THE UNIVERSITY OF MICHIGAN COLLEGE OF ENGINEERING Department of Engineering Mechanics Technical Report THE RANGE OF THE YIELD CONDITION IN STABLE, IDEALLY PLASTIC SOLIDS R. M. Haythornthwaite ORA Project 04403 under contract with: DEPARTMENT OF THE ARMY ORDNANCE TANK-AUTOMOTIVE COMMAND DETROIT ORDNANCE DISTRICT CONTRACT NO. DA-20-018-ORD-23276 DETROIT, MICHIGAN administered through: OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR April 1961

TABLE OF CONTENTS Page LIST OF ILLUSTRATIONS v SUMMARY 1 I. INTRODUCTION 2 II. MATERIAL INSENSITIVE TO MEAN STRESS 3 A. Bounding Yield Surfaces 3 B. A Plane-Stress Example 6 III. MATERIAL SENSITIVE TO MEAN STRESS 12 A. Bounding Yield Surfaces 12 B. Plane Strain 16 C. Plane-Strain Example 18 IV. DISCUSSION 21 REFERENCES 22 iii

LIST OF ILLUSTRATIONS Table Table Page 1 Expressions for the Yield Criteria Illustrated in Fig. 4 14 Figures Figure Page la Material insensitive to mean stress, Bounding yield criteria when tensile test results are available, 4 lb Material insensitive to mean stress, Bounding yield criteria when simple shear-test results are availableo 4 2 Alternative escribed yield criteria when tensile test results are available, 7 3 The reduced stress criterion for plate bending (plane stress). 7 4a Material sensitive to mean stress. Bounding yield criteria when tensile tests results are available. 13 4b Material sensitive to mean stress, Bounding yield criteria when simple shear-test results are availableo 13 5 Punch indentation (plane strain) for a material sensitive to mean stress, 20 v

SUMMARY The property of stability in an ideally plastic solid is employed to define bounding yield conditions which remain valid when test results are available for only one stress state, such as simple tension or simple shear. For material in which yield is insensitive to changes in mean stress, the bounding yield conditions are shown to be the maximum shearstress criterion of Tresca and a new criterion designated the maximum reduced stress criterion, Use of both criteria enables the best possible bounds to be obtained for the carrying capacity of a structure in the absence of further information. Parallel results are obtained for material in which yield strength is a linear function of the mean stress, and some examples are given, 1

I. INTRODUCTION The powerful variational principles of the theory of ideally plastic solids have been employed mainly in the development of approximate solutions to well-defined problems in which the yield properties of the material are supposed to be known exactly. The carrying capacity may then be bounded from above and below by means of the theorems of limit analysis, developed by Gvozdevl and by Drucker, Prager, and Greenberg,2 or by the use of escribed and inscribed yield conditions.3 On the other hand, the question of finding approximate solutions when information about the yield condition is incomplete does not seem to have been investigated, despite its technological importance. In this paper, the problem of finding upper and lower bounds to the yield load of a body is discussed for the case when the yield strength of the material is known exactly for only one or two stress states. An isotropic material in which yield is independent of the mean stress is considered first, and then the analysis is extended to material in which the yield criterion is linearly dependent on the mean stress. The former model is widely used in the analysis of metals, while interest in the latter springs from recent attempts to apply the theory of plasticity to soils. 4,5 2

