ANALYTICAL INVESTIGATION AND PARAMETRIC STUDY OF LATERAL IMPACT BEHAVIOR OF PRESSURIZED PIPELINES AND INFLUENCE OF INTERNAL PRESSURE

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1 DRAFT Proceedings of the ASME 014 International Mechanical Engineering Congress & Exposition IMECE014 Noveber 14-0, 014, Montreal, Quebec, Canada IMECE ANALYTICAL INVESTIGATION AND PARAMETRIC STUDY OF LATERAL IMPACT BEHAVIOR OF PRESSURIZED PIPELINES AND INFLUENCE OF INTERNAL PRESSURE Yangqing Dou Departent of Mechanical Engineering Mississippi State University Mississippi State, MS 3976, USA Yucheng Liu Departent of Mechanical Engineering Mississippi State University Mississippi State, MS 3976, USA ABSTRACT This paper provides a cobined coputational and analytical study to investigate the lateral ipact behavior of pressurized pipelines and inspect all the paraeters such as the outside diaeter and internal pressure affects such behavior. In this study, quartic polynoial functions are applied to forulate the axiu crushing force (F), axiu peranent displaceent (W), and absorbed energy (E) of the pressurized pipelines during the ipact proble. The effects of the diaeter and pressure on F, W, and E are therefore illustrated through analyzing those functions. Response surfaces are also plotted based on the generated quartic polynoial functions and the quality (accuracy) of those functions are verified through several techniques. Keywords: pressurized pipeline, quartic polynoial, paraetric study, FEA, response surface ethod INTRODUCTION The authors have conducted a coputational study to thoroughly investigate lateral ipact behavior of pressurized pipelines and inspect effects of internal pressure and outside diaeter of the pipelines on their ipact responses. 7 ipact siulations were carried out using 3D dynaic nonlinear finite eleent analysis (FEA) through LS-DYNA to predict the ipact response of ild steel pipelines with different diaeters and internal pressure levels, which subjected to lateral struck at id-span and one quarter span positions. The obtained coputational results were verified by coparing with soe published experiental results. Based on the results achieved fro the preliinary analysis, one objective of this study is to eploy nuerical ethods to establish analytical odels to nuerically show how the ipact paraeters (internal pressure and outside diaeter) affect the ipact response of the pipelines during low-speed lateral ipact. PROBLEM DESCRIPTION AND FEA RESULTS In preliinary coputer siulations, a rigid indenter ipacted a nuber of pressurized pipelines at the id-span position and the one quarter span position. The pressurized pipes were ade fro sealess cold drawn ild steel with outside diaeters of, 4, 60, 80, 100, and 10 with a fixed ratio of L/D = 10, where L is the distance between the two supports and D is the outside diaeter. The selected ratio of L/D = 10 is currently being used in ost research laboratory and industry plant as the largest unsupported pipe length ratio [1]. The cold worked ild steel pipes have a wall thickness, and the echanical properties in the axial direction of the pipe are: static uniaxial yield stress σ y = 663 MPa, static ultiate tensile stress σ u = 83 MPa, and static uniaxial rupture strain ε r = 6 ~ 7%. In this study, the internal pressure p varies fro 0 to 150 bar. Specifically, 6 different pressures, 0, 30, 60, 90, 10, 150 bar were applied on inner surface of the pipelines separately in order to achieve a coplete understanding of how the internal pressure affects the lateral ipact behavior of the pipelines. In order to siulate those ipact tests, 7 FEA odels were created along with appropriate boundary, loading, and initial conditions. The generated FEA odels include 1000 to 1 Copyright 014 by ASME

