial symmetry For circular footing problems only 2 13 and 2 18 of the plane strain finite difference equations need to be modified For example equation 2 138 becomes cf equation 2 8 Rk R x Oc E ea 0 tang Xp YxBC tanq J z Zp 2 19 where cf equation 2 9 Ry Ro cos 05 0 1
8. locked 26 332 go se cu Off sub subdivisions on or off 2 8 3 3 4 SETTINGS AdjustTol 1 00000E 09 Tol for Adjust 2 9 3 4 3 CalcABTol 1 00000E 09 Tol for CalcAB 2 4 3 4 4 0 1 00000E 04 x B 2 Xo 3 4 6 BEARING CAPACITY Qu 345 518 kN Q 2 10 3 2 3 4 6 qu 439 927 kPa 4 2 10 3 2 3 4 6 2004 C M Martin ABC v1 0 EDGE OF FOOTING X 7 Sigma theta Sigmaxx Sigmazz Lax Ix Tz INMOST POINT X Se Z sigma theta Sigmaxx Sigmazz tauxz zu Ix LZ OTHER OUTPUT xMis 4 thetaMis T dxMin 0 Crossing CPUTime 3 6 Messages 3 6 1 Advisory messages 0 500000 0 00000 9 74856 62 2 0000 12 999 6 54138 4 58033 A 56035 6 54138 00000E 05 U 904999 2002550 26687862100 OS T1 3229 46 UU ILOESUS 77262652 1491240 200 000 91020E r 2 88687 10 00508999 False 0 209909 x B x B kPa deg kPa kPa kPa kPa kPa x B x B kPa deg kPa kPa kPa kPa kPa x B deg x B S 41 solution point at end of degenerate a char 5 SNA Qa QgGeaNns N 2122 7 2112522 2 15 2 2 2101 22 2 2 2 2 2 2 2 10 2 10 solution point at end of final a char 5 N F x misclose misclose smallest d or d subdivision crossing characteristics CPU time excl drawing ABC ZA 22 2 142 2 2 15 22 2 1 22 22 22 22 2 10 2 10 2 9 5 1 219 92 9 5 2 9 3 2 3 43 1 2 3 6 2 3 3 1 3 3 4 The
9. using equations 2 1 2 2 and 2 3 2 6 In Once is known approximately it is more efficient to apply the iterative recalculation to just the final i e outmost a characteristic 2004 C M Martin ABC v1 0 22 equation 2 28 it 1s assumed that 15 traversed in an outward direction starting from the axis of symmetry and finishing at the edge of the footing so that dx 0 and dz lt 0O for all panels of the trapezoidal integration To clarify C C C in Figure 2 64 C C C in Figure 2 7a and gt C gt C in Figure 2 7b At each solution point on C the components of the traction vector T as plotted in green by ABC are calculated from T 0 Tos n 6 n where since compression is positive n and n are the components of the unit inward normal to the soil beneath C The integrals in equation 2 28 are based on the fact that T ds n to n 4 2 30 While this is not the only way to calculate the bearing capacity see for example the elegant method of Salencon amp Matar 1982a it is convenient in ABC because the solution points on C are stored for drawing purposes and hence are readily available 2 11 Implementation details Front end abc exe Microsoft Visual Basic v6 0 Back end abc dll Compaq Visual Fortran v6 6 Tested on Microsoft Windows 98 2000 XP Pro Dynamically allocated arrays are used extensively generally speaking these a
10. 2 25 confirms that as expected these terms are consistent with small and large values of F 2004 C M Martin ABC v1 0 14 a Solution type 2 no a characteristics progress to footing Solution type 2 example Adjusted Mesh qu 532 8 kPa Tractian Major Princ b Solution type 3 some a characteristics progress to footing Solution type 3 example Adjusted Mesh qu 8 397 kPa LT HA Major Princ Figure 2 7 Solution types 2 and 3 2 6 Adaptivity In Figures 2 6 and 2 7 the a characteristics originate at equally spaced intervals along the soil surface Although this approach is straightforward it is not always satisfactory Consider for example a smooth strip footing problem with c k 20 409 y 20 kN m B 10 m and q 0 1kPa such that 2000 in equation 2 25 Figure 2 8a shows how the use of 20 equally spaced a characteristics gives a mesh in which there is very poor resolution of the stress field near the edge of the footing In particular there is a large jump in 0 where the first non degenerate a characteristic leaves the fan and steps onto the footing and this defect is passed downstream to all subsequent characteristics Even allowing for the coarseness of the mesh the calculated bearing capacity is very inaccurate q 5 628x10 kPa is 29 6 higher than the converged value of q 4 344x10 kPa that is obtained eventually when this mesh 1s re
11. x 2 jump lim jump have been found to be suitable in practice in this case the cutoff governs when 0 is less than about 3 6 The same values are used in both plane strain and axial symmetry When stepping an characteristic onto the footing CalcA is only invoked if equation 2 27 is satisfied If the proposed jump in is too large the construction of the current a characteristic is abandoned and a new attempt is made halving the interval along the soil surface and adding two new equally spaced a characteristics This strategy is applied recursively It transpires however that if in equation 2 25 and 15 greater than a certain threshold this depends on the footing roughness and the number of fan divisions adopted no amount of recursive subdivision will allow equation 2 27 to be satisfied To avoid this numerical difficulty the maximum value of F is restricted to 10 see Section 3 1 Although this precludes a direct solution of the N problem in its pure form c 2 Kk 20 gt 0 gt 0 4 0 such that F the value of N for a particular combination of geometry roughness and friction angle can nevertheless be evaluated to any desired precision by adopting a sufficiently large value of F which reduces to yB q in the special case of a cohesionless soil Experience has shown that to evaluate an N factor correct to n significant digits using ABC a choice of parameters such that F 10 is usually adequat
12. 15kPa k 0 0 y 18kN m B 22 5 m and g 10kPa The bearing capacity is 4 87 12kPa Results from ABC Adjust Mesh 87 12 1 000 90 00 Double Up 1 87 12 1 000 90 00 Double Up 2 87 12 1 000 90 00 The correct q is calculated immediately without any need for mesh refinement This is because the a characteristics in the fan zone are circular arcs so the treatment of each curved segment as a chord does not introduce any error Even the coarsest possible solution with subdivision counts of 1 and 1 gives the correct result Initial mesh of characteristics ABC Validation 4 2 2 Adjusted Mesh qu 87 12 kPa 2004 C M Martin ABC v1 0 46 4 2 3 Smooth strip footing on homogeneous weightless cohesive frictional soil k 20 y 20 The bearing capacity is given by the analytical expression N 1 cot dJa CoN N where N e tan 4 2 This result due to Hencky 1923 is obtained from a type 1 solution in which d B JN 2 Consider a specific example with 5 k 0 38 0 B 2 5m and q 10kPa The bearing capacity is q 796 1 kPa and N 2 3 498 Results from ABC The correct 4 is not calculated immediately this is because the a characteristics in the fan zone are logarithmic spirals so the treatment of each curved segment as a chord introduces a small error As the mesh is refined however the approximate solution converges to the analytical
13. 2 kN m B 3m and q 15x0 5 7 5kPa Since the geometry is plane strain and the footing is rough the problem is entered into ABC as follows note that if a field is left blank that entry is taken to be zero Soil Data Footing Data Strip Smooth Circular Hough B kPa k g de hoz 5 kPa H q 3 2 Analysing a problem The example above will be used to illustrate the usual semi automated solution procedure Further examples including one requiring user intervention are given in Section 3 3 Having entered the soil and footing data clicking on the Auto Guess button causes the fields in the Solution Specification frame to be filled in automatically Solution Specification Auto Guess Type 71 1473 decx dil P d fo c0000 xB Dive Bias dz 3 351 14 xB Divs 4 i ef 1o4 deg Dive 59 t 2004 C M Martin ABC v1 0 23 Auto Guess has determined that the problem requires a type 2 solution with estimated values for d and O as shown the terminology is defined in Figure 2 7a All of the fields related to d are greyed out because no a characteristics progress to the footing in solution type 2 The subdivision counts suggested by Auto Guess are appropriate for a fairly coarse initial mesh that will subsequently be refined in the manner described below These counts can of course be overridden e g if producing schemat
14. 2004 C M Martin ABC v1 0 37 3 4 3 Settings Adjust Settings Adjust Adjust is the subroutine that varies the tral mesh size parameters 81 42 Theta until x 0 and theta 0 at the inmost solution paint This iterative procedure that uses the MINPACK subroutine HYBRD Tolerance Max iterations 500 Reject if amp BS5 x misclose exceeds 0 001 Reject if SBS theta misclose exceeds 0 001 rad Cancel The first two settings control the MINPACK subroutine HYBRD that is used to adjust the mesh size parameters d d and see Section 2 9 The convergence criterion in HYBRD is based purely on relative error for further details see Mor et al 1980 or the source code at www netlib org e Tol default 10 10 7 10 e Maxit default 500 min 50 max 5000 The other two settings are used to confirm that a newly adjusted mesh appears to be sensible These checks are required because of a bug in HYBRD that sometimes causes it to abandon the iterations and report successful convergence when in fact it has not converged properly This only tends to happen if the initial estimates of the mesh size parameters are very poor or if the chosen solution type for a rough footing problem is incorrect Section 3 3 5 A simple reality check on the final values of x and 0 allows any such misbehaviour to be detected e Acceptable lx default 0 001x min 0 max n a
15. 4 3 6 Rough circular footing on normally consolidated undrained clay c 20 9 0 The bearing capacity 15 independent of y and 15 given as for the smooth case by qu 6 9 This result due to Salencon amp Matar 1982a is obtained from a type 3 solution in which d d 0 As discussed in Section 2 7 this problem cannot be solved directly 1 e with c 0 using ABC It is however possible to approach the analytical solution in the limit as kB c gt oo subject to the permitted maximum of 10 that applies when 1 Consider a series of analyses with fixed values of k lIkPa m 0 0 any other value 1 and 4 0 but with various values of 0 1 0 05 0 02 0 01 kPa Converged results from ABC 100 200 500 The results exhibit the expected behaviour approaching 0 1667 kPa 0 and 0 When kB c the false head region vanishes and the analytical pressure distribution on the footing 15 linear with slope k commencing from zero at the footing edge Salencgon amp Matar 1982a In the ABC analyses it is indeed found that the false head becomes smaller as KB c increases the ratio d d 0 and the distinctive linear pressure distribution is beginning to become apparent when kB c 1000 ABC Yalidation 4 3 6 kKB cO 1000 Adjusted Mesh qu 0 2219 kPa CECI Traction Major Princ 2004 C M Martin
16. ABC v1 0 a No bias ABC Bias example Adjusted Mesh qu 0 3607 kPa Traction Major Princ b With bias Bias example Adjusted Mesh qu 0 3600 kPa Traction Major Princ SOE 9 SX OIRR Es T o CS RC EE n sem It b xd oe n XLS OKO Figure 2 10 Biasing of d subdivisions consider the problem of a rough circular footing with 0 and F kB c 500 Figure 2 11 shows a mesh in which the final d subdivision has been partitioned into 6 sub subdivisions arranged in a geometric progression to ensure that the size of the final sub subdivision matches that of the lone d subdivision The use of this special approach allows the problem to be solved with a well conditioned mesh even though the false head region is tiny d is over 10000 times smaller than d Activation of the sub subdivision strategy is automatic as explained in Section 3 3 4 2004 C M Martin ABC v1 0 20 ABC Sub subdivision example Adjusted Mesh qu 0 2424 kPa Traction Major Princ Figure 2 11 Use of sub subdivisions when d lt lt d 2 9 Adjustment of mesh size parameters Once the soil and footing data for a problem have been entered into ABC the Auto Guess facility Section 3 2 can be used to determine e solution type e initial estimates of the relevant mesh size parameters and or d and or e appropriate subdivision cou
17. ABC v1 0 66 4 3 7 Smooth circular footing on sand with no surcharge c k 20 4 0 The bearing capacity 15 given by 4 2 where f The factor of 1 2 is traditional This problem has been studied previously by Bolton amp Lau 1993 and Cassidy amp Houlsby 2002 As discussed in Sections 2 6 and 3 3 2 problem cannot be solved directly 1 with 4 0 using ABC It is however possible to approach the no surcharge condition in the limit as y8 q gt oo subject to the permitted maxima of 10 when 21 and 10 when 1 It is convenient to fix k 0 y kN m B 2m and let 0 the average bearing pressure q then corresponds directly to the bearing capacity factor N The analyses below were performed with q 10 kPa giving yB q 22x10 which is ample for evaluating to 4 digit precision in fact it is ample for 6 digit precision Converged results from ABC with numerical results from other studies n ABC 5 0 05975 0 06 0 062 The ABC N values generally agree quite closely with the other results Most of the Bolton amp Lau 1993 results are slightly high and this could be because e they used a relatively large surcharge giving a yB q value of just 2000 e their analyses were performed with fixed numbers of a and characteristics no mesh refinement was performed similarly it seems that Cassidy amp Houlsby 2002 did not perform any case by case mesh refinement checks
18. Mesh procedure outlined above it is simpler to make use of the Double Up facility Each time the Double Up button 15 clicked all active subdivision counts are doubled and a new adjustment of the active mesh size parameters in this case 2004 C M Martin ABC v1 0 2 and begins taking the converged values from the previous mesh as initial estimates After two Double Ups of the initial solution the main form looks like ABC No Title Tile Display Settings About Solution Specification Guess Type 1 ff 2 C3 di 0 00000 Divs fo Bias de 3 351 46 xB Divs 5 2182 deg Divs 272 Footing D ata i Strip Smooth Circular f Rough B 3 m q 7 5 kPa Bearing Capacity Trial Mesh Clear qu previous 2 C Force 1 93077 qu previous 1 330 040 p Average 311888834 ds qu current E it Adjust complete 420 722 1552 54 rtauxz 1 8 l33E l17Z kPa Tx 373 185 kPa Tz 716 887 kPa OTHER OUTPUT F 4 08000 I xMis 1 93077E 16 x thetaMis 9 11888E 14 deg dxMin O 0209466 x EB Crossing False CPUTime 1 05005 s The bearing capacity q is evidently converging from above 930 131 930 040 930 017 kPa With further optional Double Ups the bearing capacity converges to 930 009 kPa and the mesh size parameters d and also approach stable value
