Buckling analysis of PVC sheets with RIB

Buckling Analysis of PVC Sheets with RIB

Paulo Carlos Kaminski

Escola Politécnica da Universidade de São Paulo. Departamento de Engenharia Mecânica

Av. Prof. Mello Moraes 2231

05508-900 São Paulo. SP. Brazil

Luiz Bandeira de Mello Laterza

AQUEDUTO

Rua Fradique Coutinho 1896

05416-002 São Paulo. SP. Brazil

This study has a technical and applied character. A PVC structured wall pipe can be produced by spirally winding a ribbed sheet having a male-female lock, chemically welded by an adhesive. These pipes are "flexible" and are used mainly in underground installations, to convey fluids in free duct regime. Initial studies have indicated that the buckling resistance of the ribs from the sheet coiling to the tube manufacturing is the critical design parameter. This study presents the theoretical analytical development in order to obtain the critical buckling moment of these sheets. This analysis uses concepts initially developed to calculate buckling resistance in monosymmetrical profiles that are very used in the metallic structure industry. Since the material used was PVC, that has different mechanical properties than steel and aluminum, it was necessary to consider the differences in the analytical treatment. It is important to emphasize that the results obtained are product of the co-operative work of engineers from industry and university.

Keywords: Buckling, structured wall pipe, PVC

Introduction

The structural wall pipe is produced by spirally winding a PVC ribbed sheet having a male-female lock, chemically welded by an adhesive. These sheets are produced continuously by extrusion process and stored in steel reel with 1500 mm of diameter.

Theses pipes are used in underground installations and designed for resisting permanent and accidental loads. They are indicated to convey fluids in free or forced duct regime in low pressure, with emphasis in pluvial drainage. The great advantage for this type of pipes is its great stiffness/weight relation. This is accomplished by its rib disposal (Figures 1 and 2).

The sheet main dimensions used for pipe construction are schematized in Figure 2.

Nomenclature

Aa = web area (mm²)

Aa = web area (mm²)

Af = flange area (mm²)

Al = sheet area (mm²)

Ar = rib area (mm²)

b = spacing between ribs (mm)

E = elasticity modulus (N/mm²)

Er = reduced elasticity modulus (N/mm²)

ea = web thickness (mm)

eb = base thickness (mm)

ef = flange thickness (mm)

Gr = shearing elasticity modulus (N/mm²)

h = height between flange and base center (mm)

ha = web height (mm)

hr = rib height (mm)

If = flange index (mm)

It = torsional inertia momentof rib (mm)

Iw = warping moment (mm⁶)

Ix = area inertia moment (mm⁴)

Kt = pipe linear rigidity (kPa)

Lf = buckling length (mm)

lf = flange width (mm)

Me = winding moment (Nmm)

Mcrit’ = buckling critical moment(Nmm)

y_C.G. = neutral line height (mm)

Greek Symbols

DDy = diameter reduction in the y direction (mm)

ee = induced deformation (adim)

fcar = medium diameter of reel (mm)

fi = pipe diameter (mm)

m = index of webs area (adim)

n = Poisson coefficient.

r = monosymmetric index (mm)

se = stress (N/mm²)

From the analysis of Figure 2 we obtain that the number of sheet ribs, called n, is the ratio between the profile width and the spacing between the ribs. Therefore, the following expression is always valid:

Figure 3 shows in a simple way the dimensions of the typical rib of a ribbed sheet.

Geometric Properties of Sheet and RIB

Section Area and Position of Neutral Line

Denominating: Af (lf.ef) flange area; Ab (b.eb) base area; and Aa (ha.ea) web area, the section area section of the rib (Ar) and the sheet (Al) are respectively:

Equation (3) is approximated due of the male-female lock. Empirically the area increment is approximately 10% to the actually utilized profiles.

