The plane-strain general yielding of notched members due to combined axial force and bending—III

Int. J. Mech. Sci. Vol.22,pp. 431--440 PergamonPressLtd., 1980. Printedin GreatBritain

THE PLANE-STRAIN GENERAL YIELDING OF NOTCHED MEMBERS DUE TO COMBINED AXIAL FORCE AND BENDING--Ill COMPARISON OF THEORIES WITH E X P E R I M E N T MASAKI SHIRATORI Department of Mechanical Engineering, Yokohama National University, Tokiwadai, Hodogaya-ku, Yokohama, Japan and BRADLEY DODD Department of Engineering Science, University of Oxford, Parks Road, Oxford, England (Received 18 D e c e m b e r 1979)

Summary--The plane-strain general yielding of plates with deep single notches is studied using slip-line field analysis, Rice's upper bound solution and finite element analysis. When the plates are subjected ~to arbitrary combinations of axial force and bending moment, it is possible to calculate the corresponding constraint factors and to map them on the constraint factor plane. It is shown that there is a change in the mode of general yielding from that predicted by slip-line field analysis to that predicted by Rice's upper bound solution. This mode transition occurs when the axial force becomes dominant in the first and third quadrants of the constraint factor plane. The transition is confirmed by finite element analysis and also experiments on notched steel plates.

a

r /3

k T M Tz Mt R &E err W

NOTATION ligament length radius of notch flank semi-angle of notch shear yield stress axial force per unit width bendingmoment per unit width constraintfactor due to the axial force ( = T]2ka) constraintfactor due to the bending moment ( = M/½ka 2) radius of shear lines in Rice's upper bound (Fig. 2) angles in Rice's upper bound (Fig. 2) rate of rotation (Fig. 2) tensile yield stress distance from centre of pins to lower surface in the steel specimens (Fig. 7)

INTRODUCTION Slip-line field solutions for the plane-strain general yielding of plates with deep notches in bending were first proposed by Green[l]. Green calculated the constraint factors in bending for a number of notch geometries. For shallower notches other slip-line field solutions have been proposed by Ewing and Hill [2] and Ewing[3] in which the effect of the plate surfaces is taken into account. In practice, it is important to be able to estimate the general yielding load of notched structural members subjected to combinations of axial force and bending moment. Using generalizations of Green's slip-line fields for pure bending the authors [4, 5] have derived loci which describe general yielding under arbitrary combinations of axial force and bending moment. For a given notch geometry it is possible to calculate the constraint factor due to bending, M~, and that due to the axial force, T~. M~ is defined as the bending moment taken about the mid-point of the ligament length divided by the limit bending m o m e n t for a flat plate of thickness equal to the ligament length. The constraint factors which have been derived from such calculations are mapped on the ( T I - M O constraint factor plane. These mappings are referred to by the authors as general yielding loci. 431

432

M. SHmATorland B. DODD

Fig. l(a) shows the general yielding loci for a circular notch with H a = 1.5; this type of solution is admissible over a range of r/a values as shown in Ref. [5]. Fig. l(b) shows the general yielding loci for a single wedge-shaped notch with flank semi-angle /3 = 60°; this type of solution is admissible for flank semi-angles within the range 90 ° i>/3/> 57.3 °. The lower bound loci shown in these figures is derived for a flat plate subjected to combined bending and axial force [6]. The equations of the lower bound are; T12 + MI = 1

for

M~/-- 0

T~2 - Mi = 1

for

M] ~<0 .

and

(1)

Both of the sets of loci shown in Fig. 1 do not form closed loops and therefore there may be other upper bound solutions which do result in closed loops. One of these has been proposed by Rice [7] and Ewing and Richards [8] and is shown in Fig. 2. This upper bound solution is independent of notch geometry and it may be reformulated to give results in terms of bending moment and axial force in the following manner [9];

ra = R sin 6,

M + T ( r - ½)a <- k R 2 ( 6 - e),

(2)

(r - 1)a = R sin E,

(3)

2(6 - ~) = tan 3 - tan e.

