A numerical method for solving the equations of stability of general slip surfaces
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1 A numerical method for solving the equations of stability of general slip surfaces By N. R. Morgenstern* V. E. Pricef 1. Introduction A method has been developed by Morgenstern Price (1965) for analyzing the stability of earth slopes with regard to slips along general surfaces which can be considered as a sequence of planes. The method is believed to be the first in which all the equations of statical equilibrium are satisfied. It has been used satisfactorily to determine factors of safety for several thous slip surfaces on more than forty different cross-sections of slopes. A set of equations was derived which determine the factor of safety F, an overall force ratio scaling constant A, the internal forces. In this paper we describe the numerical technique for solving these equations, which has been used in a program for a digital computer. The equations are equivalent to a pair of non-linear simultaneous equations in the unknowns A F, the principles of the method of solving them may be applied to any set of non-linear simultaneous equations, when the first derivatives with respect to the unknowns can be evaluated. Many of the preliminary experiments were performed on the Deuce computer using the Alphacode language, but most of the numerical results have been obtained on an I.C.T using a program which was written in a dialect of FORTRAN similar to FORTRAN II. The program is at present being converted to FORTRAN as specified by the American Stards Association. In 2 there is a summary of the equations which are to be solved, in 3 the principles of the numerical method of solving the equations are described, the detailed formulae being given in 4. It is necessary to evaluate some integrals which involve some slight numerical complexity, the method for these is described in 5. A practical method of solving the equations of statical equilibrium to obtain the factor of safety of earth slopes, with slip surfaces of arbitrary shape, is described. The basic mathematical problem is the solution of a pair of. simultaneous non-linear equations which arise from the boundary conditions on the solution of an ordinary differential equation. Fig. 1. L V=Vt(l) y = y(x) Potential sliding mass shearing forcej respectively in a general vertical plane with coordinate x, which intersects the slip surface at (x, y) the resultant E at (x, y,) then L=Xn Zx + L)E) - X= XfE F de A tan 4>' F~. tan. tan (1) (2) (3) (4) (5) (6) 2. Summary of the equations We consider a cross-section of the slope of unit thickness, take axes horizontal such that sliding tends to occu: ill the negative x direction Oy vertically downwa. ds, the origin being arbitrary (see Fig. 1). If E X denote the resultant total normal force (7) t The most recent version of the program works in terms of effective E X forces also incorporates earthquake loading represented by a horizontal body force expressed as some percentage of gravity. These differences only complicate the coefficients in equation (2) in no way alter the material presented here. * Lecturer in Civil Engineering, Imperial College of Science Technology, Exhibition Road, London, S.W. 1. t Reader in Numerical Analysis, The City University, St. John St., London, E.C.I. 388
2 Slip surfaces / = kx + m (8) - (9) Hence dx The unknowns in these equations are the constants A F, the functions E, X y t. All the other quantities are known functions of x depending on the chosen slip surface, the geometry of the slope, the pore pressures in the soil mass the properties of the types of soils from which it is constructed. Equation (3) is an assumption which is made concerning the internal forces which makes the problem statically determinate. A discussion of the choice of the function f(x), which must be specified, was given by Morgenstern Price (1965). By assuming that the boundaries of the different soil types are linear, that f{x) may be approximated by a sequence of linear functions, that the slip surface consists of a sequence of planes, then the potential sliding mass may be divided into n slices by vertical planes with coordinates x 0, x u..., x n such that in any slice K, L, N P are constants. We do not assume that any of these slices are necessarily thin. The boundary conditions to be satisfied are normally: E = = 0 E = = 0 M = 0 M-- For any values of A F, equation (2) may be integrated to determine E, starting with E = 0 at x 0, then X may be determined from equation (3) y, from equation (13). Thus the values of E(x n ) M(x n ) may be regarded as functions of A F, we wish to determine those values of A F which satisfy equations (11) (12). 