19. above e x default 0 0001x B 2 min 0 max 1 x B 2 3 4 7 About This displays information including the version number the address of the ABC homepage and the email address of the author Comments bug reports requests for future enhancements etc are always welcome Note that regardless of the value of xo the average pressure g is always calculated with respect to the gross footing area 4 2004 C M Martin ABC v1 0 3 5 Text Output The text output that appears after a successful Adjust Mesh or Double Up contains an echo of the input and a summary of the results Relevant sections of this manual are indicated on the right ABC Analysis of Bearing Capacity Version 1 0 Build 1 version number 3 4 7 TEPLE Example problem title description 3 4 1 SOIL DATA eU 0 00000 kPa Co 2 1 3 1 k 0 00000 kPa m k 2 1 3 1 ph 35 0000 deg 2 1 3 1 20 0000 kN m3 y 22 351 FOOTING DATA Geometry Circular geometry strip or circular 22 501 Interface Rough roughness smooth or rough 2 3 2 3 1 B 1 00000 m B 2 5 3 1 q 0 100000 kPa q 2 3 1 3 1 SOLUTION SPECIFICATION SolnType 3 solution type 23 22 d1 0 183240 x B d 2 5 92 1 9 d subdivisions main 2 95 3 2 diDivs 3 d subdivisions adaptive 2 6 3 3 2 diBias 1 00000 d bias ratio 2272293 2 1 39782 x B d 2 5 32 d2Divs 31 d subdivisions 2 3 92 Theta 152 500 deg Q 239 22 ThetaDivs 68 subdivisions 2225222 Adaptivity On adaptivity on or
20. and Adv in Geomech Tucson Vol 2 pp 1417 1428 Martin C M 2003 New software for rigorous bearing capacity calculations Proc British Geotech Assoc Int Conf on Foundations Dundee pp 581 592 Martin C M 2004 Discussion of Calculations of bearing capacity factor using numerical limit analyses by B Ukritchon A J Whittle amp C Klangviyit J Geotech Geoenv Eng ASCE Vol 130 No 10 pp 1106 1107 2004 C M Martin ABC v1 0 E Michalowski R L 1997 An estimate of the influence of soil weight on bearing capacity using limit analysis Soils and Foundations Vol 37 No 4 pp 57 64 J J Garbow B S amp Hillstrom 1980 User guide for MINPACK 1 Tech Rep ANL 80 74 Argonne National Laboratory Powell M J D 1970 A hybrid method for nonlinear algebraic equations In Numerical methods for nonlinear algebraic equations ed P Rabinowitz pp 87 114 London Gordon and Breach Prandtl L 1921 ber die Eindringungsfestigkeit Harte plastischer Baustoffe und die Festigkeit von Schneiden Zeit angew Math Mech Vol 1 pp 15 20 Salencon J amp Matar M 1982a Capacit portante des fondations superficielles circulaires J de M canique th orique et appliqu e Vol 1 No 2 pp 237 267 Salencon J amp Matar M 1982b Bearing capacity of circular shallow foundations In Foundation engineering ed G Pilot pp 159 168 Paris Presses de l ENPC
21. becomes extremely sensitive to small variations in d and d presenting a considerable challenge to both the Auto Guess algorithm and the library routine HYBRD that is used to adjust the mesh until it satisfies the symmetry conditions see Section 2 9 One way to obtain better resolution near the axis of symmetry is to bias the partitioning of d in such a way that the subdivisions near the outside of the mesh are smaller than those near the edge of the footing This counteracts the tendency of the a characteristics to spread apart as they approach the axis of symmetry an effect that is most pronounced in axial symmetry when 0 and F is large An example is given in Figure 2 10 rough circular footing 0 F kB c 2100 Figure 2 10a shows the mesh obtained if d is partitioned into equal subdivisions There is poor resolution near the axis of symmetry and furthermore there is a sharp disparity in size between the final d subdivision and the lone subdivision These factors combine to make the mesh extremely ill conditioned for the iterative adjustment process described in Section 2 9 Figure 2 10b shows the effect of a simple biasing strategy in which the subdivisions of d are arranged in a geometric progression The result is a mesh with much better resolution near the axis of symmetry and an inmost cell that is approximately square a feature that from a conditioning point of view 1s highly desirable in axial symmetry it is not
22. d B f 2 Divs y 10 2 m3 q 1 amp 3 kPa 5 deg Dive 40 Bearing Capacity qu previous 2 2 kPa qu previous 1 2 kPa re Sverage qu curent kPa pressure Misclose Trial Mesh Clear Force 445081E 11 0000 deg Exit 2004 C M Martin ABC v1 0 30 In addition to the 40 main a characteristics a further 66 have been added adaptively near the edge of the footing see Figures 2 1 and 2 6a for the distinction between a and p characteristics The extra characteristics are drawn light blue as are the extra B characteristics that originate from the start and finish of each one This can be seen more clearly by double clicking on the thumbnail drawing and inspecting the expanded version Adaptlivity example Trial Mesh OQ ASTA REOS RS AS SESE SEE SESE UU EET UPS ANN n P 1 F i K 5 5 e A A ee IX AA Lk LA X ay ay PN UA FA AA PW UA PR ay d AA PW UP UA UR RAN AX t AS e hi h V PR AN RN Ph di h L F Tractian Major Princ The arrow indicates how the fan zone has almost since F oo degenerated into a single p characteristic this is typical of all type problems Clicking on Adjust Mesh starts the adjustment of d af
23. e Acceptable default 0 001 rad min 0 max n a misclose misclose misclose misclose 3 4 4 Settings CalcAB Settings CalcAB CalcAB is the subroutine that calculates each new solution point in the body of the soil where an alpha line intersects with a beta line This iz an iterative procedure that uses 1 a fixed point algorithm followed by 2 the MINPACK subroutine HTBRD if 1 has Failed to converge Tolerance 1 07 1E 08 iterations stage 1 Max iterations stage 2 100 Cancel 2004 C M Martin ABC v1 0 38 These settings control the behaviour of the subroutine CalcAB described in Section 2 4 Only the convergence tolerance can be varied by the user In general the calculated bearing capacity 15 surprisingly insensitive to this tolerance particularly once a mesh has been through several stages of refinement using Double Up For high quality work however this sensitivity should be explored to confirm that the bearing capacities obtained are indeed accurate to the desired number of significant figures A reduction in Tol leads to a small increase in CPU time since every call to CalcAB requires a few additional iterations e Tol default 10 min 10 7 10 The iteration counts 1 fixed point iteration stage and Max t2 HYBRD stage are hard wired to suitable values that have been determined empirically Although a single it
24. in CalcAB see Section 2 4 In a smooth footing problem solution type 1 Figure 2 6a there is only one mesh size parameter to be adjusted namely d The adjustment is iterative and continues until the x coordinate of C the inmost solution point of the mesh is approximately zero Each iteration involves recalculation of the whole mesh for a different value of d and although this is clearly inefficient it is consistent with the approach used for solution types 2 and 3 The adjustment of a type 1 mesh can be viewed as the mises 0 in one unknown d Note that there is no need to enforce explicitly the symmetry requirement that 0 when 0 since is automatically zero at all solution points on the smooth interface numerical solution of one nonlinear equation x In a rough footing problem there are two mesh size parameters to be adjusted either and solution type 2 Figure 2 7a or d and d solution type 3 Figure 2 7b Again the adjustment is iterative continuing until the conditions x 0 and 00 are satisfied approximately at the inmost solution point C Each iteration involves global recalculation of the mesh and although this is computationally arduous it does have the advantage of simplicity Schemes that use local iteration or interpolation to locate the inmost point are difficult to implement in axial symmetry as meshes that cross into the negative radius region x lt 0 are vulnerable to numerical
25. kPa Ti Ao MOX Meo o a Hee rn v LS IAE IE MESI IN ER D Rr AE E_T Traction Major Princ 2004 C M Martin ABC v1 0 70 4 3 10 Rough circular footing on non homogeneous undrained clay 0 The bearing capacity 1s independent of y and is given by q cN q where N f kB c This problem has been studied previously by Salengon amp Matar 1982a Houlsby amp Wroth 1983 Martin 1994 see also Houlsby amp Martin 2003 and Tani amp Craig 1995 It is convenient to fix cy 1 0 0 or any other value B 21m and 4 0 then choose k to give the desired kB c the average bearing pressure q then corresponds directly to the bearing capacity factor Converged results from ABC with numerical results from other studies kB c H amp W 1983 Martin 1994 T amp C 1995 0 604 1 696 69 2 7626 828 820 92332 6 95 96 8 1058 10534 In general there is close agreement between and the other results For small KB c the results of Tani amp Craig 1995 are a little high but their procedure for integrating the stresses acting on the false head is suspect Martin amp Randolph 2001 Unfortunately the results of Salengon amp Matar 1982a are only presented in chart form but they agree with ABC to within curve reading accuracy
26. on Trial Mesh Adjust Mesh etc If the footing is rough these messages may indicate a need to change from solution type 2 to solution type 3 or vice versa see Section 3 3 5 ABC ABC These errors only arise in circular footing problems They indicate that if construction of the current mesh were to continue it would cross into the forbidden region x 0 Reduce one or more of the mesh size parameters d d before clicking again on Trial Mesh Adjust Mesh etc ABC AN Convergence failure in CalcAB AN Negative strength encountered These errors usually indicate convergence difficulties near the axis of symmetry in a circular footing problem In this case reduce one or more of the mesh size parameters d d before clicking again on Trial Mesh Adjust Mesh etc If the cause of the error is elsewhere it should be apparent from the drawing of the mesh There may be a need to change solution types see Section 3 3 5 ABC IN Adjust nat making good progress This is an error issued by the MINPACK subroutine HYBRD see Section 2 9 If there is no obvious problem visible in the drawing it is usually worth clicking on Adjust Mesh to restart the adjustment 2004 C M Martin ABC v1 0 44 If problems persist perturb one or more of the mesh size parameters d d before clicking again on Trial Mesh Adjust Mesh etc If the footing is rough this message may indicate a need to change from
27. there is much poorer resolution near the axis of symmetry ABC Bias example Adjusted Mesh qu 48 4608 kPa Traction Major Princ This can lead to problems during the Adjust Mesh and Double Up stages in the form of numerical ill conditioning see Section 2 7 for a more detailed discussion In summary if Auto Guess suggests a Bias that is less than 1 it is strongly recommended that this value be adopted 3 3 4 Problems requiring sub subdivisions Figure 3 4 shows a modified version of the previous example the footing diameter B is even larger Not to scale S profile B 200m 0 2 kPa mM gt Rough circle 2 kPa m ut variable 0 0 of 20 kN m Figure 3 4 Example problem requiring sub subdivisions 2004 C M Martin ABC v1 0 33 Although this is not a realistic problem it serves to illustrate a type 3 solution where d lt lt such that the sub subdivision technique described in Section 2 8 is needed to maintain adequate resolution Footing Data near the axis of symmetry The problem 15 specified as Strip Smooth Circular f Rough B 200 m EE The dimensionless ratio F 15 now 333 3 well into the range that would be regarded as difficult for a rough circular footing on soil with 0 see Sections 2 7 and 2 8 The initial solution proposed by Auto Guess which in this instance takes several seconds to perform its calculations involves both a heavy bias and
28. 0 y 0 1 or any other value and q 1kPa the average bearing pressure 4 then corresponds directly to the bearing capacity factor N Converged results from ABC with numerical results from Bolton amp Lau 1993 B amp L 1993 1 705 2 955 9 618 140 359 1 666 x10 1103 ABC predicts N values that are greater than those for the smooth case Section 4 3 3 whereas Bolton amp Lau 1993 envisage that roughness has no effect this 1s discussed below The solutions for higher friction angles indicated by involve crossing characteristics see Section 3 6 2 Initial mesh of characteristics for 30 note crossing p characteristics ABC Validation 4 3 4 phi 30 deg Adjusted Mesh qu 37 20 kPa 2004 C M Martin ABC v1 0 63 The discrepancy between ABC and Bolton amp Lau 1993 can be resolved by seeking a third opinion from Salengon amp Matar 1982a who take the alternative and totally equivalent view that the bearing capacity for this problem is given by 9 c N qN here qu Co C q q W PAL Salencon amp Matar define a shape factor Y N c circular N c strip but unfortunately they do not give numerical results for either v just a plot of v as a function of This is shown below together with the corresponding curves obtained from ABC and from Bolton amp Lau 1993 10 20 30 40 50 deg The comparison w
29. 1982a bearing in mind the limited precision of the latter see footnote On the other hand the results of Bolton amp Lau 1993 are very high as noted previously by Cassidy amp Houlsby 2002 Although the comments made in Section 4 3 7 are equally applicable here Bolton amp Lau s 1993 results for the rough case are based on an incorrect treatment of the false head region and this is likely to be a more serious source of error The results of Cassidy amp Houlsby 2002 are broadly comparable with the converged ABC values though unlike the smooth case Section 4 3 7 there is now a systematic pattern of disagreement ranging from 17 at small to 20 at large 0 The initial mesh of characteristics for 30 is shown at the bottom of the next page Converted from the values of N coto plane strain and the shape factor v that appear in the original e g for 30 these values are 25 55 and 1 03 giving 25 55 x tan 30 1 03 15 2 2004 C M Martin 66 30 note adaptively added characteristics Smooth circular footing on sand with no surcharge Initial mesh of characteristics for ABC v1 0 Traction Major Princ Traction Major Princ N UR AY DA S ect ib chee num m NE ANS x 30 note adaptively added characteristics Per CEE TEE ni A rpm 459 HEALTH
30. 3 2 Adjusted Mesh qu 6 047 kPa 2004 C M Martin ABC v1 0 61 4 3 3 Smooth circular footing on homogeneous weightless cohesive frictional soil k 20 y 20 The bearing capacity 15 given by N 1 cot dJa CoN qN where pnt N f 4 This problem has been studied previously by Cox et al 1961 and Bolton amp Lau 1993 It is convenient to fix cj 2 0 0 0 1 or any other value and q 1kPa the average bearing pressure 4 then corresponds directly to the bearing capacity factor N Converged results from ABC with numerical results from other studies O Lucy IEitcp sal ABC 5 1 650 1 65 1 65 3591 59 n09xif 10 There is very close agreement between ABC and the other results The solutions for higher friction angles indicated by involve crossing p characteristics see Section 3 6 2 Initial mesh of characteristics for 309 ABC Validation 4 3 3 phi 30 deg Adjusted Mesh qu 29 46 kPa Converted from the values of that appear in the original 2004 C M Martin ABC v1 0 62 4 3 4 Rough circular footing on homogeneous weightless cohesive frictional soil k 20 y 20 The bearing capacity 15 given by N N 1 cot dJa CoN qN where jn N f 4 This problem has been studied previously by Bolton amp Lau 1993 and indirectly by Salencon amp Matar 19822 It is convenient to fix cj 2 0
