김경종
23 KiB
not converge. After performing MITER iteration cycles the program will then stop.
NOUTP (1) This parameter controls the output of the unconverged results after the first iteration. In order to examine the convergence process the user can vary the frequency of output for each load increment:
1—Print the displacements only after the first iteration.
2—Print the displacements and nodal reactions after the first iteration.
3—Print the displacements, reactions and stresses after the first iteration.
NOUTP (2) This parameter controls the output of the converged results:
1—Print the final displacements only.
2—Print the final displacements and nodal reactions.
3—Print the final displacements, reactions and stresses.
The loading to which the structure is subjected is monitored by the arrays ELOAD (IELEM, IEVAB) and TLOAD (IELEM, IEVAB). The total loading applied to the structure at any stage of the analysis is accumulated in the TLOAD array. On the other hand ELOAD contains the loading to be applied to the structure for each iteration of the solution process. Initially (the first iteration of the first load increment) ELOAD contains the first increment of applied load. For the second and subsequent iterations ELOAD contains the residual nodal forces which must be redistributed as described in Section 3.7. After convergence has occurred, the next increment of load is assimilated into ELOAD, so that at this stage ELOAD contains the new applied load increment together with any residual forces still remaining after convergence of the solution for the previous load increment. These residual forces should be negligibly small if the convergence tolerance factor, TOLER, is correctly chosen. However, since any residual forces are retained in ELOAD and applied as nodal forces during the next load increment, it is noted that equilibrium is maintained at every stage of the computation process.
The final role of this subroutine is to insert appropriate values in the fixity array to control any prescribed displacements. As described in Section 3.3, in order to arrive at the correct value of a displacement whose value is prescribed for a load increment, it is necessary to prescribe the given value for equation solution during the first iteration and then prescribe a zero value for all subsequent iterations. Since the displacements occurring during each iteration accumulate to give the total displacement then clearly the prescribed value will be obtained by this process.
Subroutine INCREM will now be presented and explanatory notes provided.
SUBROUTINE INCREM(ELOAD, FIXED, IINCS, MELEM, MEVAB, MITER, INCR 1 . MTOTV, MVFIX, NDOFN, NELEM, NEVAB, NOUTP, INCR 2 . NOFIX, NTOTV, NVFIX, PRESC, RLOAD, TFACT, INCR 3 . TLOAD, TOLER) INCR 4 C********** INCR 5 C INCR 6 C*** THIS SUBROUTINE INCREMENTS THE APPLIED LOADING INCR 7 C INCR 8 C********** INCR 9 DIMENSION ELOAD(MELEM, MEVAB), FIXED(MTOTV), INCR 10 . IFFIX(MTOTV), INCR 11 . NOUTP(2), NOFIX(MVFIX), INCR 12 . PRESC(MVFIX, NDOFN), RLOAD(MELEM, MEVAB), TLOAD(MELEM, MEVAB) INCR 13 WRITE(6, 900) IINCS INCR 14 900 FORMAT(1H0, 5X, 17HINCREMENT NUMBER, I5) INCR 15 READ(5, 950) FACTO, TOLER, MITER, NOUTP(1), NOUTP(2) INCR 16 950 FORMAT(2F10.5, 3I5) INCR 17 TFACT=TFACT+FACTO INCR 18 WRITE(6, 960) TFACT, TOLER, MITER, NOUTP(1), NOUTP(2) INCR 19 960 FORMAT(1H0, 5X, 13HLOAD FACTOR =, F10.5, 5X, INCR 20 .24H CONVERGENCE TOLERANCE =, F10.5, 5X, 24HMAX. NO. OF ITERATIONS =, INCR 21 . I5, //27H INITIAL OUTPUT PARAMETER =, I5, 5X, 24HFINAL OUTPUT PARAMETINCR 22 .ER =, I5) INCR 23 DO 80 IELEM=1, NELEM INCR 24 DO 80 IEVAB=1, NEVAB INCR 25 ELOAD(IELEM, IEVAB)=ELOAD(IELEM, IEVAB)+RLOAD(IELEM, IEVAB)FACTO INCR 26 80 TLOAD(IELEM, IEVAB)=TLOAD(IELEM, IEVAB)+RLOAD(IELEM, IEVAB)FACTO INCR 27 C INCR 28 C* INTERPRET FIXITY DATA IN VECTOR FORM INCR 29 C INCR 30 DO 100 ITOTV=1, NTOTV INCR 31 100 FIXED(ITOTV)=0.0 INCR 32 DO 110 IVFIX=1, NVFIX INCR 33 NLOCA=(NOFIX(IVFIX)-1)*NDOFN INCR 34 DO 110 IDOFN=1, NDOFN INCR 35 NGASH=NLOCA+IDOFN INCR 36 FIXED(NGASH)=PRESC(IVFIX, IDOFN)*FACTO INCR 37 110 CONTINUE INCR 38 RETURN INCR 39 END INCR 40
INCR 14–15 Write the number of the load increment which is being currently solved.