IIo MATERIAL INSENSITIVE TO MEAN STRESS Ao BOUNDING YIELD SURFACES Consider an ideally plastic, isotropic material in which the yield criterion is independent of the mean stresso In view of the property of isotropy, the orientation in space of the principal stress directions is unimportant and the state of stress may be represented completely by the principal stress components ia, a2, a30 It is then possible to represent the yield criterion by means of a surface in principal stress space having components al, a2, 3a as co-ordinates. In the case of yield criteria independent of the mean stress, this surface will be formed by generators at right angles to the planes 1a + a2 + 3a = const,, i.e., parallel to the axis making equal angles with the co-ordinate directions, and any intersection of the surface with a plane al + a2 + c3 = const. will be typical. Two cases of intersections of this type are shown in Fig. 1o Consider first the case where the results of tensile and compressive tests are available (Fig. la). In the case of metals, tensile and compressive yield strengths are often nearly equal, suggesting the additional assumption, which will be made here, that reversal of the sign of a stress does not alter the stress magnitude at yield. Test points are represented by the six small circles in Fig. la, because any one of the three principal stresses can be taken as the nonzero component. Alternatively, considering a case where the result of a shear test (e.go, simple torsion) is available, the six points indicated by the small circles in Fig. lb will then have been determined experimentally. We now consider the question: What are the largest and smallest yield surfaces that can be drawn through the test data in the two cases? When finding the smallest surfaces, it is sufficient to note the requirement of convexity for any yield surface associated with a stable plastic material,6 in order to arrive at the two hexagons shown. The largest surfaces are found by noting in addition that the cross sections shown must have 30~ symmetry, so that reversal of all three stress components does not influence yield. The hexagons marked 1 in Figo 1 will be recognized as representing the familiar maximum shear stress criterion of Tresca: maxo (a1 1- a 21,10 2 31,13 - a11) = aO0 = 2To (l) 5

3 33 1 1~~ Fig. la. Material insensitive to mean Fig. Ib. Material insensitive to mean stress. Bounding yield criteria when stress. Bounding yield criteria when tensile test results are available. simple shear-test results are available.

where o, is the yield stress in simple tension and T0 the yield stress in simple shear. The hexagons marked 2 in Fig. 1 represent criteria in which a restriction is placed on the value of the maximum reduced stress (which in Fig. 1 is measured parallel to the stress axes): 2 max. ( ic - al|, o,2 - al) = To = o (2) where ao, To are defined as for Eq. (1) and a = (ao + a2 + a3)/3. The hexagons shown in Fig. 1 and represented analytically by Eqs. (1) and (2) owe their significance to the property that all possible yield conditions must lie between them. A lower bound obtained by the use of the inscribed hexagon will be a lower bound for all stable materials irrespective of the details of the yield condition, and an upper bound obtained by the use of the escribed hexagon will be similarly an upper bound in all cases. Hill7 has suggested Eq. (2) as a suitable linear approximation to v.Mises' yield surface when the stress state is close to a uniaxial compression. Ivlev8 introduced the hexagons shown in Fig. la and designated them as bounding hexagons, but did not make this property dependent on the availability of tensile test data. Neither author anticipates the particular use of the criteria which is suggested here, When determining upper bounds, the computations are greatly simplified if a compact expression can be found for the rate of energy dissipation in terms of the strain rates. The theory of the generalized plastic potential9 indicates that the associated strain-rate vectors coincide with the outward drawn normal to the yield surface when corresponding axes are superimposed. Thus n Cij = 2 i (3) p=l ij where %p are positive constants and the fp are regular functions, the equations of various parts of the yield surface being fp(aij)= 0. It is then a simple matter to show that the rate of energy dissipation is D = Elmax.o (4) 5

in the case of the maximum shear-stress criterion, and D = yI1max. (5) in the case of the maximum reduced stress criterion. Still further restrictions on the range of the yield criterion can be made if the results of more than one type of test are available. If, for example, results for both tension and torsion tests were known and these results were consistent with the maximum shear-stress criterion, an examination of Fig. 1 reveals that the criterion must hold exactly for all other stress states. On the other hand, if these tests were consistent with the maximum reduced stress criterion, that criterion would be the only possible one. In both cases the range of possible criteria is reduced to a band of zero width. In other cases, it is evident that additional results from only a very few combined stress tests would be sufficient to confine the yield surface for a stable material to an extremely narrow range. As might be expected, application of the yield criteria developed above leads to bounds on the carrying capacity which in some cases are much closer than can be obtained by a simple factoring process based on inscribed and escribed yield surfaces of the same shape. If tensile test values are available, an upper bound based on the Tresca yield criterion must employ the outer hexagon shown in Fig. 2 while an upper bound based on the maximum reduced' stress criterion can employ the inner hexagon. The latter will always be at least as favorable because, as a consequence of the lower bound theorem of limit analysis, increasing the yield strength of a body in any zone can never reduce the collapse load. An example follows. B, A PLANE-STRESS EXAMPLE Consider a circular plate of radius R and thickness t << R, simply supported around the edge and subjected to a lateral pressure p uniformly distributed over the top surface. Supposing the yield stress of the material in tension to be given, it is desired to find the closest upper and lower bounds to the carrying capacity. For the lower bound, a solution using the Tresca yield criterion will be appropriate. This has been given by Hopkins and Prager.10 The corresponding solution for the reduced stress criterion will now be developed. 6

3 Mt Mr Fig. 2. Alternative escribed yield Fig. 5. The reduced stress criterion criteria when tensile test results are for plate bending (plane stress). available.