3. Solution of the equations The basis of the technique for solving the equations is the Newton-Raphson method together with certain controls on the values of A F used. Suppose that E n {\, F) M n {\, F) denote the values of E M when x = x n. Starting with values A o F o, suppose the required solution is A o -f SA.F 0 + 8F. If 8A 8F are small, then, approximately, E n {\ 0 + 8X, F o + 8F) = E n {\, F o )+ SA^" + SA, F o + 8F) = M n (\ 0, F o ) + SA-^- when x x 0 when x = x n when x = x n the derivatives are evaluated at (A o, F o ). (10) (11) (12) (13) <^-" = 0 (14) ^F=0 05) 389 SA = SF = DA IF DF DA F " DA DA DF DF DA (16) (17) Thus starting with estimated or arbitrary values of A F, by evaluating E n, M n their derivatives with respect to A F, new estimates may be found, so on, until the changes in A F are sufficiently small. However, if this iterative process is used without additional controls to the values of A F, then even with reasonable initial approximations the process will in many cases either not converge, ox converge to give values of A F which are physically unacceptable. There are three additional methods of controlling the values of A F as follows. (a) If (L + Kx) is zero for any value of x, then it may be seen from equation (2) that E is in general infinite at that point, which is physically unacceptable. This occurs if (L + Kx) changes sign within any slice. In addition, it is reasonable to expect that (L + Kx) will not change sign at any of the points x u x 2,..., x n _i it is discontinuous, since any discontinuity in the soil characteristics, or the slope of a line, may be considered as the limit of a rapidly varying continuous quantity. Thus (Z, + Kx) should be of the same sign for the complete range of x. It has been found in practice that this function should be positive, because the contribution of 1 to L in equation (5) is often the dominant term. Thus we define a feasible region for (A, F) such that for all points in this region the value of (Z, + Kx) is positive for all x. By restricting the values of (A, F) to the feasible region, unique solutions have been found. This has been tested in several experiments by starting with different initial values of A F, always the solution has been independent of the starting values. In some of the preliminary experiments, when this control was not applied, it was found that there were often several real solutions to the equations, but that not more than one solution was physically feasible. If in any iteration (L + Kx) becomes negative for any value of x, then the current values A[ F t are replaced by K*o + A,) K^o+^i) (A 0,F 0 ) is the previous value in the feasible region. It is necessary that the initial values of A F should be in the feasible region. We have found in practice that a satisfactory starting point in general is F= 1-5 A = 0-3/f max. If during the first iteration, it is found that this point is not feasible, then, keeping the value of A fixed, an attempt is made to find a feasible value of F. If this is not possible, then A is set to zero in which case it is always possible to find a feasible F.
3 From equations (4), (5) (8) we have:.,tan <b' XfA. Hence, in order that L + Kx > 0 for all x, then F must lie in the range F max >F>F min (18) ((A - A/) tan <f>' F min = max 1 + XfA > 0 ^ (19) 1 +XfA A/) tan 1 +XfA Slip surfaces will be reduced, i.e., a value of v(< 1) can be found such that: (D(A 0 + vsa, F o + vsf) < O(A 0, F o ). (30) (20) Usually it is the lower limit to F which is most critical, since for most values of x, 1 + XfA > 0. Hence if the upper limit does not exist or if F max > F min +0-1, then we set: F=F m/n (21) KF max < F min + 0-1, then we set: F = i(f min + F max ). (22) If F mi/i > F max, then it is not possible to find a feasible F with this value of A, so we set A = 0, in which case there is no upper limit to F we set: F = max {A tan <f>') (23) (6) A restriction is applied to the magnitude of the steps SA SF obtained from equations (16) (17). If either SA > 0-5 or SF >0-5 then the subsequent values A( F] are given by A, = A o + vsa (24) F t = F o + v8f (25) v = 0-5/max ( SA, SF ). (26) In this way very large increments are prevented, which would otherwise occur usually in the early iterations, when the matrix of first derivatives is ill-conditioned. (c) The function O given by O(A, F) = E 2 n + CM 2 (27) C is any positive constant, satisfies the inequality (28) SA 8F satisfy equations (14) (15) the derivatives are evaluated at (A o, F o ). Hence if <D(A 0 + SA, F 0 + SF)> < (Ao, F o ), (29) then it is possible to reduce the size of the step so that O 390 This is done by comparing the values of O(Ai, F t ) with C>(A 0, F o ) if O has increased then X t F x are replaced by KA 0 + A,) K^o+Fi), respectively, the process repeated. The quantity C is introduced so that the contributions El CM% to <]> are of the same order of magnitude. E M are of different dimensions if, as is usual, units of lb ft are used, then on practical slopes M 2 is much greater than E 2. In these circumstances, if C is not introduced, i.e. C = 1, then the number of iterations would be significantly increased as follows. Normally within a few iterations the point (A o, F o ) would be close to the curve for which M n = 0, although the point (A o + SA, F o + 8F) would be closer to the final point both E n M n = 0, the value of M\ would be larger at (A o + SA, F o + SF) than at (A o, F o ) hence because M\ would be the dominant contribution to O the values of SA SF would be repeatedly halved until condition (30) is satisfied. At this stage (A o + vsa, F o + vsf) is very close to (A o, F o ) so several iterations would have been made giving only a slight improvement. Thus the accepted values (A, F) would follow very closely with small steps the curve M = 0 until finally E n = 0. Thus we need to use a value of C which is positive with dimensions of E 2 /M 2. When comparing <5(A,, F,) with <E>(A 0, F o ) then C is given by with the derivatives evaluated at (A o, F o ). The value of C is kept constant until a point (Ai,F,) is found for which <J> has decreased. When this occurs the value of C O are re-evaluated using the derivatives at the newly accepted point. With these three additional controls on the values of A F, in approximately 80% of practical cases the process converges in less than 10 iterations, giving values of A F correct to 3 decimal places. An upper limit is imposed in the program of 20 iterations of the type so far described. If this limit is reached, then further iterations are performed, restarting with the initial estimates for F A, in which only F is varied until the equation M n = 0 is satisfied. This is performed using the usual single variable Newton-Raphson process given by (32) If this process converges within 20 iterations, then the value of F obtained the initial estimate for A is used as the starting point for further iterations of the original type, with a limit again of 20 iterations imposed. The results of a case in which the original 20 iterations did not converge, but which subsequently converged