INCR 16-23 Read and write the load increment control parameters. Note that the incremental load factor, FACTO, is input whereas the total load factor, TFACT, is output.
INCR 24–27 Accumulate the incremental loading into array ELOAD for equation solution and also into TLOAD to record the total load applied to the structure.
INCR 31–32 Zero the global vector of prescribed displacements.
INCR 33–38 Insert any prescribed displacement values, factored by the load increment factor, into the appropriate position in the global vector.
6.5.4 Solution convergence monitoring subroutine CONVER
This subroutine monitors convergence of the nonlinear solution iteration process. It is almost identical to subroutine CONUND for one-dimensional
applications described in Section 3.10.3. Since for two-dimensional and plate bending problems we have more than one degree of freedom per nodal point, summation in (3.27) must now be made over the total number of degrees of freedom in the structure. As an additional check on the nonlinear solution process we also arrange to evaluate the maximum individual residual force \psi_{i}^{r} existing in the structure.
Subroutine CONVER is now presented and can be understood with the aid of Section 3.10.3.
SUBROUTINE CONVER(ELOAD,IITER,LNODS,MELEM,MEVAB,MTOTV,NCHEK,CONV 1
NDOFN,NELEM,NEVAB,NNODE,NTOTV,PVALU,STFOR,CONV 2
TLOAD,TOFOR,TOLER)CONV 3
C*****
THIS SUBROUTINE CHECKS FOR CONVERGENCE OF THE ITERATION PROCESS
CONV 4
C
CONV 5
C*****
DIMENSION ELOAD(MELEM,MEVAB),LNODS(MELEM,9),STFOR(MTOTV),CONV 6
CONV 7
C*****
CONV 8
DIMENSION ELOAD(MELEM,MEVAB),LNODS(MELEM,9),STFOR(MTOTV),CONV 9
TOFOR(MTOTV),TLOAD(MELEM,MEVAB)CONV 10
NCHEK=0CONV 11
RESID=0.0CONV 12
RETOT=0.0CONV 13
REMAX=0.0CONV 14
DO 5 ITOTV=1,NTOTVCONV 15
STFOR(ITOTV)=0.0CONV 16
TOFOR(ITOTV)=0.0CONV 17
5 CONTINUECONV 18
DO 40 IELEM=1,NELEMCONV 19
KEVAB=0CONV 20
DO 40 INODE=1,NNODECONV 21
LOCNO=IABS(LNODS(IELEM,INODE))CONV 22
DO 40 IDOFN=1,NDOFNCONV 23
KEVAB=KEVAB+1CONV 24
NPOSI=(LOCNO-1)*NDOFN+IDOFNCONV 25
STFOR(NPOSI)=STFOR(NPOSI)+ELOAD(IELEM,KEVAB)CONV 26
40 TOFOR(NPOSI)=TOFOR(NPOSI)+TLOAD(IELEM,KEVAB)CONV 27
DO 50 ITOTV=1,NTOTVCONV 28
REFOR=TOFOR(ITOTV)-STFOR(ITOTV)CONV 29
RESID=RESID+REFOR*REFORCONV 30
RETOT=RETOT+TOFOR(ITOTV)*TOFOR(ITOTV)CONV 31
AGASH=ABS(REFOR)CONV 32
50 IF(AGASH.GT.REMAX) REMAX=AGASHCONV 33
DO 10 IELEM=1,NELEMCONV 34
DO 10 IEVAB=1,NEVABCONV 35
10 ELOAD(IELEM,IEVAB)=TLOAD(IELEM,IEVAB)-ELOAD(IELEM,IEVAB)CONV 36
RESID=SQRT(RESID)CONV 37
RETOT=SQRT(RETOT)CONV 38
RATIO=100.0*RESID/RETOTCONV 39
IF(RATIO.GT.TOLER) NCHEK=1CONV 40
IF(IITER.EQ.1) GO TO 20CONV 41
IF(RATIO.GT.PVALU) NCHEK=999CONV 42
20 PVALU=RATIOCONV 43
WRITE(6,30) NCHEK,RATIO,REMAXCONV 44
30 FORMAT(1HO,3X,18HCONVERGENCE CODE =,I4,3X,28HNORM OF RESIDUAL SUM CONV 45
.RATIO =,E14.6,3X,18HMAXIMUM RESIDUAL =,E14.6)CONV 46
RETURNCONV 47
ENDCONV 48
6.6 Problems
6.1 Using the subroutines described in this chapter devise programs to evaluate the stiffness matrices and load vectors for 4-, 8- and 9-node quadrilateral isoparametric elements for plane stress, plane strain, axisymmetric and Mindlin plate applications.
