Tresca Versus von Mises

5.53) Substituting eq(5.53) into eqs(5.52a,b,e) provides the triaxial stresses in the elastic zone;

5.6.4 Tresca Versus von Mises

A numerical, von Mises solution to the stress distribution within a non-hardening, elastic- plastic, solid disc is also possible. The elastic zone stresses are again given by eq(5.79a,b) for the same boundary condition: a

r

= 0 for r = r

0

. Moreover, eqs(5.79a,b) must satisfy the von Mises yield condition at the interface radius, i.e. where a

9

and er

r

must equal the corresponding plastic zone stresses ar r

v

. Within the inner plastic core, these two stresses must satisfy the von Mises yield criterion and the equilibrium equation simultaneously:

4 - ff

B

a

r

+ o? = F

2

(5.84a)

^ 2% ( 5 §4 b )

Using Nadai's eqs(5.67a,b) to separate c^and tr

r

in eq(5.84a) and then substituting these into eq(5.84b) leads to a first order differential equation in r and d [6]:

d r

We may then solve eq(5.85) using a Runga-Kutta method, taking a single starting value for Q, depending upon whether the disc is solid or hollow (i.e. 0e = nfl or nl6 respectively).

The von Mises solid disc solution shows that both ar and ff9must remain very nearly equal to y in this zone, in contrast to the Tresca solution (see Figs 5.15a,b).

-0.25

-0.25 L

(a) (b)

Figure 5.15 (a) von Mises and (b) Tresca stresses far solid disc at similar speeds

The hollow disc solution reveals a similar ar distributions in Figs 5.16a and b but the von Mises oe is not constant, increasing above F in the plastic zone interior rt<r< rv.

aJY

-0.5 -0.5 -

-1.0L -1.0'-

(a) (b)

Figure S.16 (a) van Mises, (b) Tresea stress distributiais for a tallow rotating disc

ELASTIC-PERFECT PLASTICITY 159

The greater differences arising in cte and <?„ from applying different yield criteria to a solid disc, are reflected in their residual stress estimates, am and om, as shown. For example, on applying ax = o ~ oE to each zone in Fig. 5.15b, the Tresca residuals follow from eqs(5.74a,b), (5.78a) and (5.79a,b) as:

0 < r < rm

(5.86b)

^ .

^fl

- A -

⁽³

r2

where a, b and 0)^ are gi¥en by eqs(5.8Oa,b) and (5.82). Equations (5.86a-d) were applied to a solid, steel disc with F = 310 MPa, v = 0.28, p = 7750 kg/m* and ra = 250 mm, following a speed JV = 13000 rev/min. Figures 5.15a,b show these Tresca's residuals are greater and with different sign in the outer zone, compared to von Mises residuals [6]. This contrasts with the conservative nature of Tresca, which predicts a greater spread of plasticity at similar rotational speeds.

References

1. Nadai A. Theory of Flow and Fracture of Solids, 1950, McGraw-Hill, London.

2. Timoshenko S. and Goodier J. N. Theory of Elasticity, 1951, McGraw-Hill, New York.

3. Grassland B. and Bones J. A. Proc. I Meek E, 1958,172,777.

4. Grassland B. Proc. I. Meek E, 1954,168,935.

5. Franklin G. J. and Morrison J. L. Proc. I. Meek E, 1960,174,947.

6. Rees D. W. A. Zeit Angew Math Mech, 1999,79,281.

Exercises

Elastic-Perfectly Plastic Beams

5.1 A simply supported beam of length I, with a rectangular section breadth b and depth d, carries a uniformly distributed load w/unit length. Determine the ultimate moment, the collapse load and the plastic hinge length, given that the yield stress Y is constant,.

[Answer: MHl, = bcfYM, w = 2W2 J7/2, /„ = /A/3]

5.2 Determine the collapse load and the length of the plastic hinges when an encastre beam of length /, with rectangular section b x d, carries a single concentrated load W, that divides this length into p and f, i.e. p + q = l.

[Answer: MMll = (M2/lV(2pf), p/6, q/b, 1/6]

53 Show that fee collapse moment for a J- section, made from equal rectangles each of length o and thickness t, is given by Ma(l = Yah(a + h)/2.

5.4 Examine the manner in which a cantilever of length I will collapse when carrying a uniformly distributed load w/unit length, with a prop to prevent deflection at its free end. Show that the initial yield loading is wr = 8M/22 and the collapse loading is wF = 11.73M,/{ ? Hence fund the ratio between these loads for a rectangular section fa x d. Hint: The prop reaction is given by 3*rf/8.

Elastic-Peifectfy Plastic Torsion Bars

5 J A bar of diameter d and length £ is bored to diameter d/2 over half its length. If te outer diameter of the solid shaft reaches its yield point under an applied torque, show that the diameter of the elastic- plastic interface dMf, within the hollow shaft section, may be found from the solution to the quartic equation: <*„* ~d*dv + 3dil6 = 0. Show mat the ratio between the angular twists for the hollow and solid shafts is given by d/dv.

5.6 Derive and plot the normalised torque-twist relationship for a hollow bar. Show the accumulation of normalised residual strain on this plot in a similar manner to that given in Fig. 5.9 for a solid bar.

5.7 What value of torque is required for an elastic-plastic interface to lie at the mean radius in a tube of inner and outer diameters 25 and 100 mm respectively? Determine the residual stress distribution and the residual twist when this torque is subsequently removed. Take k = 230 MPa and G = 78 GPa.

Elastic-Perfeetly Plastic Cylinders and Discs

5.8 What is the maximum, limiting diameter ratio of an annular disc beyond which it is not possible to achieve a full spread of plasticity when radial pressure is applied to the inner diameter? At what pressure does this occur? [Answer: 2.963, 2 FA/3]

5.9 An annular disc, with inner and outer radii 25 mm and 62.5 mm respectively, is machined from an alloy steel with a yield stress 500 MPa. Determine the internal pressures necessary to: (a) initiate yielding, (b) produce a fully plastic disc and (c) produce partial plasticity to the mean radius.

[Answer: 2.415 kbar, 5.045 kbar, 4.584 kbar]

5.10 Compare the residual stress distributions in open-end and closed-end thick walled cylinders resulting from an autofrettage pressure sufficient to penetrate an elastic-plastic interface to coincide with the mean wall radius. Employ a von Mises yield criterion and normalise the stresses with the yield stress Y for a cylinder of diameter ratio 3.

5.11 Show that the speeds necessary to irritate yielding within solid and hollow discs are each independent of the yield criterion and are respectively:

rfd v) + r

2

Plrfd - v) + ro2(3 + v)]

8F

(3 + v)pr,

²

5.12 Using the Nadai's parameter approach, determine the von Mises stress distributions in an elastic- plastic, hollow disc, r, = 50 mm and ro = 250 mm, rotating at a speed of 11000 rev/min. Derive from these the residual stresses distribution for when the disc is brought to rest. The following conditions apply to the disc material: Y= 310 MPa, v= 0.28, and p= 7750 kg/m'. Compare and contrast each distribution with the corresponding Tresca solution. [Answer is given in Figs 5.16a,b]

161

CHAPTER 6