2 ore than 5,000 shell eleents. Figs. 1 and present two ipact scenarios, where the indenter ipacted a pipeline odel with outside diaeter of 60 and internal pressure of 60 bar at its center and one quarter span, respectively. Fig. 1. FEA odel showing an indenter ipacts on center of a pressurized pipeline Fig. FEA odel showing an indenter ipacts on one quarter of a pressurized pipeline After the ipact siulations, iportant FEA results (axiu ipact force F, peranent displaceent W, and absorbed energy E) were collected and tabulated in Tables 1 and for pipeline odels which were struck at iddle and one quarter span position, respectively. No. D () p (bar) W () F (kn) E (kj) No. D p W F E () (bar) () (kn) (kj) Table 1. FEA results for the pipes struck at the iddle position No. D p W F E () (bar) () (kn) (kj) Copyright 014 by ASME

3 No. D p W F E () (bar) () (kn) (kj) Table. FEA results for the pipes struck at one quarter span position RESPONSE SURFACE METHOD [] In odern industry, RSM is extensively applied in developing, iproving, forulating, and optiizing processes. During a design process, such ethod is used to deterine potential influences of several input variables (independent variables) on the perforance or quality of the entire syste so as to acquire optiized responses fro that syste. This ethod has also been extensively applied by the author in optiu design of thin-walled coluns in order to optiize their energy absorption capacity during crash analysis [3-8]. In this study, RSM is eployed to deterine how the internal pressure (p) and outside diaeter ( of the ild-steel pipelines affect their ipact response, including ipact force (F), peranent deforation (W), and absorbed energy (E). In this study, the ipact response of the pipelines (it can be the ipact force, deforation, and absorbed energy) is approxiated using a series of the basic functions in a for of y n (x) F( p,, W ( p,, ore( p, βii (p, (1) i1 where n represents the nuber of basic functions φ i(p,. In this paper, the polynoials are used to build up these basic functions to forulate F, W, and E. In Eq. (1), the β i, known as the regression coefficients, are estiated using the ethod of least squares. Suppose we have ( > n) observations (obtained fro FEA) for the yielded response y i (y 1 y ) based on the sapling design points (p, i, the least squares function is therefore expressed as n L ε i yi β j j(p, () i1 i1 j1 where the design points (p, i are selected fro the specified design space, ε i is the error between the response y i observed at these points, and the RS approxiation at that point. Afterwards, the coefficient vector b = (β 1, β,, β n) can be deterined byl / 0, which is b = (Ф T Ф) -1 Ф T y (3) where Ф denotes the atrix consisting of basic functions evaluated using sapling points, which is 1(p, 1 n (p, 1 (4) Φ 1(p, n (p, By substituting Eq. (3) into (1), the response surface odel is created and the response functions (F(p,, W(p,, and E(p, ) then can be fully deterined. The accuracy of the developed response surface odel can be verified through several techniques. The relative error (RE) between the observed response at those sapling points y(x) and the original response ŷ(x) is RE = [ŷ(x) y(x)]/y(x) (5) Other two iportant properties in evaluating the odel s accuracy are the su of squares of the residuals (SS E) and the total su of squares (SS T), which are SS (y ŷ (6) E i i ) i1 T i yi ) i1 SS (7) (y where y i is the ean value of y i. The odel s fitness can be evaluated based on the F statistic, coefficient of ultiple deterination R, adjusted R statistic, and root ean square error (RMSE) respectively, which are calculated as (SS T SS E )/n F (8) SS E/( n 1) SS E R 1 (9) SS T 1 R adj 1 (1 R ) (10) n SS RMSE E (11) n 1 3 Copyright 014 by ASME