31. 7557 kPa and the mesh size parameters converge to d B 20 0038 and 8 872 The complementary situation occurs when a type 3 mesh is adjusted and a negative value of d is obtained this indicates that solution type 2 is in fact the correct choice for the current level of mesh refinement ABC 3 Adiusted lt 0 consider switching to solution type 2 The corrective action required in this case is simpler The solution type needs to be altered from 3 to 2 but the rest of the fields in the Solution Specification frame can be left unchanged prior to embarking on the usual the sequence of Trial Mesh Adjust Mesh Double Up Double Up Note that in either situation the prompt to change solution type does not have to be followed it is also possible to continue with the Double Up process ignoring all messages and seeing whether as refinement continues the adjusted mesh size parameters do eventually become valid for the current solution type It is however extremely rare to encounter a case where this is necessary to obtain the bearing capacity 3 4 Menu options 3 4 1 Title Title Enter title description for analysis Cancel The text entered here appears in the title bars of the main ABC form the expanded drawing window and at the head of the text output Section 3 5 3 4 2 Display Units Display Precision These options are self explanatory The defaults are Metric and 6 significant figures
32. ABC with numerical results from Salengon amp Matar 1982b 4 kPa Problem ABC S amp M 1982b 11 37 18 33 2 517x10 2 52x10 T analytical T analytical There is very close agreement between the two sets of results Note that exact agreement would not be expected because Salengon amp Matar 19826 used the method of characteristics to produce a series of charts in terms of and in the present notation F Their results for the example problems were based on factors read back from these charts and not on a direct solution of each problem using the method of characteristics as is done in ABC Problems 1 to involved a soil layer of limited depth 2004 C M Martin ABC v1 0 73 APPENDIX A HOW ABC WORKS REALLY It is useful to define the following constants q tanp A 1 k ytand A 2 Instead of developing the solution in terms of x z o and 0 as described in Chapter 2 the back end of ABC works in terms of the normalised variables X Z and 0 where X y A 3 3 2 7a A 4 5 A 4 5 Co In equation A 5 denotes the mean stress relative to the geostatic stress field o 6 9 Equations analogous to 2 1 and 2 2 can be written as follows sind 7 where 2 8 It is also useful to define normalised Mohr s circle radius Pa L Co _ sin A 9 Co 1 gt 2 5 sing In th
33. Although most of their results are within a few percent of the converged ABC values their result for 50 is high by about 14 The initial mesh of characteristics for 30 is shown at the top of the next page but one 2004 C M Martin ABC v1 0 67 4 3 8 Rough circular footing on sand with no surcharge cy k 20 4 0 The bearing capacity 15 given by q yBN 2 where N f 0 The factor of 1 2 is traditional This problem has been studied previously by Salengon amp Matar 19822 Bolton amp Lau 1993 and Cassidy amp Houlsby 2002 As discussed in Sections 2 6 and 3 3 2 problem cannot be solved directly 1 with 4 0 using ABC It is however possible to approach the no surcharge condition in the limit as y8 q gt oo subject to the permitted maxima of 10 when 21 and 10 when 1 It is convenient to fix k 0 y kN m B 2m and let 0 the average bearing pressure q then corresponds directly to the bearing capacity factor N The analyses below were performed with q 10 kPa giving yB q 22x10 which is ample for evaluating to 4 digit precision in fact it is ample for 6 digit precision Converged results from ABC with numerical results from other studies ah L anc SEM 0984 nar 199 c 000 0 0 3224 0 266 15 09m 2g 079 2 416 2 160 25 en 6o BS 5270 The ABC N values generally agree quite closely with those of Salen on amp Matar
34. BC G 1 5 689 5 689 5 689 5 689 5 689 20 08 20 10 20 32 22 39 38 81 49 28 49 4 50 50 60 59 140 8 164 7 165 5 172277 22727 754 9 There is very close agreement between the two sets of results Some of the solutions for 40 indicated by involve crossing D characteristics see Section 3 6 2 p 2004 C M Martin ABC v1 0 Vz 4 3 12 Rough circular footing on general cohesive frictional soil salengon amp Matar 1982b considered a series of problems involving a rough circular footing on a general soil The parameters for these problems including the dimensionless ratio F defined in equation 4 1 were as follows sies raum scr vow me eub F ME GENE UN The A and C problems can be solved without difficulty but the B problems require special treatment because For problem 0 and so the analytical solution q kB 6 q is applicable see Section 4 3 6 Problems B2 and also have F but now gt 0 so the situation is analogous to that of the problem considered in Section 4 3 8 To solve these problems using ABC it is first necessary to introduce a nominal c or a nominal q to make F very large but not greater than the permitted maximum of 10 that applies when 621 The analyses below were performed with c 10 kPa giving F values of 6 9x10 for problem B2 and 1 4x10 for B3 Converged results from
35. BC v1 0 1 1 INTRODUCTION 1 1 Program description ABC is a computer program that uses the method of stress characteristics also known as the slip line method to solve the classical geotechnical bearing capacity problem of a rigid foundation resting on a cohesive frictional soil mass loaded to failure by a central vertical force Figure 1 1 shows the terminology used Figure 1 1 Problem definition Soil e The soil is modelled as a rigid perfectly plastic Mohr Coulomb material assumed to be isotropic and of semi infinite extent e The cohesion c can vary linearly with depth c kz e The friction angle and unit weight y are taken to be constant Footing e Plane strain strip footing and axially symmetric circular footing analyses can be performed e The soil footing interface can be modelled as smooth or rough e Auniform surcharge pressure q can be applied to the soil adjacent to the footing Solution e The mesh or net of characteristics is constructed in an interactive environment e A sequence of increasingly accurate calculations each one involving a finer mesh can be used to obtain a converged solution e A variety of automated strategies including mesh adaptivity maintain the accuracy and efficiency of the method when solving difficult problems Results e bearing capacity is reported as a force and an average pressure q The mesh of characteristics can be annotated to show th
36. Divs 80 61 000 deg Dive 42 9 Although it is permissible to use zero as an initial guess for d it is generally better 1 e faster and more stable to enter a small positive value such as the 0 00001 shown above It is not permissible to have a zero subdivision count so this entry should be changed to 1 as shown All of the other When d lt lt d it is counterproductive to have more than one d subdivision since this results in a mesh with even more elongated cells along the perimeter of the false head region The presence of such cells near the inside of the mesh can be a source of ill conditioning particularly in axial symmetry This is also the reason that when Double Up is applied to a type 3 solution with d lt lt dh the subdivision count for d remains static until the inmost cell of the mesh has become more or less square cf Section 2 7 2004 C M Martin ABC v1 0 36 entries in the Solution Specification frame can be left unchanged As expected clicking on Trial Mesh and Adjust Mesh without for once first clicking on Auto Guess produces a valid type 3 solution Solution Specification but Guess Type 71 2 amp 3 d2 lt lt d1 di 0 00225967 Divs Bias dz 8 84830 xB Divs 80 E 60 000 deg Drs 142 The ensuing Double Up sequence confirms that solution type 3 is indeed the correct choice for this problem the bearing capacity converges to 4
37. Footing D ata Strip Smooth Circular t Rough B 1 m 2004 C M Martin ABC v1 0 35 Clicking on Auto Guess Trial Mesh and Adjust Mesh gives Solution Specification but Guess Type f 2 73 deci di o onono xB Bias dz 8 79330 Divs 40 193 831 deg This is a valid type 2 solution but the fan aperture O is very close to the maximum possible value of 160 recall from Section 2 5 2 that 31 4 0 2 On the first application of Double Up the following mesh size parameters are obtained accompanied by an advisory message Solution Specification but Guess Type 71 tf 2 di o 0nono xB Divs Bias dz 8 84873 xB Divs 80 161 000 deg Drs 142 ABC Adjusted Theta gt 160 000 consider switching to solution type 3 Now that the mesh has been refined and re adjusted exceeds O In other words the fan of the type 2 solution has wrapped around so far that near the edge of the footing there are solution points that lie just above the surface of the soil This indicates that the problem actually requires a type 3 solution with a small value of d The situation can be rectified by manually changing the solution type to 3 then editing the data pertaining to Solution Specification bu Guess Type C2 d2 lt lt d1 di 0 00001 Divs Bias dz 8 84873 xB
38. Izd x In equations 2 9 y and y denote actual body forces per unit volume in the x now radial 2 directions which as above are taken to be 0 and y respectively The variables y and y can be viewed as fictitious plane strain body forces that have been modified to incorporate the effect of axial symmetry this is simply a convenience that allows concise expression and coding of the equations Note that y and y are not constants they depend on the local values of x and 0 as well as through R on the local values of c and o 2004 C M Martin ABC v1 0 An important feature of equations 2 9 1s that they become singular on and highly nonlinear in the vicinity of the axis of symmetry x 0 Furthermore they would give meaningless results if the radius x were to become negative during the construction of a trial mesh of characteristics see Section 2 9 For these reasons the solution of a bearing capacity problem using the method of characteristics is considerably more difficult in axial symmetry than in plane strain 2 3 Boundary conditions 2 3 1 Soil surface As shown in Figure 1 1 the soil adjacent to the footing 1s subjected to a uniform surcharge q and is by assumption in a state of passive failure The values of o and at the soil surface are therefore 5 2 4 05 passive 1 sin 2 10 O ive 2 The first of these equations can be obtained from the relevant Mohr s circle con
39. User Guide for ABC Analysis of Bearing Capacity Version 1 0 C M Martin Department of Engineering Science University of Oxford OUEL Report No 2261 03 Initial draft September 2003 Revised for v1 0 October 2004 ABC v1 0 ii DISCLAIMER The program and documentation are provided as 1s without warranty of any kind either expressed or implied including but not limited to the implied warranties of merchantability and fitness for a particular purpose The entire risk as to the quality performance and application of the program lies with the user TECHNICAL SUPPORT Address Dr C M Martin Department of Engineering Science University of Oxford Parks Road Oxford OX1 3PJ U K Email chris martin eng ox ac uk Web http www civil eng ox ac uk people cmm software abc ACKNOWLEDGEMENTS For their help with the beta testing e Dave White University of Cambridge and his students in 4D5 Foundation Engineering e Mark Randolph University of Western Australia For thinking up the program s name e Geraldine Johnson University of Oxford 2004 C M Martin ABC v1 0 CONTENTS 1 INTRODUCTION 1 1 Program description 1 2 Status of solutions 1 3 Possible applications 2 HOW ABC WORKS 2 1 Introduction 2 2 Governing equations 2 2 1 Plane strain 2 2 2 Axial symmetry 2 3 Boundary conditions 2 3 1 Soil surface 2 3 2 Underside of footing 2 4 Finite difference formulation 2 4 1 Plane stra
40. a plastic rigid material J Mech Phys Solids Vol 2 pp 43 53 Bolton M D amp Lau C K 1993 Vertical bearing capacity factors for circular and strip footings on Mohr Coulomb soil Can Geotech J Vol 30 pp 1024 1033 Caquot A amp Kerisel J 1953 Sur le terme de surface dans le calcul des fondations en milieu pulv rulent Proc 3 Int Conf on Soil Mech and Found Eng Zurich Vol 1 pp 336 337 Cassidy M J amp Houlsby G T 2002 Vertical bearing capacity factors for conical footings on sand G otechnique Vol 52 No 9 pp 687 692 Cox A D 1962 Axially symmetric plastic deformation in soils II Indentation of ponderable soils Int J Mech Sci Vol 4 pp 371 380 Cox A D Eason G amp Hopkins H G 1961 Axially symmetric plastic deformation in soils Proc R Soc London Ser A Vol 254 pp 1 45 Davis E H amp Booker J R 1971 The bearing capacity of strip footings from the standpoint of plasticity theory Proc Australia New Zealand Conf on Geomech Melbourne pp 276 282 Davis E H amp Booker J R 1973 The effect of increasing strength with depth on the bearing capacity of clays G otechnique Vol 23 No 4 pp 551 563 Eason G amp Shield R T 1960 The plastic indentation of a semi infinite solid by a perfectly rough circular punch J Appl Math Phys ZAMP Vol 11 pp 33 43 Etzel M amp Dickinson K 1999 Digital Visual Fortran programmer s guid
41. activation of the sub subdivision strategy The latter is indicated by the appearance of a check in the d lt lt d box Solution Specification bu Guess Tye Ci 3 d 0 0260340 Divs 39 dz 3 752746 06 xB Divs 35 000 deg Dive 60 Bias 0 02 Note that the estimated value of d is indeed much smaller than d by a factor of about 7000 suggesting that the false head region of the type 3 solution OC C in Figure 2 7b will be extremely small Clicking on Trial Mesh and Adjust Mesh confirms that this 1s the case ABC Sub subdivision example Adjusted Mesh qu 114 235 kPa Traction Major Princ The sub subdivisions occur in the final i e outmost d subdivision and are visible as dotted characteristics As explained in Section 2 8 the spacing of the sub subdivisions is determined using a local version of the bias technique applied to the main subdivisions of d The extra a characteristics spawn extra D characteristics when they reach the footing giving a mesh that is able to resolve the tiny false head region in a well conditioned manner This can be seen more clearly by expanding 2004 C M Martin ABC v1 0 34 the thumbnail drawing right clicking on it selecting Settings making the x range say 0 1 to 0 1 and also unchecking Show Tractions ABC Sub subdivision example Adjusted Mesh qu 114 235 kPa Traction Major Princ Sol
42. al soils only exhibit such behaviour in the special case of undrained shearing 0 calculations based on associativity remain an important point of reference e g the bearing capacity design methods in codes and standards are invariably based on factors N and that pertain to soil with an associated flow rule Investigations into the effect of non associativity on the bearing capacity of shallow foundations are ongoing see the references cited in Martin 2003 1 3 Possible applications Rigorous checks of traditional N calculations such as those prescribed in design codes These methods rely on the superposition of three separate bearing capacities a technique that 1s inherently conservative but they also rely on tabulated or curve fitted values of the bearing capacity factor N which may be unconservative Further approximations are introduced if the footing is circular multiplicative shape factors are used to modify the plane strain values of N N and N or if the soil is non homogeneous calculations must then be based on some representative strength By contrast ABC constructs a numerical solution from first principles without resorting to superposition shape factors or any other form of approximation Establishing benchmarks for the validation of other calculation methods To validate the performance of say a commercial finite element package that is to be used for bearing capacity c