In the analysis, the usual assumptions of plate theory will be adopted. If the effects of shear are neglected, and if the lateral pressure is small compared with the stresses in the plane of the plate, then the state of the plate will be approximately one of plane stress, the nonzero principal stresses lying in the plane of the plate. Adopting the maximum reduced stress criterion, the yield curve drawn in terms of the radial moment per unit length Mr and the circumferential moment per unit length Me will be as shown in Fig. 3. Since Mr is zero at the outer edge and equal to Me at the center (by symmetry), the state points are expected to lie in the range ABC in Fig. 3. The equation of AB is Me - JM = Mo (6) and that of BC is 1 1 Me + Mr = Mo (7) Eliminating the shear force from the radial equilibrium equation by means of the vertical equilibrium equation, we obtain d 1 2 dr (rM) - Me + pr 0 (8) where the signs have been assigned so that positive bending moments tend to induce sag of the plate in the same direction as the action of the pressure p. Substituting Eqs. (6) and (7) in (8) and integrating the resultant first-order ordinary differential equations, 1 Mr = 2Mo - 5 pr + clr (9a) Me = 2Mo - 1 pr + cjr (9b) on AB and Mr = Mo - pr2 + c2r2 (lOa) Me = Mo + pr2 - c2r2 (lOb) on BC. 8

Substituting the boundary conditions Mr = 0, Me = Mo at r = R and Mr = Me = MO at r = 0 and equating the resultant expressions for Mr at r = b = PR, the following transcendental equation for P is obtained 6p2~5 - 15P2 +4 = 0 (11) The root in the range 1 P >, 0 is = 0.6242 which on substitution in Eqs. (9) and (10) leads to p = 6.852 (12) R2 The above pressure is statically admissible and so is a lower bound. It can be shown to be the actual collapse load by associating a velocity field, The velocities are determined to within an arbitrary constant through the position of the state point in Fit, 3, According to the theory of plasticity for generalized stresses, the generalized strains corresponding to Mr, Mg will form a vector coinciding with the outwards drawn normal to the yield criterion, when corresponding axes are superimposed, The generalized strain rates corresponding to the moments are the curvature rates d2w 1 dw Kr =; Ke = - (13) dr2 rdr where w is the vertical velocity. The normal to the line described by Eq. (6) leads to the curvature components 1 K = r = 2 so Ke + 2Kr = 0 and hence 9

d2w 1 dw + r-,. = O dr2 2r dr On integration, this becomes W = c3r2 + c4 (14) and in a similar fashion the normal to Eq. (7) leads to w = csr2 + c6 (15) Setting the deflection zero at the supports and noting that there must be continuity of deflection and slope at r = b, we obtain w = 2.45(1 - p2)wo 1 p > (16a) w = (1 - 1.240 p2)wo > p >0 (16b) where wo is the deflection rate at the center. A velocity distribution has been associated with the statically admissible load, Eq. (12), and the solution is complete. We are now in a position to compare the estimates of collapse pressure obtained using the two criteria. In the case of Trescals yield criterion, the collapse pressure given by Hopkins and Prager1 is p. 6Mo (17) R2 and this will be the best lower bound. Comparing the expressions (12) and (17), pR2 6.852,P 6 (18) Mo and the pressure is bounded to within 14o2%. In contrast, if the Tresca yield criterion were used to obtain the 10

upper bound, the outer hexagon shown in Fig. 2 would be employed and pR2 8 > MO-. 6 (19) The pressure is bounded to within 33.3%. 11

