Influence of gap size on the dynamic behaviour of piping systems

This paper addresses the effects of dynamic events induced by support motion on piping systems with snubbers having variable gap sizes. The investigation consists of 3 parts: (i) Mathematical examination of a linear I DOF mass-spring-snubber system with gap size zero or infinity. (ii) Numerical analysis of a piecewise linear I DOF mass-spring-snubber system with varying gap sizes, by means of a simple computer code. (iii) Numerical analysis of a realistic three-dimensional piping system with 3 snubbers, each having 4 different gap sizes, with the aid of a non-linear F.E. code.     

Analysis of piping with hysteretic supports using response spectra

This paper introduces a response spectra-based method for analyzing piping with hysteretic nonlinear supports. The method is developed to be as simple and versatile as possible, yet accurate enough to model the essential nonlinear behavior of the supports. The required data is the hysteresis loops of the supports, the linear properties of the piping, and the linear acceleration response spectra. The supports are modeled by equivalent linear stiffness and damping, and the combined piping/support system is analyzed using complex modal properties that account for high-damping effects. The final peak response is obtained by a mode combination rule which is a new generalization of Complete Quadrature Combination (CQC) that accounts for nonlinear properties and complex modes. Sensitivities that indicate the degree of nonlinear behavior and support interaction are also determined. The method is used to analyze two three-dimensional piping systems with multiple nonlinear supports, which have been tested on a shaking table. Comparisons between experimental and analytical results show good agreement.     

Inelastic analysis methods for piping systems

The analysis of pipework systems which operate in an environment where local inelastic strains are evident is one of the most demanding problems facing the stress analyst in the nuclear field. The spatial complexity of even the most modest system makes a detailed analysis using finite element techniques beyond the scope of current computer technology. For this reason the emphasis has been on simplified methods. It is the aim of this paper to provide a reasonbly complete, state-of-the-art review of inelastic pipework analysis methods and to attempt to highlight areas where reliable information is lacking and further work is needed.     

Outline of dynamic analysis for piping systems

The paper deals, in a coherent mathematical manner, with a number of theoretical and practical problems involved in the solution of dynamic structural problems by the modal superposition method. In particular, problems related to piping system structures subject to single or multi-level support movement loading are treated.     

Stress analysis of non-uniform thickness piping system with general piping analysis software

Most general piping analysis software can only perform ASME design stage type code compliance analysis with uniform pipe wall thickness. However, non-uniform wall thickness, commonly on elbows or bends, can be found in many industrial applications. A typical example is thinned non-uniform thickness at bends or elbows caused by flow accelerated corrosion (FAC). In this paper, an analysis procedure is introduced to enable a general piping software to conduct ASME III class 1 piping analysis with non-uniform wall thickness. The demonstration is performed on CANDU (Canadian Deuterium Uranium) feeder pipes, which have been subjected to FAC caused wall thinning. The results are compared with both conventional uniform thickness piping analysis and non-uniform thickness solid finite element analysis. The comparison shows the validity of the proposed “average-minimum-average” approach by employing the general piping analysis software. The approach remains conservative compared to the benchmark solid finite element analysis results. Meanwhile it provides lower acceptable thickness than the conventional piping analysis.     

Combined analysis and evaluation of piping systems using the computer

The organization and theoretical aspects of a computer program designed primarily for the analysis and evaluation of power piping with particular emphasis on nuclear class I and II piping are described. The program includes a one-dimensional (radial coordinate in a cylindrical system) thermal analysis which computes the required entries in eqs (10), (11) and (13) of section NB 3600. Six types of loadings may be considered simultaneously: deadweight, sustained design mechanical loads, thermal expansion, thermal anchor movement, earthquake-response spectra or time history, and other anchor movement. For dynamic analysis the generalized Rayleigh-Ritz method using unit loads to generate the trial vectors is utilized. This method provides a consistent reduction of both the stiffness matrix and the mass matrix. The Ritz method is not without its problems, however, in the selection of the so-called ‘master nodes’. While getting enough master nodes is generally not a problem (easier than lumping masses) it is easy to over-specify and therefore introduce ill conditioning. To overcome this shortcoming a method is described for checking the trial vectors for their degree of parallelism in kinetic energy space, and for automatically removing those degrees of freedom which may lead to difficulties. Sample solutions with the program are presented.     

