Research Article Calculation of Reaction Forces in the ...

Research ArticleCalculation of Reaction Forces in the Boiler SupportsUsing the Method of Equivalent Stiffness of Membrane Wall

Josip SertiT DraDan Kozak and Ivan SamardDiT

Mechanical Engineering Faculty in Slavonski Brod Trg Ivane Brlic-Mazuranic 2 35 000 Slavonski Brod Croatia

Correspondence should be addressed to Josip Sertic josipserticsfsbhr

Received 29 August 2013 Accepted 29 December 2013 Published 15 May 2014

Academic Editors S Kou and K-L Ou

Copyright copy 2014 Josip Sertic et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The values of reaction forces in the boiler supports are the basis for the dimensioning of bearing steel structure of steam boiler Inthis paper the application of the method of equivalent stiffness of membrane wall is proposed for the calculation of reaction forcesThemethod of equalizing displacement as themethod of homogenization of membrane wall stiffness was applied On the exampleof ldquoMilanordquo boiler using the finite element method the calculation of reactions in the supports for the real geometry discretized bythe shell finite element was made The second calculation was performed with the assumption of ideal stiffness of membrane wallsand the third using themethod of equivalent stiffness of membrane wall In the third case themembrane walls are approximated bythe equivalent orthotropic plateThe approximation of membrane wall stiffness is achieved using the elasticity matrix of equivalentorthotropic plate at the level of finite element The obtained results were compared and the advantages of using the method ofequivalent stiffness of membrane wall for the calculation of reactions in the boiler supports were emphasized

1 Introduction

Nowadays the steam boilers of waste incineration powerplants are made specifically according to the requirementsof investors Due to the location where the boiler will beset up the amount and type of fuel to be burned in agiven time and technical parameters of produced steamthe production of steam boilers is mainly individual Thisrepresents an exceptional effort for the designers The pipesand sheet metal are the basic structural elements used formaking boilers Larger structural units called the membranewalls are obtained by welding these elementsThemembranewalls make the basic structure of boiler and have a doublefunction The heat exchanger that is evaporator is theprimary function and the secondary is transfer of loadingThe steam boiler membrane walls are loaded by the pressureof working fluid (watersteam) their ownmass and the massof other structural units that rely on the membrane walls themass of working fluid the mass of insulation and refractoryand the foulingmass that is deposited during the combustionprocessThe loading is transferred across themembranewallsto the boiler supports and then to the bearing steel structure

Since the water tube boilers are large structures consistingof a large number of pipes connected to the membrane wallsit is very difficult to accurately calculate the reactions in thesupports In practice there are two approaches to calculatethe reactions in the steam boiler supports The first approachis based on the assumption that themembrane stiffness of theboiler membrane wall is very high so the boiler is calculatedas an ideal rigid bodyThe second approach takes into accountthe finite stiffness of the membrane walls using the finiteelementmethodThe discretization of real geometry of boiler(membrane walls) requires an extremely large number offinite elements so this approach is rarely used eventuallyfor smaller boilers Most often the membrane wall stiffnessis approximated by the equivalent stiffness of orthotropicplateshell which allows a multiple reduction of degrees offreedom of a numerical model

It is difficult to find more detailed research papers onthe topic of homogenization of steam boiler membranewall In 2012 Milosevic-Mitic [1] with a group of authorsproposed a procedure for calculating the strength of watertube boiler using the finite element of orthotropic plate

Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 392048 12 pageshttpdxdoiorg1011552014392048

2 The Scientific World Journal

The homogenization ofmembrane wall stiffness is performednumerically using the method of equalizing displacement

When calculating the deflection of membrane walls dueto the influence of underpressureoverpressure of flue gasesthe equivalent wall thicknessmethod ismost commonly usedin practice The bending stiffness of membrane wall pipe isequal to the bending stiffness of equivalent wall thickness forthe axis perpendicular to the longitudinal axis of membranewall pipe and the bending stiffness of membrane wall tapeis equal to the bending stiffness of equivalent wall thicknessfor the axis parallel to the longitudinal axis of membrane wallpipe

The steam boiler membrane wall represents a structurallyorthotropic plate The stiffened plates that are frequentlyused both in building and in the processingpower plantswhere it is necessary to achieve the optimal structuralsolution considering the mass of structure also belong tothis class of structures The earliest research on the topicof stiffness of stiffened plates was most likely done byHuber [2ndash4] His investigations were inspired by the lackof exact expressions for solving the problems of bendingand buckling of stiffened structures by using the theoryof homogeneous orthotropic plates Similarly Flugge [5]published the first paper on the topic of structurally stiffenedshell in 1932 and 15 years later so did van der Neut [6]Over time the understanding of the theory of deformation ofstructurally orthotropic plates and shells has progressed Theterms ldquoequivalent stiffnessrdquo and ldquoequivalent wall thicknessrdquobecame the basis for the analysis of deformation of stiffenedor structurally orthotropic platesshells Furthermore Daleet al [7] applied the equivalent stiffness in the researchof buckling of pressure loaded sandwich plates Smith etal [8] proposed an improved formulation that takes intoconsideration the change of position of the neutral surfacethat is connected with the local interaction between theplate and stiffener Pfluger [9] Gomza and Seide [10] andLibove and Hubka [11] applied in their research the methodof equivalent stiffness and thickness of stiffened plateshellIn the papers [12ndash14] the theory of symmetric sandwichplates with the included transverse shear stiffness is appliedIn 1956 Huffington Jr [15] analyzed the equivalent stiffnessof orthogonally stiffened plate without the stiffener eccen-tricity During the last fifty years the equivalent plate orshell approach was still being used as a first approxima-tion method [16ndash19] For the experimental determinationof elasticity constants the method of testing the load-deformation characteristics of repeating structural part of thestructurally orthotropic plate was most commonly used [20]In 1969 Soong [21ndash23] presented a derivation of the stiffnessexpressions for orthotropic cylinders based on a deformationenergy approach Throughout the following years specialattention was given to the interaction between the plate andstiffeners [24] for a different layout or grid of stiffeners [25ndash27] and different combinations of loads In 1995 Jaunky etal [28 29] presented a refined smeared-stiffener theory forgrid-stiffened laminated-composite panels based upon thework presented by Smith et al [8] Similarly Wodesenbetpresented an improved smeared-stiffener theory for grid-stiffened laminated-composite cylinders in 2003