43. alculations a series of test problems could be specified and solved using ABC These problems could then be analysed using the FE package ensuring that the settings adopted were consistent with ABC Mohr Coulomb yield criterion associated flow rule etc Assuming that the comparisons were satisfactory the FE package could then be applied with confidence to more complex bearing capacity problems perhaps involving a different yield criterion or flow rule non vertical loading 3D footing geometry etc Education Students learning about the theory of bearing capacity will find ABC a useful tool for understanding the method of characteristics and for visualising the stress fields obtained with various combinations of parameters In the small number of axially symmetric problems where crossing characteristics arise see Section 3 6 2 the calculated bearing capacity has no formal status not even as an incomplete lower bound collapse load 2004 C M Martin ABC v1 0 3 2 HOW ABC WORKS 2 1 Introduction If it is assumed a priori that the soil is at yield the two dimensional stress state at a point x z can be fully specified in terms of two auxiliary variables namely the mean stress o and the orientation 0 of the major principal stress together with a function R x 2 6 0 that defines the radius of Mohr s circle of stress and thus the strength of the soil The sign conventions adopted for x 2 o and 0 are indicated i
44. application of CalcA noting however that 15 now obtained from equation 2 11r rather than 2 11s The remaining a characteristics resemble those of solution type 2 they are terminated in mid soil without being stepped onto the footing The solution type that is applicable to a particular rough footing problem depends on the geometry plane strain or axial symmetry the friction angle and the dimensionless ratio kB yB tan 2 25 Cy q tano Indeed as noted by Salen on amp Matar 1982a these three factors control the form and extent of the whole mesh Appendix A shows how the governing equations and boundary conditions can all be recast in terms of 0 F and normalised variables alone For a given geometry and friction angle problems with small values of F have solution type 2 and those with large values of F have solution type 3 When 0 the transition from type 2 to type 3 occurs when F 1 193 plane strain and F 0 715 axial symmetry The corresponding thresholds when 30 are about 10 98 and 5 58 It should be pointed out that the naming of the solution types in ABC 1s quite arbitrary there is no generally accepted terminology For example Davis amp Booker 1971 1973 emphasise the role of the footing breadth B the soil properties and surcharge being assumed given they therefore refer to solution types 2 and 3 as narrow rough footing and wide rough footing solutions Equation
45. artin ABC v1 0 8 Although straightforward subroutine CalcAB 15 critical because it is called many millions of times during the construction of a highly refined mesh of characteristics This piece of code has therefore been optimised quite carefully As indicated in Step 3 simple convergence checks based on absolute error are used for the quantities where this is appropriate x z 0 The convergence check for o which can vary by many orders of magnitude within certain stress fields 15 based on relative error The method used to obtain the initial approximations and depends on the availability of a known solution point diagonally opposite C in the current cell of the mesh Figure 2 4 If there is such a point then good initial approximations can be found by linear extrapolation 2 15 0 20 9 If no C point is available as is the case when a new a characteristic has just been initiated at the soil surface an alternative procedure must be used This is explained in Section 2 5 1 As well as calculating new solution points in the body of the soil it 1s also necessary to perform one legged calculations in which a characteristics are stepped onto the underside of the footing This is illustrated in Figure 2 5 As before there are four unknowns to be found x z o and 0 at C Two of the necessary equations come from the integration in finite difference form of equations 2 40 and 2 5
46. cteristic to be determined analytically thus providing a base characteristic from which the rest of the mesh be built up numerically in the manner described above Note that the fixed quantities o and 0 are given by equation 2 10 passive passive 2004 C M Martin ABC v1 0 13 2 5 2 Rough footings solution types 2 and 3 When constructing the stress field for a rough footing problem the symmetry requirement that 0 0 when x 0 major principal stress direction vertical on z axis means that the fully mobilised roughness condition of equation 2 11r and Figure 2 3b cannot apply over the whole of the soil footing interface Instead there are two possibilities and these are shown schematically in Figure 2 7 In solution type 2 full roughness is not mobilised at any point of the interface no a characteristics progress to the footing so no p characteristics become tangential to it In solution type 3 full roughness is mobilised on part but not all of the interface there is a defined region C C away from the axis of symmetry where the a characteristics do progress to the footing spawning p characteristics that originate tangentially as per Figure 2 3b In both of the rough footing solution types the bearing capacity can be found without extending the stress field into the blank false head region OC C As mentioned in Section 1 2 however an admissible extension into this region as well as into the soil o
47. d to construct a well conditioned solution As discussed in Section 2 7 the simplest method of achieving this effect is to distribute the starting points of the characteristics in geometric progression rather than spacing them equally along the free surface The Bias entry in the Solution Specification frame sets the ratio of the final 1 e outmost to the initial 1 e inmost subdivision of d from which the parameters for the geometric progression can readily be worked out If bias is required a suitable ratio will be chosen by Auto Guess and in this instance 0 1 1s proposed Solution Specification puto Guess Type 71 C2 3 d 0 0587744 xB Divs 39 da i a1 O26E 04 xB Divs 35 000 deg E d2 lt lt d1 Bias 11 After clicking on Trial Mesh and Adjust Mesh the initial solution can be inspected by double clicking on the thumbnail drawing When 0 the bearing capacity is independent of y but the direct stresses at solution points with z gt 0 will only be calculated correctly if the true value of y is entered For example if weightless soil is used the direct stresses will be low by yz throughout the mesh and this will affect the plotting and reporting of the stresses 2004 C M Martin ABC v1 0 24 ABC Bias example Adjusted Mesh qu 48 3984 kPa Traction Major Princ If the recommended Bias is overridden with 1 such that the d subdivisions are equally spaced
48. e TOM Equations 2 24 relating o and 0 on the degenerate characteristic become SG Oase 2 passive 9 if 0 passive B cott Siene e Um coth ifo 0 passive The average bearing pressure q is evaluated from 2004 C M Martin A 18 A 19 ABC v1 0 75 4 9 26 c dx T dz plane strain 2 20 4 4 8 axial symmetry C The curve C is as defined in Section 2 10 If x 0 see Section 3 4 6 the expression for axial symmetry becomes q L sx l q 8o xdz A 21 C where cf equation A 3 x EB A 22 In equations A 20 and A 21 the stress components are evaluated from the second and third of the following cf equations 2 3 2 6 G 6 R 20 c o R cos20 A 23 tT R sin20 Having determined q the bearing capacity as a force can now be obtained from Q q B plane strain 2 A 24 q 1B 4 axial symmetry Once the mesh has been constructed and the bearing capacity has been calculated the actual values of and t at certain solution points need to be recovered for drawing and reporting purposes These can be found from the normalised stress components of equation A 23 via q YBZ 0 q 25 Cot XZ 2004 C M Martin ABC v1 0 76 REFERENCES Bishop J F W 1953 On the complete solution to problems of deformation of
49. e Oxford Digital Press Hansen B amp Christensen N H 1969 Discussion of Theoretical bearing capacity of very shallow footings by A L Larkin J Soil Mech and Found Div ASCE Vol 95 No 6 pp 1568 1572 Hencky H 1923 Uber einige statisch bestimmte Falle des Gleichgewichts in plastischen K rpern Zeit angew Math Mech Vol 3 pp 241 251 Houlsby G T 1982 Theoretical analysis of the fall cone test G otechnique Vol 32 No 2 pp 111 118 Houlsby G T amp Wroth C P 1983 Calculation of stresses on shallow penetrometers and footings Proc IUTAM IUGG Symp on Seabed Mech Newcastle upon Tyne pp 107 112 Houlsby G T amp Martin C M 2003 Undrained bearing capacity factors for conical footings on clay G otechnique Vol 53 No 5 pp 513 520 Lundgren 1953 Discussion of Session 4 Proc 3 Int Conf Soil Mech and Found Eng Zurich Vol 3 pp 153 154 Lundgren H amp Mortensen K 1953 Determination by the theory of plasticity of the bearing capacity of continuous footings on sand Proc 3 Int Conf on Soil Mech and Found Eng Zurich Vol 1 pp 409 412 Martin C M 1994 Physical and numerical modelling of offshore foundations under combined loads D Phil thesis University of Oxford Martin C M amp Randolph M F 2001 Applications of the lower and upper bound theorems of plasticity to collapse of circular foundations Proc 10 Int Conf on Comp Meth
50. e and 10 is certainly adequate Examples are given in Sections 3 3 2 4 2 7 4 2 8 4 3 7 and 4 3 8 2 7 Bias For any given friction angle bearing capacity analyses using the method of characteristics or any other numerical method become progressively more difficult as the dimensionless ratio F in equation 2 25 increases When 6 1s greater than about 1 adaptively added characteristics can be introduced Note that a mesh with 35 equally spaced a characteristics only gives a modest improvement on the one with 20 equally spaced a characteristics qu 5 131 107 kPa which is still 18 1 too high 2004 C M Martin 17 m i co 4 is Lu n T n wa 1 P KB cO 20 4 0 1553 d B 0 0161 Hough stip phi ABC v1 0 a F Tractian 2 zi Eur n inn Duos Lane eT Lj lanes T Aem f ir 7 Major Princ is n n Li a 1 0 KB cO Hough stip phi b F 50 d B 0 1020 d B 0 0041 Traction Major Princ Traction Major Princ kB cy 0 and increasing values of 2004 C M Martin m E ur r ll cr we n n La gz az ll
51. e bearing capacity 15 given by 4 yBN 2 where N f e The factor of 1 2 is traditional This problem has been studied previously by Lundgren amp Mortensen 1953 Caquot amp Kerisel 1953 Hansen amp Christensen 1969 Davis amp Booker 1971 Salengon amp Matar 1982a and Bolton amp Lau 1993 among others As discussed in Sections 2 6 and 3 3 2 the problem cannot be solved directly i e with 4 0 using ABC It is however possible to approach the no surcharge condition in the limit as yB q gt oo subject to the permitted maxima of 10 when 21 and 10 when 1 It is convenient to fix 0 y kN m B 22m and let 0 the average bearing pressure q then corresponds directly to the bearing capacity factor N The analyses below were performed with q 10 kPa giving yB q 22x10 which is ample for evaluating to 4 digit precision in fact it is ample for 6 digit precision Converged results from ABC with numerical results from other studies s oem 15 ri 29 37 L 50 wm Jp ce JJ e o 104 The ABC N values show very close agreement with those of Salencon amp Matar 1982a There is also close agreement with the one off result presented by Lundgren amp Mortensen 1953 namely 14 8 for 6 30 On the other hand the results of both Caquot amp Kerisel 1953 and Bolton Lau 1993 are very high in fact most of their propo
52. e footing tractions as used to obtain the bearing capacity and or the principal stresses in the soil 2004 C M Martin ABC v1 0 2 1 2 Status of solutions In the terminology of limit analysis a converged solution obtained using ABC is classified as a partial or incomplete lower bound collapse load see e g Bishop 1953 This is because only part of the stress field at collapse namely the part needed to compute the bearing capacity is constructed For many problems in both plane strain and axial symmetry it has been shown that the bearing capacity obtained in this manner is identical to the exact collapse load but various additional calculations not currently performed by ABC are needed to establish this formally for a particular combination of parameters Specifically these calculations involve proving that the partial lower bound stress field can e be extended throughout the rest of the soil mass without violating equilibrium or yield e be associated with a velocity field that gives a coincident upper bound collapse load It is important to realise that the bound and uniqueness theorems of limit analysis are only valid for ideal materials that exhibit perfect plasticity i e no post yield hardening or softening and an associated flow rule The latter requirement means that in the case of a Mohr Coulomb soil the theorems are only applicable if the dilation angle y is equal to the friction angle 6 Even though re
53. e last line the dimensionless ratio He i 2 A 10 Co has been introduced Equations A 1 and A 2 show that this 1s the same as the ratio referred to throughout the manual namely p A 11 cy qtano Noting that equation A 7 is equal to R in equation 2 1 it is possible to recast the equations of Chapter 2 in dimensionless form using just F and the normalised variables As before the problem is assumed to be static so that the body forces are y 0 and y y 2004 C M Martin ABC v1 0 In plane strain equations 2 4 and 2 5 along the characteristics become S tan 0 8 and do uq F Fdx In axial symmetry equations 2 7 and 2 8 along the characteristics become tan 0 8 and OR do COS where R 20 1 X The first of equations 2 10 defining the mean stress the free surface becomes assive 1 It is also easy to express equation 2 21 in terms of the normalised variables d0 xFdx v x y tan dx v 7 tan dz 74 A 12 A 13 A 14 A 15 A 16 A 17 When converting the finite difference formulation in Section 2 4 it 1s largely a matter of making the replacements 207 and R9 R along with appropriate modifications to the finite difference equations For example equations 2 13 become ci CRO E Oc O0 m a e