III. MATERIAL SENSITIVE TO MEAN STRESS A. BOUNDING YIELD SURFACES Consider an ideally plastic, isotropic material in which the stress level at yield is a linear function of the mean stress. This constitutes a first step in the generalization of the analysis in Section II. There is evidence that it may be appropriate for certain soils, and also possibly for some concretes and cast metals. Intersections of the yield surfaces with the planes a1 + 02 + a3 = const. are shown in Fig. 4. Figure 4a shows the case where the results of tensile and compressive tests (represented by the circles) are available and Fig. 4b the case where shear tests are available, The limiting lines are obtained by arguments parallel to those used in Section IIL When expressing the bounding criteria mathematically, it is convenient to introduce the cohesion c and the angle of internal friction / in the fashion in which they are commonly employed in statements of Coulomb's yield criterion. Any line which cuts the octahedral axis can then be defined in terms of c and $ values appropriate to the generalized Coulomb yield criterion which contains the line. The algebra is straightforward and will not be given here: the resulting expressions for the various bounding criteria are summarized in Table I. When tensile and compressive tests are available, these are not necessarily consistent with the same generalized Coulomb yield surface (see, for instance, the test results for sand quoted in Ref. 5), so values c, $ corresponding to the compressive test results and c', I' corresponding to the tensile test results are introduced. The ratio a = $/$' is limited by the requirement of convexity to the range 2 - sin > a ( - sin ) (20) 2 the extreme values representing the attainment of triangular cross sections to the yield surfaces A and B. When a = 1, surface A reduces to the Coulomb yield surface (see Table I). The flow equations in the last column of Table I are obtained by substituting in Eq. (3) the analytic expressions for the yield surfaces given in the first column. 12

3 ^^ / \ p Fig. 4a. Material sensitive to mean Fig. 4b. Material sensitive to mean stress. Bounding yield criteria when stress. Bounding yield criteria when tensile test results are available. simple shear-test results are available.

TABLE I EXPRESSIONS FOR THE YIELD CRITERIA ILLUSTRATED IN FIG. 4 Sur- Yield Criteria Yield Criteria Strain-Rate Equations face __________ A (a+sin ) al+(l-c) a2-(l-sin) a3 = 2c cos (1-sin0) (Cl+c2)+(l+sin) e3 = 0 where 2-sin >/a ) (1-sino)/2 (ai = a2 > a3) (ai > a2 > a3; Oa = sin~/sin') (l-sin0') cl+(l+sin0') (2+C3) = 0 (ai > a2 = 3) el:E2: 3 = a+sin0:l-a:sin0-l (al > a2 > a3) = 2-sino (l-sin^)(el+c2)+(l+sin0)e3 = 0 r_ r~max (al-a,a2-oa,a3-a) = 2(c coso-a sin~)/(5-sino) (al =c 2 > 03) or al = (a2+a3)(1-sin0)/2+c cos - c1:2~:c = 2/(l-sino):-1:1 (al > 0G2,a3) (al > a2 > 03) = (l-sino)/2 (1-3 sin1) el+(l+sin^) (e2+3) = 0 in(al-ao,a(2-a,a3-a) = 4((a sing-c coso)/(3-sin) (al > C2 = 3) 1:C2:eC3 = l:l:-2(1-sin) (l+sin0) (al > a2 > a3) = 1 1: C2: 3 = (l+sin) (l-sini):0:-1 ai-a3+( o1+a3) sinO = 2c coso (1,> >a2 )a3)

TABLE I (Concluded) fSur- Yield Criteria Strain-Rate Equations face~" B (l+sin-)( 1a+a2) /2-(l-sin) as = 2c cosa (3-sino)(l-sino') el+ (a2 > C2 = [(2a-l+sinO) ol+(2-a-sino) 03]/(14a),) [4 cosocoso'-(l+sin) (l-sino') ]e2+ (l-sin=')(al+a2)/2-(l+sin')cr3 = -2c cos| (l+sino) (3+sino') e3 = 0 (a2 < aC) (a2 = a2) ( CI > a2 ) a3;a = sino/sin') 1:E2:e3 = l:l:-2(1-sinO)/(l+sinO) (cx2 ~ ac) El:2:e3 = l:l:-(l+ainO')/2(l-sinP') (Cr2 ~ a) C max.(Iorl-a|, 1a2-a|,|aI3-ca) = c cos-oa sin| (l+sin) el+e2+(l-sinZ)es = 0 H-' ~ or (2-sin$) al-(l+sinY)(a2+aCr3)+3e cos = 0 D ax (-Co,!~,iro2 > (aC1+a3)/2) D max. ( l-cr21, c2-a3 1, 3-a1) = 3(c cos-a csing) (l-sin)(E1+e) 2+(l+2sinZ) e3 = 0 or (l+sin$) al+a2sinn-(l-sin)-ao3 = 3c cos (al = a2 > a3) (o1 > a2 > a3) (l-2sin) el+(l+sin) (e2+e3) = 0 (al > c2 = a3) ~l:~2:E 3 = l+sin:sinO:sinO-l (O > c2 > a3)

