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Pipelines have been acknowledged as the most reliable, economic and efficient means for the transportation of gas and other commercial fluids such as oil and water. The designation of pipeline system as “lifelines” signifies that their operation is essential in maintaining the public safety and well being. A pipeline transmission system is a linear system which traverses a large geographical area, and soil conditions thus, is susceptible to a wide variety of hazards. This pa-per is concerned with the dynamic behavior of buried town gas pipelines. A computer model with a finite number of nodes is created to simulate the behavior of the real gas pipeline. The dynamic susceptibility method is applied for twenty mode shapes of this model, which utilizes the stress per velocity method and is an incisive analytical tool for screening the vibration modes of the system. It can be readily identified, which modes, if excited, could potentially cause large dynamic stresses. This paper discusses also two of the piping dynamic analyses, namely the effect of the response spectrum of an earthquake and the time history analysis of a truck crosses the pipeline.

Gas distribution systems are one of six broad categories of infrastructure grouped under the heading “lifelines”. Together with electric power, water and liquid fuels, telecommunications, transportation and wastewater facilities, they provide the basic services and resources upon which modern communities have come to rely, particularly in the urban context. Disruption of these lifelines through damage can therefore have a devastating impact, threatening life in the short term and a region’s economic and social stability in the long term. Pipeline system consists of buried and above ground pipelines, above ground facilities such as pumping stations, storage tanks and miscellaneous terminal facilities. However the term pipeline in general implies a relatively large pipe spanning a long distance.

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The liquefied soil will do great damage to buried pipeline during an earthquake. It causes a floating force and leads to floating response that is a dynamic process varying with time. [

In addition to the modal frequencies and mode shapes, the pipeline modal analysis includes new outputs called dynamic stresses and dynamic susceptibility. These new outputs are based on modal analysis method by a PC computer program. The dynamic susceptibility method is essentially a post processor to fully exploit the modal analysis results from the piping system [11,12]. The pipeline designer does not have either the specific requirements, or the analytical tools and technical references which are typically available for other plant equipment such as rotating machinery. Piping vibration problems only become apparent at the time of commissioning and early operation, after a fatigue failure or degradation of pipe supports.

The underlying theoretical basis for the Stress/Velocity method is a deceptively straightforward but universally-applicable relationship between kinetic energy and potential (elastic) energy for vibrating systems. Stated simply, for vibration at a system natural frequency, the kinetic energy at maximum velocity and zero displacement must then be stored as elastic (strain) energy at maximum displacement and zero velocity [

At the design stage, the dynamic susceptibility method allows the designer to quickly identify and correct features that could lead to large dynamic stresses at frequencies likely to be excited. The method provides a quick and incisive support to efforts of observation, measurement, assessment, diagnosis, and correction. Furthermore, it reveals which features of the system layout and support are responsible for the susceptibility to large dynamic stresses. The dynamic stresses are the dynamic bending stresses associated with vibration in a natural mode [

Buried gas pipelines structures, when subjected to loads or displacements, behave dynamically. If the loads or displacements are applied very slowly then the inertia forces can be neglected and a static load analysis can be justified [

The underlying theoretical basis for the Stress/Velocity method is a deceptively straightforward but universallyapplicable relationship between kinetic energy and potential (elastic) energy for vibrating systems. Stated simply, for vibration at a system natural frequency, the kinetic energy at maximum velocity and zero displacement must then be stored as elastic (strain) energy at maximum displacement and zero velocity. Since the strain energy and kinetic energy are respectively proportional to the squares of stress and velocity, it follows that dynamic stress, ó, will be proportional to vibration velocity, v. For idealized straight-beam systems, consisting of thin-walled pipe and with no contents, insulation or concentrated mass, the ratio ó/v is dependent primarily upon material properties (density ρ and modulus E), and is remarkably independent of system-specific dimensions, natural-mode number and vibration frequency.

For real continuous systems of course, the kinetic and potential energies are distributed over the structure in accordance with the respective modes shapes. However, integrated over the structure, the underlying energyequality holds true. Provided the spatial distributions are sufficiently similar, i.e. harmonic functions, the rms or maximum stress will still be directly related to the rms or maximum vibration velocity.