54. eration of the fixed point algorithm is much faster than a single iteration of HYBRD the fixed point method sometimes requires many more iterations to reach convergence e g when close to the z axis in axial symmetry CalcAB therefore spends a maximum of 50 iterations in the fixed point loop before switching to HYBRD which is quadratically convergent in the neighbourhood of a solution If this back up strategy fails to converge within 100 iterations a fatal error is notified see Section 3 6 3 3 4 5 Settings Drawing Settings Drawing 11 Tic Marks e Show Scale length to 0 025 Trachons Show Scale largest to 0 1 xB Major Principal Stresses Show Scale largestto 0 05 Inmost solution point only Se faris Apply Cancel As already noted this form can also be accessed by right clicking the drawing itself whether thumbnail or expanded The settings are all self explanatory and their effects are best explored by experimentation When producing a series of drawings that are related it can be useful to uncheck Auto and set xMin and xMax manually to ensure that the plots all have a common x range Once the various drawings have been copied and pasted into a word processor or spreadsheet they can then be stacked vertically to facilitate comparison Figure 2 9 was created in this way To show major principal stresses for the whole mesh the default setting nmost solution
55. esion at footing level gt Rate of increase of cohesion with depth Footing properties are entered in the Footing Data frame EN Geometry plane strain or axial symmetry Roughness smooth or rough Width of strip or diameter of circle Surcharge pressure at footing level At least one of k and must be non zero Also the dimensionless ratio kB Cy q tan F Otherwise the soil has no shear strength 2004 C M Martin 23 3 1 ABC v1 0 24 is subject to various restrictions It must be defined not 0 0 and must satisfy F x10 if 1 3 2 lt 10 if 2 1 9 2 These restrictions only come into play for certain problems of theoretical interest such the evaluation of the bearing capacity factor N practical problems have F values that fall well within the permitted limits The reasons for the somewhat arbitrary choices of 10 and 107 are discussed later in the chapter see also Sections 2 6 2 7 and 2 8 Figure 3 1 Example problem with self weight and surcharge Consider the drained bearing capacity problem shown in Figure 3 1 Drained analysis 1s performed in terms of effective stresses so because the soil below footing level is saturated y must be taken as the effective unit weight y On the other hand the overburden soil is dry so the effective surcharge is equal to the total surcharge The numerical data are c k 0 35 20 9 8 10
56. f In X 3 M Em 4 i wa uu n a I n T my ER 08 m T 1 Y ABC Yaldation 4 3 7 phi 30 Adjusted Mesh qu 7 102 kPa Rough circular footing on sand with no surcharge Initial mesh of characteristics for 2004 C M Martin ABC v1 0 69 4 3 9 Smooth circular footing on non homogeneous undrained clay 0 The bearing capacity 1s independent of y and is given by q cN q where f KB c This problem has been studied previously by Houlsby amp Wroth 1983 Martin 1994 see also Houlsby amp Martin 2003 and Tani amp Craig 1995 It is convenient to fix c 1kPa 0 0 or any other value B 1 and 4 0 then choose k to give the desired kB c the average bearing pressure q then corresponds directly to the bearing capacity factor Converged results from ABC with numerical results from other studies kB c H amp W 1983 Martin 1994 T amp C 1995 0 54689 1 6426 65 y 2 6m 7134 716 7 556 6 830 fF 83 8 409 004 There is very close agreement between ABC and the other results Initial mesh of characteristics for kB c 6 ABC Yalidation 4 3 9 kB 6 Adjusted Mesh qu 8 298
57. fined using Double Up see Section 3 2 2004 C M Martin 15 Tractian Major Princ Tractian Major Princ the tactical insertion simple recursive a a No adaptivity ABC v1 0 XX QU PU im F A Ma NV MM NBT TY t et i I LA m Lr ET ur c un ll Img n X T n Lad Lu T E m n E x aimed m a m T az az ABC Adaptivity example Adjusted Mesh qu 4 348E 03 kPa a 2522222 2 522252227 LR 2222 riu i rr 222222222225 m 552222220222 E g e Ea a rac acr gt rre Dos ee p MM 5 i ERR 2 2004 C M Martin Figure 2 8 Adaptively added characteristics the quality of the solution can be greatly improved by of some additional characteristics while the mesh is being constructed In ABC 9 subdivision technique described in more detail below is used to restrict the jump in 0 that can occur As shown in Figure 2 8b ABC v1 0 16 when an a characteristic is stepped onto the footing i e when CalcA is applied In this example 15 additional characteristics have been generated adaptively These give better resolution of the stress field near the edge of the footing which in turn
58. first message 1s issued when Double Up is first applied to a mesh containing adaptively added characteristics see Section 3 3 2 This 1s a reminder that during the ensuing Double Up sequence it not permissible to perform any manual editing of the d d or subdivision counts If any such edits are made the second message is issued indicating that the next Trial Mesh will be a fresh adaptive calculation Sections 2 6 3 3 2 2004 C M Martin ABC v1 0 42 ABC G Adjust max iterations reached Assuming that progress towards the solution appears to be satisfactory the adjustment can be resumed with a new batch of iterations by clicking on Adjust Mesh If the message is received regularly the value of Maxit in Settings Adjust Section 3 4 3 should be increased ABC Eh Q Calculation cancelled If the adjustment was cancelled accidentally it can be resumed by clicking on Adjust Mesh ABC G Mesh toa large to draw To prevent overflow problems with the Visual Basic drawing routines meshes with unrealistic dimensions perhaps arising because of numerical difficulties during an adjustment are not drawn 3 6 2 Warning messages ABC ABC Adjusted Theta gt 855555 deg Adjusted dl lt 0 consider switching to solution type 3 consider switching to salutian type 2 User intervention may be required Section 3 3 5 gives details of how to proceed in each case ABC Warning mesh co
59. he bearing capacity is independent of y and is given by Gd N q where 5 69 Shield 1955 was the first to obtain this numerical result with a type 1 solution in which d B 0 29 It is convenient to fix 1 Kk 2 0 0 0 or any other value or any other value 4 0 the average bearing pressure q then corresponds directly to the bearing capacity factor N Results from ABC Adjust Mesh 5 689 0 2872 Double Up 1 5 689 0 2872 Double Up 2 5 689 0 2871 The ABC values agree with those of Shield 1955 Initial mesh of characteristics d ABC Yalidation 4 3 1 Adjusted Mesh qu 5 689 kPa 2004 C M Martin ABC v1 0 60 4 3 2 Rough circular footing on homogeneous undrained clay k 20 0 The bearing capacity 1s independent of y and is given by Gd CoN q where N 6 05 Eason amp Shield 1960 were the first to obtain this numerical result with a type 2 solution in which d B 0 44 and 116 It is convenient to fix 1kPa k 0 0 y 0 or any other value B or any other value 4 0 the average bearing pressure q then corresponds directly to the bearing capacity factor N Results from ABC Adjust Mesh 6 047 0 4405 116 1 Double Up 1 6 048 0 4401 116 1 Double Up 2 6 048 0 4400 116 1 Double Up 3 6 048 0 4399 116 1 The ABC values agree with those of Eason amp Shield 1960 Initial mesh of characteristics ABC Validation 4
60. ic diagrams like those in Figure 2 7 it 1s better to use even fewer subdivisions The next step 15 to click on the Trial Mesh button which constructs the initial mesh of characteristics and displays it in the thumbnail drawing window The main form then looks like ABC No Title X Title Display Settings About Solution Specification Auto Guess Tope os dess di P d 0 00000 Divs fo Bias 82 3 351 1 xB Divs 40 5 27184 deg Divs Es Footing Data Strip Smooth Circular f Rough Bearing Capacity Trial Mesh Clear qu previous 2 KPa Force 116202E42 xB qu previous 1 Pa e Average 8 onus deg qu current 2 ix Adi Notice how e thexand 0 miscloses at the inmost solution point see Figure 2 1 for the definitions of x and 0 are very small indicating that the Auto Guess estimates of d and are accurate e bearing capacity has been calculated e text output has been generated Clicking on the Adjust Mesh button causes the iterative adjustment of the mesh size parameters Section 2 9 to commence After each iteration the current values of d and the two miscloses are displayed and the thumbnail drawing 15 updated For speed only the outline of the mesh is drawn at each update but this 1s sufficient for monitoring the progress of the solution During the iterations the Adjust Mesh button beco
61. in 2 4 2 Axial symmetry 2 5 Stress field construction 2 5 1 Smooth footings solution type 1 2 5 2 Rough footings solution types 2 and 3 2 6 Adaptivity 2 7 Bias 2 8 Sub subdivisions 2 9 Adjustment of mesh size parameters 2 10 Calculation of bearing capacity 2 11 Implementation details 3 USING ABC 3 1 Specifying a problem 3 2 Analysing a problem 3 3 Further examples 3 3 1 Convergence behaviour 3 3 2 Problems requiring adaptivity 3 3 3 Problems requiring bias 3 3 4 Problems requiring sub subdivisions 3 3 5 Problems requiring user intervention 3 4 Menu options 3 4 1 Title 3 4 2 Display Units Display Precision 2004 C M Martin lil NO Ww Oo nan BBW W Ut NO NO NO N e O Oo oO A 23 23 24 23 27 28 3l 32 34 36 36 36 ABC v1 0 3 4 3 Settings Adjust 3 4 4 Settings CalcAB 3 4 5 Settings Drawing 3 4 6 Settings Misc 3 4 7 About 3 5 Text output 3 6 Messages 3 6 1 Advisory messages 3 6 2 Warning messages 3 6 3 Error messages 4 VALIDATION 4 1 Introduction 4 2 Plane strain 4 2 1 Smooth strip footing on homogeneous undrained clay 4 2 2 Rough strip footing on homogeneous undrained clay 4 2 3 Smooth strip footing on homogeneous weightless cohesive frictional soil 4 2 4 Rough strip footing on homogeneous weightless cohesive frictional soil 4 2 5 Smooth strip footing on normally consolidated undrained clay 4 2 6 Rough strip footing on normall
62. instability and in any case give results that have no physical meaning The adjustment of a type 2 or type 3 mesh can be viewed as the numerical solution of two nonlinear equations x 0 0 in two unknowns misclose misclose It is obviously harder to solve a pair of nonlinear equations than a single one and the degree of nonlinearity is much greater in axial symmetry than it 15 in plane strain A further complication in axial symmetry is the need for special procedures to abandon and recover from iterations where the mesh of characteristics if continued would have entered the region x 0 of these factors influence the time that it takes for ABC to reach a converged solution In crude terms the relative difficulty of adjusting the mesh size parameters can be summarised as follows 2 10 Calculation of bearing capacity The bearing capacity is integrated over a curve C that connects the endpoints of the a characteristics For rough footings solution types 2 and 3 the weight of the soil within the false head region must be deducted to obtain the net bearing capacity available to the foundation This 1s the purpose of the terms involving y in the line integrals QU 2 oO dx t dz yzdx plane strain C 2 28 2nd xdz yzxdx axial symmetry C Integration is performed using the trapezoidal rule With the solution ES 2 6 0 known at each point on C it is easy to recover the stresses and
63. ints on the previous a characteristic have been used up this is the situation shown in Figure 2 6e The current characteristic is then completed by stepping it onto the footing using a single application of CalcA see Figures 2 5 and 2 6f Further characteristics in this case only two more are added until the mesh is complete The method for choosing d correctly such that the final a characteristic intersects the footing at x 0 is discussed in Section 2 9 The characteristic used to initiate the construction is degenerate in the sense that all of its solution points have the same x z coordinates namely B 2 0 Equally spaced 0 values are assigned to these solution points varying from 1 2 at the start of the characteristic to 0 at the end see equations 2 10 and 2 11s and the corresponding o values can then be worked out Because dx dz 0 for each of the degenerate segments between successive solution points equation 2 50 simplifies to 2R COs do 40 0 2 22 where This same simplification applies to equation 2 80 in axial symmetry Now because z 0 so do 2 cotan dO 0 2 23 which after separating variables can readily be integrated to give passive s 2 Ce 0 if 0 2 24 2 tanb 0 cotd o _ Cy cot if o gt 0 passive Equations 2 24 allow the solution Lx at every point on the degenerate a chara
64. irectly i e with 4 0 using ABC It is however possible to approach the no surcharge condition in the limit as y8 q gt oo subject to the permitted maxima of 10 when 21 and 10 when 1 It is convenient to fix k 0 y kN m B 2m and let 0 the average bearing pressure q then corresponds directly to the bearing capacity factor N The analyses below were performed with q 10 kPa giving yB q 22x10 which is ample for evaluating to 4 digit precision in fact it is ample for 6 digit precision Converged results from ABC with numerical results from other studies 1965 amp L 1993 The ABC N values show very close agreement with those of Sokolovskii 1965 Most of the Bolton amp Lau 1993 results are slightly high and this could be because e they used a relatively large surcharge giving a yB q value of just 2000 e their analyses were performed with fixed numbers of a and characteristics no mesh refinement was performed Unfortunately the values of Hansen amp Christensen 1969 and Davis amp Booker 1971 are only presented in chart form but they agree with ABC to within curve reading accuracy The initial mesh of characteristics for 30 is shown at the top of the next page but one Converted from the values of 2N that appear in the original 2004 C M Martin ABC v1 0 53 4 2 8 Rough strip footing on sand with no surcharge cj k 0 4 0 Th
65. is close agreement between and the other results For small KB c the results of Tani amp Craig 1995 are a little high but their procedure for integrating the stresses acting on the false head is suspect Martin amp Randolph 2001 Unfortunately the results of Davis amp Booker 1973 and Salencon amp Matar 19822 are only presented in chart form but they agree with ABC to within curve reading accuracy Initial mesh of characteristics for kB c 6 ABC Validation 4 2 10 KBZcl 6 Adjusted Mesh qu 10 42 kPa eh Nene cR Cert MUT ij er D a nds cca TEE EET ira I EE 7 i Traction Major Princ 2004 C M Martin ABC v1 0 JA 4 2 11 Smooth strip footing on general cohesive frictional soil Cox 1962 considered a series of problems involving a smooth strip footing on a general soil He defined a relative cohesion c which in the present notation is equivalent to qtang the term that appears in the denominator of F in equation 4 1 To characterise the relative importance of self weight and cohesion surcharge he used a dimensionless ratio g 18 2 Since Cox was only concerned with homogeneous soil k 0 there 15 a simple relationship between his the dimensionless ratio F used here namely F 2G tano In Cox 1962 the results are presented as average bearing pressures n