6.2 Use the shape functions L_{t}^{(e)}(\xi, \eta) from the 9-node Lagrangian quadrilateral isoparametric element to devise a new family of 8-node Serendipity quadrilateral element shape functions N_{t}^{(e)}(\xi, \eta) of the form
N _ {i} ^ {(e)} = L _ {i} ^ {(e)} + a L _ {9} ^ {(e)} \quad i = 1, 3, 5 \text { and } 7 \text { (corner nodes) },
N _ {i} ^ {(e)} = L _ {i} ^ {(e)} + b L _ {9} ^ {(e)} \quad i = 2, 4, 6 \text { and } 8 \text { (midside nodes) },
where L_{9}^{(e)} is the shape function of the central node of the Lagrangian element. What limits are there on a and b ?
6.3 Determine some further diagnostic checks on the input, other than those described in Sections 6.4.13 and 6.4.15. Apart from the check on the Jacobian determinant given in Subroutine JACOB2 in Section 6.4.4, are there any other checks which could be incorporated into the program after the input has been successfully read and checked?
6.4 Determine the consistent nodal forces for the case when a point load with components P_x, P_y acts at an arbitrary point along an element edge defined by Cartesian coordinates (x_P, y_P) , which correspond to local coordinates (\xi, \eta) = (\xi_P, -1) .
6.7 References
- HILL, R., The Mathematical Theory of Plasticity, Oxford University Press, 1950.
- TIMOSHENKO, S. P. and GOODIER, J. N., Theory of Elasticity, McGraw-Hill, New York, 1951.
- HUGHES, T. J. R., COHEN, M. and HAROUN, M., Reduced and selective integration techniques in the finite element analysis of plates, Nucl. Eng. Design, 46, 203–222 (1978).
- HINTON, E. and OWEN, D. R. J., Finite Element Programming, Academic Press, London, 1977.
- IRONS, B. M. and AHMAD, S., Techniques of Finite Elements, Ellis Horwood, Chichester, 1980.
- BATHE, K. J. and WILSON, E. L., Numerical Methods in Finite Element Analysis, Prentice-Hall, Englewood Cliffs, New Jersey, 1977.
Chapter 7 Elasto-plastic problems in two dimensions
7.1 Introduction
In this chapter we consider the elasto-plastic stress analysis of solids which conform to plane stress, plane strain or axisymmetric conditions. Most of the problems encountered in engineering can be approximated to satisfy one of these classifications.
The basic laws governing elasto-plastic material behaviour in a two-dimensional solid must be presented before the numerical aspects of the problem can be considered and to this end new concepts, such as the plastic potential and the normality condition will be introduced. Only the essential expressions will be provided in this text and the reader will be directed to other sources for a more complete theoretical treatment.
The situation is complicated by the fact that different classes of materials exhibit different elasto-plastic characteristics. In this chapter four different yield criteria are employed. The Tresca and Von Mises laws, which closely approximate metal plasticity behaviour, are considered and the Mohr-Coulomb and Drucker-Prager criteria, which are applicable to concrete, rocks and soils, are presented.
In the latter sections of this chapter a computer code is developed to allow the solution of practical problems. Many of the subroutines required for elasto-plastic solution have been reviewed in Chapter 6. In this chapter the additional subroutines are developed and assembled to provide a working program.