As stated above, for idealized pure beam systems the stress-velocity ratio will depend primarily upon material properties. For real systems, the spatial patterns of the mode shapes will depart from the idealized harmonic functions, and the stress-velocity ratios accordingly increase above the theoretical minimum or baseline value. System details causing the ratios to increase would include the three-dimensional layout, large unsupported masses, high-density contents in thin-walled pipe, susceptible branch connections, changes of cross section etc. The more “unfavorable” is the system layout and details, the larger will be the ó/v ratios for some modes.

Thus, the general susceptibility of a system to large dynamic stresses can be assessed by determining the extent to which the ó/v ratios for any mode exceed the baseline range. Furthermore, by determining which particular modes have the high ratios, and whether these modes are known or likely to be excited, the at-risk vibration frequencies and mode shapes are identified for further assessment and attention. This is the basis of the Stress/Velocity method of analysis and its implementation as the “dynamic susceptibility” feature in [

There are various general and application-specific acceptance criteria based upon vibration velocity as the quantity of record. Some, in order to cover the worst case scenarios, are overly conservative for many systems. Others are presented as being applicable only to the first mode of simple beams, leading to the misconception that the Stress/Velocity relationship does not apply at all to higher modes. In any case, there are real and perceived limitations on the use of screening acceptance criteria based upon a single value of vibration velocity.

The dynamic susceptibility method turns this apparent limitation into a useful analytical tool! Specifically, large Stress/Velocity ratios, well above the baseline values, are recognized as a “warning flag”. Large values indicate that some feature(s) of the system make it particularly susceptible to large dynamic stresses in specific modes.

The Dynamic Susceptibility method is essentially a post processor to fully exploit the modal analysis results of the system. Mode shape tables of dynamic bending stress and vibration velocity are searched for their respective maxima. Dividing the maximum stress by the maximum velocity yields the “ó/v ratio” for each mode. That ratio is the basis for assessing the susceptibility to large dynamic stresses. Larger values indicate higher susceptibility associated with specific details of the system.

The Stress/Velocity method has been implemented as additional analysis and output of the CAEPIPE modal analysis. The modal analysis load case now includes additional outputs and features as follows:

Dynamic Stresses: This output provides the “mode shapes” of dynamic bending stresses, tabulated along with the conventional mode shape of vibration magnitude.

Dynamic Susceptibility: This output is a table of s/v ratios, in psi/ips, mode by mode, in rank order of decreasing magnitude. In addition to modal frequencies and s/v ratios, the table also includes the node locations of the maxima of vibration amplitude and bending stresses.

With the dynamic susceptibility output selected, the animated graphic display of the vibration mode shape includes the added feature of color spot markers showing the locations of maximum vibration and maximum dynamic bending stress. These outputs will assist the designer through a more-complete understanding of the system’s dynamic characteristics. They provide incisive quantified insights into how specific details of components, layout and support could contribute to large dynamic stresses, and into how to make improvements.

The Stress/Velocity method of assessment, and its implementation in CAEPIPE as dynamic susceptibility, is based entirely upon the system’s dynamic characteristics per se. Thus the vibration velocities and dynamic stresses employed in the analysis, although directly related to each other, are of arbitrary magnitude. There is no computation of the response to a prescribed forcing function, and no attempt to calculate actual dynamic stresses. Thus the dynamic susceptibility results do not factor directly into a pass-fail code compliance consideration. Rather, they assist the designer to assess and reduce susceptibility to large dynamic stresses if necessary, in order to meet whatever requirements have been specified.

The model is created based on an actual pipeline used to transmit natural gas to three buildings. This model was created using software package CAEPIPE. The structural analysis performed by this software is in compliance with a standard piping code ASME B31.3 (pressure process piping code) [^{3}.

The model is buried. The ground level, i.e. the height of soil surface is one meter over pipeline. The soil is cohesion less type and has the density of 1922 kg/m^{3}. The angle of friction between soil and pipe (delta) is 20 degrees and the coefficient of horizontal soil stress (K_{s}) is 0.3. It is assumed that all fittings and pipeline components (anchors, valves, tees, bends, and reducers) are

made of the same material of pipeline (SS A53 Grade A). It is also assumed that the three ends of the pipeline are fixed at nodes 60, 80 and 120.

