Plastic yielding of single notched bars due to bending

YLASTIC

YIELllING

By

OF

SINGLE

TO

BENDING

G. LIASIS

NOTCHED

and II.

C‘ity and Guilds College, Vnivrrsity

BARS

DUE

FOHI) of I,o~~tlou

.\ GIINI:IIAI.m&hod is drveloped to evaluate the constraint factor of single notched bars sul),jwted to pure bending under conditions of plane strain. Upper and lower bounds for the yield moment. are obtained. Bounds for the constraint factor are calculated for (a) rectangularly, (b) traprzoidaly, and (c) \:-notched bars with circular fillets. The theory is checked experimentally for both thr overall yield loads and the actual stress distribution, for the above shapes and for circular and T-notches for which GREEN has worked A new technique to obtain slip-lint fields experimentally is also out an upper bound solution. developed.

1.

ISTR~DLTCTIOS

WIIES a bar, notched in the middle, is subjected to pure bending gradually ing from zero, it is initially stressed elastically throughout. With increasing

increasloading

the bar first becomes locally plastic around the notch due to the stress concentration. The corresponding instance With

load is known as the Elastic limit and it could be calculat,cd

by relaxation increasing

for

methods.

loading

the

elastic-plastic

interface

spreads

into

the

material. For a single notched bar, at some stage of the loading another region starts from the flat side and with further loading the two plastic

elastic plastic regions

eventually join together. ,4t this stage. large strains are free to develop with little or no further change of load and plastic flow is no longer contained. This load is defined

as the limit or collapse

The bar is subjected

load.

to pure bending and is supposed to be of suflicient

length fol

the collapse moment to be independent of the precise distribution of the surface tractions at the ends. The bar is wide compared with the thickness under the notch so that plane strain conditions can be assumed. According to the theory of Limit Analysis an incomplete plastic the slip-line and velocity fields provides an upper bound for the while a statically admissible stress field provides a corresponding In the following, upper and lower bound solutions are obtained for single notch of any shape, symmetrical about an axis normal to the bar, and applications to particular shapes of notches are illustrated. on

solution based yield moment. lower bound. a bar having ;I top face of thtl

2

(G. ILOIIS

Slip-hw

field (upprr

H. Foxtn

SOILJTION

hh-b:RAI.

2. il.

and

bound)

GREEN (1953) has shown that his method of obtaining solutions is applicable to any shape of notch, but he does not give the general solution. Depending on the geometry of the notcah there are two possible solutions. Solution, I (Fig. 1). WC have the so-called second boundary-value since the hydrostatic pressure is known along the surface of the notch 2f = angle

of

of directim

(-

problem k) and

change at

W”

Pig. 1. Slip-line Field Solution 1. the surface of the flat side (+ 12). Using the step-by-step method (HIM, 1950, pp. 144-145) the slip-line field can be extended as far as the other conditions of the problem allow. It consists of two parts : (a) the lower part NHH’ where the slip-lines are straight and there is a uniform compression - SC parallel to the surface ; (b) the upper part Z I. T D F N F’ II’ T’ L’ 27, the shape of which depends on the geometry of the notch. If the tangents are discontinuous at Y and P’ these points are singular points. The field is extended around P and P’, starting from PA and P’A’, through an angular span f equal to half the jump of tangent direction on both sides of the singularities. The position of 2 and therefore of N is defined hj the equilibrium condition. i.e. the stresses across the minimum cross-section must be equivalent to a pure couple. If a9 is the tensile stress normal

to Ohr the equilibrium

from which zN can IX, fout~d. Then the yield moment

condition

is

per unit thickness

of bar is

The constraird fa,ctor L is defined in the following as the ratio of the yield moment of the notched bar to the yield moment of a bar without notch having a crosssection equal to the minimum cross-section under the notch root. Therefore :

N is a point of stress discontinuity. The jump Ap of the hydrostatic pressure in iV must not exceed the value 2k times the value of the angle of rotation of a distontinuous slip-line around N (KILL, ~6%~). In our case the angle of rotation is $ rr and t,herefore, for the present solution to be valid,