At the point al = 02 = a3 = c cot i, which is the apex for all the yield surfaces, the rate of energy dissipation is D = aij eij = c cot ~ (el + c2 + c3) (21) Equation (21) also applies to all other points on the surfaces. The apex can be considered as belonging to every side and on any one side the projection of stresses in the direction normal to the side is always the same. In the case of surface C, the rate of energy dissipation can be written in the form D = Y cmax, c cos (22) where [7Ymax is the value of the largest shear strain rate. Equally simple expressions do not appear to exist for the other surfaces and Eq. (21) has to be used. B. PLANE STRAIN In the case of plane strain, all the bounding criteria introduced above can be expressed in terms of equivalent Coulomb criteria, with a suitable choice of the constants. A solution which has employed the Coulomb criterion can then be readily adapted to apply to any of the other criteria by a simple factoring process. In plane strain, one principal strain rate is to be set equal to zero. The ordering e1 > C2 > c3 is already established as a consequence of the ordering al > a2 ) a3 adopted in Table I, and it remains to determine which strain rate is to be zero. The dilatation is everywhere positive, so e1 + 0. If ~3 = 0, then Es and C2 are both greater than zero, but by the flow equations in Table I they are both of opposite sign in every case, which is a contradiction; hence C2 = 0, For surface A, Fig. 4, substitution of C2 = 0 in the flow equations, Table I, corresponding to the two types of corner, gives (1 - sin ) el + (1 + sin ) E3 = 0 (23a) when ao = a2 > as3 and 16

(1 - sin 0')e1 + (1 + sin't)C3 = 0 (23b) when al > a2 = a3. Examination of the plane stress cross section (not shown) reveals that Eq. (23a) defines a vector lying between normals to adjacent flats providing i > i' and Eq. (23b) defines a similar vector providing i (< i'. Substituting the stress conditions associated with (23a) and (23b) in the equation for the yield surface given in Table I reduces the latter to (1 + sin >) al - (1 - sin ) as = 2c cos i (24a) when ~ ) O' and (1 + sin O')ai - (1 - sin O')a3 = 2c'cos O' (24b) when B < ~'o Thus, for plane strain, surface A reduces to Coulomb's criterion, the constants c,, being used when t >' and c',' when For surface B, Fig. 4, there is only one type of corner. Substituting the corresponding stress a2 = CC in the yield criterion, Table I, we obtain after some reduction an equivalent Coulomb yield criterion in which the effective values i* and c* are defined by sin * = 2(sin i + sin Of) (25a) 3 + sin - sin' + sin i sin + s c* _ 2(1 + sin t') \ 1 - sin ^ 1 2 2(1 + s ~ c sin L(3 + sin')( - sin )( sin )()J (25b) For surface C, Fig, 4, the yield criterion in Table I reduces immediately to the Coulomb criterion when the value a2 = (al + a3)/2, which holds at a corner, is substituted. This is as expected, since this value of a2 represents the state of simple shear for which the experimental data were available in the first place. For surface D, Fig. 4, substitution of e2 = 0 in the flow equations, 17

