Pipe finite element

[Hurricanes]

Can anyone point me to, or explain in some detail the theory of pipe elements and how they are used in pipe stress software?

I know the basic formulation from the beam on elastic foundation theory, but I am curious to see how piping programs such as AutoPIPE and Caesar use this to get the maximum total stress at a point on the cross-section of the pipe.

Do they use multiple integration points, formulate different beams every 15 degrees around the circumference of the pipe or something else?

 

[StevenHPerry]

C2 and AutoPIPE are not finite element programs. They use stick figure representations of piping systems with properties that mimic the pipe.

My understanding of AutoPIPE's general stress report is that it calculates a "real" (as opposed to code) stress at 15° intervals from the components of force and moment taken from the stick-figure analysis.

It is not looking at stress flow at transitions, and I bet it would still report a stress at a branch header wall where no metal remains.

For that kind of analysis, Paulin Research Group might be able to help you out.

http://www.prg-software.com

If you want more information, you might get some informative answers if you post to the CAESAR2 forum on the Coade site.

- Steve Perry

http://www.linkedin.com/in/stevenhperry

This post is designed to provide accurate and authoritative information in regard to the subject matter covered. It is offered with the understanding that the author is not engaged in rendering engineering or other professional service. If you need help, get help, and PAY FOR IT.

 

[Hurricanes]

Steven,

Thanks for your reply. I am not looking for any specific analysis package/capabilities, just curious on the inner workings of AutoPipe/Caesar.

You are not correct when you say they are not finite element programs, they are; using beam and curved beam elements.

I guess I am after a more in depth breakdown of these elements than I have been able to find so far.

Maybe I will repost this in the CAESAR2 forum, thanks again.

 

[ColonelSanders83]

Hello Hurricanes,

I lifted the following information from the AutoPIPE help files. Autopipe and ceaser both make use of line type beam elemets with each point having six degrees of freedom. As such an analysis must keep in mond the limitations of beams whe=n performing an analysis, for example any natural frequencies calculated over 100-150 hz will generally not be accurate, ect. I lifted the following information from the AutoPIPE help files.

"In order to study the behavior and response of a piping system subjected to various types of loading, the system must be analyzed using the commands found in the Result menu. AutoPIPE provides the latest structural analysis techniques to perform this task. These techniques were originally derived from the structural finite element program SAP IV, developed by the College of Engineering at the University of California at Berkeley. Since then, these techniques have been enhanced or complemented by other methods to provide the most efficient solution algorithms.

AutoPIPE uses the finite element method, also known as the stiffness or displacement method, to mathematically model the piping system in three dimensional space. The system is formulated into a set of linear equations, describing the system characteristics at each degree of freedom (DOF). Each point (e.g., piping, soil, beam, etc.) in the system has six (6) degrees of freedom. The exact formulation of these equations and their solution depends on the type of loading being investigated. Each option, and its formulation, is described in the following areas."

"Linear Analysis - Gaps, Friction and Soil Yielding Ignored
The linear analysis is a simplified static analysis and ignores all nonlinear effects which may or may not be defined in the piping system. This type of analysis requires the solution of a system of linear equations using matrix algebra. Equation G-1 shows the primary structure of the analysis strategy:

[K]*=[R] (G-1)

where:

K
=
structural stiffness matrix

U
=
response (or displacement) matrix

R
=
applied load matrix


The structural stiffness matrix [K] is derived by assembling the equilibrium equations of each element in the structural system. The load matrix [R] is constructed from the loads acting on the elements and points in the system. In general, [R] may contain more than one load case.

The solution for the displacement matrix is obtained by using the Gauss elimination method. This matrix is then used to determine element end forces and moments. In turn, this information is used to calculate stresses at each defined point."

"Modal Analysis Theory
Dynamic loadings have a tendency to increase the response of the structure beyond the response obtained if the same load was applied statically. To determine this response, the undamped frequencies and modes of vibration of the structure are required. This is achieved by the solution of generalized eigenvalue problem:

[K]*[f]=[w]^2*[M]*[f] (G-2)

where:

K
=
stiffness matrix

f
=
matrix of mode shapes

w
=
diagonal frequency matrix

M
=
mass matrix


AutoPIPE lumps the mass of the pipe, components and contents, etc. at the associated node point. This assumption yields a diagonal mass matrix with no mass coupling terms. There are three mass degrees of freedom per node. Rotational mass is ignored, except for points with eccentric weights, specified using the Insert/Xtra data/Weight command). At these points there may be up to three additional rotational masses and thus three additional mass degrees of freedom.