Assuming the notch surface has an increasing angle of direction, discontinuities, (Fig. I) HEKCKY’S equations lead to jw, = - k (I + 26%

with possible

a,/k = 2 (I + 0,) < TF,

or 8, < 32” 40’

(5)

by equation (4). The limiting value of a is

where a:Nis such that 8, = 32” 40’. For values of 8, > 32” 40’ the yield criterion is violated in some part of the rigid zone around N. Solution I.2 (Fig. 2). The upper slip-line domain is completed in N and extended on either side as before. The extended domains bounded by the slip-lines NC and NC’ join with the straight line field of the flat side by two circular arcs CF and C’F’. Thus, the central hinge being considered as a fixed rigid pivot, the rigid ends of the bar rotate by sliding over CF and C’F’ as rigid bodies. N is the point where the slip-lines from 2 and Z’, having 6 = 32” W, meet the central axis. The unknowns of the problem are (a) the position of T 0x1 the not,ch surface, (b) the radius R of the arc of the central hinge. These can be found from the two equilibrium conditions, i.e. the resultant of the ;1:and y stresses across any cross-section, such as ONCFG', must be zero, Analytical expression of these two conditions in terms of the two unknowns, 0, and R, is not possible for an arbitrary shape of notch and a trial and error procedure must be adopted. In the special cases investigated below such analytical expressions are achieved and the evaluation of the unknown is obtained in terms of the geometrical dat,a of the bar. Once T and R have been found the sizes of the field are fixed and the constraint factor can be found. Velocity field. The hodograph for Fig. 2, for points of the regions (I) ant1 (II) across CF and C’F’ respectively, is a circular arc around the pole. Assuming a unit angular velocity, the radius of this arc is equal to the radius R of the sliplines CF and C’F’, and its angular span equal to X (Fig. 3). Across NC u =: O, hence zc = cons& and NC in the hodograpl~ is a circular arc Ci, Ni, of angular span (h - f n). It can easily be proved that for a unit angular velocity the images

I’htic 22X?’

is reached.

straight

yielding of single notched lxm

The domains

lines having

lengths

I*71 and I’ll1 equal

5

due to hendittg

have their images ~*oincid~ng in two

to the relevant

lengths

of the physical

plane

since ~$4 = 0 across them. Another velocity discontinuity on both sides of SG (v) and SG’ (u), equal to R, is demanded from the continuity of the normal velocities in S. The domain SG’G has an exactly similar criss-cross image. the positions of the points G. G’ being interchanged

in the hodograph.

We shall give now a general upwards symmetrical

proof

of the following

r&o&hhaving monotonictrtty

possible ~~sco~t~~~~~t~e~~ the stress ad velocif+y q&em itz the plastiic regioiz ,}zegaf~zlerate ~~~~ast~~ wo&. I ~r~(~: The rate of plastic work is given (C~EES

or in terms of the radii of curvature S’, (hodograph

plane)

(i) ?L.‘py,rrre,dion (Fig. 2,). principal direction trajectory,

Since images other

For a COJKW~ (with

of Figs. 2 nMl 8 Cm&wtt.O7&fP 1953) by

RI S, of the slip-lines

(ph+cal

plane) and E,

b> r’(>=kj

Therefore

statement:

i~racreasi~tgtnrtgmt direction

_q;

_

(8)

;).

It ran be shown that, because the notch surface is n the angle #Xincreases in the positive r. and /3dir&ions.

:

on both

lines 41 increases

are therefore lines of the

concave

they

are both

concave

upwards.

to the right and left respectively,

two domains.

Therefore,

t,he curvatures

Their

velocit>-

and so are all the.

of the hodograph

art‘

(,)I From

(a) and (b) it follows that S’/S > 0

and

-

R’/R

> 0.