Consideration of uncertainties in seismic analysis of coupled building piping systems

In this paper, we present an analytical study for incorporating the effect of uncertainties in modal properties of uncoupled primary and secondary systems in the seismic analysis of non-classically damped coupled systems such as building piping by response spectrum method. Monte Carlo simulation is used to illustrate that the secondary system design response when defined at a non-exceedence probability of 0.84 over the individual responses obtained from multiple response spectrum analyses by considering uncertainties in modal parameters is excessively higher than the design response specified at the same non-exceedence probability over the responses obtained from multiple time history analyses. This is so because the earthquake input in a response spectrum method is characterized by a design spectrum which by itself is specified at a non-exceedence probability of 0.84 over the multiple time histories with normalized peak ground acceleration. Accurate evaluation of design response at a non-exceedence probability of 0.84 in the response spectrum method requires that the individual modal responses be defined at appropriate probability levels that may be different than the conventionally used non-exceedence probability value of 0.84. The required probability values are evaluated by using first order reliability method. It is shown that the modal responses, when defined at a non-exceedence probability of 0.84, would give relatively accurate values of design response only if the individual modes are perfectly correlated or a single mode contributes to the particular response quantity of interest. For all other cases, the design response would be excessively high. The accurate probability values needed to specify each modal response evaluated using the first order reliability method cannot be incorporated directly in a response spectrum analysis due to computational inefficiency. Two simplified methods, based on total probability theorem, are developed in this paper to overcome this limitation. It is shown that these methods give design response values that are very close to the true values obtained from multiple time history analyses.     

Comparative methods for analysis of piping systems subjected to seismic motion

The dynamic analysis of a three-dimensional piping system of a nuclear power plant is conveniently performed through a finite element method. When the modal analysis is used, only the first few modes of vibration are computed for practical purposes. In this paper is proposed a method of residues which evaluates the neglected modes and combines them with the first calculated modes to estimate the total seismic response of the piping. This methods emphasizes the importance of the selected modes. When the approach is made through a time history input function, this latter is usually characterized by a combination of several recorded accelerograms, e.g. El Centro, San Francisco and Taft. The response of a particular piping has been evaluated by means of these two methods: the use of the modal approach will be strongly recommended due to its inherent advantage of economy and also computation time and reliability.     

Standardization of piping supports in nuclear power plants using pretested components

The article reports on the “standard piping-support catalog”, which was used in connection with the Convoy-nuclear-power-plants in Fed. Rep. Germany, under special consideration of: (i) cost reduction for planning and installation, (ii) suitability tests (type testing for standardized supports), and (iii) application of CAD. The advantages using the piping support catalog are summarised.     

An implicit three-dimensional finite-element formulation for long-duration structural analysis of piping systems

This paper describes an implicit three-dimensional finite-element formulation for the structural analysis of reactor piping systems. The numerical algorithm considers hoop, flexural, axial, and torsion modes of the piping structures. It is unconditionally stable and can be used for calculation of piping response under static or long duration dynamic loads.The method uses a predictor-corrector, successive iterative scheme which satisfies the equilibrium equations. A set of stiffness equations representing the discretized equations of motion are derived to predict the displacement increments. The calculated displacement increments are then used to correct the element nodal forces. The algorithm is fairly general, and is capable of treating large displacements and elastic-plastic materials with thermal and strain-rate effects.The implicit-time integration scheme described herein has been incorporated into the three-dimensional piping code SHAPS. Two sample problems are presented to illustrate the analysis. The first problem deals with a dynamic analysis of a pipe-elbow loop. The second problem studies the piping response to seismic excitation. The results are discussed in detail.     

Shakedown analysis of elastic-plastic structures

Behavior of elastic-plastic structures under repetitive and fluctuating loads is considered in this paper. Plastic deformation either stabilizes after a finite number of cycles or continues during the cycling. In the first case the structure is said to have shaken down to the boundary at the loading program. If plastic deformation does not stabilize an elastic-plastic structure becomes unserviceable due to either alternating plasticity when yielding occurs repeatedly in the opposite senses or accumulation of plastic strains and progressive increase of permanent displacements. This paper attempts to survey the shakedown theory, including the accommodation of a structure to the prescribed loading range as well as inadaptation and unserviceability.The notion of shakedown, incremental collapse, ratchetting and alternating plastic deformation are first illustrated with examples. The fundamental theorems on shakedown and inadaptation are presented next, attention being directed to generalizations of the classical Melan and Koiter theorems. Applications of the theorems are given. Available methods for determining the shakedown range on generating self-equilibrated stress fields are discussed. Special techniques applicable to problems involving both dead and fluctuating loads are invoked. Recent studies accounting for inertia forces, geometrical changes, material hardening, variation of material properties with temperature, displacement assessment, etc. are referred to.     

