Stress analysis of a welded joint

Engineering Frocfure Mechanics Vol. 48, No. 5, pp. 619-627, 1994 Elsevier Science Ltd

Pergamon

0013-7944(94)EOO26-D

STRESS

ANALYSIS

OF A WELDED

N. S. RAGHAVENDRANTG School

of Engineering

Abstract-The use of the across a butt-welded joint dimensional photo-elastic demonstrated. The results of an embedded porosity

and Applied

Printed in Great Britain. 0013-7944/94 %7.00+ 0.00

JOINT

and M. E. FOURNEY

Science, University of California CA 90024, U.S.A.

at Los Angeles,

Los Angeles,

two-dimensional finite element method (FEM) in finding the stress distribution is illustrated for two types of loading conditions, viz. tension and bending. Two analysis (PEA) of a butt weld under plane stress conditions has been of FEM (for plane strain) and PEA (for plane stress) are discussed. The effect on the stress distribution under bending load is highlighted.

NOMENCLATURE constant in Peterson’s equation strain displacement matrix stress strain matrix global stiffness matrix fatigue reduction factor stress concentration factor actual strain concentration actual stress concentration notch sensitivity notch root radius peak stress nominal stress peak stress nominal stress maximum shear stress

INTRODUCTION of notches and changes of shape lead to stress concentration in components and can be understood by visualizing the stress flow lines (analogous to the lines of electric flux) that are distributed in the member. In the vicinity of a discontinuity, stress flow lines would normally be denser and deflected around the discontinuity, and the density of lines is a measure of the intensity of stress. It is of importance to note that it is not necessary for the change of shape to involve a reduction in cross-sectional area for the stress concentration to be produced. A local increase in area also has the same effect. In the case of a butt-welded joint the cross-sectional area increases from plate to reinforcement and hence increases the stress concentration factor (KN), defined as follows: THE INFLUENCE

crp= KNo,

(1)

where gP = peak stress, on = nominal stress and KN = stress concentration factor. Yamaguchi et al. [1] showed that one of the most important factors in determining KN was the curvature at the toe of the projection in the case of a butt weld. Conventional methods of determining the stress distribution include photo-elastic model analysis, strain gauges and Moire fringes to name a few. They do not readily lend themselves to the analysis of an arbitrary shape like the cross-section of the butt weld, even for plane stress conditions, without a perfect experimental setup. Hence the finite element method was resorted to as an alternative to the plane strain analysis of the butt-welded joint. 619

620

N. S. RAGHAVENDRAN

and M. E. FOURNF.1

The fatigue life of a component is in general determined by the geometric stress risers [2). Lawrence et al. [3] have shown the effect of notch sensitivity, fatigue reduction factors and Neuber’s rule in their estimation of fatigue crack initiation life. Notch sensitivity is defined as: K,-

q=f(,-

1

(2)

1,

where: K, = Fatigue

reduction

factor =

Unnotched Notched

fatigue fatigue

strength strength

’

KN = stress concentration factor. q can take values from 0 (no notch effect) to 1 (fully notch sensitive). An empirical relationship due to Neuber [3] for Kr is given by: KN - 1 (3)

Kf=l+I+J&.

where r is the notch root radius and u is a material constant obtained from long-life fatigue analysis. Neuber has also developed a relationship between nominal stress-strain and local stress-strain for a notch which may be written as: KN = (K,,K,)“,5,

(4)

where KN is the linear elastic stress concentration factor, K, is the actual stress concentration or plastic) and K, is the actual strain concentration (elastic or plastic).

TOE

ANGLE

1 56

L UNIFORM

PRESSURE

OF

1 kg /cm2

I

POROSITY OF DIAMETER

Fig. l(a). Finite element configuration

for butt weld

3 mm

(elastic

Stress analysis of a welded joint

I

_

;b

1

-16

I

I

-12

- 6

I

I

-b

I

0’

b

I 6

621

I 12

x’ STRESS

MAGNITUDE

Fig. l(b). Illustration of section X-X’ across which stress values are plotted.

Fatigue analysis of welded joints One of the main features of fatigue analysis of welded structures is the reduction in fatigue life caused by notch sensitivity (q) due to stress concentration at the toe of the weld reinforcement. Numerous researchers [2] have made reference to how the toe angle 8 (Fig. 1) affects the fatigue strength. Partial penetration in butt and fillet welds has been prohibited by the existing codes for welded structures. Besides the toe angle, welding defects like porosity (gas packet), inclusions, incomplete fusion, etc., are known to be the potential stress risers. Also, although the KN (SCF) depends only on the geometry of the joint, the notch sensitivity index q and K,, the fatigue reduction factor depends on sizes as well, meaning that they (q and Kf) are not material properties. Thus, the stress rising effect of the notch sizes also plays an important role in determining the fatigue life of a welded component [3].

