Simplified limit load estimation using mα-tangent method for branch pipe junctions under internal pressure and in-plane bending

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21st European Conference on Fracture, ECF21, 20-24 June 2016, Catania, Italy

Simplified limit load estimation using m-tangent method for branch XV Portuguese Conference on Fracture, PCF 2016, 10-12 February 2016, Paço de Arcos, Portugal pipe junctions under internal pressure and in-plane bending Thermo-mechanical ofa,aWang highMiao pressure turbine a, Jae-Min Gim a, Yun-Jae Sang-Hyun Kimmodeling Kima,*blade of an airplane gas turbine engine Dept. of Mechanical Engineering, Korea University, Anam-Dong, Sungbuk-Ku, Seoul 136-701, Korea a

P. Brandãoa, V. Infanteb, A.M. Deusc* AbstractaDepartment of Mechanical Engineering, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1, 1049-001 Lisboa, b

Portugal

IDMEC, Department of Mechanical Institutolimit Superior Universidade de Lisboa, Av. Pais, 1, 1049-001 The m-tangent method is a simpleEngineering, way to estimate loadTécnico, for mechanical components. TheRovisco method is based on a Lisboa, linear Portugal elastic finite element analysis to estimate the limit loads. Present work reports limit loads for branch pipe junctions under internal c CeFEMA, Department of Mechanical Engineering, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1, 1049-001 Lisboa, pressure and in-plane bending which is determined by ma-tangent method. All results are compared with published closed-form Portugal solutions and also FE results. The FE results can be found by small-strain three-dimensional finite element (FE) limit load analyses using elastic–perfectly plastic materials. Various branch pipe geometries are considered to verify the accuracy of the ma-tangent method. Abstract

Copyright © 2016 operation, The Authors.modern Published by Elsevier B.V.components This is an openare access article under the CC BY-NC-ND licenseoperating conditions, aircraft engine subjected to increasingly demanding © During 2016 Thetheir Authors. Published by Elsevier B.V. (http://creativecommons.org/licenses/by-nc-nd/4.0/). especially under the high pressure turbine blades. Such conditions cause these parts to undergo different types of time-dependent Peer-review responsibility of Scientific the(HPT) Scientific Committee of ECF21. Peer-review under responsibility of the Committee of ECF21. degradation, one of which is creep. A model using the finite element method (FEM) was developed, in order to be able to predict the creep behaviour of HPT Flight Keywords: ma-tangent method, Limit blades. load, branch pipedata records (FDR) for a specific aircraft, provided by a commercial aviation company, were used to obtain thermal and mechanical data for three different flight cycles. In order to create the 3D model needed for the FEM analysis, a HPT blade scrap was scanned, and its chemical composition and material properties were obtained. The data that was gathered was fed into the FEM model and different simulations were run, first with a simplified 3D 1. Introduction rectangular block shape, in order to better establish the model, and then with the real 3D mesh obtained from the blade scrap. The overall expected behaviour in terms of displacement was observed, in particular at the trailing edge of the blade. Therefore such a Determination of limit is predicting importantturbine in structural integrity Traditionally, Limit loads are determined model can be useful in theloads goal of blade life, given aanalysis. set of FDR data.

by analytical method or numerically. A number of analytical and numerical papers can be found in literature which gives closed limitPublished load solutions. ButB.V. these are performed for simple geometry and loading condition. For the © 2016 Theform Authors. by Elsevier more, inelasticunder FE analysis requires for computation. Peer-review responsibility of thenumerous Scientific time Committee of PCF 2016. R. Seshadri and M.M. Hossain, reports the m tangent method which can be rapid limit load. The method is based High Pressure Blade; Creep; to Finite Elementthe Method; Model; Simulation. onKeywords: a linear elastic finiteTurbine element analysis estimate limit3D loads.

* Corresponding author. Tel.: +82 2 3290 3372; fax: +82 2 926 9290. E-mail address: [email protected] (Y.-J. Kim). 2452-3216 © 2016 The Authors. Published by Elsevier B.V. Peer-review underauthor. responsibility the Scientific Committee of ECF21. * Corresponding Tel.: +351of218419991. E-mail address: [email protected]

2452-3216 © 2016 The Authors. Published by Elsevier B.V.

Peer-review under responsibility of the Scientific Committee of PCF 2016.