It can be concluded that the current research is mainlybased on two systematic analytical methods that take ordo not take into account the transverse shear stiffnessFirst there is a direct equilibrium-compatibility methodThe first method uses equilibrium and compatibility in adirect manner for plates reinforced by stiffeners The secondone is called the basic-cell energy-equivalence method Thesecond method is based on defining the equivalence betweenthe deformation energy of the basic repeating cell and thedeformation energy of the equivalent orthotropic plate Thismethod is suitable for the determination of the members ofelasticity matrix for grid-stiffened plates and sandwich plates

Unlike the stiffened plates a steam boiler membrane wallconsists of a homogeneous plate with attached stiffenersTherefore the two mentioned methods are not entirelysuitable for the determination of equivalent stiffness ofmembrane wall that is the equivalent elasticity constantsThe process of homogenization of steam boiler membranewall is presented in the papers [30ndash32] The elasticity con-stants for the membrane and bending stiffness of mem-brane wall are determined theoretically by equalizing thedeformation of membrane wall with the deformation ofequivalent orthotropic plate In these papers the transverseshear stiffness is not taken into consideration

2 Steam Boiler Membrane Wall asthe Structurally Orthotropic Plate

The steam boiler membrane wall can be approximated asthe equivalent orthotropic plate that is the orthotropicplate which will have the same elastic properties as the realmembrane wall Using Kirchhoff-Love theory of shells [3334] the constitutive equations of equivalent orthotropic plateor shell can be written as follows

120590 = D sdot 120576 (1)

where 120590 is vector of section forces D is matrix of elasticityand 120576 is vector of deformation According to [35] it can besaid that the constitutive equations for the analysis of platesand shells are analogThematrix expression (1) represents thesix constituent equations ofmembranewall as the structurallyorthotropic plate or its equivalent orthotropic plate whichconnect the section forces with the corresponding deforma-tions

[

[

[

[

[

[

[

[

1198731

1198732

11987312

1198721

1198722

11987212

]

]

]

]

]

]

]

]

=

[

[

[

[

[

[

[

[

11986011

11986012

0 0 0 0

11986022

0 0 0 0

11986033

0 0 0

11986311

11986312

0

sym 11986322

0

11986333

]

]

]

]

]

]

]

]

sdot

[

[

[

[

[

[

[

[

1205761

1205762

12057612

1205811

1205812

2 sdot 12058112

]

]

]

]

]

]

]

]

(2)

11987311198732 and119873

12are the section forces (Figure 1) and119872

11198722

and11987212are the sectionmoments (Figure 2) in themembrane

wall that is the equivalent orthotropic plate 11986011 11986012 11986022

and 11986033

are the members of elasticity matrix for membraneloads and 119863

11 11986312 11986322 and 119863

33are the members of

The Scientific World Journal 3

N12N12N21 N21N1

N1N2 N2

120572212057221205721 1205721

AmAp

Am equivAp

Figure 1 Section forces in the membrane wall that is the equivalent orthotropic plate with the membrane stiffness of membrane wall A119898

equivalent to the membrane stiffness of orthotropic plate A119901

M2

M1M12M21

120572212057221205721 1205721

DmDp

DmequivDp

M2M21 M1

M12

Figure 2 Section moments in the membrane wall that is the equivalent orthotropic plate with the bending stiffness of membrane wallD119898

equivalent to the bending stiffness of orthotropic plateD119901

elasticity matrix for bending loads of equivalent orthotropicplate 120576

1 1205762 and 120576

12are the membrane deformations and 120581

1

1205812 and 120581

12are the curvatures of neutral surface of structurally

orthotropic plate that is the equivalent orthotropic plateThe members of elasticity matrix referring to the mem-

brane stiffness [30 34] of membrane wall can be calculatedusing the following expressions

11986011

=

1198641198981

sdot ℎ

1 minus ]11989812

sdot ]11989821

11986012

=

1198641198982

sdot ℎ sdot ]11989812

1 minus ]11989812

sdot ]11989821

11986022

=

1198641198982

sdot ℎ

1 minus ]11989812

sdot ]11989821

11986033

= 11986611989812

sdot ℎ

(3)

where1198641198981

is modulus of elasticity for themembrane stiffnessin the direction of axis 120572

1(the axis parallel to the axis of

pipe in the membrane wall) 1198641198982

is modulus of elasticity forthe membrane stiffness in the direction of axis 120572

2(the axis

perpendicular to the axis of pipe in themembrane wall) ]11989812

is Poissonrsquos ratio for the membrane stiffness in the directionof axis 120572

1 ]11989821

is Poissonrsquos ratio for the membrane stiffnessin the direction of axis 120572

2 and 119866

11989812is shear modulus for

the membrane stiffness The members of elasticity matrix

referring to the bending stiffness [29 33] of membrane wallcan be calculated using the following expressions