7.2 The mathematical theory of plasticity
The object of the mathematical theory of plasticity is to provide a theoretical description of the relationship between stress and strain for a material which exhibits an elasto-plastic response. In essence, plastic behaviour is characterised by an irreversible straining which is not time dependent and which can only be sustained once a certain level of stress has been reached. In this section we outline the basic assumptions and associated theoretical expressions for a general continuum. For a more complete treatment the reader is directed to Refs. 1–3. In order to formulate a theory which models elasto-plastic material deformation three requirements have to be met:
- An explicit relationship between stress and strain must be formulated to describe material behaviour under elastic conditions, i.e. before the onset of plastic deformation.
● A yield criterion indicating the stress level at which plastic flow commences must be postulated.
- A relationship between stress and strain must be developed for post-yield behaviour, i.e. when the deformation is made up of both elastic and plastic components.
Before the onset of plastic yielding the relationship between stress and strain is given by the standard linear elastic expression.*
\sigma_ {i j} = C _ {i j k l} \epsilon_ {k l}, \tag {7.1}
where \sigma_{ij} and \epsilon_{kl} are the stress and strain components respectively and C_{ijkl} is the tensor of elastic constants which for an isotropic material has the explicit form
C _ {i j k l} = \lambda \delta_ {i j} \delta_ {k l} + \mu \delta_ {i k} \delta_ {j l} + \mu \delta_ {i l} \delta_ {j k}, \tag {7.2}
where \lambda and \mu are the Lamé constants and \delta_{ij} is the Kronecker delta defined by
\delta_ {i j} = \left\{ \begin{array}{l l} 1 & \text { if } \quad i = j \\ 0 & i \neq j. \end{array} \right. \tag {7.3}
7.2.1 The yield criterion
The yield criterion determines the stress level at which plastic deformation begins and can be written in the general form
f (\sigma_ {i j}) = k (\kappa), \tag {7.4}
where f is some function and k a material parameter to be determined experimentally. The term k may be a function of a hardening parameter \kappa discussed later in Section 7.2.2. On physical grounds, any yield criterion should be independent of the orientation of the coordinate system employed and therefore it should be a function of the three stress invariants only
J _ {1} = \sigma_ {i i}
J _ {2} = \frac {1}{2} \sigma_ {i j} \sigma_ {i j}
J _ {3} = \frac {1}{3} \sigma_ {i j} \sigma_ {j k} \sigma_ {k i}. \tag {7.5}
Experimental observations, notably by Bridgeman, ^{(4)} indicate that plastic deformation of metals is essentially independent of hydrostatic pressure. Consequently the yield function can only be of the form
f (J _ {2} ^ {\prime}, J _ {3} ^ {\prime}) = k (\kappa), \tag {7.6}
- In the indicial notation employed, Einstein's summation convention is invoked, whereby it is implicitly assumed that a summation from 1 to 3 is performed over any index which is repeated in any term of an expression. Also indices 1, 2, 3 refer to Cartesian components x, y, z respectively. Note that \sigma_{11} = \sigma_{xx} = \sigma_x, \sigma_{12} = \sigma_{xy} , etc.
where J_{2}' and J_{3}' are the second and third invariants of the deviatoric stresses,
\sigma_ {i j} ^ {\prime} = \sigma_ {i j} - \frac {1}{3} \delta_ {i j} \sigma_ {k k}. \tag {7.7}
Most of the various yield criteria that have been suggested for metals are now only of historic interest, since they conflict with experimental predictions. The two simplest which do not have this fault are the Tresca criterion and the Von Mises criterion.
The Tresca yield criterion (1864)
This states that yielding begins when the maximum shear stress reaches a certain value. If the principal stresses are \sigma_{1} , \sigma_{2} , \sigma_{3} where \sigma_{1} \geqslant \sigma_{2} \geqslant \sigma_{3} then yielding begins when
\sigma_ {1} - \sigma_ {3} = Y (\kappa), \tag {7.8}
where Y is a material parameter to be experimentally determined and which may be a function of the hardening parameter \kappa . By considering all other possible maximum shearing stress values (e.g. \sigma_{2}-\sigma_{1} if \sigma_{2}\geqslant\sigma_{3}\geqslant\sigma_{1} ) it can be shown that this yield criterion may be represented in the \sigma_{1}\sigma_{2}\sigma_{3} stress space by the surface of an infinitely long regular hexagonal cylinder as shown in Fig. 7.1. The axis of the cylinder coincides with the space diagonal, defined by points \sigma_{1}=\sigma_{2}=\sigma_{3} , and since each normal section of the cylinder is identical, (a consequence of the assumption that a hydrostatic stress does not influence yielding), it is convenient to represent the yield surface geometrically by projecting it onto the so-called \pi plane, \sigma_{1}+\sigma_{2}+\sigma_{3}=0 as shown in Fig. 7.2(a). When the yield function f depends on J_{2}' and J_{3}' alone it can be
text_image
π plane σ₁ + σ₂ + σ₃ = 0 Space diagonal σ₁ = σ₂ = σ₃ Von Mises Tresca
Fig. 7.1 Geometrical representation of the Tresca and Von Mises yield surfaces in principal stress space.