Copyright © 2016 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer review under responsibility of the Scientific Committee of ECF21. 10.1016/j.prostr.2016.06.323

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Nomenclature P PL y max VT eq mL m0 m mT Po Pom Poeq MoIB Mom Moeq

Applied load Limit load Yield strength Maximum stress at component Total volume of component The Von Mises equivalent stress Lower bound multiplier Upper bound multiplier m multiplier m tangent multiplier Limit pressure Limit pressure which is estimated by m tangent method Limit pressure which is estimated by closed form solution Limit moment for in-plane bending Limit moment which is estimated by m tangent method Limit moment which is estimated by closed form solution

Present work reports limit loads for branch pipe junctions under internal pressure and in-plane bending which is determined by ma-tangent method. All results are compared with published closed-form solutions and also FE results. The FE results can be found by small-strain three-dimensional finite element (FE) limit load analyses using elastic–perfectly plastic materials. Various branch pipe geometries are considered to verify the accuracy of the mtangent method. 2. The m-tangent method. 2.1. Classical Lower bound Multiplier A lower bound multiplier (mL) can be obtained by applying the lower-bound theorem of plasticity. The classical lower bound limit load multiplier, mL is expressed as:

mL 

y  max

(1)

Where y and max is yield strength and maximum stress of component. Then, lower bound limit load can be obtained by:

PL P  mL

(2)

Where PL and P is limit load and applied load. 2.2. Upper bound Multiplier Based on the “integral mean of yield” (Mura et al., 1965) criterion, the upper bound limit load multiplier m0 can be obtained as (R. seshadri and S. P. mangalaraman, 1997):



Sang-Hyun et al. / Procedia Integrity (2016) 2583–2590 Author nameKim / Structural IntegrityStructural Procedia 00 (2016)2000–000

 y VT

m0 



VT

2585 3

(3)

( eq ) 2 dV

Where VT and eq is total volume of component and The Von Mises equivalent stress. 2.3. m Multiplier The m multiplier method was developed by R. seshadri and S. P. mangalaraman, (1997). The m method is simple way to estimate the lower bound limit load. The issue of lower boundedness of m method has been discussed by W. D. Reinhardt and R. Seshadri (2003). 2

2

 m0   m0  m0   m0  m0     1 2  2 1 1 2      mL  mL   mL  mL  mL   m  2m0 2 2   m0    m0          2 5 2 5   mL    mL     

(4)

2.4. m tangent Multiplier The m tangent multiplier method was developed by R. seshadri and M. M. Hossain (2009). m tangent method is improved to estimate the limit load by concept of reference volume. Ma tangent multiplier could be obtained by lower bound and upper bound limit multipliers. And these multipliers are determined from elastic analysis. The m tangent multiplier are determined by two different case.

m0 When peak stresses is negligible then:   1 2 mL

m0 m  1  0.2929(  1) T

When peak stresses cannot be negligible then: 

mT 

(5)

m0  1 2 mL

mi 0 1  0.2929( f  1)

(6)

Where,

f  (1  0.2929(i  1)  (1  0.2929(i  1)) 2  1

(7)

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3. Finite element analysis 3.1. Material properties In this paper, elastic analysis was considered. The Young’s modulus, E and Poisson’s ratio were used to be E = 200 GPa and ν = 0.3, respectively. For plastic properties which is used for calculate Limit load by m tangent method, the yield strength was assumed to be σy = 200 MPa. 3.2. Geometry 3-D elastic FE analyses of the branch junction, depicted in Fig. 1, were performed using ABAQUS. It is assumed that the branch junction has no weld or reinforcement around the intersection. The half-length of the run pipe is denoted as L and the length of the branch pipe as l. The geometric variables (R, T, r, t, L, l) were systematically varied (table. 1), within the ranges 0.2≤ (r/R= t/T) ≤0.6 and 2.0≤R/ T≤20.0.

Fig. 1. Schematics of branch junctions with relevant geometric variables.

Table 1. Analysis parameters considered in this work. R/T 2 5 10 20

r/R=t/T

L/R= l/r

0.2 0.4

100

0.6

3.3. FE analysis Symmetry conditions were fully utilized in FE models to reduce the computing time. To avoid problems associated with incompressibility, reduced integration elements (element type C3D20R within ABAQUS) were used. Fig. 2 and Fig. 3 depicts typical FE meshes, employed in the present work; one for internal pressure and the other for in-plane bending. For all cases, five elements are used through the thickness, and the resulting number of elements and nodes



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in typical FE meshes are 4,860 elements and 47,397 nodes for half model, and 3,240 elements and 9.720 nodes for quarter model. Fig. 4 depicts mesh sensitivity results. Regarding loading conditions, both internal pressure and in-plane bending moment were considered. For internal pressure, pressure was applied as a distributed load of 1MPa to the inner surface of the FE model, together with axial tensions equivalent to the internal pressure applied at the end of the branch and run pipes to simulate closed ends. Due to symmetry, only a quarter model was used. For in-plane bending cases, the nodes at the end of the branch pipe were constrained through the MPC (multi-point constraint) option within ABAQUS and 1MN.m bending moments was applied (rotation)