Experimental

determination

of K,

@lane stress analysis)

Two-dimensional photo-elastic model analysis [4] of the butt weld was used to obtain KN. The model studied was one which is used in mass production of agricultural and transportation equipment. It involves butt welding of plates with proper edge preparation before welding to avoid defects like partial penetration. The question then was how, in relation to plate thickness, weld reinforcement size and hence toe angle 8 can have a deleterious effect on the stress distribution across the transverse cross-section of the welded joint. The model was developed by considering the toe angle (0 ) of the weld reinforcement (Fig. 1). The material used was Araldite. The final model was for a 0.5 in. thick plate with an initial toe angle of 90”. The weld toe angle (0) was then gradually adjusted to take values of 120” and 150”. The tests were conducted to find KN (SCF) for uniaxial tension. The case of pure bending necessitated a delicate fixture for the model and hence could not be tried out. Since the fatigue cracks were normally initiated at the weld toe, the stress distribution across the plane section

N S. RAGHAVENDRAN

622

Table

and M. E. FOURNE\

I. Table ot SCL Tension

Loading Approach FEM (plane ~7(notch sensitivity) PEA (plane stress)

strain)

Bending

90

I20

IS0

90

120

150

1.04

0.91

0.22

1.30 0.95

1.05 0.90

0.39 0.83

1.093

0.625

‘tThe values of 4 are found from the expression 4 = (K, - I)/(&, - I). where KC is found from ref. [2]. The values in the rows FEM and PEA are the values of K, by the respective methods. Also the KN found here are based on stress distribution across the section X-X’ passing through the weld toe. Therefore KN based on the entire two-dimensional stress field can be different from those reported here. But given the fact that almost all the equistress lines [7] pass through the section X-X’. this difference should be negligible.

passing through by:

the weld toe (X-X’) was found. Thus the elastic stress concentration

KN =

Max stress across the section X-X’ passing Nominal

stress or average

through

factor is given

toe

stress (5)

Sufficient experimental points were determined to demonstrate the effect of toe angle (0) on KN. Figure 2 shows the experimental setup used. Figure 3a and b represents the typical fringe pattern obtained for the cases of toe angles of 90” and 120”, respectively. The effect of the stress magnitude along the section X-X’ was used in finding the KN . Although cr,,, can take a maximum value anywhere in the field of analysis of fringes. the objective in this work was to find the effect of the stress magnitude across X-X’. The stress magnitude on the surface of the joint adjacent to the weld toe is usually the deciding factor in the initiation of fatigue cracks. While the geometry effect and influence of the weld defect (porosity) on this stress is studied using the FEM, the effect of residual stress on the weld metal due to constraints by the plates being welded is not incorporated, i.e. no thermal stress effect has been considered. Finite element

analysis

The butt-welded welded to be parallel

joint is symmetrical with respect to the Y-axis if we take the plates being to the X-axis. Hence, the stress distribution is also going to be symmetrical

-*

X’

if

61” ( NORMALISED)

(a) Fig. 4. Stress distribution

6YY (NORMALISED) (b)

across

the section

X--X’ through

0

i’

L

6 T:

( NOR”iyL (C)

the toe of the butt weld in tension

ISEO)

12

I 16

Stress analysis

Fig. 2. Experimental

of a welded joint

set-up

for photo-elastic

623

analysis

Fig. 3(a). Fringe

pattern

for butt weld (0 = 90 ).

Fig. 3(b). Fringe

pattern

for butt weld (B = 120 ).

Stress analysis of a welded joint

X’

6X&

625

6uY

(NORMALISEO)

=XY

f NORWALISED)

IN~RMAFISED)

(b)

(a)

(cl

Fig. 5. Stress distribution across the section X-X’ through the toe of the butt weld in bending mode.

-

71

12%

0 VI *

Ih 4

t 12

I 8

I 16

I -20

6 Ill

(3

8

-

WlTll

=

90’

I

I -6

/I-\

-4

0

I 4

r 9

X’ .Ib)

WELD

WITHOUT

I -12

6y v ( NORMALISBD)

( NORYALISLD)

(a)

o-

I -16

TOE

POROSITY POROW?Y

(cl

( N&iiiALISED)

Fig. 6. Effect of porosity of size 3 mm on the stress distribution across section X-X’ through the toe of the butt weld in bending mode.