4 According to the classical RSM theory, the larger the values of R and R adj, and the saller the value of RMSE, the better the odel fit. RSM MODELS AND ASSESSMENT As shown in Tables 1 and, six pressures and diaeters were selected for odeling and siulation and totally 36 cobinations were presented. Thus, in Eqn. (4) the atrix Φ has 36 rows ( = 36), which corresponds to the 36 cobinations of p and D (No. 1 to 36 in Tables 1 and ). In this study, quartic polynoial is used as the basic function because it provides the best fitness to the real probles [9]. Thus, the basic functions φ i are ters in a full quartic for, which are 1, p, D, p, pd, D, p 3, p D, pd, D 3, p 4, p 3 D, p D, pd 3, D 4. Substituting p and D values into Eqn. (4), a atrix Φ then can be created. Next, response surface odels (quartic polynoials) are created for F, W, and E separately. Based on the FEA results listed in Tables 1 and, the regression coefficients are deterined using Eqn. (3) and then the corresponding quartic polynoial functions are achieved as: Mid-span ipact F(p, = D P D 0.005DP P D D P DP P D D 3 P D P DP P 4 (1) W(p, = D P D DP P D D P DP P D D 3 P D P DP P 4 (13) E(p, = D P D DP P D D P DP P D D 3 P D P DP P 4 (14) One quarter-span ipact W(p, = D P D DP P D D P DP P D D 3 P D P DP P 4 (15) F (p, = D 0.077P D DP P D D P DP P D D 3 P D P DP P 4 (16) E(p, = D P D DP P D D P DP P D D D P DP P 4 (17) The approxiation of the responses obtained fro the response functions and the FEA results are then substituted into Eqns. (6) to (11) to evaluate the fitness of the developed RS odels. In those equations = 36 and n = 15 (nuber of basic functions in a full quartic polynoial for). The values of RE, R, R adj, and RMSE are calculated and displayed in Table 3. The generated response surfaces with respect to variables pressure and diaeter are plotted in Figs. 3 and 4. Fig 3. Quartic response surfaces of F; E under id-span ipact 4 Copyright 014 by ASME

5 internal pressure p as the only variable. Curves are then plotted fro those analytical odels and copared with the results obtained fro FEA siulations. (As displayed in Figs. 5 and 6) Fig 4. Quartic response surfaces of F; Ep under one quarter span ipact Fro figure 3 and 4, it can be observed that the different ipact positions did not largely change the characteristic lateral ipact response of the pressurized pipelines. The response surfaces of axiu ipact force, and absorbed energy obtained fro both ipact scenarios showed siilar shapes. Fig 5. Effects of p on axiu ipact force, Mid-span ipact one quarter span ipact Table 3. Evaluation of the developed RS odels RS odel R R adj RMSE F id-span W id-span E id-span F one quarter span W one quarter span E one quarter span PARAMETRIC STUDIES Influence of Internal Pressure In order to reveal the effects of internal pressure on the pipelines lateral ipact response, as well as validate the accuracy of the developed response surface odels, we substitute D =, 4, 60, 80, 100, and 10 into Eqns. (1-17) to obtain a series of siplified analytical odels with the 5 Copyright 014 by ASME

6 Fig 6. Effects of p on absorbed energy, id-span ipact one quarter span ipact Influence of Outside Diaeter Siilarly, in order to study the effects of outside diaeter on the pipelines lateral ipact response, p = 0, 30, 60, 90, 10, 150 are substituted into Eqns. (1-17) separately to create a series of analytical odels with the outside diaeter as the only variable. Those analytical odels are then copared with the FEA results and the coparison results are shown in Figs. 8 and 9. Fig 7. Effects of D on axiu ipact force, Mid-span ipact one quarter span ipact Fig 8. Effects of D on absorbed energy, Mid-span ipact one quarter span ipact Discussion It sees the influences of internal pressure and diaeter on the ipact response of those pipelines provide siilar tendencies under both id-position ipact and one quarterposition ipact (Figs. 5 to 8). Meanwhile, different ipact positions really affect the values of axiu ipact force, and absorbed energy. Figs 5 and 5 discuss that the axiu ipact forces yielded during the one quarter-position ipact were apparently higher than those generated during the id-position ipact. This is because that the one quarter position is closer to fixture and therefore is ore rigid than the iddle position. However, as reflected fro Figs. 6 and 6, the ipact energy absorbed during the id-position ipact was slightly lower than that absorbed during the one quarter-position ipact. As for the influences of the internal pressure and outside diaeter, Figs 5 and 5 reveal that when the internal pressure increased, the axiu ipact force also increased, even such tendency was not apparent when the outer diaeter was low. It is believed that the internal pressure will enhance ipact resistance of the pipeline odel, therefore leads to higher ipact forces. Meanwhile, those figures also tell us that under the sae pressure, the pipeline odels with larger outside diaeter ay be subjected to higher ipact force. Figs. 6 and 6 show that when the internal pressure increased, the ipact energy absorbed by the pipelines decreased (still, such tendency was not that evident for pipelines with a sall outside diaeter). Siilar finding was also 6 Copyright 014 by ASME