Residual load method for modal analysis of piping systems subjected to seismic excitation

Due to the fact that piping systems normally possess a large number of unknowns, the computation of eigenvectors and frequencies is limited. It does not seem to be reasonable to calculate all modes because of the wellknown phenomenon that the accuracy of the higher modes is deteriorating the more the higher the frequency of these eigenforms. When the modal-analysis method is used, this means, that the complete response of piping systems remains unknown.In this paper two methods which pretend to take the neglected higher modes into account are discussed. These are the residual-load-method and the modal-acceleration-method. Error investigations on both methods reveal that the residual-load-method might be given preference.Two examples are presented in which piping systems are dealt with the residual-load-method. It seems surprising, that even for branched systems it is sufficient to compute a few eigenvectors and then apply the residual load.     

Loads of piping systems due to malfunctions of snubbers

The aim of the numerical investigations reported here was to extend our analyses in order to obtain a broader basis for evaluations of the behavior of piping systems which are subjected to anticipated malfunctions of snubbers. The investigated piping systems were assumed to be supported by mechanical snubbers and hydraulic snubbers, as well. For assessment purposes stress and force/moment criteria were applied. The effects of malfunctions were studied for several configurations of snubbers, assuming clamping of these devices under normal operational conditions in some cases and postulating failure due to limited dynamic motions due to earthquake excitation in others. The analyses were performed with hot-running and cold systems as well. In case of the investigated hot-running piping systems it was found that clamping of snubbers can result in substantial load increases, especially in elbows. Failure to lock rapidly varying displacements in these systems was found to be less severe. In the latter case higher stress levels were observed only at some isolated positions. The analysis of the cold system revealed that the postulation of non-responding snubbers raised the stress level considerably in the piping system due to shifts of eigenfrequencies towards the region of higher values of the underlying floor response spectrum. In this case restraining of earthquake induced deformation by less vulnerable systems than snubbers would be preferable. The investigation has shown that appropriate variations of the position, the effective direction, and the number of snubbers can lead to considerable stress reductions in a piping system.     

Pipe rupture and steam/water hammer design loads for dynamic analysis of piping systems

The design of restraints and protection devices for nuclear Class I and Class II piping systems must consider severe pipe rupture and steam/water hammer loadings. Limited stress margins require that an accurate prediction of these loads be obtained with a minimum of conservatism in the loads. Methods are available currently for such fluid transient load development, but each method is severely restricted as to the complexity and/or the range of fluid state excursions which can be simulated. This paper presents a general technique for generation of pipe rupture and steam/water hammer design loads for dynamic analysis of nuclear piping systems which does not have the limitations of existing methods. Blowdown thrust loadings and unbalanced piping acceleration loads for restraint design of all nuclear piping systems may be found using this method. The technique allows the effects of two-phase distributed friction, liquid flashing and condensation, and the surrounding thermal and mechanical equipment to be modeled. A new form of the fluid momentum equation is presented which incorporates computer generated fluid acceleration histories by inclusion of a geometry integral termed the “force equivalent area” (FEA). The FEA values permit the coupling of versatile thermal-hydraulic programs to piping dynamics programs. Typical applications of the method to pipe rupture problems are presented and the resultant load histories compared with existing techniques.     

Assessment of coupling effects of coolers and piping systems

In the dynamic analysis of large systems there is a great incentive to perform the calculations separately for smaller, decoupled subsystems. This simplification has to be justified, as best by using appropriate decoupling criteria. In practice, however, this procedure is difficult to accomplish thoroughly due to the unsatisfactory state-of-the-art at present, as shown in this paper on the basis of two investigations.In these investigations (see A and B below) linear elastic FEM-calculations have been carried out for both the large joint (coupled) systems and the decoupled subsystems. The resulting coupled and uncoupled loads at significant nodal points are evaluated.In investigation A which deals with coolers and connected piping systems it is shown that an uncoupled analysis of the subsystems and a superposition of the influence of the piping systems to the coolers will yield adequate loads at the supports and the nozzles.The investigation B for piping systems with different diameter ratios of the main system to the subsystem leads to the following results: Uncoupled analysis for the main system without considerations of any effects of connected subsystems results in a satisfactory accuracy only for diameter ratios higher than 3:1, the comparison of the coupled and uncoupled analysis for a diameter ratio of 2:1 shows a necessary correction factor for the uncoupled analysis of about 7.     

