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The Influence of Types of Supports on
the Stress Distribution on Pipe Reducer

Darmawan Harsokoesoemo
Mechanical Engineering Department, Institut Teknologi Bandung
Gatot Santoso
Mechanical Engineering Department, Universitas Pasundan Bandung

Abstract
This paper presents numerically calculated stress distribution in the region of the junction of the cylindrical and conical part of a reducer pipe and then compare them with stresses at the junctions obtained from analytical formulas.
Significant differences between the two are disclosed.

1. Table of Notation
D = large pipe nominal diameter
D = small pipe nominal diameter
T = large pipe thickness
t = small pipe thickness
tc = cone shell thickness
? = half of apex angle
p = internal pressure
? = stress in tension or compression
? = shear stress

2. Introduction
Stresses at the junction of conical and cylindrical shells were investigated by Boardman (1960), Langer (1971), Zaremba (1959) and others using analytical methods. Stress distributions in the region of the junctions was investigated by John and Orange (1961), also using analytical method. Their works were cited in the books by Roark and Young (1976) and Bednar (1981). This solutions were in the forms of analytical formula, including formulas for stresses at the junctions.
This paper investigates the same problem, however, using numerical method, e.i. finite element method, as the method to calculate the stresses in the region of the junction of conical and cylindrical shells, which will subsequently be called reducer pipe. Comparison will be made with previously obtained results.
As in the case of piping system components, there are five broad categories of variables that are involved in the study stress distribution in the region of reducer pipes, i.e. : (1) the reducer pipe dimensions (D, d, T, t, tc, ?, (2) the types of loading, (3) the types of pipe support, (4) methods of manufacture, and (5) reducer pipe materials.
This paper investigates the influence of the type pf pipe supports to the stress distribution in the region of reducer pipe subjected to internal pressure. Details of the reducer pipe and its supports are discussed in the following paragraphs.

3. Procedure
3.1. The Reducer Pipe’s Model and Type of Loading
For purposes of obtaining a general idea of the stress distribution in region of a reducer pipe’s junctions, one model is used, i.e. 8 inch nominal diameter, 0.1 inch thickness and 20 inch long large cylindrical part and 4 inch nominal diameter, 0.1 inch thichness and 23 inch long small cylindrical part and conical part of half apex angle of 450 and 0.1 inch thickness (figure 1).
The reducer pipe is subjected to internal pressure. Two cases of reducer pipes are considered : (1) the large and small ends being open, e.i. the reducer pipe is considered to be a part of a large piping system, (2) both ends being closed.
The reducer pipe is assumed to be of the same isotropic, homogeneous mild steel.

3.2. The finite Element Method and Model
The finite element model is constructed in three steps. The first step is the determination of the nodal points as indicated by small circle in figure 2.
The number of the chosen nodal points is forty and the resulting number of element is thirty six. The second step consist of refinement of the twelve element (see fig. 2). Each of twelve element is divided further into 5 element, except the conical element which is subdivided into 2 element (see fig. 3). The third step is the final refinement of each element that results from the first two steps. Ninety-six nodal points are added to each of the elements. The addition of nodal points in the third step are automatically done by MECHANICA.
The reason for dividing the twelve elements further is because from initial calculations it was found out that large stresses occurred there. Hence small element sizes in the region will locate the highest stress location more accurately. The distance between two nodal points of the smallest elements is now 0.1 inch.

3.3. The Types of Pipes Supports
 Four types of pipe supports are considered in the analysis, i.e. :
(1) the reducer pipe is simply supported at both ends
(2) the reducer pipe is fixed at both ends
(3) the reducer pipe is fixed at the large end
(4) the reducer pipe is fixed at the small end
 
 

3.4. The Finite Element Output
 MECHANICA computes the six stress components, ?x, ?y, ?z, ?xy, ?yz, ?xz, the principal stresses ?1, ?2, ?3, the displacements x, y, z and the rotations ?1, ?2, ?3, for each chosen and automatically generated nodal points. MECHANICA does not automatically print out all the computed stress and displacement values. It will print out computed stress and displacement values selected nodal point only.
 The further processing purposes of this paper, the six component stresses at line A0-B0-C0-D0 and A1-B1-C1-D1 of each case of pipe support type, at selected nodal points are read and noted. The lines just mentioned are shown on Figure 4 that follows. The shear stress values at all nodal point at the two lines above are practically zero, indicating that the normal stresses are principal stresses. Furthermore, one of the three principal stresses is zero, leaving the other two to be tangential and meridional/longitudinal stresses.

3.5. The Data Proccessing
 The Stresses, read and noted above, are then drawn as curves in twelve figures (i.e. Fig. 5 – Fig. 16) as function of location.
 For purposes of drawing and reading the stress vs. location diagrams, the following line and diagram definitions are described.

A0-B0-C0-D0  : Line of intersection of the reducer pipe outer surface and (any)   plane passing through the reducer pipe axis
A1-B1-C1-D1 : Line of intersection of the reducer pipe inner surface and (any) plane passing through the reducer pipe axis

 Location of points on the line of intersection are measure from the left or large end of the reducer pipe.