66. is given by the analytical expression q kB 4 4q This result due to Davis amp Booker 1973 is obtained from a type solution in which d 0 As discussed in Section 2 7 this problem cannot be solved directly i e with c 20 using ABC It is however possible to approach the analytical solution in the limit as kB c oo subject to the permitted maximum of 10 that applies when 1 Consider a series of analyses with fixed values of k 1kPa m 0 0 any other value B 1m and 4 0 but with various values of 0 1 0 05 0 02 0 01 kPa Converged results from ABC The results exhibit the expected behaviour approaching 0 25 kPa and 0 When kB c the analytical pressure distribution on the footing is linear with slope k commencing from zero at the footing edge Davis amp Booker 1973 In the ABC analyses it is indeed found that this type of pressure distribution is beginning to become apparent when amp 1000 ABC Yalidation 4 2 5 kB cO 1000 Adjusted Mesh qu 0 2912 kPa Traction Major Princ 2004 C M Martin ABC v1 0 51 4 2 6 Rough strip footing on normally consolidated undrained clay 0 0 The bearing capacity 15 independent of y and is given as for the smooth case by q 4 4 This result due to Davis amp Booker 1973 is obtained from a type 3 solution in which d d 0 As discussed in Section 2 7 this problem cannot be solved direct
67. ith Salencon amp Matar 1982a confirms albeit indirectly that the ABC derived values of N should be preferred to those of Bolton amp Lau 1993 in the table above 2004 C M Martin ABC v1 0 64 4 3 5 Smooth circular footing on normally consolidated undrained clay cy 0 0 The bearing capacity 15 independent of y and is given by the analytical expression q kB 6 4q This result due to Salencon amp Matar 1982a is obtained from a type 1 solution in which d 0 As discussed in Section 2 7 this problem cannot be solved directly i e with c 20 using ABC It is however possible to approach the analytical solution in the limit as kB c oo subject to the permitted maximum of 10 that applies when 1 Consider a series of analyses with fixed values of k 1kPa m 0 0 any other value B 1m and 4 0 but with various values of 0 1 0 05 0 02 0 01 kPa Converged results from ABC The results exhibit the expected behaviour approaching 0 1667 kPa and 0 When kB c the analytical pressure distribution on the footing is linear with slope k commencing from zero at the footing edge Salencon amp Matar 1982a In the ABC analyses it is indeed found that this type of pressure distribution is beginning to become apparent when amp 1000 ABC Validation 4 3 5 kKBZcl 1000 Adjusted Mesh qu 0 2052 kPa Traction Major Princ 2004 C M Martin ABC v1 0 65
68. leads to better resolution in the downstream region of the mesh that was previously sparse the extra characteristics spawned by the extra a characteristics greatly improve the concentration of solution points in this area As might be expected from a qualitative comparison of Figures 2 8a and 2 8b the use of adaptivity also gives a dramatic improvement in the accuracy of the calculated bearing capacity 4 4 348x10 kPa is just 0 1 above the converged solution obtained with a very fine mesh This is an excellent result for a mesh with just 35 a characteristics When the adaptive mesh is refined only a few applications of Double Up are needed for the bearing capacity to stabilise at 4 4 344 x 10 kPa There is another compelling reason for avoiding a large jump in when stepping an a characteristic onto the footing and this 1s related to the finite difference approximation employed in CalcA In plane strain for example when equation 2 18 is solved for o the result is poe y 2 26 This expression is exact when 0 Og in which case the segment AC is straight At the other extreme the expression becomes singular when 0 To preserve a reasonable degree of accuracy in the finite difference approximation the jump in is restricted according to Jump 0 0 lt 7 lt 4 2 27 where f and are empirically chosen constants The values f 0 1 and
69. ly 1 e with c 0 using ABC It is however possible to approach the analytical solution in the limit as kB c gt oo subject to the permitted maximum of 10 that applies when 1 Consider a series of analyses with fixed values of k 1kPa m 0 0 any other value B 1m and 4 0 but with various values of 0 1 0 05 0 02 0 01 kPa Converged results from ABC 100 200 500 The results exhibit the expected behaviour approaching 0 25 kPa 0 and 0 When KB c oo the false head region vanishes and the analytical pressure distribution on the footing is linear with slope k commencing from zero at the footing edge Davis amp Booker 1973 In the ABC analyses it 1s indeed found that the false head becomes smaller as kB c increases the ratio d d 0 and the distinctive linear pressure distribution is beginning to become apparent when kB c 1000 ABC Validation 4 2 6 SEDES 1000 Adjusted Mesh qu 0 3101 kPa Traction Major Princ 2004 C M Martin ABC v1 0 22 4 2 7 Smooth strip footing sand with no surcharge c k 20 4 0 The bearing capacity 15 given by 4 yBN 2 where N f 0 The factor of 1 2 is traditional This problem has been studied previously by Sokolovskii 1965 Hansen amp Christensen 1969 Davis amp Booker 1971 and Bolton amp Lau 1993 among others As discussed in Sections 2 6 and 3 3 2 problem cannot be solved d
70. mes a Cancel button allowing the calculation to be terminated at the end of the current iteration if needed When the iterative adjustment has finished the main form should resemble Some of the output values notably the miscloses may not be identical to those shown because floating point arithmetic 1s processor dependent of the examples in this manual were run on a Pentium III machine 2004 C M Martin ABC v1 0 26 ABC No Title fel Tite Display Settings About Solution Specification but Guess f 1 ff 2 C3 dee di mi di 0 00000 Dive 0 Bias dz 3 351 14 xB Divs 40 i ef 154 deg Dive Footing Data Strip Smooth Circular Rough Bearing Capacity Misclose Trial Mesh Clear qu previous 2 SS 74 x B qu previous 1 is 8 211075545 deg Exit Adjust complete Average pressure qu current 930 1 3l kPa 420 756 1552 77 4 l 701S8E 14 373 242 716 990 OTHER OUTPUT F 4 08000 xMisz 3 97435E 16 theraMis Z ll1075E 15 dxMin 0837792 Crossing False O 109985 Notice how the x and miscloses are to all intents and purposes zero e bearing capacity has been calculated and displayed e text output has been generated The thumbnail drawing can be expanded by double clicking on it or by right clicking on it and selecting Expand The expanded drawing ini
71. n Figure 2 1 which also shows the yield criterion used in ABC 2 1 This 15 the two dimensional form of Mohr Coulomb criterion By definition a Mohr Coulomb soil is isotropic because at a given point 2 the strength does not depend on the orientation of the principal stresses equation 2 1 is independent of If the strength parameters c and 6 are constant the soil is described as homogeneous if c and or vary with position the soil is non homogeneous though still isotropic as defined above For simplicity ABC only allows vertical non homogeneity in c via the linear equation C C 2 2 The friction angle 15 taken to be constant throughout the soil When the stresses at yield of Figure 2 1 are combined with the equations of equilibrium a pair of coupled partial differential equations is obtained spatial variables x and z field variables o and 0 Standard techniques can be used to show that this equation system 15 hyperbolic and hence there are two distinct characteristic directions here denoted and along which the partial differential equations reduce to coupled ordinary differential equations The relevant equations for plane strain and axial symmetry are summarised in the next section For a concise summary of the derivation using the same terminology as that adopted here see Martin 2003 2 2 Governing equations In the tables below 1 4 6 2 deno
72. ntains crossing beta lines calculated bearing capacity may be inaccurate The mesh contains crossing D characteristics indicating that a stress discontinuity analogous to a shock wave should have been introduced while the mesh was being constructed to maintain the admissibility of the solution These calculations are not performed in the current version of ABC Although they will be implemented in future it is worth pointing out that crossing characteristics only arise in a fairly small class of circular footing problems see Sections 4 3 3 4 3 4 4 3 11 2004 C M Martin ABC v1 0 43 Furthermore if the area of crossing 1s localised the error in the calculated bearing capacity may not be too serious Houlsby 1982 Certainly it appears that many previous researchers including Cox et al 1961 Cox 1962 Salencon amp Matar 1982a and Bolton amp Lau 1993 have turned a blind eye to the occurrence of crossing characteristics in their meshes if indeed they were aware of it at all 3 6 3 Error messages ABC ABC AN Adjusted eMisclose gt prescribed threshold AN Adjusted thetaMisclase gt prescribed threshold The thresholds referred to are those prescribed in Settings Adjust Section 3 4 3 If there is no obvious problem visible in the drawing it is usually worth clicking on Adjust Mesh to restart the adjustment If problems persist perturb one or more of the mesh size parameters d d before clicking again
73. nts for an initial i e fairly coarse mesh e a suitable degree of bias if any e whether sub subdivisions are required These specifications can also be entered manually if desired though it 15 anticipated that users will prefer to use Auto Guess on most occasions Various aspects of the algorithm are still being refined notably the provision of faster initial guesses for difficult rough footing problems where the friction angle and the dimensionless ratio F in equation 2 25 are both large A description of how Auto Guess works will therefore be deferred until the algorithm has stabilised in a future release The initial estimates of d d and O are adjusted using the MINPACK subroutine HYBRD Mor et al 1980 This routine implements the hybrid algorithm of Powell 1970 and the source code in Fortran 77 is freely available from the Netlib repository at www netlib org HYBRD has proven to be extremely robust as a general purpose code for the solution of nonlinear equations even when analytical derivatives are unavailable clearly the case here or when non smooth behaviour is 2004 C M Martin ABC v1 0 21 encountered during the iterations often an issue in axial symmetry see below As an indication of its enduring popularity HYBRD still forms the core of the nonlinear equation solvers in many commercial subroutine libraries including NAG and IMSL As already noted HYBRD is used elsewhere in ABC as a back up strategy
74. one Initial mesh of characteristics ABC Yalidation 4 2 3 Adjusted Mesh qu 796 7 kPa 2004 C M Martin ABC v1 0 49 4 2 4 Rough strip footing on homogeneous weightless cohesive frictional soil k 20 y 20 The bearing capacity is given as for the smooth case by N 1 cot dJa CoN N where N e tan 4 2 This result due to Prandtl 1921 is obtained from a type 2 solution in which d B JN and 2 Consider a specific example with c 5kPa k 0 38 y 0 B 2 5m and q 10kPa The bearing capacity is 4 796 1 kPa and N 6 995 Results from ABC Adjust Mesh 796 3 6 996 90 00 Double Up 1 796 1 6 995 90 00 Double Up 2 796 1 6 995 90 00 Double Up 3 796 1 6 995 90 00 The correct 4 is not calculated immediately this is because the a characteristics in the fan zone are logarithmic spirals so the treatment of each curved segment as a chord introduces a small error As the mesh is refined however the approximate solution converges to the analytical one Compared with the smooth case Section 4 2 3 fewer Double Ups are required because the initial number of fan divisions selected by Auto Guess is greater Initial mesh of characteristics gt ABC Validation 4 2 4 Adjusted Mesh qu 796 3 kPa 2004 C M Martin ABC v1 0 50 4 2 5 Smooth strip footing on normally consolidated undrained clay cy 0 09 0 The bearing capacity 15 independent of y and
75. ormalised with respect to c TABLE 2 VALUES OF THE MEAN YIELD POINT PRESSURE p c FOR DIE INDENTATION G 10 10 1 1 10 0 5 14 514 5 14 5 14 10 8 35 8 42 9 02 13 6 20 14 87 152 179 37 8 30 30 29 316 429 127 40 16 13 83 0 139 574 To reproduce these analyses in ABC it is convenient to fix 1kPa Kk 20 B 2m and g 0 the unit weight y then corresponds directly to G and the average bearing pressure q corresponds directly to the normalised pressure in Cox s table Converged results from ABC _ 10e J n j 3 1 5 142 5 142 5 142 5 142 5 142 G 14 83 14 87 15 17 17 69 37 76 30 14 30 29 31 61 42 87 126 7 795 941 76 13 83 05 139 0 573 3 There is very close agreement between the two sets of results The results in the first row 0 the first column G 0 of each table can of course be obtained analytically see Sections 4 2 1 and 4 2 3 but the other 16 entries can only be obtained numerically 0 2004 C M Martin ABC v1 0 58 4 2 12 Rough strip footing on general cohesive frictional soil salencgon amp Matar 19826 considered a series of problems involving a rough strip footing on a general soil The parameters for these problems including the dimensionless ratio F defined in equation 4 1 were as follows sies kam scr vow meu umb ME GENE UN The A and C problems can be solved without difficulty b
76. particularly important in plane strain A convenient way to specify the degree of bias is via the ratio of the final 1 e outmost d subdivision to the initial 1 e inmost d subdivision In Figure 2 10b this ratio is 0 05 in the plane strain examples of Figure 2 9 the biasing is somewhat milder ratios 0 5 0 5 0 2 When using ABC the choice of an appropriate bias ratio is handled automatically as explained in Section 3 3 3 2 8 Sub subdivisions When is small or zero and F 1s greater than about 100 rough footing problems in both plane strain and axial symmetry have type 3 solutions where d can be many orders of magnitude smaller than d In these circumstances the biasing technique described in the previous section is no longer effective on its own if too many characteristics are concentrated near the perimeter of the mesh there is a loss of resolution near the edge of the footing A more effective approach is to introduce a small number of additional a characteristics as sub subdivisions of the final 1 e outmost d subdivision These can be arranged to provide a smooth transition in spacing between the d characteristics and the 4 characteristic s To give an example Note that in the limiting case of F there is no need to use ABC at all there are simple analytical solutions for both strip and circular footings on soil with 0 and F see Sections 4 2 5 4 2 6 4 3 5 and 4 3 6 2004 C M Martin