4 after finding a value of F for which M n = 0 is given in Table 1. From the table it may be seen that the first four iterations took the point (F, A) near to a local minimum of O at about (1-15, 1-2). The other sixteen iterations of the first batch produced only two more acceptable points, viz. (11513, ) (11537, 11883) with lower values of <1>. After the first batch had been unsuccessful, the values of F A were returned to their initial values F varied to make M n zero. This converged in four iterations to the point (1-203, 0-3) from which the Newton process applied to both variables converged in a further four iterations to the point (1-134, 0-491). As will be explained in 5, the last two iterations were needed to refine these values of A F using accurate formulae for the evaluation of M its derivatives. In a few cases the triple system of iterations described above has not converged to give solutions. Several of these have been investigated in detail it is almost certain that in these cases a solution in the feasible region does not exist. This conclusion was reached by considering a set of slip surfaces gradually changing from one for which a physically acceptable solution had been found to the required surface for which a solution could not be found. In this way the solution was traced was found to approach the boundary of the feasible region finallyattempted to cross it. 4. The calculation of E, M their derivatives We wish to obtain the formulae for determining E n, -r-p, -z-j-, M n, -J^TT, -y~^ for any values of A f. We will denote by K n L,, JV, P, the constant values of K, L, N P in the slice *,_, < x < x ; for i= 1,2,..., n. If in the /th slice x is measured from x ; _, then the solution of equation (2) is Hence E = r. a,,-, + P,x + injx 2 }. (33) L, Pfi, (34) 6, = x, *,_,. (35) Differentiating equation (34) with respect to F gives Slip surfaces Table 1 Values of F, X, E n M n for successive iterations in case which did not converge within the first batch of 20 iterations F L t I 1693 I ] A E n x 10-s M n X The derivatives of K, L, N P can easily be obtained from equations (4), (5), (6) (7). By starting with E = YF = 3A = (38) equations (34), (36) (37) may be used to determine differentiating with respect to A gives From equation (13) we have M, = M,_ x + J (X - A,E)dx, (39) Using equations (3), (8) (33) we have IL, 391 x _ \{k t x + ntj) Li + K,x ; ' -' (40)
5 hence A,} SY//> surfaces mate values, based on the trapezium rule of integration, can be obtained much more quickly. From equations (39), (3) (8) we have, approximately, (,,_, + P,x + in,x 2 )dx. (41) Differentiating equation (41) with respect to F A, ~i>f t»f J o L-, + -C (L, + K ix ) 2 (}h 4_? ', (42) Jo ~T 1 v^ {L,E, _ [ Lij +- A,-.X {AfA'/X + «7,-) /4,} J o (L, u,^-l (43) By changing the scale of x in these integrals so that the range of integration is from 0 to 1, letting H; = K,b,IL, (44) then we obtain the following recurrence relations: f 1 1 "' = "'-'+J 0 (TH 3 (45) (46) 'jx'dx, (47) the coefficients 7}, C/ y F y are independent of x. Since only values of A F are used for which L s L t + Kfi; are positive then H-, > 1 all the integrals are convergent. The evaluation of the basic -1 x^dx r x^dx integrals j Q ^ - ^ J Q fppr^a will be described in the next section. Mj = Mi_ x + ibi{(\i(kib, + m,) Ai)E t + (A,w,- A i )E i _ l }, (48) IM, 5. The evaluation of the integrals The calculation of M its derivatives takes approximately \ of the time of each iteration, but their approxi- 392 ib ' /) ~ J>E, ;)^F i -l + (A,m ; - Ai)^:- (49) (Am,- (50) These approximate formulae are used until the iterative process converges, then the exact formulae are used to refine the values of F A. Normally, about 6 of the simpler iterations are followed by 1 or 2 of the iterations in which M its derivatives are evaluated accurately. Thus there is an overall saving of approximately 50% of the computing time by using the approximate formulae first. In order to evaluate M n its derivatives accurately, it was shown in 4 that it is necessary to evaluate integrals of the following form: dx 1 + Hx 9(\+Hxf (51) (52) for values of H > 1 for; = 0, 1,..., 4. For non-zero values of H there are analytical formulae for these integrals which may easily be generated by the following recurrence relations: Bn = ~TT l g (1 + ") V 53 ) C n = 1 +H C r / f> -y+i (54) (55) (56) When H = 0, it is obvious that these formulae cannot be used. Also, for small values of H, their use introduces serious rounding-off errors since each integral is expressed as the relatively small difference between two large numbers. For small values of H, however, it can be seen that the integrs may be exped as polynomials in x, so Gaussian numerical integration formulae may be satisfactorily used.