7 reported in Ng and Shen s work, which were obtained through experients. Since the internal pressure enhances the rigidity of the pipeline, that way the indenter will rebound at a coparatively high speed after ipacting the pipeline and less ipact energy will be absorbed by the pipeline. Meanwhile, fro both figures it is observed that the absorbed ipact energy decreased when the outside diaeter increased. Those effects of outside diaeter on the pipelines lateral ipact response are reflected in Figs. 7 and 7 as well as Figs. 8 and 8. Different ipact position did affect axiu ipact force even those curves see siilar and have sae tendency as shown in Figs 7 and 7. The pipeline with ipact one quarter position is near to boundary condition and therefore is ore rigid than the iddle position. Those figures also reveal that outside diaeter increased while the axiu ipact force increasing as well. The slope of each curve is different and gradually increased. It is reasonable to have larger value of axiu ipact force with higher internal pressure which could increase the ipact resistance of pipeline odel. Figs. 8 and 8 are used to illustrate influences of diaeter on the ipact response of those pipelines under different ipact position scenarios which is center position and quarter span separately. It is obviously to deterine diaeter increased as ipact energy absorbed energy decreased. With increasing of diaeter and internal pressure, there will have ore resistance force to avoid deforation and weaker ability to absorbed energy. CONCLUSIONS This paper provides a cobined coputational and analytical study to investigate the lateral ipact behavior of pressurized pipelines and inspect all the paraeters such as the outside diaeters and internal pressures affect such behavior. In this study, the effects of the diaeter and pressure on axiu force (F), peranent deforation (W), and absorbed energy (E) are illustrated through analyzing those functions which is listed above. 1. In this study, quartic polynoial is built and used as the basic function and provides the best fitness. Therefore basic functions φi are ters in a full quartic for, which are 1, p, D, p, pd, D, p3, pd, pd, D3, p4, p3d, pd, pd3, D4.) A atrix Φ is created in order to deterine the influence.. Response surfaces are plotted based on the generated quartic polynoial functions and the quality (accuracy) of those functions are verified through several techniques. 3. When the internal pressure increases, the axiu ipact force increases, the axiu transverse displaceent decreases, and the absorbed ipact energy also decreases. 4. When the outside diaeter increases, the axiu ipact force increases, the axiu transverse displaceent decreases, and the absorbed ipact energy decreases also. This paper could be used as a guidance to deterine the suitable polynoial functions which are verified through statistic ethods. With the purpose of predication the influences of internal pressure and diaeter on the ipact response of those pipelines under both id-position ipact and one quarterposition ipact, a quartic polynoial is chosen to represent those odels and a group of detailed D curves are selected to reveal the relationship between the paraeters. However in order to get a fully explanation the influences of the internal pressure and outside diaeter found fro this study, a coplete experiental analysis is needed to be as a positive coparison. ACKNOWLEDGMENTS This study was sponsored by Louisiana Board of Regents (LA BoR) with contract No. LEQSF(01-15)-RD-A-8. The financial support is greatly appreciated. REFERENCES [1] B.N. Leis, R.B. Francini, R. Mohan, D.L. Rudland and R.J. 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