11986311

=

1198641198871

sdot ℎ

3

12 sdot (1 minus ]11988712

sdot ]11988721

)

11986312

=

1198641198872

sdot ℎ

3sdot ]11988712

12 sdot (1 minus ]11988712

sdot ]11988721

)

11986322

=

1198641198872

sdot ℎ

3

12 sdot (1 minus ]11988712

sdot ]11988721

)

11986333

=

11986611988712

sdot ℎ

3

12

(4)

where the elasticity constants for the bending stiffness areindicated by index 119887 The expression for the wall thicknessof equivalent orthotropic plate [30] can be obtained fromthe equality of moment of inertia of the cross-section ofmembrane wall considering the plane perpendicular to thelongitudinal axes of the membrane pipes (Figure 3) and thecross-section of equivalent orthotropic plate Consider

ℎ =radic

3 sdot 120587

16 sdot 119863 sdot 119896

sdot (119863

4minus (119863 minus 2 sdot 120575)

4) +

(119896 minus 119863) sdot 119905

3

119863 sdot 119896

(5)


h

k

120601D times 120575

1205722

1205722

1205723

1205723

t

b2

Membrane wallkmdashstep of pipe in the

Equivalent orthotropic platehmdashwall thickness of the equivalent orthotropic plate

membrane wall

Figure 3 Geometry of cross-section of membrane wall and equivalent orthotropic plate

1

2

3

4

5

6

7

8

9

10

11

25

24

23

22

21

20

19

18

17

16

15

14

13

12

X

Y

Z

Figure 4 Real calculation geometry model


850

2500

2100

1925

1030

1080

4915

1790

890

11701440

X

Y

Z

1980

Figure 5 Calculation geometry model approximated as the equivalent orthotropic plate (dimensions are in mm)

The elasticity constants of equivalent orthotropic platefor the membrane stiffness can be calculated using theanalytical expressions suggested in the paper [31 32] Theseexpressions are obtained by equalizing the deformations ofmembrane wall and equivalent plate (thickness ℎ) with theequivalent membrane load

1198641198981

=

119864

119896 sdot ℎ

[

120587

4

sdot (119863


2) + (119896 minus 119863) sdot 119905]

1198641198982

=

119896 sdot 119864

ℎ sdot [3 sdot (119863120575 minus 1)

3sdot (1205878 minus 1120587) + 119887119905]

11986611989812

asymp

radic1198641198981

sdot 1198641198982

2 sdot (1 +radic]11989812

sdot ]11989821

)

]11989821

= ]11989812

sdot

1198641198982

1198641198981

(6)

where119864 is modulus of elasticity and ]11989812

= ] is Poissonrsquos fac-tor for the isotropic material of membrane wall The expres-sions for the elasticity constants of equivalent orthotropicplate and for the bending stiffness of membrane wall areproposed in the paper [31 32] Same as for the membranestiffness these expressions are also obtained by equalizing


Figure 6 3D-disposition of biomass incineration power plant ldquoMilanordquo

the deformations of membrane wall and equivalent plate(thickness ℎ) with the equivalent bending load

1198641198871

=

119864

119896 sdot ℎ

3sdot [

3 sdot 120587

16

sdot (119863


4) + (119896 minus 119863) sdot 119905

3]

1198641198872

= 119864 sdot (

119905

ℎ

)

3

sdot

(119899 sdot 119896)

4

sum

119899

119894=1(119894 sdot 119896)

4minus (119894 sdot 119896 minus 119887)

4

11986611988712

asymp

radic1198641198871

sdot 1198641198872

2 sdot (1 +radic]11988712

sdot ]11988721

)

]11988721

= ]11988712

sdot

1198641198872

1198641198871

(7)

where 119899 is number of pipes in the membrane wall and ]11988712

=

] is Poissonrsquos factor for the isotropic material of membranewall After the homogenization of steam boiler membrane

wall the stiffness matrix of shell element can be calculatedaccording to the expression

k = int

119860

B119879DB 119889119860 (8)

where 119860 is surface of shell element and B is operator ofboundary magnitude

3 Calculation of Reactions in the Steam BoilerSupports for a Real Model of MembraneWall Geometry for the MembraneWalls Approximated by the EquivalentOrthotropic Plate and for the Ideal RigidModel of Boiler Geometry

Thefinite element method using Abaqus 69-3 will be appliedfor the calculation of reactions in the supportsThe real geom-etry of membrane walls (Figure 4) and the membrane wallsapproximated as the equivalent orthotropic plate (Figure 5)


360

100

2120

100

2120

100

100

1035118013851315

1 2 3 4 5

1740

Superheater 1

Superheater 2

Superheater 3

(SH 1)

(SH 2)

(SH 3)

Seco

nd p

ass

Firs

t pas

s

Third

pas

s

A

A

A

A

Evaporator(EV)

Figure 7 Position of boiler supports and heat exchangers (dimensions are in mm)

will be discretized by a four-node doubly curved thin shellelement using the reduced integrationwith hourglass controlThe finite element has the codename S4R5 and it is applied forsmall deformations and has five degrees of freedom per node

The load of the mass of boiler structure will be taken intoconsideration for the calculation of reactions in the supportsof ldquoMilanordquo steam boiler (Figure 6) Other loads such as themass of working fluid the fouling mass that is depositedon the heating surfaces during the combustion process

the mass of refractory and the mass of insulation will not beconsidered here Pipematerial ofmembranewalls is P235GHmaterial of membrane tape and buckstays is S235JRG2 andmaterial of chambers is 16Mo3 All three steels belong tothe group of low-carbon steels with le 03 carbon Sincethe boiler membrane walls are evaporator heating surfacesthe calculation temperature is 120599 = 275