‘rI~~refore, everywhere in the upper region, y = 0. (ii) Lnx~r rq$oi~ Since l/B, I/R’, l/6’. l/S’, me all zero (7) becomes + -

arc

as,

i

av 2s;

From thr hodograph it follows that anywhere in the region, 3~/&9~ > 0. Jvi’>s, > 0. Consequently y > 0 in the lower region. (iii) Uiscontinuities NZ, NZ’. Since, as we cross NZ moving in the positive dc - direction, the jump of u is positive Av == + R, &J/‘&Y,= + co, i, -;7 4.. a. At the discontinuities FG’ and F’G, as we cross FG’ in the positive /&direction, 11changes from _- R to 0, ilzc = + R, &1/i.~~ = -j- co, Jo -j- rx). The above analysis also holds for Solution I where N, C, F, S, F’, C’, coincide in the hodograph. The field remains the same in the regions NZ’Z and NH’N and the discontinuity R disappears.

6

(;.

hANIS Wd tf.

PORD

R. hYress field (lower bound) It can be shown that the discontinuous stress system shown in Fig. 4 is a,statically admissible field if /3 < 80”. With this restriction, /S is defined in each case so a,s to yield a maximum value to the constraint factor. Once the value of /I has been chosen, p (Fig. 4) can be evaluated in terms of 13and the dimensions of the bar. since .4C is tangential to Lhc notch surface.

..-

-

_._..._L_-

-----i-----

-

L._-_--

.._. - _._.-

II_------

‘I%(: rwnstraint factor is

p being defined so as to maximize (IO), i.e.

The slip-hue field for rectangular notches constructed aeeording to the method given in Section 2B has already been described (LIANIS and FORD, 1957). Them are two solutions depending upon whether the ratio Zb/cz is greater or less than 0.575. Lower bound. From Fig. 4 p _ (I -- b/a tati j3) (12) -------p-> u 12 + sin 8) the stress trapezoid being tangential to the corner of the notch. The lower bound for the constraint factor is therefore L __ 2 (1 --b/u

tan fi) (2 + 3 sin #I _i- sin2 8) (2 + sin p)2

(13)

PIastic

yielding of single notched

bars due to bending

7

Fig. 5 SIIOWthat /3 is determined in terms of itb/a so ESto maximize L in (13). the mean value does I& differ from the bow& by more than 5 245 per cent ; the maximum total stress under the notch in the plastic region, acc.ccordiK~g to the upper bound solution, is also plotted. where

Fig. (5. Slip-line field, Solution

I for tmpezoidal

notch.

L

I’i;riiti~yieiding of &y$e not.otchedbars due to hcndinp 8.F < y < 57.8”.

II 0x1

From the two e~u~libriu~~lequations along U~~~~~~~~~~ we End R?j’a and C_‘(I irt terms of 2b!a for each value of y. Since c/u must bc positive, Solution II far rectangular notches holds (i.e. the field does not extend to the oblique faces) for S/a smaller than that for which c,‘a = 0. By taking the moments of the stresses acting along 0AZ)ER"'FGabout K, the ronstraint factor is calculated and plotted in Fig. 7.

IQ& 8. Slip-line Geld. Solution 11, trapezoidal notch.

Lamer bound. The stress trapezoid is tangential to t.hr corner of the ncttch and ,Q is independent of y pFoGded that p < 30” and f $ T - y_ Therefore, for values of p smaller than 4 a - y, L is cafeulated by means of f13f. ft remains constant and independent of y for y < 4 77 - &,,, w4xTe @,,,,, is calculated by rna~~rn~~in~

1,

2 (1 -r

.__~___’

bi’2 cot y) (2 -I- 3 -~ --..I._

C’OS y _______

-I-

cc+ y) .

!______

(2 + (‘(11;y)2

t

0

Plastic yielding of single notched bars due tn benning

11

in PP, and it is co~~~eted as shown in Fig. 11. It can he shown that sueh a family futfils all the properties of slip-lines. This solution is valid if t.he yield criterion is not violated in N. The jump in the hydrostatic pressure is Ap = IP -

PINl = k (2 + n - ay) < kn,

for

+ ?7> y > 1 = 57.3”.