3.6. Data Presentation
 The stresses vs. location for points on various intersection lines are then drawn on twelve diagrams described below :
Figure 5: Tangensial stress on reducer pipe outer surface points at intersection line A0-B0-C0-D0 vs. location for the case of type of pipe support: simply supported and fixed at both ends.
Figure 6: Tangensial stress on reducer pipe outer surface points at intersection line A0-B0-C0-D0 vs. location for the case of type of pipe support: fixed at large end and fixed at small end.
Figure 7: Tangensial stress on reducer pipe inner surface points at intersection line A1-B1-C1-D1 vs. location for the case of type of pipe support: simply supported and fixed at both ends.
Figure 8: Tangensial stress on reducer pipe inner surface points at intersection line A1-B1-C1-D1 vs. location for the case of type of pipe support: fixed at large end and fixed at small end.
Figure 9: Meridional stress on reducer pipe outer surface points at intersection line A0-B0-C0-D0 vs. location for the case of type of pipe support: simply supported and fixed at both ends.
Figure 10: Meridional stress on reducer pipe outer surface points at intersection line A0-B0-C0-D0 vs. location for the case of type of pipe support: fixed at large end and fixed at small end.
Figure 11: Meridional stress on reducer pipe inner surface points at intersection line A1-B1-C1-D1 vs. location for the case of type of pipe support: simply supported and fixed at both ends.
Figure 12: Meridional stress on reducer pipe inner surface points at intersection line A1-B1-C1-D1 vs. location for the case of type of pipe support: fixed at large end and fixed at small end.
Figure 13: Tangensial stress on reducer pipe outer surface points at intersection line A0-B0-C0-D0 vs. location for the case of type of pipe support: fixed at small end open end and fixed at small end close end.
Figure 14: Tangensial stress on reducer pipe inner surface points at intersection line A1-B1-C1-D1 vs. location for the case of type of pipe support: fixed at small end open end and fixed at small end close end.
Figure 15: Meridional stress on reducer pipe outer surface points at intersection line A0-B0-C0-D0 vs. location for the case of type of pipe support: fixed at small end open end and fixed at small end close end.
Figure 16: Meridional stress on reducer pipe inner surface points at intersection line A1-B1-C1-D1 vs. location for the case of type of pipe support: fixed at small end open end and fixed at small end close end.

 There are three points of discussion, i.e. : (1) the influence of the reducer pipe’s types of support on the stress distribution along the outer and inner lines of intersection between reducer pipe outer and inner surfaces and any plane that passes through the reducer pipe axis, (2) comparison of stress values at the reducer pipe’s two junctions as obtained from numerical computation and from analytical formulas, and (3) differences of stresses values obtained from numerical computation for two cases, i.e. whether both reducer pipe ends are open or closed.

4.1. Influence of Reducer Pipe’s Types of Support
 Stress distributions along the outer and inner lines of intersection between the reducer pipe outer and inner surface and a plane that passes through the reducer pipe axes for the cases of simply supported reducer pipe and reducer pipe with both ends fixed are almost the same. The maximum tangensial and meridional stresses in the simply supported case are slightly higher than those in the case with both ends fixed. The maximum tangensial or meridional stress values do not necessarily occur at the reducer pipe junctions. If that is the cases they occur at the conical part of the reducer pipe.
 The reducer pipe with the small end fixed seems to experience the largest maximum or largest minimum (compresive) tangensial and meridional stresses compared to those of the other three cases with different types of support. These are as follows:
- The largest positive tangensial stress occurs in the reducer pipe with 8.18 ksi at the outer surface.
- The largest negative tangensial stress occurs in the reducer pipe with –15.42 ksi at the inner surface.
- The largest positive meridional stress occurs in the reducer pipe with 15.11 ksi at the outer surface.
- The largest negative meridional stress occurs in the reducer pipe with -22.77 ksi at the inner surface.

4.2. Comparison of Stress Values at the reducer Pipe Junctions Obtained from Numerical Computation and Analytical Formulas
 The stress values at the reducer pipe junction as obtained from numerical computation and analytical formulas are very different. Further investigation are necessary to find cause of it. It may happen that the assumptions used in the derivation of the analytical formulas (see Roark and Young (1976), page 499 and 500) were not realistically acceptable. It may also be that the package program MECHANICA does not yield the accurate values. In this case, another better finite element program should be used. Experimental investigation, especially stress (strain) measurements on the conical part, will further help clarify the cause of the differences.

4.3. Cases of Both Reducer Pipe Ends Open or Closed
 Only the case of reducer pipe with small end fixed are considered for the purpose of eliminating the difference between the two cases.
 The difference in the stress distribution along the upper and inner surface of the reducer pipe with both ends closed and both ends open is in the meridional stress distribution only, while for the tangensial stress distribution they are practically the same.
 The meridional stresses in the large cylindrical part of the reducer pipe with both ends open are zero and with smaller cylindrical part negative, while the meridional stresses for the reducer pipe with both ends closed are positive in the large cylindrical part as well as in the small cylindrical part.

Acknowledgment
 The research was funded by the director for research, Ministry of Education and Culture of the Republic of Indonesia, Contract Number 016/HTPP/URGE/1995.

References
1. Boardman, H.C., 1960, “Stresses at Junction of Cone and Cylinder in Tanks with Cone Bottoms or Ends,” Pressure Vessel and Piping Design Collected Paper, ASME, New York.
2. Bednar, Henry H, 1981, “Pressure Vessel Design Handbook,” Van Nostrand Reinhold Company, New York.
3. Johns, R.H., and T.W. Orange, 1961, “Theoretical Elastic Stress Distribution Arising from Discontinuities and Edge Loads in Several Shell-Type Structures,” NASA Tech. Report R-103.
4. Langer, B.F., 1971, “Design Stress Basis for Pressure Vessels,” Experimental Mechanical Journal Soc. Experimental Stress Analysis, Vol. 11 No. 1.
5. MECHANICA, 1994, “Model Reference: Structure and Thermal Release 7,” Rasna Corporation, San Jose, California.
6. Roark, R.J. and W. C. Young, 1975, “Formulas for Stress and Strain,” McGraw-Hill International Book Company
7. Zaremba, W.A., 1959, ”Elastic Interactions at the Junction of an Assembly of Axy-Symmetric Shells,” Journal Mechanical Engineering Science, Vol. 1 No. 3.
 

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