Fig. 7.2 Two-dimensional representations of the Tresca and Von Mises yield criteria. (a) \pi plane representation. (b) Conventional engineering representation.
written in the form f(\sigma_{1}-\sigma_{3},\sigma_{2}-\sigma_{3}) and a two-dimensional plot of the surface f=k is then possible as shown in Fig. 7.2(b). It can be shown generally (1,2) that yield surfaces must be convex (except for local flat areas, possibly) and that they must contain the stress origin.
The Von Mises yield criterion (1913)
Von Mises suggested that yielding occurs when J_{2}^{\prime} reaches a critical value, or
(J _ {2} ^ {\prime}) ^ {\ddagger} = k (\kappa), \tag {7.9}
in which k is a material parameter to be determined. The second deviatoric stress invariant, J_{2}' , can be explicitly written as
\begin{array}{l} J _ {2} ^ {\prime} = \frac {1}{2} \sigma_ {i j} ^ {\prime} \sigma_ {i j} ^ {\prime} = \frac {1}{6} [ (\sigma_ {1} - \sigma_ {2}) ^ {2} + (\sigma_ {2} - \sigma_ {3}) ^ {2} + (\sigma_ {3} - \sigma_ {1}) ^ {2} ] \\ = \frac {1}{2} \left[ \sigma_ {x} ^ {\prime 2} + \sigma_ {y} ^ {\prime 2} + \sigma_ {z} ^ {\prime 2} \right] + \tau_ {x y} ^ {2} + \tau_ {y z} ^ {2} + \tau_ {x z} ^ {2}. \tag {7.10} \\ \end{array}
Yield criterion (7.9) may be further written as
\bar {\sigma} = \sqrt {3} (J _ {2} ^ {\prime}) ^ {\frac {1}{2}} = \sqrt {3} k, \tag {7.11}
where
\tilde {\sigma} = \sqrt {(3 / 2) \left\{\sigma_ {i j} ^ {\prime} \sigma_ {i j} ^ {\prime} \right\} ^ {\frac {1}{2}}}, \tag {7.12}
and \bar{\sigma} is termed the effective stress, generalised stress or equivalent stress. Some physical insight into the definition of \bar{\sigma} will be apparent later from Section 7.2.4 where the case of uniaxial yielding is considered. There are two physical interpretations of the Von Mises yield condition. Nadai (1937) introduced the so-called octahedral shear stress \tau_{oct} , which is the shear stress on the planes of a regular octahedron, the apices of which coincide with the
principal axes of stress. The value of \tau_{oct} is related to J_{2}' by
\tau_ {\mathrm{oct}} = \sqrt {(2 J _ {2} ^ {\prime} / 3)}. \tag {7.13}
Thus yielding can be interpreted to begin when \tau_{oct} reaches a critical value. Hencky (1924) pointed out that the Von Mises law implies that yielding begins when the (recoverable) elastic energy of distortion reaches a critical value.
Fig. 7.1 shows the geometrical interpretation of the Von Mises yield surface to be a circular cylinder whose projection onto the \pi plane is a circle of radius \sqrt{(2)}k as shown in Fig. 7.2(a). The two dimensional plot of the Von Mises yield surface is the ellipse shown in Fig. 7.2(b). A physical meaning of the constant k can be obtained by considering the yielding of materials under simple stress states. The case of pure shear ( \sigma_{1} = -\sigma_{2}, \sigma_{3} = 0 ) requires on use of (7.9) and (7.10) that k must equal the yield shear stress. Alternatively the case of uniaxial tension ( \sigma_{2} = \sigma_{3} = 0 ) requires that \sqrt{(3)}k is the uniaxial yield stress.