This solution is restricted more by the condition tl > 0. It can be shown (see Fig. 11) that d -_= 1 -1’(&&lP~--1) (22) (2 + 4). a (I

Fig.

11,

Slip-line field, Solution I for V-notch with circular fillet.

Fig. 13. Constr;tint Fwtors, k-notch with circular fillet.

The equation for the constraint factor is too complicated to express an~iytiea~ly, but it is plotted in Fig. 12. For values of ~,/(a + T) smaller than those for which d/a = 0, Solution I for circular notches holds.

t

Q

Phtic

yielding of sin&e notched hrs

due to bending

1.3

(ii) Ilu7c~r~~~~~~. The stress trapezoid remains tal~gential to the circular root of the notch when y < 4 w - 8. For such angles the lower bound is calculated from (21) and remains constant for y < $ rr - ,&itX where ,f3,,,,,is given by maximizing (21). For Y > t m - PI,,,, the sides of the trapezoid coincide with t,he oblique face of the notch and fi = b 7p- y. Whencr L=2

1

i

_:(I a

--ind2.-.cos y

I

(2 i_ 3 cos y

in Fig. 14 both bounds and the cxperimcntd plot,tcd.

$

cos2 y)/;( 2 _t cos y)2.

(23)

results for constant, y = 30” arc

Two groups of tests have been carried out to confirm tlte theory experimentally. The first group was to measure the constraint factor, the second to check the slipline field. Test pieces of high conductivity copper and commercially pure aluminium cold rolled to 30 per cent reduction, were used. Fig. 15 shows the type of stressstrain curve obtained. The metal showed a rapid change from elastic to plastic condition with a low workhardening rate. The yield stress in simple tension was taken as (a) copper, Y = 22.8 tons,‘in2 (b) aluminium, Y = 8.9 tons/in2. The Mises criterion, which is an approximation for copper and aluminium, was assumed. Both values of Y agree with the yield stress found by plane strain tests (WATTS and FORD 1%X?-53) for 30 per cent redwhon.

Stmin,

%

Fig. 15. Yield stress curve, aluminium.

The dimensions of the bars used were 3 in. x 1 in. for copper and 3 in. x 1) in. for aluminium. The notches were sufhciently deep so that plane strain conditions were ensured and the plastic zone did not spread to the un-notched part of the upper surface of the bar. The specimens were subjected to four-point loading to give pure bending. The rig consisted of two pairs of steel rollers between which the test piece was placed, the defiexions being measured by dial gauges ( LIANISand FORD 19%). The moment

Materiul

Jlr/lllbOl

IR

Copper

1

IR

2

IR

:s

IR

4

-__~ 2b

-,a

O.Of34 .___

0.793

0.067 ___

0~53G

j

0.130

0~087

__

0.7!‘3

_-

2bj’rt --__-

-I-

0.127.5 0.1G.k _-._-_

___

IX

3

IR

(i

IR

7

IR

10

__ .-

I__~___

_~__-.-----.~ ( w:ll23

/

0.685

_~. _ Error I’tteor. E
(?o)

I .l.k5

1.14) ~. ~~-

1,184

1.12 1~10’3

8mJ

/ x.35

I.114

946

( 0~20

1.100

I-la’2

0-C

, -1.0

I____.~ - 0.5 __-~-

0.7

.__-. 8.13 (

8.10

1~080

_____II’

4~00

6

IO,13 AIUWI.

i 4.X I

110.1

1.070

1+75 1 .O66

1,060 I.037 -___- --. 1 ak

-

0.53

-

0.80

.- ___-

0.7

.___.--

__-_IT

04.5

.-

-I-

1 +k7

-

-l--~

9

Copper

1.036

0

Plastic

yielding

1~~s due to bending

15

derived from the measurements.

As soon as the.

of single notched

and deflexion

could be accurately

limit moment

was reached large deflexions

occurred.