∘C Assuming thatthe mechanical properties of boiler structure are isotropic[36] the modulus of elasticity for all three materials is


Table 1 List of positions according to Figure 4

Pos Type Dimensions Pos Type Dimensions

1 Shell and end of drum 1206011200 times 35mm 14 Collector header ofside wall 1206011397 times 125mm

2 Downcomers 1206011683 times 16mm 15 Collector header ofrear wall of third pass 1206011397 times 125mm

3 Connecting pipe 120601889 times 88mm 16 Side wall of secondpass

Pipe 12060157 times 4mmTape 119905 = 6mmStep 119896 = 90mmNumber of pipes

119899 = 12

4 Collector header ofrear wall of first pass 1206011397 times 125mm 17 Rear wall of second

pass


119899 = 23

5 Front wall of first pass


119899 = 23

18 Side wall of third pass


119899 = 15

6 Side wall of first pass


119899 = 21

19 Rear wall of third pass


119899 = 23

7 Rear wall of first pass


119899 = 23

20 Buckstays HE-B120

8 Collector header ofwall of hopper 1206011397 times 125mm 21

Distributor header ofrear wall of second

pass1206011397 times 125mm

9 Connecting pipes 120601483 times 4mm 22 Wall of hopper


119899 = 23

10 Distributor header ofside wall 1206011683 times 20mm 23 Distributor header of

rear wall of first pass 1206011397 times 125mm

11 Distributor header offront wall 1206011397 times 125mm 24 Distributor header of

wall of hopper 1206011397 times 125mm

12Collector header offront and rear wall of

second pass1206011397 times 125mm 25 Distributor header of

rear wall of third pass 1206011397 times 125mm

13 Side wall in transitionzone


119899 = 34

approximately equal and amounts to 119864 = 18712GPa For thevalue of Poissonrsquos factor ] = 03 is used

Due to the single boiler symmetry for the calculationusing the finite element method it is sufficient to model onlyone-half of the boiler geometry (Figures 4 and 5) The boileris supported by ten supports five on the distributor chamber

of the left side wall and five on the distributor chamber of theright side wall (Figure 7) In the third pass of the boiler theheat exchangers were installed that is one evaporator andthree superheaters The heat exchangers were supported onthe side walls of the third pass Due to the weight of heatexchangers the load of the left and right side wall of the third


Table 2 Masses of constituent structural elements for half geometry of ldquoMilanordquo boiler

Pos Mass kg Pos Mass kg Pos Mass kg Pos Mass kg Pos Mass kg1 1967 7 639 13 315 19 786 25 432 595 8 43 14 197 20 668 SH1 11853 23 9 17 15 43 21 43 SH2 11854 43 10 402 16 702 22 157 SH3 5275 893 11 43 17 721 23 43 EV 536 1184 12 43 18 1018 24 43

Table 3 Elasticity constants and wall thickness of equivalent orthotropic plate for membrane walls of ldquoMilanordquo boiler

Membrane wall(position)

ℎmm

1198641198981

GPa

1198641198982

GPa

11986611989812

GPa ]

11989821

1198641198871

GPa1198641198872

GPa11986611988712

GPa ]

11988721

5 7 17 19 22 2348 76492 1367 4915 0005361 454159 8064 29095 00053276 2348 76492 1367 4915 0005361 454159 8023 29025 000530013 2348 76492 1367 4915 0005361 454159 8203 29336 000541916 2348 76492 1367 4915 0005361 454159 7690 28439 000508018 2040 88592 3880 8722 0013138 442956 9583 31199 0006490

Detail A

A

X

Y

Z

Figure 8 Finite element mesh-calculation model APPROX

pass is identical For the calculation of reactions in the boilersupports half of the weight of individual heat exchanger willbe used as the equivalent tangential surface pressure on theside wall of the third pass This pressure is applied on surface

Table 4 Members of elasticity matrix of equivalent orthotropicplate for membrane walls of ldquoMilanordquo boiler for membrane stiffness


11986011

Nmm11986012

Nmm11986022

Nmm11986033

Nmm5 6 7 13 16 1719 22 1799282 9646 32152 115438

18 1814749 23843 79476 177966

Table 5Members of elasticitymatrix of equivalent orthotropic platefor membrane walls of ldquoMilanordquo boiler for bending stiffness


11986311

Nsdotmm11986312

Nsdotmm11986322

Nsdotmm11986333

Nsdotmm5 7 17 19 22 491000497 2615341 8717804 314048896 490996567 2602240 8674134 3132927213 491014120 2660750 8869166 3166540416 490964096 2494004 8313348 3069652418 314156438 2038906 6796354 22084218

Table 6 Reactions in the boiler supports for the influence of totalmass

Reactions inboiler supports 119865

1198771 N 119865

1198772 N 119865

1198773 N 119865

1198774 N 119865

1198775 N

REAL 22748 27020 32657 38515 12676APPROX 24183 25992 31015 40026 12499RIGID 27234 22608 31088 36609 16077Δ119860 631 minus380 minus503 392 minus140

Δ119870 1972 minus1633 minus480 minus495 2683

A of membrane wall on which the carriers of heat exchangersare welded (Figure 7) Bending of the side membrane wallsof the third pass due to the load of heat exchangers was nottaken into consideration


Table 7 Reactions in the boiler supports for the influence of mass of boiler without the mass of heat exchangers

