REFERENCE: Moore, I.D., “Profiled HDPE Pipe Response To Parallel Plate Loading”, Buried Plastic Pipe Technology: 2nd Volume, ASTM STP 1222, Dave Eckstein, Ed., American Society for Testing and Materials, Philadelphia, 1994.
ABSTRACT : The parallel plate load test is used t o measure ‘pipe stiffness’ for HDPE pipe. Pipe stiffness is employed as a measure of pipe resistance t o bending deformation as well as a quality control index for the manufacturing process. Unfortunately, the parallel plate test induces a complex state of stress and strain in the pipe, and interpretation of the test results is not straightforward. Simple analysis for a thin circular ring or shell is generally used for these products, but in reality materials like high density polyethylene are viscoelastic (modulus is time and load path dependent) and the depth of the pipe profile may be a significant proportion of the diameter. This paper introduces a three dimensional viscoelastic finite element analysis for HDPE pipe, testing the computational method through comparisons with laboratory data. The analysis is used t o examine the nature of pipe response during the parallel plate test. The local distributions of stress and strain through the profile are considered, as well as the effect of loading rate on the pipe response. Conclusions are drawn regarding the ability of conventional thin ring theory t o predict circumferential stress and strain, and the implications for pipe design are briefly discussed.
KEY WORDS: finite element analysis, high density polyethylene, pipe, viscoelasticity, local strain, stiffness.
‘Pipe stiffness’ is generally defined t o be the total vertical load applied t o a pipe segment, divided by the pipe length and change in vertical diameter. A simple theoretical relationship exists between pipe stiffness PS and the pipe radius T , modulus E and second moment of area I ,
This relationship is based on the initial bending stiffness of a thin elastic circular ring responding under two dimensional (plane stress) conditions.
Observations made during HDPE pipe tests clearly confirm that equation (1) is highly idealised. Firstly, HDPE exhibits a viscoelastic (time dependent) response which
‘Associate Professor, Geotechnical Research Centre, Faculty of Engineering Science, The University of Western Ontario, London, Ontario, N6A 5B9, Canada
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26 BURIED PLASTIC PIPE TECHNOLOGY
pipe
i
circumferential r
radial
FIG. 1 - Finite element model for three dimensional profiled pipe analysis makes the apparent value of E a function of the rate of loading and the time after application of loading at which load and deflection are measured. Secondly, many HDPE pipes have profile depth which is a significant proportion of the diameter, so that thin ring theory may be inappropriate. Finally ring deformations may be geometrically nonlinear, that is the ring under two point loading becomes noncircular and this affects the stiffness, [2].
A three dimensional finite element analysis has recently been developed by Moore [4]. This analysis can be used to examine the response of profiled HDPE pipe in the parallel plate test. It can be used to check a number of important design assumptions, both in relation to the parallel plate test and the behaviour of profiled HDPE pipe in the field.
The finite element analysis is described with details provided of the viscoelastic material model used. Elastic predictions are made of local strain distributions for one particular HDPE pipe profile, and these are compared with specific measurements made in the laboratory for HDPE pipe under parallel plate loading. The analysis is then used to make viscoelastic predictions of the load-deflection response of the pipe during a load- unload test, and these are also compared with experimental measurements. Distributions of circumferential and axial stress are predicted and discussed in relation to estimates based on thin ring theory. The effect of loading rate on pipe response is also considered. The paper concludes with a general discussion of the findings.
MOORE ON PARALLEL PLATE LOADING 27 A N A L Y S I S O F T H E P A R A L L E L P L A T E T E S T
The finite element method (e.g. Zienckiewicz, [6]) is an ideal computational pro- cedure for analysing problems in engineering mechanics involving complex geometries and material characteristics. However, the full three dimensional analysis of solids using con- ventional three dimensional finite element methods is a formidable task since it involves the generation of the three dimensional finite element mesh and the formulation and solution of huge numbers of equations. For corrugated pipes with annular (not helical or spiral) design, use can be made of the axisymmetric geometry to simplify the analysis. A two dimensional finite element mesh is then used to model the geometry and strain fields in the r,z plane, Figure 1, and a Fourier series is used to model variations around the pipe circumference.
Pipe response to each Fourier harmonic around the pipe is determined independently, and the full pipe response is assembled using superposition from each of these separate com- ponents. The result is a linear three dimensional analysis requiring relatively modest data preparation and computations, Moore [4].
The parallel plate test involves a short length of pipe subjected to load across the vertical pipe diameter, Figure 2a. The actual boundary condition at the crown and invert is one of prescribed displacement, since the pipe rests on a very stiff horizontal surface at the invert and the crown is deformed through a stiff steel plate mounted under the top load platten. Only short segments at the outside of corrugation crests attract load, both at the crown and invert positions.
To simulate the parallel plate test, the short length of annular pipe is modelled and small patches of pressure are applied at the outside of corrugation crests at crown and invert, Figure 2b. In all the analyses reported in this paper, the pipe loading and pipe response is assumed to be symmetric about the horizontal and vertical pipe diameters. The two dimensional finite element mesh used for the HDPE pipe is shown in Figure 3, featuring 1200 six noded linear strain triangular elements. The mesh represents one half of the pipe length 'cut' through the pipe perpendicular to the pipe axis. For the calculations reported here, response was evaluated using one hundred Fourier terms and it was assumed that the plates at crown and springline each made contact over an angle ~ measured from the pipe axis of 0.1 radians, Figure 2b. These choices are discussed in detail elsewhere, [3].
C O N S T I T U T I V E M O D E L
Linear viscoelastic finite element analysis is employed, based on conventional rhe- ology utilising sets of ~springs' and ~dashpots', Figure 4a. Unfortunately there is a dearth of test clata for HDPE in the literature. However, Chua [1] has reported and used a power law model. This model defines the uniaxial 'secant' relaxation modulus E -- r (in psi) at 70~ as
E(t) = 7630 + 99507t -~176 (2)
where time is expressed in minutes, Figure 4b. Undertaking a simple curve-fitting exercise for a multi-Kelvin model with one independent spring and nine Kelvin elements in series, the independent spring is found to have modulus 1000 MPa. The Kelvin springs have moduli 3580MPa, 0.85x3580MPa, 0.852x3580MPa, ..., 0.85Sx3580MPa. The Kelvin dashpots have viscosities 0.503MPa.days, 5.03MPa.days, 50.3MPa.days, up to 5.03xl0~MPa.days. This conventional rheological model is used to predict the time dependent response of the pipe.
j rigid steel