The load remained

but the spring of the testing machine caused instability.

To overcome

constant,

this difficulty

the load was increased a little above the desired value to cause rapid rotation of the indicator of the dial gauges and then left to decrease slowly until the rotation stopped.

For deflexions

well in the plastic

from the dial gauges under constant deflexion

curve and extrapolating

M, was found.

The constraint

region,

however,

load were taken.

continuous

By plotting

readings

the moment

back to meet the elastic line the limit moment

factor

is given by

(b) Preparation of test pieces Test pieces of length 1 ft were cut from 3 in. wide bars of copper and aluminium. Care was taken to ensure the accuracy of the dimension of the notches. A series of V-cutters having semiangles 30”, 45”, 50”, 55”, 60°, 65”, 70” and 75” was used for the V, trapezoidal and V-notches with circular fillets together with radius cutters R = _k. 4, 2, 8, 1 in. For each case of single notched bars examined in the theory 9 - 10 specimens were prepared from both copper and aluminium. The exact notch dimensions

were measured

by a travelling

microscope.

The bending direction was reversed in the single notched bars so that the notch was first opened and then closed to check the Bauschinger effect. No Bauschinger effect was found. For tests on trapezoidal

and fillet notches

the angle of the notch

was held

constant and the ratios 2b/a and u,/(a + r) were varied since specimens with constant ratios and varying angles would be difficult to prepare to close tolerances. The experimental

results

for

the constraint

and Figs. 5, 9, 10 and 14, and generally results.

factor

are given

in Tables

1- 4

lie within 14 per cent of the slip-line field

These results confirm that the upper bound solution is the correct solution

that the points lie slightly below the upper bound could easily be explained

;

by the

yield stress being a little overestimated. As far as the deformation

pattern is concerned

another qualitative agreement with It has been shown in velocity

the theory Has been found in single notched specimens. field analyses

that on the middle part GG’ (Fig. 2) of the flat side of the bar the

vertical

component

mation

therefore

of the velocity

is constant

(Fig. 3).

For a small initial defor-

the plane GG’ must remain plane with two kinks at the points

G’, G, H’ and H. Such a deformation picture has been detected in almost all the specimens, being particularly clear in Solution I where H’H remained plane after a pronounced deformation with two distinct, kinks on either side. The flat part of the rectangular and trapezoidal notches also remained plane in accordance with the hodograph. experimentally

A more critical confirmation of the theory is found by examining t,he slip-line field as described below.

(c) Slip-line field tests Preparation of the specimens ; grid measurements. If the bar is wide compared with the notch size, plane stress conditions are closely realized in the middle plane of the bar. It is necessary therefore to cut the test bar on its middle plane and scribe a reference on t,he cut face, afterwards joining the two pieces together to restore the continuity. The joint must be sufficiently strong to sustain t,he stresses

/

/

I

ASjjttrbol/

.lltrturird

I--.

Jli~tttot.sious (in.)

--, - .-- ---

*‘I,, j

(tons.

iu.)

I.

I

_I.

I

---

-- i

. .-i--

1.16lJ’1.13fJ~ ---0.7 I

_.(

.-

1~0x0

_I ._ _-. /

1 a;7

’

;

--

OG

17

--~_----

_--- --_

I

,‘;:

I

<.

I i I i

Plastic

yielding of single notched

bars due to bending

IZEFERENCES BISHOP, J. F. W. UI~OP, J. F. W., GREEN, A. P. and EIXLL. R. PORO, If. GREEN, A. F. ITma, R. IJANIS. 6. and FORD, H. \‘.‘\‘*ms, A. R. and I.-mm, Il.

1053

.I.

1956 1952-53 1953 1956 1950 1951 1957 1952

4, 150. Proc. Ins&n. Meek. Enprs. 1 B, 448. f&art. J. Mech. _A_mL Maths. 6, 223. .7.

Mech. Phys. Meek.

Phyr.

,Solids Solids

2, 43.