Table I, corresponding to the two types of corner, gives (1 - sin ) ei + (1 + 2 sin ) c3 = 0 (26a) when al = c2 > as and (1 - 2 sin ~)e1 + (1 + sin ) E3 = 0 (26b) when al > a2 = as. Examination of the plane-stress cross section of the yield surface shows that the first equation is inadmissible because it is not the normal to a supporting plane of the yield surface. Substituting a2 = as in the yield criterion, Table I, an equivalent Coulomb yield criterion is obtained in which the effective values -** and c** are defined by sin ** 3 sin (27a) 2 - sin pi c** 3 1 - 2 sin 2 2 L 1 - sin 2J As an alternative approach, the same effective values of c and f for substitution in an equivalent Coulomb yietd criterion can be found by comparing the corresponding flow equations given in Table I. All problems of plane strain have now been shown to reduce to equivalent problems formulated in terms of the Coulomb yield criterion. In the case of materials insensitive to mean stress, the parallel result is well known. It must be recalled, however, that in the presence of dilatation the material is no longer in a state of simple shear and the argument used in the latter case is not applicable. C. PLANE-STRAIN EXAMPLE Consider the indentation of a half space by a long, rigid punch in the particular case when the tensile and compressive strengths have been measured and have been found to be consistent with the same constants in 12 Coulomb's law, so that a = 1. As stated by Prandtl, the indentation pressure is 18

p = c cot i(eO tan2m tan2(. + 4) - 1) (28) Prandtl's solution is complete in the sense than an incipient velocity field can be associated with the stresses in the deformable zone and the material can be shown to be necessarily rigid in the remainder of the half space (see Shieldl31l)l1 In plane strain, surface A, Fig. 4, reduces to the Coulomb criterion with the original constants c and 4, while surface B requires the new constants defined by Eq. (25) after setting At = 4. Surface C reduces to the Coulomb criterion with the original constants and surface D requires the new constants defined in Eqso (27). Equation (28) can be applied equally well using the modified values * and c*, or ~** and c** because the resultant stresses and velocities are consistent with the various yield surfaces of Fig. 4, in the planestrain case, The resulting upper and lower bounds are shown in Fig. 5 for various 4, the range between the upper and lower curves representing the maximum variation in pressure which can occur with any convex, conical yield surface. 19

Upper bounds: from rriaxial compression rests from shear tests 80 P 6O0 C 40 lower bound [Coulomb] 20 / 0 10 0 10 20 30 40 50 Fig. 5. Punch indentation (plane strain) for a material sensitive to mean stress. 20

IV. DISCUSSION Certain yield criteria have been established as intrinsically suitable for engineering analysis because they lead to upper and lower bounds, as the case may be, without any reliance on a detailed knowledge of the actual yield criterion. All the criteria are piecewise linear, with the attendant simplifications of the analytical work which are well known. It is perhaps worth emphasizing that results from a few tests may serve to define the yield criterion exactly or nearly so if the results fall close to one of the bounding criteria, If, for example, a metal develops yield strengths in tension and torsion consistent with Trescats criterion, no further testing is necessary to identify this as the correct criterion providing there is sufficient other evidence to establish the metal as an ideally plastic material, A similar remark applies for the Coulomb criterion in the case of a material for which yield is a linear function of the main stress. 21

REFERENCES 1. A. Ao Gvozdev, Proc. Confo Plastic Deformations, 1936 (Academy of Sciences, U.SS.R., 1938), 19; English trans: Into J. Mech. Scio, 1 (1960), 322. 2. D. C. Drucker, W. Prager, and Ho J. Greenberg, Q. App. Math,, 9 (1952), 381. 3. R. Hill, Phil. Mag., 42 (1951), 868. 4. D. C. Drucker, R, E. Gibson, and D, J. Henkel, Proc. Am. Soc. Civil Eng., 81 (1955), 791. 5. R. M. Haythornthwaite, Proc. Am. Soc. Civil Eng,, 86SM5 (1960), 35. 6. D. C. Drucker, Proc. 1st. U. S. Nat, Cong. App. Mech. (Am. Soc. Mech. Eng., 1951), 487. 7. R. Hill, Phil. Mag., 41 (1950), 7335 8.. D. D. Ivlev, Prikl. Mat. Mekh. (Academy of Sciences, UoS.SoR.), 22 (1958), 850; English trans. Q, Appo Math, Mecho 9. W. T. Koiter, Q. App. Math., 11 (19553), 350. 10. H. Go Hopkins and W. Prager, J. Mech. Phys. Solids, 2 (1953), 1. 11. W. Prager, Proc. 8th. Int, Cong. Appl Mech., 1952 (Istanbul, 1955), 2, 65. 120 L. Prandtl, Nachr. Koeniglichen Gesello Wiss. Goettinger, Matho Phys. K,. (1920), 74. 13. R. T. Shield, Q. A. Math., 11 (1953), 61, 14. R. T. Shield, J. App. Mecho, 21 (1954) 193. 22