N S. RAGHAVENDRAN

626

and M. E F’OURNt-\

WELD

i’ Fig. 7. Maximum

shear stress across

7;max

(NORMALISED

the section

1

X-X’ through

-

the toe of the butt weld in tension

as long as the loading conditions are similar. Thus, only the portion to the left of the Y-axis is taken for analysis. A CO,-welded specimen was used for the reference profile and was discretized into a number of quadrilateral elements. The nodes that fall on the Y-axis (line of symmetry) were given zero displacement in the X-direction and were free to move in the Y-direction. The node at the bottom of the Y-axis is pinned to arrest the rigid-body motion. The plate thickness was subjected to a tensile load of unity (directed along the X-axis) for the case of tensile loading. For the bending mode a moment of 1 kg cm was applied as in Fig. I. The response of the joint and hence the stress distribution were found independently for these two cases, i.e. the tension and bending modes. The stiffness matrix used for plane-strain analysis, i.e. K = jjBrDB dx d-v, was computed by an existing program SAPIV [5]. For most of the analyses, the time taken by an IBM/370 system was around 3 min. In the previous expression, K is the global stiffness matrix. B is the strain displacement matrix and D is the stress-strain matrix.

RESULTS

AND DISCUSSIONS

The finite element analysis was applied to butt-welded joints with and without an embedded porosity (gas packet). The pressure exerted by the entrapped gas is assumed to be negligible in comparison with the loadings of the weld joint. The latter is of interest, as the effect of porosity on the stress distribution will give an idea about the size of porosity acceptable, the effect of multiple porosity interactions on stress distribution, etc., which are discussed in detail elsewhere [5]. The study here is restricted to finding the effect of a single porosity of size (dia) 3 mm on the stress distribution when the joint is subjected to bending moment. The toe angles (0 ) considered were 90”, 120“ and 150”. Finite element analysis for plane-strain conditions and photo-elastic analysis for plane-stress conditions were carried out. In the case of PEA the corresponding stress distributions were computed using the applicable formulae [5]. The stress distribution across the section X-X’ passing through the weld toe have been plotted (Figs 4-7) for plane-stress as well as plane-strain conditions.

CONCLUSIONS

(1) Maximum (2)

shear stress (tmax) trend for plane-stress as well shear stress is very different SCF increases as 8 decreases

variation along the section passing through X-X’ shows a similar as plane-strain conditions, although the magnitude of maximum in both the cases (Fig. 7). and approaches 90”.

Stress analysis

of a welded joint

627

(3) The presence of single porosity increases the shear stress magnitude by several times at the weld toe, under bending load. (4) The shear stress rising effect of the weld toe is more pronounced in bending than in tension. (5) From Peterson’s equation (3) for K, we have the following expression for q, the notch sensitivity: q-r

=a, 4

where r is the notch root radius and a is a constant. on: (i) (ii) (iii) (iv)

This expression

for Q tells us that a depends

weld toe radius r; conditions of analysis (plane stress or plane strain); K,, the fatigue reduction factor which depends on notch size; and Kh, the SCF which tells us the stress rising effect of the toe angle 8.

Hence the interpretation that a is a material constant seems to be less appropriate. Our analysis here shows that with so many working variables affecting the parameter a, the appropriate definition for a should depend on the above four factors. The findings of this analysis therefore support the interpretation for a as the initiated but non-propagating crack length given in ref. [3] for long-life fatigue analysis.

REFERENCES [I] L. Yamaguchi, [2] [3] [4] [5] [h] [7]

Y. Terada and A. Nitta, On the fatigue strength of steels for ship structures. IIW Document XIII, pp. 425426, London (1966). T. R. Gurney, Fatigue of Welded Smcfures. Cambridge University Press, London (1968). F. V. Lawrence and P. K. Mazumdar, An analytical study of fatigue notch size effect. F.C.P. Report 36, University of Illinois (1981). J. W. Dally and W. F. Riley, Experimental Sfress Analysis. McGraw-Hill, New York (1965). N. S. Raghavendran. Stress analysis of welded joints. M.Tech Thesis, Indian Institute of Technology. Madras (1978). G. Sines and J. L. Waisman, Metal Fatigue. McGraw-Hill, New York (1959). Use of ANSYS to find equistress plots for buttwelds. Private communication with Hindustan Computers Ltd, India. (Received 6 July 1993)