Reactions in boiler supportswithout the mass of heat exchangers 119865

1198771 N 119865

1198772 N 119865

1198773 N 119865

1198774 N 119865

1198775 N

REAL 24329 29007 24633 21190 5521APPROX 25790 27681 23752 21822 5574RIGID 27397 21905 34552 10957 9869Δ119860 601 minus457 minus358 298 096

Δ119870 1261 minus2448 4027 minus4829 7875

B

Detail B

X

Y

Z

Figure 9 Finite element mesh-calculation model REAL

The masses of individual structural elements of boilermarked according to Figures 4 and 7 and Table 1 are givenin Table 2 The inhomogeneity of material of boiler structurewill not be taken into consideration here It is assumed thatthe geometry of boiler is ideal without the tolerances ofproduction and assembly as well as the tolerances of semifin-ished rolled products built into the boiler It is assumed thatthe mechanical properties of material of boiler structure areisotropic that is that anisotropy of the welded joints andsemifinished rolled products does not influence significantlythe reactions in the boiler supports

The mass of half of boiler without the heat exchangersis 119898

1015840= 10671 kg and with the heat exchangers it is 119898 =

13621 kg The membrane walls of ldquoMilanordquo boiler are made

with the step of 90mm Two dimensions of the cross-sectionof pipe (12060157 times 4mm and 120601483 times 45mm) are appliedas well as the membrane tape (thickness 6mm) For theboiler membrane walls it is possible to calculate the elasticityconstants of equivalent orthotropic plate (Table 3) accordingto expressions (2) and (7) The members of elasticity matrix(Tables 4 and 5) which will be applied to the finite element ofequivalent orthotropic plate will be calculated according toexpressions (3) and (4)

In this paper the results of six different calculationsof reactions in the supports of ldquoMilanordquo boiler will bepresented the calculation of real geometry of boiler (REAL)discretized by the shell finite element the calculation withthe approximation of membrane walls as the plateshell ofequivalent stiffness (APPROX) and the calculationwhere theboiler is considered as a rigid body (RIGID) Each calculationis made with the load of the mass of boiler with and withoutthe mass of heat exchangers

The calculation model for the real geometry of boilerconsisted of 18943590 variables that include the degrees offreedom and Lagrange multipliers and for the approximatedgeometry of boilermembranewalls as the plateshell of equiv-alent stiffness the calculation model consisted of 1619526variables (degrees of freedom and Lagrange multipliers) Inboth cases 4-node finite element S4R5 was predominantlyused and an extremely triangular element was created by thecollapse of one node of 4-node element S4R5 (Figures 8 and9)

The reaction forces in the supports for the influence oftotal mass of boiler are given in Table 6 and the reactionforces for the influence of mass of boiler without the massof heat exchangers are given in Table 7 The reaction forcesin the boiler supports are denoted according to Figure 7In Tables 6 and 7 the deviations of reaction forces forcalculation APPROX (compared to calculation REAL) werecalculated and the deviation was expressed as a percentageand marked as Δ

119860 In the same way the deviation of reaction

forces for calculation RIGID (compared to calculation REAL)was calculated and it was expressed as a percentage andmarked as Δ

119870 The deviations were calculated according to

the following expressions

Δ119860

= (

119865Ri (APPROX)

119865Ri (REAL)minus 1) sdot 100

Δ119870

= (

119865Ri (RIGID)


(9)


4 Conclusion

The investigation of the influence of approximation of steamboilermembranewalls using the equivalent orthotropic platefor the calculation of reactions in the boiler supports ispresented in this paper The expressions for the elasticityconstants that have been published in previous papers [30ndash32] were applied

(i) The reactions in the boiler supports were calculatedwith the assumption that the structure of boiler isideally rigid

(ii) The calculations were carried out for the load of themass of boiler with and without the mass of heatexchangers in the third pass of boiler

(iii) The number of variables for the calculation modelREAL was more than 11 times higher than for thecalculation model APPROX Therefore the calcula-tion of node values of model REAL lasted for 1 h23min and 50 s and for the model APPROX 1minand 56 sThe calculation was carried out on a PCwithCPU Intel(R) Core (TM) i7 24GB RAM and 64-bitoperating system

(iv) The deviations of results for the reactions in thesupports for the calculation APPROX compared tothe calculation REAL are max 631 for the load ofthe total mass of boiler and 601 for the load withoutthe mass of heat exchangers

(v) The deviations of results for the calculation RIGIDcompared to the calculation REAL are 7875 for theload without the mass of heat exchangers

(vi) If the results of calculation REAL are accepted asapproximately accurate then it can be concludedthat the application of calculation model APPROX isacceptable for practical use Due to the huge errorthe calculationmodel RIGID is not recommended forpractical use

(vii) Using themethod of equivalent stiffness ofmembranewall more accurate results of calculation could beacquired if when calculating themodulus of elasticity1198641198872 the membrane pipe would be observed as a solid

body and not as an ideal rigid body [32]

(viii) Additional improvements could be achieved by tak-ing into consideration the transverse shear stiffness

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors would like to thank PLANUM doo fromSlavonski Brod in Croatia for usage of technical data whichwere needed for this research

References

[1] V Milosevic-Mitic B Gacesa N Anđelic and T ManeskildquoNumerical calculation of the water-tube boiler using finiteelement of the orthotropic platerdquo Structural Integrity and Lifevol 12 no 3 pp 185ndash190 2012

