The Sensitivity Analysis of Expression for Calculation of Stress Concentration Factor in Butt Welding Joints
The stress concentration factor is the ratio of maximum stress,
σmax and nominal stress,
σnom, i.e.,
The surface of the weld is irregular with a number of local form changes that have a significant impact on the value of the geometric stress concentration factor [
39]. Ushirokawa and Nakayama [
28] suggested the expression for the calculation of the geometric stress concentration factor at the butt-welded joint in the following form:
where:
φ is the toe radius;
θ is the weld toe angle;
W is the weld width;
t is thickness of the base material; while
h denotes the reinforcement height.
From Equation (A2) it can be concluded that each of the five geometric quantities does not influence the stress concentration factor in the same way. Therefore, the sensitivity analysis is carried out in order to determine the level of impact of each single geometric quantity on the stress concentration factor [
29].
Two models of the sensitivity analysis were carried out. A partial derivation of the expression (A2) with respect to each geometric quantity was carried out in model A. In model B, two geometric quantities that have the highest impact on the stress concentration factor were first singled out, then the expression (A2) was analyzed with respect to these two geometric quantities.
Model A—Complete Derivation of the Expression
The sensitivity analysis is carried out by the partial derivative of Expression (2) with respect to each geometric quantity, i.e.,
In Equations (A3a)–(A3e), the partial derivative describes the rate of change of the function regarding the change of the independent variable. The positive value of the derivative expresses the positive sensitivity of the independent variable to the result [
30].
For each influential geometric quantity, the span of expected quantity value has been determined first and the sensitivity analysis for values within that area is carried out. For the two geometric quantities with the highest impact (the weld toe angle and the toe radius), the sensitivity analysis is carried out for four values, while for the other three geometric quantities it is carried out for three values,
Table A1.
Table A1.
The area of sensitivity analysis and of values in which the sensitivity is carried out.
Geometric Quantity | Values in which the Sensitivity was Analyzed |
---|
Thickness of the base metal | 6, 10, and 20 mm |
Weld toe angle | 5°, 10°, 20°, and 50° |
Toe radius | 0,1, 0.5, 1, and 3 mm |
Reinforcement height | 1, 2, and 4 mm |
Weld width | 20, 25, and 30 mm |
Once the sensitivity analysis has been performed, four areas of influence of geometric quantities on the stress concentration factor are obtained:
Area of negligible influence:
Areas of small influence:
Areas of significant influence:
weld toe angle with the toe radius of up to 1 mm;
toe radius with the radius from 0.5 mm to 1 mm;
Area of great influence:
toe radius with the radius of up to 0.5 mm.
The graphs in
Figure A1 show the areas of significant and great influence of geometric quantities. It can be concluded that the toe radius has a great influence on the stress concentration factor, so that the optimization of the welding procedure will be directed to this geometric quantity. The graphs in
Figure A1 also show the influence of the toe radius change, that is, the influence of the change of the weld toe angle on the stress concentration factor. They were obtained according to expressions (A3b) and (A3c).
For a better understanding of the graphs in
Figure A1, points “A” and “B” are selected to be specially analyzed herein. Point “A” is shown on the graph that presents the effect of the change of the toe radius, while the value of
Uφ is −53.71. The value is negative, which means that by lowering the toe radius, the stress concentration factor increases. The value of –53.71 is the coefficient of the direction of the tangent Equation (A2) in point (
φ = 0.1mm,
t = 10 mm,
W = 20 mm,
h = 4 mm and
θ = 50°).
The value of
Uθ in point “B” is 16.87. It is found in the
Figure A1 that shows the influence of the change at the weld toe angle. The value is positive, which means that by increasing the weld toe angle, the stress concentration factor is increased. The value 16.87 is the coefficient of the direction of the tangent in the Equation (A2) in the point (
φ = 0.1 mm,
t = 10 mm,
W = 20 mm,
h = 4 mm, and
θ = 5°).
Figure A1.