r —----------- -T —- ----— _ —---- AD Accession No. UNCLASSIFIED AD Accession No. UNCLASSIFIED The University of Michigan, Office of Research Administra- The University of Michigan, Office of Research Administration, Ann Arbor, Michigan. tion, Ann Arbor, Michigan. I The Range of the Yield Condition in Stable, Ideally Plas- The Range of the Yield Condition in Stable, Ideally Plastic Solids, R. M. Haythornthwaite tic Solids, R. M. Haythornthwaite Report No. 04403-1-T, April 1961, 22 pp., 1 table, Project Report No. 04403-1-T, April 1961, 22 pp., 1 table, Project 04403 (Contract No. DA-20-O18-ORD-23276) UNCLASSIFIED 04403 (Contract No. DA-20-018-ORD-23276) UNCLASSIFIED The property of stability in an ideally plastic solid is The property of stability in an ideally plastic solid is employed to define bounding yield conditions which remain employed to define bounding yield conditions which remain valid when test results are available for only one stress valid when test results are available for only one stress state, such as simple tension or simple shear. For ma- state, such as simple tension or simple shear. For material in which yield is insensitive to changes in mean terial in which yield is insensitive to changes in mean stress, the bounding yield conditions are shown to be the stress, the bounding yield conditions are shown to be the maximum shear-stress criterion of Tresca and a new cri- maximum shear-stress criterion of Tresca and a new criterion designated the maximum reduced stress criterion. terion designated the maximum reduced stress criterion. Use of both criteria enables the best possible bounds to Use of both criteria enables the best possible bounds to be obtained for the carrying capacity of a structure in be obtained for the carrying capacity of a structure in (over) UNCLASSIFIED (over) UNCLASSIFIED UNCLASSIFIED UNCLASSIFIED | I the absence of further information. Parallel results the absence of further information. Parallel results are obtained for material in which yield strength is a are obtained for material in which yield strength is a linear function of the mean stress, and some examples linear function of the mean stress, and some examples are givare given. IUNCLASSIFIED II D IUNCLASSIFIED UNCLASSIFIED

r - _ _ __AD Accession No. UNCLASSIFIED AD Accession No. UNCLASSIFIED The University of Michigan, Office of Research Administra- The University of Michigan, Office of Research Administration, Ann Arbor, Michigan. tion, Ann Arbor, Michigan. I The Range of the Yield Condition in Stable, Ideally Plas- The Range of the Yield Condition in Stable, Ideally Plastic Solids, R. M. Haythornthwaite - tic Solids, R. M. Haythornthwaite Report No. 04403-1-T, April 1961, 22 pp., 1 table, Project Report No. 04403-1-T, April 1961, 22 pp., 1 table, Project 04403 (Contract No. DA-20-018-ORD-23276) UNCLASSIFIED 04403 (Contract No. DA-20-018-ORD-23276) UNCLASSIFIED The property of stability in an ideally plastic solid is The property of stability in an ideally plastic solid is employed to define bounding yield conditions which remain employed to define bounding yield conditions which remain valid when test results are available for only one stress valid when test results are available for only one stress state, such as simple tension or simple shear. For ma- state, such as simple tension or simple shear. For material in which yield is insensitive to changes in mean terial in which yield is insensitive to changes in mean stress, the bounding yield conditions are shown to be the stress, the bounding yield conditions are shown to be the maximum shear-stress criterion of Tresca and anew cri- maximum shear-stress criterion of Tresca and a new criterion designated the maximum reduced stress criterionterion designated the maximum reduced stress criterion. Use of both criteria enables the best possible bounds to Use of both criteria enables the best possible bounds to be obtained for the carrying capacity of a structure in be obtained for the carrying capacity of a structure in (over) UNCLASSIFIED (over) UNCLASSIFIED I UNCLASSIFIED |UNCLASSIFIED the absence of further information. Parallel results the absence of further information. Parallel results are obtained for material in which yield strength is a are obtained for material in which yield strength is a linear function of the mean stress, and some examples linear function of the mean stress, and some examples are givare given. L _ I_ _ _, aran I UCNI I II L UNCLASSIFIED IUNCLASSIFIED I