The resulting general yielding loci determined from this upper bound are shown in Fig. 3(a) and (b), because of symmetry only the first and fourth quadrants are shown. As can be seen from these mappings, there should be a transition in the mode of general yielding depending on the combinations of T1 and M1. To elucidate this transition a finite element technique was used and also some general yielding experiments have been carried out. fl = 6 0 °

r/o

=1.5

M1

,1.0

\

0.0

-1.0

~t---

'ID

-

I

--

~

T

~

Slip line sol.

I

--

Slip une

.~:fl.

. . . . I.ow~ I ~ u ~ l

(b)

(a)

FIG. 1. Slip-line field and lower bound general yielding loci for (a), a plate with a single circular notch and (b), a plate with a single wedge-shaped notch.

-

Cl

~

N'k

T

FIG. 2. Rice's upper bound solution for general yielding.

The plane-strain general yielding of notched members

433

M11

M1

1.5~.__.....~ \,

1.5

r/o =1.5

....

~

13= 60 °

1.0 O.5 ~

0.0 . . ~ f o

0.5 ""' '~" i ~

\

. .

1.5 T,

0.0 . ~ , ~ ~ )

-0.5 -1.0 -1.5

1.5 1"1

-0.5

jeJ

- - - upper bound - - Sliplinesol . . . . Lowerbound

-1.0 y

(a)

-1.5

- - - Upper bound --

Slip line sol.

.... Lowerbound (b)

FIG. 3. General yielding loci for (a), a plate with a circular notch and (b), a plate with a wedge-shaped notch. FINITE ELEMENT ANALYSIS In order to investigate the transition in mode of general yielding, a plane-strain elastic-plastic analysis has been completed with a finite element method. In the FEM program the material is assumed to yield according to the von Mises criterion and to obey the Prandtl-Reuss flow rule. Triangular elements with linear shape functions were utilized. Because of symmetry it is sufficient to analyse only half of the specimen as shown in Fig. 4. Also shown are the dimensions of the specimen and the state of element breakdown. The Young's modulus of the material was taken to be 205.9 GN/m 2 and Poisson's ratio 0.3. The elastic-plastic analysis was carried out by the incremental method and general yielding was defined as the point at which the plastic zone spreads across the ligament length from the notch to the lower surface. Proportional loading paths were chosen in which the ratio MI/TI was zero, +0.25, -+0.5, -+1, -+2 and -+4, where TI was chosen to be positive.

I0 I0

I

(a)

(b)

FIG. 4. State of element breakdown for (a), plate with a circular notch and (b), a plate with a wedge-shaped notch.

434

M. SHIRATOR1and B. DODD

Therefore the first and fourth quadrants of the T~-M~ plane were considered. As the second and third quadrants are symmetrical about the origin with respect to the fourth and first quadrants respectively, then generality is not lost by neglecting the former pair of quadrants. From the general yielding load resulting from the finite element analysis, the constraints factors can be obtained through the following equations;

2ka

and

MI = ~

(where k =

.

(4)

The circle points shown in Fig. 5(a) were obtained by using this technique. In the figure the shapes of the plastic zones at the general yielding points are shown where the positive and negative signs indicate, respectively, tensile and compressive stress fields. In the fourth quadrant of the figure the results of the finite element analysis agree closely with those of the slip-line field solutions. Also, in the first quadrant, when the bending moment is dominant the finite element results agree with those of the slip-line field solutions. However, when the tensile axial force is dominant the finite element results coincide with those of Rice's upper bound solution. Thus there is a clear transition in the mode of general yielding from the slip-line field solution to Rice's upper bound solution. This is a verification of the conclusion reached in the previous section. Fig. 6(a) shows the stress distributions across the ligament for five MdTI ratios. For MI/T~ ratios between 2 and 0.5 it can be seen that the finite element results agree closely with the slip-line field results. However for MJTj = 0.25 the finite element results differ markedly from the slip-line field solutions. This is further evidence for the transition from the slip-line field to Rice's upper bound solution. Figs, 5(b) and 6(b) are the corresponding results for a wedge-shaped notch and the same transition in mode of general yielding can be observed. EXPERIMENTS For the experiments nitrogenized low carbon steel with the following composition in weight percent was used; C 0.08%,

Mn 0.32%,

N 0.019%,

P 0.01%,

S0.015%.