6 By evaluating the most complicated integral C 4, for a sequence of values of H starting with 0-95, using both the recurrence relations (53),..., (56) Gaussian integration formulae of various orders, it was found that, using floating-point arithmetic with a precision of about 9 decimal places, the integrals could be obtained correct to at least 6 significant decimal places by using the analytical formulae for H < 0-4 H > 0-5, for 0-4 < #< 0-5 using the Gauss formula of order Discussion The methods described in this paper have been incorporated in a computer program wh ; ch has been used by a number of civil engineers. A description of the practical aspects of the program of some of the problems associated with it will be reported else. The method of specifying the data for an analysis is similar to that for the slip-circle program described by Little Price (1958). It is well known that the Newton-Raphson technique may be used for solving non-linear simultaneous equations, it is known that extra controls on the variables are often essential to make the process converge. The control (c) described in 3, that the sum of the squares of the residuals should be reduced by each iteration is References Slip surfaces of general application for one or more non-linear equations. It is less arbitrary than the control (b) of 3 which is advocated in N.P.L. (1961) which requires an arbitrary fixed limit to the change, or proportional change, of any of the variables. However, it should be noted that squares of the residuals should be scaled if necessary so that they are of the same order of magnitude, otherwise the restriction can increase the number of iterations considerably. The scaling may conveniently be performed by dividing each residual by the sum of the squares of its derivatives with respect to the unknowns. 7. Acknowledgement The authors gratefully acknowledge the assistance of members of the staff of the English Electric London Computing Service, in particular, Mrs. J. Skinner, with the programming of the preliminary experiments of these calculations. The authors are grateful to Messrs. Binnie Partners for permission to publish the results in Table 1, for many helpful discussions encouragement with the project. This study was supported in its early stages by a research grant awarded by the Department of Scientific Industrial Research. LITTLE, A. L., PRICE, V. E. (1958). "The Use of an Electronic Computer for Stability Analysis," Ceotechnique, Vol. 8, 3 p MORGENSTERN, N. R., PRICE, V. E. (1965). "The Analysis of the Stability of General Slip Surfaces," Geotechnique Vol 15 1, p. 79. N.P.L. (1961). Modern Computing Methods, 2nd Edition, chapter 6, 23, London, H.M.S.O. To the Editor, The Computer Journal. Sir, Many algorithms have been published in which values of 7T, 277, 7T 2 etc., are written as constants, commonly with about 9 significant figures. Sometimes comments are included that these constants are intended as approximations to 77 etc., that the number of significant figures should be adjusted to the capacity of the computer. But it is not necessary to use any explicit approximation to 77 or related values, since in ALGOL 60 the appropriate value can be generated by the statement pi: = 4 X arctan (1) Correspondence the variable pi may then be used as a component of arithmetic expressions. The accuracy with which pi approximates to n is limited only by the accuracy of the arctan function ( possible roundoff). For instance, in FOR- TRAN II-D on the IBM 1620, the corresponding statement PI = 4 0*ATANF (10) (with 28 decimal digits in the mantissae of floating-point numbers) does indeed give 77 correctly to 28 significant figures. Likewise, of course, if e is required then the statement e : exp (1) can be used. Yours faithfully, G. J. TEE University of Lancaster, 108 St. Leonardgate, Lancaster 24 November
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