A node refers to a connecting point between elements such as pipes, reducers, valves, and so on. A node has a numeric designation. For bend nodes, the node number is followed by a letter such as A/B. A and B nodes (e.g., 110A, 110B) designate the near and far end of a bend node. Node number refers to the location of the tangent intersection point (TIP); it is not physically located on the bend. Its purpose is only to define the bend. Soil modeling is based on Winkler’s soil model of infinite closely spaced elastic springs. Soil stiffness is calculated for all three directions at each node. Pressure value in the load is suitably modified to consider the effect of static overburden soil pressure [

An anchor, is a type of support used to restrain the movement of a node in the three translational and the three rotational directions (or degrees of freedom; each direction is a degree of freedom) [

The key analytical step is to determine, mode by mode, the ratio of maximum dynamic stress to maximum vibration velocity. This ratio will lie in a lower baseline range for uncomplicated systems such as classical uniformbeam configurations. For more complex systems, the Stress/Velocity ratio will increase due to typical complications such as three-dimensional layout, discrete heavy masses, changes of cross-section and susceptible branch connections. System modes with large stress-velocity ratios are the potentially susceptible modes [

The dynamic stresses table (like

There are two main types of analyses. The first is conceptual, where the structure does not yet exist and the analyst is given reasonable leeway to define geometry, material, loads, and so on. The second analysis is where the structure exists, and it is this particular structure that must be analyzed [

In the following cases time history and response spectrum analysis is performed for the pipeline model. Time history force is a function of time at all changes in directions (bends/tees). These separate force-time histories are then applied separately as Time Varying Loads in CAEPIPE at the corresponding nodes in the piping model. Time functions describe the variation of the forcing function with respect to time. A study of an earthquake effects is presented using response spectrum technique.

The earthquake safety of buried pipelines has attracted a great deal of attention in recent years. Important characteristics of buried pipelines are that they generally cover large areas and are subject to a variety of geotectonic hazards. Another characteristic of buried pipelines, which distinguishes them from above-ground structures and facilities, is that the relative movement of the pipes with respect to the surrounding soil is generally small and the inertia forces due to the weight of the pipeline and its contents are relatively unimportant. Buried pipelines can be damaged either by permanent movements of ground (i.e. PGD) or by transient seismic wave propagation.

The concept of response spectrum, in recent years has gained wide acceptance in structural dynamics analysis, particularly in seismic design. Stated briefly, the response spectrum is a plot of the maximum response (maximum displacement, velocity, acceleration or any other quantity of interest), to a specified loading for all possible single degree-of-freedom systems. The abscissa of the spectrum is the natural frequency (or period) of the system, and the ordinate, the maximum response. One of the widely used methodologies for describing the behavior of a structural system subjected to seismic excitation is the response spectrum modal dynamic analysis [

In general, response spectra are prepared by calculating the response to a specified excitation of single degree-of-freedom systems with various amounts of damping [

Since the response spectra give only maximum response, only the maximum values for each mode are calculated and then superimposed (modal combination) to give total response. A conservative upper bound for the total response may be obtained by adding the absolute values of the maximum modal components (absolute sum). However this is excessively conservative and a more probable value of the maximum response is the square root of the sum of squares (SRSS) of the modal maxima. In the SRSS method, displacements, element forces, and support loads from the three X, Y and Z accelerations are squared individually and added [

Any phenomenon that gives rise to loads that vary with time can be input into the software CAEPIPE for time history analysis to get the variation of forces or moments with respect to time at different points in the piping system [

where:

[M] = diagonal mass matrix

[C] = damping matrix

[K] = stiffness matrix

{u} = displacement vector

{u} = velocity vector

{u} = acceleration vector

{F(t)} = applied dynamic force vector

The time history analysis is carried out using mode superposition method. It is assumed that the structural response can be described adequately by the p lowest vibration modes out of the total possible n vibration modes and p < n. Using the transformation u = ΦX, where the columns in are the p mass normalized eigenvectors, Equation (1) can be written as:

where

In Equation (2), it is assumed that the damping matrix [C] satisfies the modal orthogonality condition

Equation (2) therefore represents p uncoupled second order differential equations. These are solved using the Wilson method, which is an unconditionally stable step-by-step integration scheme. The same time step is used in the integration of all equations to simplify the calculations.

A direct analytical approach to the problem of the earthquake analysis is to subject the pipeline model to accelerations as recorded in actual earthquakes [