The steel was firstly annealed at 650°C for one hour, after which the specimens were machined to the shapes shown in Fig. 7. The geometries of the notches are the same as those considered above. In the experiments the load was applied through pins, whose location relative to the mid-point of the ligament length determine the ratio MI/Tt by the following equation; MI = 4T---!(W-2).

(5)

A schematic diagram of the testing apparatus is shown in Fig. 8. From the apparatus the load-deflection curve could be obtained and the general yielding load was defined by the intersection of the elastic slope with extrapolation of the post-yield slope. In terms of the general yielding loci the results are shown in Fig. 9.

60 1.5

.5

1.0 " - - .

001

0s

/o-.~

~5

r,

/--

0.0

0.5

1.0

,0 (a)

1.5

Tt

.0 (b)

FIG. 5. Comparison of upper bound general yielding loci with F.E.M. results for (a), a plate with a circular notch and (b), a plate with a wedge-shaped notch.

435

The plane-strain general yielding of notched members

i"

~o~

,,~.--~.o",-"°i 05 1.0 0.0

-'"

0.0

ylo

2.

'

1

" ~o~ i1 ~,,,,=l.oq"1 ,,,~,-o5~ 05 yla

1.0 0.0

05 yla

1.0

o FEM ---- Slip line sol.

20t

-'.0r ~,,,,=o26 ~ -'.ol ~,,,,=o.0 o.o

ygs

~.o

21~" q

-I

°°! 05 yto

M1I1"1=0.5 oo Q5 y/a

-1.C

1.0

2- 0 -0

1

i ~1.zo ~o- ' °o." ]

1

L_

1.0 0.0

o

0.5 Y/ a (a)

~,

t

~1~.~.0 v I

0.0

~1.C

0.0

.=1.

05 yta

1.0 0.0

.9 o . o

05 y/O

1.0

----- Slip Ur~ sol.

¢%1

_

"

to

O.0

/TI=O.0 1.0 y0.5 /a

(b) FIG. 6. Comparisons of axial stress distributions across the ligament length calculated by slip-line field analysis and F.E.M. for (a), a plate with a circular notch and (b), a plate with a wedge-shaped notch.

rio =1.5

3

8 =60 °

J

~'~_ 30

L

.

120

120

(b)

(a) FIG. 7. Dimensions of steel test specimens.

After unloading, the specimens were sectioned at the mid-thickness perpendicular to the notch axis. The specimens were then aged at 2000C for half an hour and etched with Fry's reagent. Figs. 10(a)--(d) show typical patterns of the plastic zones for the two notch geometries. Here, again, the transition from the slip-line field solution to Rice's upper bound solution is clear. DISCUSSION For two notch geometries it has been found possible to produce closed-loop general yielding loci using slip-line field analysis and Rice's upper bound. These composite general yielding loci include four corners.

M. SHIRATORI and B. DODD

436

CLIP GAUGE

X-Y RECORDER

Fit;. 8, Schematic diagram of testing apparatus.

M1 1.5¸_.

TMA=4. 12 I.~ //2.10

r/0=1.5

""~'" 1.0 !