Areas of significant and great influence of geometric quantities on the stress concentration factor; (a) Influence of the weld toe radius on the stress concentration factor, (b) Influence of the weld toe angle on the stress concentration factor
Model B—Derivation of the Expression by Toe Radius and Weld Toe Angle Only
In the model B, the analysis of influence of geometric quantities was carried out in two steps.
In the first step, values of geometric quantities that are expected during the experiment are included in the expression (A2). For each geometric quantity the analysis was carried out with seven values:
The thickness of the base material (6, 10, 14, 18, 22, 26, and 30 mm);
The reinforcement height (1, 2, 3, 4, 5, 6, and 7 mm);
The weld width (2, 5, 8, 11, 14, 17, and 20 mm) and
The weld toe angle (10°, 20°, 30°, 40°, 50°, 60°, and 70°).
The toe radius was analyzed for the values from 0 to 12 mm.
Diagrams in
Figure A2 show the influence of the thickness of the base material, the reinforcement height, the weld width, and the weld toe angle on the stress concentration factor with regard to the change of the toe radius. The analysis was made for seven values of each geometric quantity.
Figure A2.
The analysis of the influence of geometric quantities (
a) thickness of base material, (
b) reinforcement height, (
c) weld width and (
d) weld toe angle on the geometric factor of stress concentration as function of toe radius after the expression suggested by Ushirokawa and Nakayama [
28].
From the aforementioned diagrams it can be concluded that the reinforcement height and the weld width have a very small influence on the stress concentration factor. The thickness of the base material is constant so that there is no use analyzing the influence of the thickness of the base material. This is why we will continue by making the comparison of the influence of the weld toe angle and the toe radius on the stress concentration factor. In the further analysis, three geometric quantities will be fixed, namely, the thickness of the base material to 10 mm, the reinforcement height to 2 mm and the weld width to 18 mm.
In the second step, the sensitivity analysis was carried out with regard to the two geometric quantities that in the foregoing analysis were established to have the biggest influence on the stress concentration geometric factor. The sensitivity analysis was carried out in order to determine the level of impact of each single geometric quantity on the stress concentration factor and to direct the further optimization of welding parameters. It was carried out in a way that the Equation (A2) was partially derived with regard to the two geometric quantities for which it was earlier established to be of the biggest influence on the stress concentration factor, i.e., Equations (A3b) and (A3c).
Graphs in
Figure A3 show the influence of the toe radius (a) and the weld toe angle (b) on the stress concentration factor. Two points, “A” and “B”, were selected in diagrams in order to have a better understanding. The value of point “A” is
Uφ = −21.48. This point is in diagram a, which presents the influence of the toe radius on the stress concentration factor. The value is negative, which means that by lowering the toe radius, the stress concentration factor increases. Value −21.48 is the coefficient of the tangent expression (A2) in point (
φ = 0.1 mm,
θ = 60°,
t = 10 mm,
h = 2 mm, and
W = 18 mm).
The value of point “B” is Uθ = 4.14. This point is found in diagram b, which presents the influence of the weld toe angle on the stress concentration factor. The value is positive, which means that by lowering the weld toe angle, the stress concentration factor is increased. Value 4.14 is the coefficient of the tangent expression (A2) in point (φ = 0.1 mm, θ = 60°, t = 10 mm, h = 2 mm, and W = 18 mm).
Figure A3.
(a) Influence of the weld toe radius on the stress concentration factor, (b) Influence of the weld toe angle on stress concentration factor.
We can conclude that it is the toe radius that has the biggest influence on the stress concentration factor. With the analysis of the results obtained by the carried-out experiments, it will be established which welding parameters have the biggest influence on the toe radius in which the smallest stress concentration will be generated.
Appendix Conclusion
It can be concluded that the toe radius has the biggest influence on the stress concentration factor, especially for very small values of the radius. When changing the toe radius for values smaller than 0.5 mm, the stress concentration factor changes significantly. During the welding of butt welded sheets it is definitely necessary to avoid the radius smaller than 0.5 mm as in a such a small radius big stress concentrations take place, which can initiate the occurrence of surface cracks.
The influence of the weld toe angle is big, however, it is not as significant as the influence of the toe radius.