The index of the web area (m) is given by the expression:

According to Figure 4 the distance between the neutral line and the base is given by the expression:

Inertia Moments

The main inertia moments of the ribs are given by the expression:

The torsional inertia moment of the rib (the profile is opened and thin by hypothesis) is given by the expression:

The monosymmetric index of the rib (r) indicate the relation between the inertia moment of the base and the sum of the web and base inertia moments around the y axis. Thus:

It is important to note that if the rib has two planes of symmetry (for example section "I") the index is 0,5; but if the rib has reversed "T" section the monosymmetric index is 1. Therefore, the index provides information about the "order" of rib monosymmetry.

The warping moment for the kind of rib section analyzed is determined in function of the monosymmetric index (Kitipornchai and Trahair, 1980), and it is given by the expression:

Criticals Parameters of Project

The basic objective requested to the construction of pipes by ribbed sheet is to have the maximum rigidity after manufactured. It is also important to avoid the buckling due to the flexion and torsion of the sheet rib. The engineering problem is to attend this objectives with the minimum sheet section area possible. Note that the sheet section area is directly proportional to the amount of material used per meter of a determined pipe dimension, so it is function of pipe manufacture price. When rib height increases the rigidity of sheet and duct increase too. Next it will be present the analytic development of the parameter that influence the mentioned phenomenon.

Deformation by Winding

The PVC sheet is produced in continuous flux by an extrusion process. In manufacturing, the PVC goes through a cooling conduct and is wound in a steel reel for storage and transportation. So the "natural" configuration of the sheet isn’t straight, but it is curved with curvature radius approximately equal to the storage reel radius.

For manufacturing of the pipe, the sheet is uncoiled and tractioned by an equipment that will wind the sheet in spiral with pipe diameter. Thus, if the pipe doesn’t have the same reel diameter, a deformation (ee) will be introduced, and it is given by (Kaminski, 1998):

where:

f i winding diameter (pipe)

f car medium diameter of reel

Reduced Elasticity Modulus

From Figure 6 it can be noticed that the PVC stress x deformation graph doesn’t have a linear behavior. Thus, the elasticity modulus can’t be taken constant; but it decreases in function of the deformation.

From data obtained empirically (Carlowitz, 1990 and Kaminski, 1998) and approximating the relation s x e for a graph of second order, it is obtained the expression:

where:

E Elasticity modulus of the material (for small deformations)

Er Reduced elasticity modulus of material

Stress and Moment by Winding

The stress by winding, considering the reduced elasticity modulus, is given by:

The rib moment due to the winding (Me) is given by (Moser, 1990):

Pipe Rigidity

The diameter reduction in the y direction, for the ring in compression, is given by (Hartog, 1977):

Therefore the ring rigidity is given by:

Thus the structure pipe linear rigidity (by length unity) is given by (Kaminski, 98):

Buckling Critical Moment

Side buckling of beams under flexure is the subject of some classical works (Timoshenko & Gere, 1961), considering just bisymmetrical beams, such as the cross rectangular one. The buckling mechanism involve torsion, when the torsional stiffness of the beam is much lower then the flexure one, which is the feature of opened thin-walled sections, such as the outline used in pipes with ribs. This phenomenon was first equationed by Wagner (1936).

Usually, the project of bi and monosymmetrical beams are based mainly in the calculation of the critical elastic stress for the buckling by flexion and torsion. Several works have documented solutions for this phenomenon for bisymmetrical beams. However, it is difficult to find one with a solution for this phenomenon for the monosymmetrical ribs (Pi & Trahair, 1992).

In this work, the sheet was considered to be the assembly of laterally localized beams (rib). The transmission of lateral effects of beams on the next in transverse was not considered. Also, the profile propeller angle by winding was despised. Thus it can be considered that the ribs plus collaborator plate of the base was curved with the pipe curvature radius.