M1 1.5

126

~.__o/,,o •

/0.97 "x" " ~ v ~ O

" •4.07 J - =13=60° j2.09 "\

1.0 ~ o ~ 1 . 1 2 70

0.5

"\

0.5

.i..~J

"

o15

O~OI

T,

----- Upper bound --

Slip line soL+

. . . . Lower bound o FEM • Experiment (a)

05

--°sl

i 10

15

T,

- - - Upper bound -Slip line sol, . . . . Lower bound o FEM • Experiment (b)

Fit;. 9. Comparison of theoretical general yielding loci with experimental results for (a), a plate with a circular notch and (b), a plate with a wedge-shaped notch. Since the slip-line field and Rice's solution are both upper bounds, it is quite possible that the true solution will have a continuous slope. Therefore it is suggested that the regions of discontinuous slope be studied further. In Fig. 5(b) it can be seen that the finite element results give constraints factors which are a little higher than those given by the slip-line field analysis for MdTI ~ 1.0, although the shape of the plastic zones are quite similar. Generally, triangular elements with linear shape functions are said to produce higher constraint factors[10], therefore more elaborate elements should be developed to obtain a more precise solution. In Figs. 9(a) and (b) the experimental results have lower constraint factors than the lower bound solution when the axial force is dominant. This may be caused by thickness effects in the test specimens, i.e. the specimens may not have been sufficiently thick to ensure plane-strain. In Fig. II the lower bound for general yielding in plane stress is plotted. In the region of the Tt-axis the experimental results approach this lower bound curve.

CONCLUSIONS For two notch geometries, the plane-strain general yielding loci obtained from slip-line field analysis and Rice's upper bound solution have been compared. It has been shown that when the axial force becomes dominant, Rice's solution gives lower constraint factors than the slip-line field solution. Because of this, a change in the mode of general yielding has been postulated. This postulation has been substantiated by finite element analysis and experiments on notched steel plates.

The plane-strain yielding of notched members

(o)

M~

T-T = 2.10

(b) T'~ = 0.28

(C)T~ =4.07

(d)

T~ =1.12

FIG. 10. Etched sections of test specimens illustrating the transition in mode of general yielding with changes in the ratio MJTI.

437

The strain-strain general yielding of notched members M1

439

M1

1

1.5~_.__.~.~..

r/o =1.5

f~ = 600

1.5 ~ . . ~ _ . _

0.5

~

•

\_ O.C

---- Upper bound -Slip line sol. . . . . Lower bound (Rane stress)

05

1.0

1"1

- - - - Upper bound -Slip line sol. - - - - Lower bound(Pkane stress)

o

F E M (Plane strain)

o

= •

F E M ( Plane stress) Experiment

'= F E M ( Plane stress)

(a)

1.5

•

FEM (Plonestroin) Experiment

(b)

FIG. 1!. Comparison of plane strain and plane stress general yielding loci with experiments for (a), a plate with a circular notch and (b), a plate with a wedge-shaped notch.

The resulting composite general yielding loci on the (TpMO constraint factor plane have four corners and it has been suggested that the true solution will result in general yielding loci with a continuous tangent.

Acknowledgements---The authors would like to express their gratitude to Dr. K. Ando for his useful suggestions concerning the experimental techniques and to Mr. M. Furuya for his help in the finite element and experimental work. REFERENCES A. P. GREEN, Quart. J. Mech. Appl. Math. 6, 223 (1953). D. J. F. EWlNG and R. HILL, 3". Mech. Phys. Sol. 15, 115 (1967). D. J. F. EWING, Jr. Mech. Phys. Sol. 16, 205 (1968). M. SH1RATOmand B. DODD, Int. J. Mech. Sci. 20, 451 (1978). B. DODD and M. SHIRATORI,Int. J. Mech. Sci. 20, 465 0978). W. PRAGER, Introduction to Plasticity, p. 51. Addison-Wesley, Reading Mass., (1959). J. R. RICE, The line spring model for surface flaws, in The Surface Crack; Physical Problems and Computational Solutions (Edited by J. L. Swedlow), p. 171, ASME, New York (1972). 8. D. J. F. EWlNG and C. E. RICHARDS,Z Mech. Phys. Sol. 22, 27 (1974). 9. M. SHIRATORIand T. MIYOSHi, Proc. o[ ICM3, Vol. 3,425, Cambridge (1979). 10. J. C. NAGTEGAAL,D. M. PARKS and J. R. RICE, Comp. Meth. Appl. Mech. Engng 4, 153 (1974). I. 2. 3. 4. 5. 6. 7.