[2] M T Huber ldquoDie grundlagen einer rationellen berechnungder kreuzweise bewehrten eisenbetonplattenrdquo Zeitschrift desOsterreichischen Ingenieur- und Architekten-Vereins vol 66 pp557ndash564 1914

[3] M T Huber ldquoDie Theorie der kreuzweise bewehrtenEisenbeton-platten nebst Anwendungen auf mehrerebautechnisch wichtige Aufgaben uber rechteckige PlattenrdquoBauingenieur vol 4 pp 354ndash360 1923

[4] M T Huber Probleme der Statik Technisch WichtigerOrthotroper Platten Nakkadem Akadenji Nauk TechnicznychWarszawa Poland 1929

[5] W Flugge ldquoDie Stabilitat der Kreiszylinderschalerdquo Ingenieur-Archiv vol 3 no 5 pp 463ndash506 1932

[6] A van der Neut ldquoThe general instability of stiffened cylindricalshells under axial compressionrdquo Report S 314 National Lucht-vaartlaboratorium 1947

[7] F A Dale R C T Smith et al ldquoGrid sandwich panels in com-pressionrdquo Report ACA-16 Australian Council for Aeronautics1945

[8] C B Smith T B Heebink and C B Norris ldquoThe effective stiff-ness of a stiffener attached to a flat plywood platerdquo Report 1557U S Department of Agriculture Forest Products Laboratory1946

[9] A Pfluger ldquoZumBeulproblem der anisotropen RechteckplatterdquoIngenieur-Archiv vol 16 no 2 pp 111ndash120 1947

[10] A Gomza and P Seide ldquoMinimum-weight design of simplysupported transversely stiffened plates under compressionrdquoNACA TN 1710 1948

[11] C Libove and R E Hubka ldquoElastic constants for corrugated-core sandwich platesrdquo NACA TN 2289 1951

[12] C Libove and S B Batdorf ldquoA general small-deflection theoryfor flat sandwich platesrdquo NACA Report 899 1948

[13] E Reissner ldquoOn small bending and stretching of sandwich-typeshellsrdquo International Journal of Solids and Structures vol 13 no12 pp 1293ndash1300 1977

[14] M Stein and J Mayers ldquoA small-deflection theory for curvedsandwich platesrdquo NACA Report 1008 1951

[15] N J Huffington Jr ldquoTheoretical determination of rigidityproperties of orthogonally stiffened platesrdquo Journal of AppliedMechanics vol 23 pp 15ndash20 1956

[16] G Gerard ldquoMinimum weight analysis of orthotropic platesunder compressive loadingrdquo Journal of the Aeronautical Sci-ences vol 27 no 1 pp 21ndash26 1960

[17] M Baruch and J Singer ldquoEffect of eccentricity of stiffenerson the general instability of stiffened cylindrical shells underhydrostatic pressurerdquo Journal ofMechanical Engineering Sciencevol 5 no 1 pp 23ndash27 1963

[18] R J Clifton J C L Chang and T Au ldquoAnalysis of orthotropicplate bridgesrdquo ASCE Journal of the Structural Division vol 89no 5 pp 133ndash171 1963

[19] W J Stroud ldquoElastic constants for bending and twisting ofcorrugation-stiffened panelsrdquo NASA TR R 166 1963

[20] R R Meyer and R J Bellefante ldquoFabrication and experimentalevaluation of common domes having waffle-like stiffeningmdashpart Imdashprogram developmentrdquo Report SM 47742 DouglasMissile amp Space Systems Division 1964


[21] T-C Soong ldquoBuckling of cylindrical shells with eccentricspiral-type stiffenersrdquo AIAA Journal vol 7 no 10 pp 65ndash721969

[22] R R Meyer ldquoComments on buckling of cylindrical shells witheccentric spiral-type stiffenersrdquo AIAA Journal vol 7 no 10 p2047 1969

[23] T-C Soong ldquoReply by author to R R Meyerrdquo AIAA Journalvol 7 no 10 pp 2047ndash2048 1969

[24] F Nishino R P Pama and S-L Lee ldquoOrthotropic plateswith eccentric stiffenersrdquo International Association of BridgeStructural Engineering vol 34 no 2 pp 117ndash129 1974

[25] W L Ko ldquoElastic constants for superplasticallyformeddiffusion-bonded sandwich structuresrdquo in Proceedingsof the AIAAASMEAHS 20th Structures Structural Dynamicsand Materials Conference AIAA paper 79-0756 1979

[26] W L Ko ldquoElastic constants for superplasticallyformeddiffusion-bonded sandwich structuresrdquo AIAA Journalvol 18 no 8 pp 986ndash987 1980

[27] W L Ko ldquoElastic stability of superplastically formeddiffusion-bonded orthogonally corrugated core sandwich platesrdquo inProceedings of the AIAAASMEAHS 21st Structures StructuralDynamics and Materials Conference AIAA paper 80-06831980

[28] N Jaunky N F Knight Jr and D R Ambur ldquoFormulation ofan improved smeared stiffener theory for buckling analysis ofgrid-stiffened composite panelsrdquo NASA TM 110162 1995

[29] N Jaunky F Knight and D R Ambur ldquoFormulation of animproved smeared stiffener theory for buckling analysis of grid-stiffened composite panelsrdquo Composites B vol 27 pp 519ndash5261996

[30] J Sertic D Kozak D Damjanovic and P Konjatic ldquoHomoge-nization of steam boiler membrane wallrdquo in Proceedings of 7thInternational Congress of Croatian Society of Mechanics Zadar2012