Fig. 2. Typical FE meshes for internal pressure

Fig. 3. Typical FE meshes for in-plane bending

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Fig. 4 Mesh sensitivity results (a) internal pressure (b) in-plane bending

4. Limit pressure Closed form limit load solution for branch pipe under internal pressure which is obtained by FE analysis results (K-H Lee, Y-J Kim, 2009) are below:

Q    0.5   0.25  0.5h1  h12  0.79h22       1  t / (2r )  ln    Po  2  min   1  t / (2r )    1  T / (2 R)      1 / (2 ) T R  3  y ln   ln   1 / (2 ) T R       1  T / (2 R)      2   3   Where,





2

r   t   2R    0.4603  0.1525    1  1.015  Q  R T     T 

0.297

4

r   2R    0.52   1.127   R    T 

(8)

0.0505

4 2  1 r  3r  1 1     ; k  1    ; f2  f1  3 3 R  2  16  R   1  t T    2 2 2   t  r  2R   t  r  2R   r  1   ; h 0.175 k 1   1   0.145k 0.3185 h1  f f 2     f2   2   1   R  T   2r  2r  R  T  R  

Limit pressure results which obtained by m tangent method for branch pipes for various r/R and R/T are presented at Fig. 5. Fig. 5 compares limit pressure using m tangent method with eq. (8), where the m results are normalized with respect to prediction using eq.(8). m results are under estimate the limit loads for various geometry. Limit pressure is increasing with increasing R/T. for thin wall branch pipes (R/T=20), limit pressure getting close to eq. (8).



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Fig. 5 Variations of normalized limit Pressures (Pom / Poeq) for branch pipes

5. Limit Moment for in-plane bending Closed form limit load solution for branch pipe under in-plane bending which is obtained by FE analysis results (K-H Lee, Y-J Kim, 2009) are below:

(9) Where,

2

3

2

r  t   r r r  2R  QIB   1.102  0.653   0.7   2.583    5.462    3.544    2.009  0.0025   R T R R R           T  4 2 1 r   3r  1 1     ; k  f1  1    ; f2  3   3 R  2 16  R  1 t T   

Also, Limit moment for in-plane bending results which obtained by m tangent method for branch pipes for various r/R and R/T are presented at Fig. 6 Fig. 6 compares limit pressure using m tangent method with eq. (9), where the m results are normalized with respect to prediction using eq. (9). The result shows dramatic difference by R/T and accuracy is increasing with increase of t/T. Over estimation of limit moment is caused by localized stress field, compare with internal pressure condition.

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Fig. 6 Variations of normalized limit moments (Mom / Moeq) for branch pipe under in-plane bending

6. Conclusion In this paper, Present work reports limit loads for branch pipe junctions under internal pressure and in-plane bending which is determined by ma-tangent method. All results are compared with published closed-form solutions and also FE results. Various branch pipe geometries are considered to verify the accuracy of the ma-tangent method. The m-tangent method is a simple way to estimate limit load for mechanical components. But it does not guarantee accurate limit load estimation. Nevertheless, m tangent method is rapid method to obtain limit load for complex geometry and loading conditions for specific cases.

Acknowledgements This work was supported by the Korea Institute of Energy Technology Evaluation and Planning (KETEP) and the Ministry of Trade, Industry & Energy (MOTIE) of the Republic of Korea (No. 2013101010170A).

References ABAQUS version 6. 14., 2015. User’s manual, Inc. and Dassault Systems. Seshadri, R., Mangalaramanan, S. P., 1997. Lower Bound Limit Loads Using Variational Concepts: The m�-Method, Int. J. Pressure Vessels Piping 71, 93–106. Reinhardt, W. D., Seshadri, R., 2003. Limit Load Bounds for the m�-Multiplier, ASME J. Pressure Vessel Technol. 125, 11–18. Seshadri, R., Adibi-Asl, R., 2007. Simplified Limit Load Determination Using the Reference Two-Bar Structure, ASME J. Pressure Vessel Technol., 129, 280–286. Lee, K. H., Kim, Y. J., Budden, P. J., Nikbin, K,. 2009. Plastic limit loads for thick-walled branch junctions. The Journal of Strain Analysis for Engineering Design 44(2), 143-148. Kim, Y. J., Lee, K. H., Park, C. Y. ,2006. Limit loads for thin-walled piping branch junctions under internal pressure and in-plane bending. International journal of pressure vessels and piping 83(9), 645-653.