The Tresca yield locus is a hexagon with distances of \sqrt{(2/3)}Y from origin to apex on the \pi plane whereas the Von Mises yield surface is a circle of radius \sqrt{(2)}k . By suitably choosing the constant Y, the criteria can be made to agree with each other, and with experiment, for a single state of stress. This may be selected arbitrarily; it is conventional to make the circle pass through the apices of the hexagon by taking the constant Y = \sqrt{(3)}k , the yield stress in simple tension. The criteria then differ most for a state of pure shear, where the Von Mises criterion gives a yield stress 2/\sqrt{(3)} (\approx 1 \cdot 15) times that given by the Tresca criterion. For most metals Von Mises' law fits the experimental data more closely than Tresca's, but it frequently happens that the Tresca criterion is simpler to use in theoretical applications.
The Mohr-Coulomb yield criterion
This is a generalisation of the Coulomb (1773) friction failure law defined by
\tau = c - \sigma_ {n} \tan \phi , \tag {7.14}
where \tau is the magnitude of the shearing stress, \sigma_{n} is the normal stress (tensile stress is positive), c is the cohesion and \phi the angle of internal friction. Graphically (7.14) represents a straight line tangent to the largest principal stress circle as shown in Fig. 7.3 and was first demonstrated by Mohr (1882). From Fig. 7.3, and for \sigma_{1} \geqslant \sigma_{2} \geqslant \sigma_{3} (7.14) can be rewritten as
- \frac {1}{2} \left(\sigma_ {1} - \sigma_ {3}\right) \cos \phi = c - \left(\frac {\sigma_ {1} + \sigma_ {3}}{2} - \frac {\left(\sigma_ {1} - \sigma_ {3}\right)}{2} \sin \phi\right) \tan \phi , \tag {7.15}
or rearranging
\left(\sigma_ {1} - \sigma_ {3}\right) = 2 c \cos \phi - \left(\sigma_ {1} + \sigma_ {3}\right) \sin \phi . \tag {7.16}
text_image
P τ F E φ c φ C D B A -σ₁ 0 c Cot φ Q σₙ -σ₂ -σₙ -(σ₁ + σ₂)/2 -σ₃
Fig. 7.3 Mohr circle representation of the Mohr–Coulomb yield criterion.
Again, as for the Tresca criterion, the complete yield surface is obtained by considering all other stress combinations which can cause yielding (e.g. \sigma_{3} \geqslant \sigma_{1} \geqslant \sigma_{2} ). In principal stress space this gives a conical yield surface whose normal section at any point is an irregular hexagon as shown in Fig. 7.4. The conical, rather than cylindrical, nature of the yield surface is a consequence of the fact that a hydrostatic stress does influence yielding which is evident from the last term in (7.14). When \sigma_{1} = \sigma_{2} = \sigma_{3} we have from (7.16) that the mean hydrostatic stress, \sigma_{m} = c \cot \phi and therefore the apex of the hexagonal pyramid, 0, in Fig. 7.4, lies along the space diagonal at the point \sigma_{1} = \sigma_{2} = \sigma_{3} = c \cot \phi . This criterion is applicable to concrete, rock and soil problems.
The Drucker-Prager yield criterion
An approximation to the Mohr–Coulomb law was presented by Drucker and Prager (1952) as a modification of the Von Mises yield criterion. The influence of a hydrostatic stress component on yielding was introduced by inclusion of an additional term in the Von Mises expression to give
a J _ {1} + (J _ {2} ^ {\prime}) ^ {\frac {1}{2}} = k ^ {\prime}. \tag {7.17}
This yield surface has the form of a circular cone. In order to make the Drucker–Prager circle coincide with the outer apices of the Mohr–Coulomb hexagon at any section, it can be shown that
a = \frac {2 \sin \phi}\sqrt {(3) (3 - \sin \phi)}, \quad k ^ {\prime} = \frac {6 c \cos \phi}{\sqrt {(3) (3 - \sin \phi)}}. \tag {7.18}
Coincidence with the inner apices of the Mohr–Coulomb hexagon is provided by
\alpha = \frac {2 \sin \phi}{\sqrt {(3) (3 + \sin \phi)}}, \quad k ^ {\prime} = \frac {6 c \cos \phi}{\sqrt {(3) (3 + \sin \phi)}}. \tag {7.19}