[31] J Sertic I Gelo D Kozak D Damjanovic and P KonjeticldquoTeoretical determination of elasticity constants for steamboiler membrane wall as the structurally ortotropic platerdquo inProceeding of 4th International Scientific and Expert Conferenceof the International TEAM Society Slavonski Brod 2012

[32] J Sertic I Gelo D Kozak D Damjanovic and P KonjaticldquoTheoretical determination of elasticity constants for steamboiler membrane wall as the structurally orthotropic platerdquoTechnical Gazette vol 20 no 4 pp 697ndash703 2013

[33] R Solecki and R J Conant Advanced Mechanics of MaterialsOxford University Press Oxford UK 2003

[34] E Ventsel and T Krauthammer Thin Plates and Shells TheoryAnalysis and Applications Marcel Dekker Basel Switzerland2001

[35] V V Novozhilov Thin Shell Theory Noordhoff GroningenGermany 1964

[36] ASME B311 1995

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



The homogenization ofmembrane wall stiffness is performednumerically using the method of equalizing displacement

When calculating the deflection of membrane walls dueto the influence of underpressureoverpressure of flue gasesthe equivalent wall thicknessmethod ismost commonly usedin practice The bending stiffness of membrane wall pipe isequal to the bending stiffness of equivalent wall thickness forthe axis perpendicular to the longitudinal axis of membranewall pipe and the bending stiffness of membrane wall tapeis equal to the bending stiffness of equivalent wall thicknessfor the axis parallel to the longitudinal axis of membrane wallpipe

The steam boiler membrane wall represents a structurallyorthotropic plate The stiffened plates that are frequentlyused both in building and in the processingpower plantswhere it is necessary to achieve the optimal structuralsolution considering the mass of structure also belong tothis class of structures The earliest research on the topicof stiffness of stiffened plates was most likely done byHuber [2ndash4] His investigations were inspired by the lackof exact expressions for solving the problems of bendingand buckling of stiffened structures by using the theoryof homogeneous orthotropic plates Similarly Flugge [5]published the first paper on the topic of structurally stiffenedshell in 1932 and 15 years later so did van der Neut [6]Over time the understanding of the theory of deformation ofstructurally orthotropic plates and shells has progressed Theterms ldquoequivalent stiffnessrdquo and ldquoequivalent wall thicknessrdquobecame the basis for the analysis of deformation of stiffenedor structurally orthotropic platesshells Furthermore Daleet al [7] applied the equivalent stiffness in the researchof buckling of pressure loaded sandwich plates Smith etal [8] proposed an improved formulation that takes intoconsideration the change of position of the neutral surfacethat is connected with the local interaction between theplate and stiffener Pfluger [9] Gomza and Seide [10] andLibove and Hubka [11] applied in their research the methodof equivalent stiffness and thickness of stiffened plateshellIn the papers [12ndash14] the theory of symmetric sandwichplates with the included transverse shear stiffness is appliedIn 1956 Huffington Jr [15] analyzed the equivalent stiffnessof orthogonally stiffened plate without the stiffener eccen-tricity During the last fifty years the equivalent plate orshell approach was still being used as a first approxima-tion method [16ndash19] For the experimental determinationof elasticity constants the method of testing the load-deformation characteristics of repeating structural part of thestructurally orthotropic plate was most commonly used [20]In 1969 Soong [21ndash23] presented a derivation of the stiffnessexpressions for orthotropic cylinders based on a deformationenergy approach Throughout the following years specialattention was given to the interaction between the plate andstiffeners [24] for a different layout or grid of stiffeners [25ndash27] and different combinations of loads In 1995 Jaunky etal [28 29] presented a refined smeared-stiffener theory forgrid-stiffened laminated-composite panels based upon thework presented by Smith et al [8] Similarly Wodesenbetpresented an improved smeared-stiffener theory for grid-stiffened laminated-composite cylinders in 2003

It can be concluded that the current research is mainlybased on two systematic analytical methods that take ordo not take into account the transverse shear stiffnessFirst there is a direct equilibrium-compatibility methodThe first method uses equilibrium and compatibility in adirect manner for plates reinforced by stiffeners The secondone is called the basic-cell energy-equivalence method Thesecond method is based on defining the equivalence betweenthe deformation energy of the basic repeating cell and thedeformation energy of the equivalent orthotropic plate Thismethod is suitable for the determination of the members ofelasticity matrix for grid-stiffened plates and sandwich plates

Unlike the stiffened plates a steam boiler membrane wallconsists of a homogeneous plate with attached stiffenersTherefore the two mentioned methods are not entirelysuitable for the determination of equivalent stiffness ofmembrane wall that is the equivalent elasticity constantsThe process of homogenization of steam boiler membranewall is presented in the papers [30ndash32] The elasticity con-stants for the membrane and bending stiffness of mem-brane wall are determined theoretically by equalizing thedeformation of membrane wall with the deformation ofequivalent orthotropic plate In these papers the transverseshear stiffness is not taken into consideration

2 Steam Boiler Membrane Wall asthe Structurally Orthotropic Plate

The steam boiler membrane wall can be approximated asthe equivalent orthotropic plate that is the orthotropicplate which will have the same elastic properties as the realmembrane wall Using Kirchhoff-Love theory of shells [3334] the constitutive equations of equivalent orthotropic plateor shell can be written as follows

120590 = D sdot 120576 (1)

where 120590 is vector of section forces D is matrix of elasticityand 120576 is vector of deformation According to [35] it can besaid that the constitutive equations for the analysis of platesand shells are analogThematrix expression (1) represents thesix constituent equations ofmembranewall as the structurallyorthotropic plate or its equivalent orthotropic plate whichconnect the section forces with the corresponding deforma-tions