It should be noted that for the eccentric weight, the coupling terms between the translational and rotational degrees of freedom are neglected. The structure is assumed to be linear and thus all gaps, friction and soil yielding is ignored.

The solution to the generalized eigenvalue problem is obtained by using the subspace iteration with progressive forward shifting technique. This procedure is an iterative process and uses trial vectors to converge to an eigenvalue. Converged eigenvectors are used as initial trial vectors for the next eigenvalue."

"Response Spectrum Analysis
A response spectrum analysis is used to determine the response of a structural system subjected to earthquake excitation. The equation of dynamic equilibrium associated with the response of the structure subjected to ground motion is:

[M]*[u'']+[c]*[u']+[K]*=[M]*[u''g](G-4)

where:

M
=
mass matrix of the structure

u
=
structural response (displacement) matrix


u'
=
structural response (velocity) matrix

u''
=
structural response (acceleration) matrix

C
=
structural damping matrix

K
=
structural stiffness matrix

u''g
=
ground acceleration matrix


A response spectrum analysis uses results from the modal analysis to obtain a solution. Thus, a modal analysis must be performed before a response spectrum analysis is initiated. All assumptions made during the modal analysis apply.

The ground excitation can consists of three independent ground accelerations, acting simultaneously in three global directions as shown in Equation G-4.

u''g=u''gx+u''gy+u''gz (G-5)

For a particular ground excitation, the response spectrum is constructed by computing the maximum response of a series of single DOF oscillators to the excitation. Using the response spectrum, the maximum response in each direction for each mode is calculated. Modal responses in each direction are combined using the specified combination method. The final response is calculated by combining the response from three directions using Square Root of the Sum of the Squares (SRSS) method. The methods described in the following subsections are available in AutoPIPE to combine the response of individual modes."

"Harmonic Analysis
A harmonic force analysis is used to analyze the effect of vibration due to oscillating loads. Harmonic forces can arise from unbalanced rotating equipment, acoustic vibrations caused by reciprocating equipment, flow impedance, and other sources. These forces can be damaging to a piping system if their frequency is close to the system's natural frequency, thereby introducing resonant conditions. The equation of dynamic equilibrium associated with the response of the structure subjected to harmonic forces is:

[M]*[u'']+[c]*[u']+[K]*=sin(wt)*[F](G-14)

where:

M
=
mass matrix of the structure

u
=
structural response (displacement) matrix


u'
=
structural response (velocity) matrix

u''
=
structural response (acceleration) matrix

C
=
structural damping matrix

K
=
structural stiffness matrix

w
=
frequency of the applied force

F
=
maximum magnitude of the applied force

t
=
time


A regular periodic force oscillates from a maximum value in one direction to the same value in the opposite direction, at regular interval. Such a forcing function may be considered sinusoidal or simple harmonic in nature. More complex forms of vibration, such as those caused by the fluid flow, may be considered a superposition of several simple harmonics, each with it own frequency, magnitude, and phase.

A harmonic analysis uses the results from the modal analysis to obtain a solution. Thus, a modal analysis must be performed before a harmonic analysis is initiated. All assumptions made during the modal analysis apply. A single damping factor must be specified for all mode shapes.

First, the maximum response of each harmonic is obtained separately. Then the total response of the system is determined by combining the individual responses. The combination method may be specified as the maximum or RMS harmonics (RMS=0.707* SRSS of all harmonics).

Limitations
In cases where the dynamic load is applied very near a support or directly at the support, the support reaction may be near zero or very much less than the actual reaction. The reason is because the mode shapes involving the movement between the applied load direction and the support point were not computed as specified by the number of modes or cut-off frequency. In order to capture the effect of these higher modes, it is recommended that the cutoff frequency should be at least 50% higher than the harmonic frequency and that the missing mass or ZPA corrections are used."

There are more types of analysis that can be performed (listed below) but I think this should be enough to satisfy your curisity.

Static Analysis

Equivalent Linear Analysis

Hanger Analysis

Modal Analysis

Response Spectrum Analysis

Harmonic Analysis

Force Spectrum

Seismic Anchor Movement Analysis

Time History

Thermal Bowing Analysis

Fluid Transient Analysis

Steam Relief Analysis















A question properly stated is a problem half solved.

Always remember, free advice is worth exactly what you pay for it!

http://www.ap-dynamics.ab.ca/