Response of piping system on friction support to bi-directional excitation

Installation of friction devices between a piping system and its supporting medium is an effective way of energy dissipation in the piping systems. In this paper, seismic effectiveness of friction type support for a piping system subjected to two horizontal components of earthquake motion is investigated. The interaction between the mobilized restoring forces of the friction support is duly considered. The non-linear behavior of the restoring forces of the support is modeled as an elastic-perfectly plastic system with a very high value of initial stiffness. Such an idealization avoids keeping track of transitional rules (as required in conventional modeling of friction systems) under arbitrary dynamic loading. The frictional forces mobilized at the friction support are assumed to be dependent on the sliding velocity and instantaneous normal force acting on the support. A detailed systematic procedure for analysis of piping systems supported on friction support considering the effects of bi-directional interaction of the frictional forces is presented. The proposed procedure is validated by comparing the analytical seismic responses of a spatial piping system supported on a friction support with the corresponding experimental results. The responses of the piping system and the frictional forces of the support are observed to be in close agreement with the experimental results validating the proposed analysis procedure. It was also observed that the friction supports are very effective in reducing the seismic response of piping systems. In order to investigate the effects of bi-directional interaction of the frictional forces, the seismic responses of the piping system are compared by considering and ignoring the interaction under few narrow-band and broad-band (real earthquake) ground motions. The bi-directional interaction of the frictional forces has significant effects on the response of piping system and should be included in the analysis of piping systems supported on friction supports. Further, it was also observed that the velocity dependence of the friction coefficient does not have noticeable effects on the peak responses of the piping system.     

Asymptotic techniques in elastic-plastic analysis of structures

Elastic-plastic structures can nowadays be analyzed with the powerful numerical procedures of the finite element method. Nevertheless, in many engineering applications, analytical expressions capable of predicting with sufficient accuracy the stress distributions, the extent of the plastic zones and the load displacement behavior could be of great practical value. For simple structures and loading stages not too far from the elastic limit, such analytical expressions may be obtained by using perturbation methods and asymptotic expansions. A small dimensionless parameter is defined as the ratio of a length characterizing the extent of the narrow plastic zone, to a conveniently chosen typical dimension of the structure. Stresses and displacements are formally expanded as asymptotic series in terms of powers of . For each order of magnitude, the exact basic relations lead to a separate set of simplified differential equations which can be integrated analytically or numerically by using standard procedures. The method is very general and can be applied to several classes of plastic behaviour and of structural problems.Three examples of very simple structures are chosen in particular to illustrate the applicability of the perturbation method to engineering problems: (1) For a simply supported beam under a concentrated load, is chosen as the ratio of the largest axial extension of the plastic zone to twice the beam length. (2) The case of a long thin cylindrical shell subject to a gradually increasing radial ring of load is algebraically more involved but can be treated similarly. (3) For a simply supported circular plate under a ring of load, is defined as the thickness ratio of the central plastic zone which is assumed to be sufficiently thin in the first stage of loading beyond the elastic limit. Both Tresca's and von Mises' yield criteria with the associated flow rule (rate theory) can be used.The method can also be applied to more involved geometries (in conjunction with linear FE-procedures) and more complex material behaviour.     

Earthquake and aircraft impact simulation testing of piping systems

Safety-related piping systems in nuclear power plants and their supports are usually designed and analysed as linear elastic structures within elastic limits of material behavior. This applies for earthquake loads (SSE) and in Germany also for the external event of aircraft impact (AI). This load case with its extremely low probability and short duration is comparable in frequency content with internal shock loads.In order to define the conservativities of realistic systems designed in that way, at high SSE and AI load levels a large series of tests with different configurations were performed.AI runs were made up to a level of more than 300% of the design level. SSE loads met a system pressurized with 45 bar up to 6 times the analytically determined dynamic capacity. In spite of high displacement (90 mm = 3.6″), response accelerations (10 g) and strains (1.3%) and of maximum loss of dowel torque of about 50%, all load levels were withstood by all configurations without leakage or loss of pressure, without loss of integrity of fixed point structures and without failure of fixtures of steel to concrete.Postcalculations with linear elastic methods yielded analytic stresses up to 10.2 times the allowable stress with an artificially weakened configuration.Safety margins found in that way are due to nonlinear behavior of systems and materials.     

Finite element elastic-plastic analysis of LMFBR components

The present effort involves the development of computationally efficient finite element methods for accurately predicting the isothermal elastic-plastic three-dimensional responses of thick and thin shell structures subjected to mechanical and thermal loads. This work will be used as the basis for further development of analytical tools to be used to verify the structural integrity of liquid metal fast breeder reactor (LMFBR) components. The methods presented here have been implemented into the three-dimensional solid element module (HEX) of the Grumman PLANS finite element program. These methods include the use of optimal stress points as well as a variable number of stress points within an element. This allows monitoring the stress history at many points within an element and hence provides an accurate representation of the elastic-plastic boundary using a minimum number of degree of freedom. Also included is an improved thermal stress analysis capability in which the temperature variation and corresponding thermal strain variation are represented by the same functional form as the displacement variation. Various problems are used to demonstrate these improved capabilities.