r7 7 —7 —7' -- - 7' AD Accession No. UNCLASSIFIED AD Accession No. UNCLASSIFIED The University of Michigan, Office of Research Administra- The University of Michigan, Office of Research Administration, Ann Arbor, Michigan. I tion, Ann Arbor, Michigan. I The Range of the Yield Condition in Stable, Ideally Pla- The Range of the Yield Condition in Stable, Ideally Plas- tic Solids, R. M. Haythornthwaite tic Solids, R. M. Haythornthwaite Report No. 04403-1-T, April 1961, 22 pp., 1 table, Project Report No. 04403-1-T, April 1961, 22 pp., 1 table, Project 04403 (Contract No. DA-20-018-ORD-23276) UNCLASSIFIED 04403 (Contract No. DA-20-018-ORD-23276) UNCLASSIFIED The property of stability in an ideally plastic solid is The property of stability in an ideally plastic solid is employed to define bounding yield conditions which remain employed to define bounding yield conditions which remain valid when test results are available for only one stress valid when test results are available for only one stress state, such as simple tension or simple shear For ma- state, such as simple tension or simple shear. For ma- terial in which yield is insensitive to changes in mean terial in which yield is insensitive to changes in mean stress, the bounding yield conditions are shown to be the stress, the bounding yield conditions are shown to be the maximum shear-stress criterion of Tresca and a new cri- maximum shear-stress criterion of Tresca and a new criterion designated the maximum reduced stress criterion. terion designated the maximum reduced stress criterion. Use of both criteria enables the best possible bounds to Use of both criteria enables the best possible bounds to be obtained for the carrying capacity of a structure in be obtained for the carrying capacity of a structure in (over) UNCLASSIFIED (over) UNCLASSIFIED I csUNCLASSIFIED UNCLASSIFIED the absence of further information. Parallel results the absence of further information. Parallel results are obtained for material in which yield strength is a are obtained for material in which yield strength is a linear function of the mean stress, and some examples linear function of the mean stress, and some examples are given. are given. _I__ "IUNCLASSIFIED UNCLASSIFIED I I I [UNCLASSIFIED I UNCLASSIFIED, 4 _,.,.., *,, _,.,, _,,..,. _~~~~~~~~~~~~~~c~~-~,.~

| AD Accession No. UNCLASSIFIED AD Accession No.UNCLASSIFIED The University of Michigan, Office of Research Administra- The University of Michigan, Office of Research Administration, Ann Arbor, Michigan. tion, Ann Arbor, Michigan. I The Range of the Yield Condition in Stable, Ideally Plas- The Range of the Yield Condition in Stable, Ideally Plas-I tic Solids, R. N. Haythornthwaite tic Solids, R. M. HaythornthwaiteI Report No. 04403-l-T, April 1961, 22 pp., 1 table, Project Report No. 04403-1-T, April 1961, 22 pp., 1 table, Project 04403 (Contract No. DA-20-018-ORD-23276) UNCLASSIFIED 04405 (Contract No. DA-20-018-ORD-23276) UNCLASSIFIED The property of stability in an ideally plastic solid is The property of stability in an ideally plastic solid is employed to define bounding yield conditions which remain employed to define bounding yield conditions which remain valid when test results are available for only one stress valid when test results are available for only one stress state, such as simple tension or simple shear. For ma- state, such as simple tension or simple shear. For material in which yield is insensitive to changes in mean terial in which yield is insensitive to changes in mean stress, the bounding yield conditions are shown to be the stress, the bounding yield conditions are shown to be the maximum shear-stress criterion of Tresca and a new cri- maximum shear-stress criterion of Tresca and a new criterion designated the maximum reduced stress criterion, terion designated the maximum reduced stress criterion. Use of both criteria enables the best possible bounds to Use of both criteria enables the best possible bounds to be obtained for the carrying capacity of a structure in be obtained for the carrying capacity of a structure in (over) UNCLASSIFIED (over)UNCLASSIFIED I UNCLASSIFIEDUNCLASSIFIED | I the absence of further information. Parallel results the absence of further information. Parallel results are obtained for material in which yield strength is a are obtained for material in which yield strength is a linear function of the mean stress, and some examples linear function of the mean stress, and some examples are given, are given. 1|I IUNCLASSIFIED