With these hypotheses, the buckling critical moment is given by (Kitipornchai and Trahair, 1980):

where c2 is:

From empirical estimation the buckling length can be considered to be L=10h. The shearing elasticity modulus is given by:

The K value is:

The d value is given by

The parameter Bx appears due to the resultant torque of traction and compression stress that appears when the beams twist during the buckling. This phenomenon is called "Wagner effect" (Hartog, 1987). In case of bisymmetrical profile the value of Bx is 0, because the resultant torque is null. However for monosymmetrical profile the torque isn’t null causing a change in torsional rigidity.

It is not possible to evaluate this parameter directly. So, mainly for project finality, it was developed an approximated formulation for the parameter Bx. The professional envelopment with the problem was interested in steel profile where the monosymmetric index is near 0,5. These approximations cause big differences in work dominion for this case (monosymmetric index within 0,9 and 1), so they aren’t adequate for this case.

Thus, the adopted approximation (Wang & Kitipornchai, 1989) was:

Flange Index

The comparison between the winding moment and buckling critical moment provides the information about the possibility or not of occurring the rib buckling. Theoretically, the buckling occurs if the winding moment is bigger than the critical moment.

The definition of flange index is very useful, and is given by:

From evaluation of this parameter it is possible to improve the model. This can be done, for example, winding a profile kind in many different diameters of pipe, and checking the practical result of buckling and the flange index value. So it is obtained a trustful cut value, in which below it there should be no conditions to work. Through this cut value it is possible to optimize the rib in relation to the mass (section area), obtaining a pipe with the necessary rigidity and with the minimum consumption of material.

Case Study

Table 1 shows the numeric results for a PVC pipe with a diameter of 400 mm. It is important to note that this is the minimum diameter that can be used with this sheet.

Conclusions

This study tried to show the procedure to obtain the necessary design parameters to the development and manufacturing of PVC structured pipes. From the initial studies, the buckling resistance of the nervures when the sheet is winding for the tube manufacturing was identified as the critical design parameter. The theoretical analytical development to obtain the buckling critical moment of these sheets has been showed. Then, it has been used develop concepts to calculate buckling resistance of monosymmetrical profiles, which are is very typical in metallic structure industries.

The mechanical behavior differences between PVC and steel and aluminum have been incorporated in the analytical development.

The development was consolidated in an electronic spreadsheet to create an agile and simple design tool which allows the study of the most adequate profiles to each kind of pipes and their applications.

The results are consequence of a co-operative work of engineers form Industry and University, which show the importance of this partnership to the Brazilian industrial sector.

Manuscript received: March 2000. Technical Editor: Paulo Eigi Miiyagi.

• Carlowitz, B., 1990, "Tabellarische Übersicht über die Prüfung von Kunststoffen", 5. Auflage, Kunststoff Verlag.
• Hartog, J.P., 1977, "Strength of Materials", Dover Publications.
• Hartog, J.P., 1987, "Advanced Strength of Materials", Dover Publications.
• Kaminski, P.C., 1998, "Memorial de Cálculo RIBLOC", Tubos e Conexões Tigre.
• Kitipornchai, S.; Trahair, N. S., 1980, "Buckling Properties of Monosymmetric I - Beams", Journal Of the Structural Division, vol 106, n. ST5.
• Moser, A.P., 1990, "Buried Pipe Design", McGraw-Hill.
• Pi, Y. L.; Trahair N. S., 1992, "Prebuckling Deflections and Lateral Buckling I - Theory", Journal of Structural Engineering, vol 118, n. 11.
• Timoshenko, S.P.; Gere, J.M., 1961, "Theory of elastic stability", 2. edition, McGraw-Hill.
• Wagner, H., 1936, "Torsion and Buckling of Open Sections", Translated Technical Memorandum No. 807, NACA - National Advisory Commitee for Aeronautics.
• Wang, C. M.; Kitipornchai, S., 1989, "New Set of Buckling Parameters for Monosymmetric Beam-Columns / Tie Beams", Journal of Structural Engineering, vol 115, n. 6.

Publication Dates

• Publication in this collection
  04 Oct 2001
• Date of issue
  2000

History

• Received
  Mar 2000