77. perfectly acceptable from a practical point of view leading to a bearing capacity of q 276 2 kPa that 1s only marginally too high It should now be clear that the limit of F x10 in equation 3 2b is not particularly restrictive in fact it is sufficient for an N problem to be solved to about 9 correct digits if desired see Section 2 6 2004 C M Martin ABC v1 0 31 Not to scale S profile B 50m 0 2 kPa I 2 kPa m s variable 0 y 20 kN m Figure 3 3 Example problem requiring bias 3 3 3 Problems requiring bias Consider the undrained bearing capacity problem shown in Figure 3 3 Undrained analysis is performed in terms of total stresses so even though the soil is saturated y must be taken as the total unit weight and 4 as the total surcharge pressure at footing level namely 20x0 5 10kPa As indicated in the figure the undrained strength s increases with depth and its value at the level of the footing base is 0 2 2x0 5 1 2 kPa The problem is therefore entered as follows Footing Data Strip Smooth Circular Rough B m EL The dimensionless ratio F which reduces to kB c when 0 is equal to 83 3 This is well inside the limit of 1000 that applies when lt 19 see equation 3 2a but it is large enough to mean that a significant degree of bias skewing the concentration of characteristics towards the outside of the mesh is neede
78. point only must be unchecked Often some trial and error adjustment of the default scale factor will then be necessary as well during which it is convenient to use Apply rather than OK gt Use Alt PrtSc to copy an image of the active window to the clipboard then Ctrl V to paste it 2004 C M Martin ABC v1 0 39 3 4 6 Settings Misc Settings Miscellaneous subdivision count d1D reg d2Drvs for storage and drawing of Full mesh Exclusion radius 0 far circular footings 0 0001 x Bie eras Cancel The first setting determines the threshold at which the calculation and drawing procedures switch from full mesh mode to outline mode in order to conserve memory and save drawing time see Section 2 11 The transition is based on the total number of a characteristics excluding the degenerate characteristic at the edge of the footing and any that have been added adaptively e MaxDivsFullMesh default 100 min 0 max 1000 The second setting is only applicable in axial symmetry and is ignored in plane strain It allows the goal of the mesh adjustment Section 2 9 to be altered from 20 and 0 at the inmost solution point to and 0 0 at the inmost solution point Setting x to some small fraction of the footing radius B 2 say 107 or 10 has the desirable effect of accelerating the adjustment process and is particularly effective when attempting to solve difficult ci
79. quite straightforward by comparison with say a series of analyses to evaluate factors for a rough circular footing 2004 C M Martin ABC v1 0 46 4 2 Plane strain 4 2 1 Smooth strip footing on homogeneous undrained clay k 20 0 The bearing capacity is independent of y and is given by the analytical expression 4 CoN q where N 2 1 This result due to Hencky 1923 is obtained from a type 1 solution in which 4 1 2 Consider specific example with c 15 k 20 0 y 2 18kN m 2 5 m and g 10kPa The bearing capacity is q 87 12kPa Results from ABC Adjust Mesh 87 12 0 5000 Double Up 1 87 12 0 5000 Double Up 2 87 12 0 5000 The correct q is calculated immediately without any need for mesh refinement This is because the a characteristics in the fan zone are circular arcs so the treatment of each curved segment as a chord does not introduce any error Even the coarsest possible solution with subdivision counts of 1 and 1 gives the correct result Initial mesh of characteristics ABC Validation 4 2 1 Adjusted Mesh qu 97 12 kPa 2004 C M Martin ABC v1 0 47 4 2 2 Rough strip footing on homogeneous undrained clay k 20 0 The bearing capacity is independent of y and is given as for the smooth case by dy CoN q where N 2 m This result due to Prandtl 1921 1s obtained from a type 2 solution in which d B 1 and 2 Consider a specific example with
80. racteristics is as follows Referring to Figure 2 4 if the solution is known at two points A on an a characteristic and B on a p characteristic it can be propagated to a new point by integrating the governing equations simultaneously along the a segment AC and the p segment BC It is sometimes possible to devise special patterns of characteristics that allow these integrations to be performed analytically notably in plane strain when the soil is either e homogenous and purely cohesive c 0 k 0 20 y 20 e homogeneous and weightless c 20 k 0 50 0 In general however an approximate numerical integration must be carried out To solve the four nonlinear equations 2 40 and 2 5 for the solution at the new point it is convenient and customary to adopt a midpoint finite difference scheme in conjunction with a fixed point iteration strategy In detail the algorithm used in ABC is as follows 4 Q e e an Figure 2 4 Calculation of new solution point in body of soil CalcAB i e the coordinates and the auxiliary variables conveniently stored as a vector x z o 0 2004 C M Martin ABC v1 0 CalcAB In Out Es Zig Gx 0 pn Z MON i Oh 0 Tol MaxItl MaxIt2 Ee Zo Oc 6 Preliminaries Calculate R c cos o sing where c C and Rp c cosd o sing where Cy Initialise
81. rcular footing problems such as the one considered in Section 3 3 4 Not only is the singularity at x 20 specifically excluded but there are fewer iterations involving trial meshes that if continued would cross the axis of symmetry into the physically meaningless and numerically capricious region of negative radius see Sections 2 2 2 and 2 9 In terms of the calculated bearing capacity an analysis with x gt 0 will give a lower converged value of 4 or Q than an analysis with x 0 so the exclusion procedure is conservative If x is small the error in the bearing capacity will be even smaller because the fraction of the total area being excluded is proportional to x though it must also be noted that in circular footing problems the bearing pressure is invariably higher near the axis of symmetry than at the edge of the footing Experience has shown that the default value of x 210 x B 2 has an utterly negligible influence on the calculated bearing capacity of the order of 10 Unsurprisingly however the mesh size parameters d d and are affected more directly with errors of order 10 For high quality work in axial symmetry the sensitivity of the results to the choice of x can always be examined by overriding the default value Although it is permissible to set x 0 this can have a detrimental effect on the rate at which Adjust Mesh and Double Up converge in certain more difficult problems as mentioned
82. re allocated in the front end and passed by reference to the back end where they are filled with data On returning to the front end therefore the arrays are instantly available for drawing the mesh and printing the text output The use of temporary disk files to transfer the data would be much slower In full mesh mode array space 15 allocated for every solution point so that the complete mesh of characteristics can be drawn In outline mode memory is conserved by only allocating enough space to hold the longest two characteristics plus the solution points on curve C Figures 2 6 2 7 Section 2 10 When using Double Up the cost of drawing the full mesh quickly becomes prohibitive so outline mode is automatically activated once a certain user defined number of a characteristics 1s reached During the adjustment of the mesh size parameters Section 2 9 a callback to the front end is made after each iteration Etzel amp Dickinson 1999 This is used to display the progress of the solution and to check whether the user has cancelled the calculation see Section 3 2 The mesh is built up using a leapfrogging approach storing only the current and most recent characteristics 2004 C M Martin ABC v1 0 3 USING ABC 3 1 Specifying a problem When the program is started a disclaimer appears followed by the main form ABC No Title Adjust Mest Soil properties are entered in the Soil Data frame Coh
83. s Note how once a specified number of subdivisions has been reached only the outline of the mesh is stored and drawn see Sections 2 11 and 3 4 6 In principle there is no restriction on the number of Double Ups that can be executed The main practical limit is that each one typically takes three to four times longer than its predecessor though for most problems the convergence of the calculated bearing capacity is sufficiently rapid that several digits of precision can be achieved within seconds as in the example above This is more than adequate for practical purposes but it is sometimes desirable to obtain results to a higher precision e g in theoretical studies or benchmarking exercises Another potential limitation when using Double Up is that each successive analysis requires more memory though the allocations required are moderate once the program has entered the outline mode referred to in the previous paragraph 3 3 Further examples 3 3 1 Convergence behaviour As discussed in Section 2 9 the computational cost of obtaining a converged solution is higher in axial symmetry than in plane strain and higher for rough footings than for smooth ones This can be illustrated by re running the example from the previous section for all four combinations of geometry and roughness The convergence histories are tabulated below Each entry shows the calculated bearing capacity q in kPa with the associated CPU time in seconds in bracke
84. sed N values can be categorically ruled out because they lie above the strict upper bound solutions of Michalowski 1997 Ukritchon et al 2003 and others Although the comments made in Section 4 2 7 are equally applicable here Bolton amp Lau s 1993 results for the rough case are based on an incorrect treatment of the false head region and this 1s likely to be a more serious source of error Coincidentally the results of Caquot amp Kerisel 1953 suffer from the same problem see Lundgren 1953 and Martin 2004 for further details Unfortunately the values of Hansen amp Christensen 1969 and Davis amp Booker 1971 are only presented in chart form but they agree with ABC to within curve reading accuracy The initial mesh of characteristics for 30 is shown at the bottom of the next page Converted from the values of N coto that appear in the original 2004 C M Martin 54 ABC v1 0 Smooth strip footing on sand with no surcharge 30 note adaptively added characteristics Initial mesh of characteristics for m m qum m r we c m z az m a em amp 4 2 E at nt o OUO bcc othe MM ih Traction m ANSA NAVAS MSS PURA AA HUN ANS PRADA AM AAA MSN NISI A
85. solution type 2 to solution type 3 or vice versa see Section 3 3 5 ABC AN Adjust tolerance too small This is another error issued by the MINPACK subroutine HYBRD Section 2 9 In Settings Adjust Section 3 4 3 Tol cannot be set to anything smaller than 10 7 so it is unlikely that this message will ever be received ABC IN Out of adaptive alpha lines During an adaptive Trial Mesh or Adjust Mesh see Sections 2 6 3 3 2 the standard allocation of adaptive a characteristics has been exceeded This error should not arise under normal conditions 1 e if Trial Mesh has been preceded by Auto Guess ABC ABC AN Unable to allocate memory VB try closing other apps AN Unable to allocate memory in DLL try closing other apps An attempt to allocate array space in the front end Visual Basic or the back end Fortran DLL has failed Closing other applications may free up sufficient memory to allow the calculation to proceed The larger the value of MaxDivsFullMesh in Settings Misc Section 3 4 6 the more likely it is that one of these errors will be received Once the calculation of the mesh has switched to outline mode see Sections 2 11 3 4 6 it is unlikely that insufficient memory will be a problem ABC AN Abnormal exit from Subroutine ame error code If a message like this is received please report it see the About menu and the circumstances in which it arose 2004 C M Mar
86. struction Figure 2 2 noting that since 2 0 and that the minor principal stress is vertical and equal to The a and characteristics are inclined at to the horizontal The second of equations 2 10 simply states that the major principal stress o 15 horizontal see Figure 2 1a R osin o 20 0 q 60 R 2 q 7 1 sin Figure 2 2 Mohr s circle for soil surface 2 3 2 Underside of footing The soil directly beneath the footing is by assumption in a state of active failure The orientation 0 of the major principal stress depends on the roughness of the underside of the footing If it is smooth the major principal stress 15 vertical 2 11s footing E If it is rough and full roughness is mobilised see Section 2 5 2 then the Mohr Coulomb criterion is satisfied on the plane of the interface hence 7 4 0 2 2 11r Mohr s circles for the two cases are shown in Figure 2 3 In the smooth case the interface shear stress is Zero and the and characteristics are inclined at e to the vertical In the rough case the footing exerts an inward shear stress on the soil with the B characteristic tangential to the interface 2004 C M Martin ABC v1 0 6 20 0 20 2 0 0 4 2 Figure 2 3 Mohr s circle for underside of footing a smooth b rough 2 4 Finite difference formulation 2 4 1 Plane strain The basis of the method of cha
87. ter which the bearing capacity is displayed q 275 7 kPa Each iteration of this initial adjustment involves a fresh adaptive calculation of the mesh When the solution is refined using Double Up however the initial pattern of recursive subdivision 15 locked in and provides the template for all subsequent doublings of the mesh this can be seen by inspecting the pattern of dark blue and light blue characteristics in successive drawings The following message 15 displayed at the start of the Double Up sequence ABC Adaptivity pattern locked for doubling As the mesh is refined the calculated bearing capacity q quickly converges to 275 9 kPa It is worth recalling from Section 3 3 1 that the corresponding result with an effective surcharge of 7 5 kPa was 4 839 0 kPa indicating the profound effect of embedment on the bearing capacity of a footing on cohesionless soil It remains to confirm that the artificial surcharge q 10 kPa is small enough to have no significant influence on the result This can be done by repeating the analysis with a larger surcharge say q 10 kPa such that F which is just yB q when c k 20 has a smaller value of 3 1x10 It is found that the bearing capacity still converges to 4 275 9 kPa slightly more quickly than before because the initial mesh prior to doubling contains fewer adaptive characteristics 47 as opposed to 66 Even a surcharge of 10 kPa such that F 3 1x10 is
88. tes the angle between the direction of the major principal stress o and the directions of the a and p characteristics Figure 2 1a Note that the characteristics coincide with the planes on which the Mohr Coulomb criterion 15 satisfied Figure 2 1b b Note X and 7 denote planes having x and z directions as normals Figure 2 1 Notation and sign conventions 2004 C M Martin ABC v1 0 4 2 2 1 Plane strain Stresses in terms of auxiliary variables 0 Rcos20 Oo 0 Rcos20 2 3 T Rsin 20 Characteristics Directions 2 4 differential equations 2 5 nm dr 0 y tand k dx y y tang dz cos dona dos y ty tan Kk dc y y tang dz In equations 2 5 y and y denote body forces per unit volume in the x and z directions In the current version of ABC it is always assumed that y 0 and y y so the governing equations can be simplified slightly 2 2 2 Axial symmetry Stresses in terms of auxiliary variables note hoop stress from Haar von Karman hypothesis 0 Rcos20 0 Rcos20 2 6 o 0 R T Rsin20 Characteristics Directions 2 7 Ordinary differential equations 2 8 2R O do d0 y tanp k dx y 7 tanp dz cos 2R m tan 0 d0 y y tang k dice y y tang dz cos where 20 1 2 9 x Rsin20
89. tially occupies the full screen but the window may be resized and repositioned if desired it will subsequently reappear in the same position To close the expanded drawing click on the control or right click on the drawing and select Reduce Right clicking on the drawing whether thumbnail or expanded and selecting Settings brings up a form that allows certain aspects of the drawing to be customised For example unchecking nmost solution point only causes major principal stresses to be plotted throughout the mesh By default the bearing capacity is displayed as an average pressure q but it can also be displayed as a force per unit run in the case of a strip footing These modes be toggled at any time by selecting the appropriate button The text output can either be viewed in situ using the scroll bar or copied and pasted into another application should a record of the analysis be required use the right mouse button to Select All and Copy Section 3 5 gives a key to the abbreviations used in the text output At present there is no facility to save input or output data directly from ABC in fact the program makes no use of disk files at all not even temporary scratch files The final stage of the solution process is to refine the mesh until the calculated bearing capacity converges While this can be achieved manually by increasing the subdivision counts in the Solution Specification frame and repeating the Trial Mesh Adjust
90. tics are introduced adaptively see Section 2 6 A literal specification of the input data 1 e B 3m lt gt c 0 b 358 Yeat 20 kN n Figure 3 2 Example problem requiring adaptivity 2004 C M Martin ABC v1 0 29 Soil Data Footing Data eti kPa Strip tv Sroooth 7 IT Circular Rough 35 deq B 3 Lm y 10 2 m3 q E kPa results in an error message when Auto Guess is attempted ABC re ES vBtanid c gtand Cannot have gt 1E 12 Because c g tang 0 the dimensionless ratio F in equation 3 1 is infinite violating the restriction lt 107 in equation 3 2b To analyse the example using ABC it is first necessary to introduce nominal surcharge e g g 10 kPa which gives a large but legal F 3 1x10 alternatively a nominal cohesion c can be introduced This tiny surcharge corresponding to an embedment of just 10 m in the saturated soil has a negligible effect on the bearing capacity this is shown below but it prevents the potential numerical difficulty described in the final paragraph of Section 2 6 After revising the entry for g to be le 9 clicking on Auto Guess and then Trial Mesh results in ABC Adaptivity example Tile Display Settings About Soil Data Footing Data Solution Specification cu 0 kPa Strip Smooth Auto Guess Type 1 C2 vere f Circular Rough k 0 kFa m di 0 883440 Divs 40
91. tin ABC v1 0 45 4 VALIDATION 4 1 Introduction In this chapter ABC is tested on a wide range of bearing capacity problems Closed form analytical solutions are invaluable for this purpose but they are only available for certain special combinations of the geometry plane strain or axial symmetry the friction angle and the dimensionless ratio kB cy qtano F 4 1 The situation 1s summarised in the following table po 0 Ys No 42 422 431432 general 0 Yes 425424 435434 o Ye 425426 455436 general 9 No 427428 437438 _ 0 general No 4294210 4394310 For problems where there is no analytical solution the only comparisons that can be made are with numerical solutions obtained by other researchers It should be noted however that many previous studies of bearing capacity using the method of characteristics do not appear to have incorporated any systematic approach to mesh refinement so the results obtained in these studies are not necessarily correct to the quoted number of significant figures It is disappointing that attention to this aspect of the solution process seems to have declined over the years despite the enormous advances in computing power that have taken place Cox et al 1961 state that their bearing capacities were computed for successively smaller mesh sizes and the final values are believed to be acc
92. to the bearing capacity factor Converged results from ABC with numerical results from other studies N ABC amp 1983 T amp C 1995 kB c 598 6660 666 666 7819 72 72 8839 84 88 978 98 0 10 10607 1067 10607 There is perfect agreement between ABC and the other results Unfortunately the results of Davis amp Booker 1973 are only presented in chart form but they agree with ABC to within curve reading accuracy Initial mesh of characteristics for kB c 6 ABC Yalidation 4 2 9 kB Adjusted Hesh qu 8 839 kPa 2004 C M Martin ABC v1 0 56 4 2 10 Rough strip footing on non homogeneous undrained clay 0 The bearing capacity is independent of y and is given by qa N q where f KB c This problem has been studied previously by Davis amp Booker 1973 Salencon amp Matar 1982a Houlsby amp Wroth 1983 and Tani amp Craig 1995 It is convenient to fix cy 0 0 or any other value B 1 and 4 0 then choose k to give the desired kB c the average bearing pressure q then corresponds directly to the bearing capacity factor Converged results from ABC with numerical results from other studies kB c H amp W 1983 T amp C 1995 0 512 6600 68 6 102 103 1049 8 us 152 In general there
93. truction procedure 2004 C M Martin ABC v1 0 12 previous section Referring to Figure 2 6c let A denote the first point of the current 1 new a characteristic and B the first point of the previous characteristic The coordinates of A are easily established while the values of o and 0 at A are obtained from equations 2 10 these are of course identical to o and at B Propagation of the solution to C the second point of the current characteristic is performed using CalcAB starting with the following initial approximations pong 1 1 2 21 Oc 2 where Ze 1 _ and km 2 The first of equations 2 21 is derived from a passive Mohr s circle construction similar to that in Figure 2 2 but with and q yz These special initial approximations are actually exact in plane strain and they provide reasonable starting values in axial symmetry Figure 2 6d shows the next application of CalcAB The second point of the current a characteristic the point C just calculated is now used as A and the second point of the previous a characteristic is now used as The first point of the previous characteristic is now available as known solution point diagonally opposite cf Figure 2 4 so the initial approximations and can be determined from the linear extrapolation equations 2 15 CalcAB 15 applied in a similar fashion until all of the po
94. ts 2004 C M Martin ABC v1 0 26 Rough strip Rough circle Adjust Mesh 619 840 0 1 930 131 0 1 838 772 0 1 1449 10 0 1 Double Up 1 619 711 0 2 930 040 0 3 838 934 0 6 1449 37 1 2 The comparative timings for the different classes of problem are reasonably typical The initial Adjust Mesh is always quite fast but only because most of the necessary work has been done in Auto Guess In all of these examples the bearing capacity converges monotonically but this is not always the case Consider the smooth circular footing problem with B modified to 1 m Footing Data Strip Smooth Circular Rough B 1 EN 5 kPa Double Up 5 597 599 50 5 It 15 clear that if high precision results are required a degree of caution is necessary In particular even if two successive Double Ups give identical bearing capacities this does not necessarily indicate convergence this is why the latest three bearing capacities are displayed in the main form Although the Double Up process might seem an obvious candidate for Richardson extrapolation forecasting the final result based on estimates obtained with stepsizes 4h 2h and h this technique can only be applied if the convergence is strictly monotonic so it is not entirely suitable for use in ABC 3 3 2 Problems requiring adaptivity The drained problem in Figure 3 2 will be used to illustrate a solution in which additional characteris
95. urate to the number of figures given This was however a painstaking process using the computers then available According to Cox 1962 it took 120 hours 5 days of running time on the R A R D E digital computer AMOS to obtain just 32 bearing capacity factors to a precision of in most cases 3 significant figures When running ABC on a typical present day PC this whole series of analyses including systematic mesh refinements using Double Up can be completed in a matter of minutes confirming incidentally that Cox s results are exemplary see Sections 4 2 11 and 4 3 11 Among the more recent studies some researchers have adopted fixed numbers of a and p characteristics for a whole series of analyses while others make no mention at all of the subdivision counts employed apart from perhaps giving a picture of a typical mesh Certain results obtained in these studies appear to be suspect though in some cases inadequate mesh refinement is compounded by other problems These issues are discussed in more detail below as they arise In this chapter all of the bearing capacities obtained using ABC are converged values that are believed to be correct to the quoted 4 digit precision The user can repeat any of these analyses by following the standard procedure of entering the data followed by Auto Guess Trial Mesh Adjust Mesh Double Up Double Up Note however that these analyses smooth footings only Fmax gt 17 are
96. ut the B problems require special treatment because For problem 0 and so the analytical solution q kB 4 q is applicable see Section 4 2 6 Problems B2 and also have F but now gt 0 so the situation is analogous to that of the problem considered in Section 4 2 8 To solve these problems using ABC it is first necessary to introduce a nominal c or a nominal q to make F very large but not greater than the permitted maximum of 10 that applies when gt 1 The analyses below were performed with c 210 kPa giving F values of 6 9x10 for problem B2 and 1 4x10 for B3 Converged results from ABC with numerical results from Salencgon amp Matar 1982b 4 kPa Problem ABC S amp 19826 12 66 20 91 1 626x 10 1 61x10 T analytical T analytical There is very close agreement between the two sets of results Note that exact agreement would not be expected because Salengon amp Matar 19826 used the method of characteristics to produce a series of charts in terms of and in the present notation F Their results for the example problems were based on factors read back from these charts and not on a direct solution of each problem using the method of characteristics as is done in ABC Problems A1 to A3 involved a soil layer of limited depth 2004 C M Martin ABC v1 0 59 4 3 Axial Symmetry 4 3 1 Smooth circular footing on homogeneous undrained clay k 0 0 T
97. utions involving bias and or sub subdivisions can be refined as usual using Double Up In this case convergence is quite slow because of the inherent difficulty of the problem cf Section 3 3 1 Adjust Mesh 114 235 0 8 Double Up 1 112 849 1 6 Double Up 2 112 516 4 1 Double Up 3 112 418 14 1 Double Up 4 112 390 59 3 Double Up 5 112 383 176 2 Rough footing problems where 0 and F kB c becomes greater than about 1000 become very difficult to solve even with the bias and sub subdivision strategies and this is the reason for imposing the restriction in equation 3 2a As with equation 3 2b however this limitation is unlikely to be important in practice witness the outlandish choice of parameters needed to achieve F 333 3 in the example above It must also be borne in mind that unlike the N problem gt 0 and F the limiting undrained problem 20 F can be solved analytically both plane strain and axial symmetry For details see Sections 4 2 5 4 2 6 4 3 5 and 4 3 6 3 3 5 Problems requiring user intervention For most rough footing problems the choice of the applicable solution type 2 or 3 is clear cut Occasionally however for problems that fall close to the borderline see Section 2 5 2 the solution type may need to be changed from 2 to 3 or vice versa as the mesh 15 refined using Double Up To illustrate such a case consider the following specially contrived problem
98. utside the mesh is one of the formalities that would need to be completed in order to establish the calculated bearing capacity as a strict rather than incomplete lower bound solution In many respects the procedure for constructing a type 2 mesh of characteristics Figure 2 7a 1s similar to that already described for solution type 1 but there are two major differences First when establishing the degenerate characteristic the final value of is now an angle lying somewhere between the extremes defined by equations 2 11s and 2 11r In practice it is simpler to specify the fan aperture which must for a valid type 2 solution lie somewhere between 1 2 and 3 4 2 Second while the remaining a characteristics are initiated and extended using CalcAB in exactly the same way as before they are not stepped onto the footing using CalcA As shown in Figure 2 7a each new a characteristic is simply terminated in mid soil when all of the solution points on the previous a characteristic have been exhausted In a type 3 solution Figure 2 7b the first step in constructing the mesh is to establish the degenerate a characteristic using the rough footing equation 2 11r to determine the final value of The construction then proceeds in two distinct phases The first characteristics to be added up to and including the one on the d d boundary resemble those of solution type 1 in that they are stepped onto the footing with a final
99. y consolidated undrained clay 4 2 7 Smooth strip footing on sand with no surcharge 4 2 8 Rough strip footing on sand with no surcharge 4 2 9 Smooth strip footing on non homogeneous undrained clay 4 2 10 Rough strip footing on non homogeneous undrained clay 4 2 11 Smooth strip footing on general cohesive frictional soil 4 2 12 Rough strip footing on general cohesive frictional soil 4 3 Axial symmetry 4 3 1 Smooth circular footing on homogeneous undrained clay 4 3 2 Rough circular footing on homogeneous undrained clay 4 3 3 Smooth circular footing on homogeneous weightless cohesive frictional soil 4 3 4 Rough circular footing on homogeneous weightless cohesive frictional soil 4 3 5 Smooth circular footing on normally consolidated undrained clay 4 3 6 Rough circular footing on normally consolidated undrained clay 4 3 7 Smooth circular footing on sand with no surcharge 4 3 8 Rough circular footing on sand with no surcharge 4 3 9 Smooth circular footing on non homogeneous undrained clay 4 3 10 Rough circular footing on non homogeneous undrained clay 4 3 11 Smooth circular footing on general cohesive frictional soil 4 3 12 Rough circular footing on general cohesive frictional soil APPENDIX A HOW ABC WORKS REALLY REFERENCES 2004 C M Martin 37 37 38 39 39 40 4 4 42 43 45 45 46 46 47 48 49 50 5 52 53 55 56 57 58 59 59 60 61 62 64 65 66 67 69 70 71 72 73 76 A

User Guide for ABC - Analysis of Bearing Capacity v1.0

Contents

Download Pdf Manuals

Related Search

Related Contents