[

[

[

[

[

[

[

[

1198731

1198732

11987312

1198721

1198722

11987212

]

]

]

]

]

]

]

]

=

[

[

[

[

[

[

[

[

11986011

11986012

0 0 0 0

11986022

0 0 0 0

11986033

0 0 0

11986311

11986312

0

sym 11986322

0

11986333

]

]

]

]

]

]

]

]

sdot

[

[

[

[

[

[

[

[

1205761

1205762

12057612

1205811

1205812

2 sdot 12058112

]

]

]

]

]

]

]

]

(2)

11987311198732 and119873

12are the section forces (Figure 1) and119872

11198722

and11987212are the sectionmoments (Figure 2) in themembrane

wall that is the equivalent orthotropic plate 11986011 11986012 11986022

and 11986033

are the members of elasticity matrix for membraneloads and 119863

11 11986312 11986322 and 119863

33are the members of


N12N12N21 N21N1

N1N2 N2

120572212057221205721 1205721

AmAp

Am equivAp



M2

M1M12M21

120572212057221205721 1205721

DmDp

DmequivDp

M2M21 M1

M12




1 1205762 and 120576


1

1205812 and 120581




11986011

=

1198641198981

sdot ℎ

1 minus ]11989812

sdot ]11989821

11986012

=

1198641198982


1 minus ]11989812

sdot ]11989821

11986022

=

1198641198982

sdot ℎ

1 minus ]11989812

sdot ]11989821

11986033

= 11986611989812

sdot ℎ

(3)

where1198641198981





2(the axis



1 ]11989821


2 and 119866




11986311

=

1198641198871

sdot ℎ

3

12 sdot (1 minus ]11988712

sdot ]11988721

)

11986312

=

1198641198872

sdot ℎ

3sdot ]11988712

12 sdot (1 minus ]11988712

sdot ]11988721

)

11986322

=

1198641198872

sdot ℎ

3

12 sdot (1 minus ]11988712

sdot ]11988721

)

11986333

=

11986611988712

sdot ℎ

3

12

(4)


ℎ =radic

3 sdot 120587

16 sdot 119863 sdot 119896

sdot (119863


4) +

(119896 minus 119863) sdot 119905

3

119863 sdot 119896

(5)


h

k

120601D times 120575

1205722

1205722

1205723

1205723

t

b2



membrane wall


1

2

3

4

5

6

7

8

9

10

11

25

24

23

22

21

20

19

18

17

16

15

14

13

12

X

Y

Z



850

2500

2100

1925

1030

1080

4915

1790

890

11701440

X

Y

Z

1980



1198641198981

=

119864

119896 sdot ℎ

[

120587

4

sdot (119863


2) + (119896 minus 119863) sdot 119905]

1198641198982

=

119896 sdot 119864

ℎ sdot [3 sdot (119863120575 minus 1)

3sdot (1205878 minus 1120587) + 119887119905]

11986611989812

asymp

radic1198641198981

sdot 1198641198982

2 sdot (1 +radic]11989812

sdot ]11989821

)

]11989821

= ]11989812

sdot

1198641198982

1198641198981

(6)






1198641198871

=

119864

119896 sdot ℎ

3sdot [

3 sdot 120587

16

sdot (119863


4) + (119896 minus 119863) sdot 119905

3]

1198641198872

= 119864 sdot (

119905

ℎ

)

3

sdot

(119899 sdot 119896)

4

sum

119899

119894=1(119894 sdot 119896)

4minus (119894 sdot 119896 minus 119887)

4

11986611988712

asymp

radic1198641198871

sdot 1198641198872

2 sdot (1 +radic]11988712

sdot ]11988721

)

]11988721

= ]11988712

sdot

1198641198872

1198641198871

(7)


=



k = int

119860

B119879DB 119889119860 (8)





360

100

2120

100

2120

100

100

1035118013851315

1 2 3 4 5

1740

Superheater 1

Superheater 2

Superheater 3

(SH 1)

(SH 2)

(SH 3)

Seco

nd p

ass

Firs

t pas

s

Third

pas

s

A

A

A

A

Evaporator(EV)













119899 = 12


pass


119899 = 23



119899 = 23



119899 = 15



119899 = 21



119899 = 23



119899 = 23







119899 = 23










119899 = 34









ℎmm

1198641198981

GPa

1198641198982

GPa

11986611989812

GPa ]

11989821

1198641198871

GPa1198641198872

GPa11986611988712

GPa ]

11988721

5 7 17 19 22 2348 76492 1367 4915 0005361 454159 8064 29095 00053276 2348 76492 1367 4915 0005361 454159 8023 29025 000530013 2348 76492 1367 4915 0005361 454159 8203 29336 000541916 2348 76492 1367 4915 0005361 454159 7690 28439 000508018 2040 88592 3880 8722 0013138 442956 9583 31199 0006490

Detail A

A

X

Y

Z





11986011

Nmm11986012

Nmm11986022

Nmm11986033

Nmm5 6 7 13 16 1719 22 1799282 9646 32152 115438

18 1814749 23843 79476 177966



11986311

Nsdotmm11986312

Nsdotmm11986322

Nsdotmm11986333

Nsdotmm5 7 17 19 22 491000497 2615341 8717804 314048896 490996567 2602240 8674134 3132927213 491014120 2660750 8869166 3166540416 490964096 2494004 8313348 3069652418 314156438 2038906 6796354 22084218



1198771 N 119865

1198772 N 119865

1198773 N 119865

1198774 N 119865

1198775 N







1198771 N 119865

1198772 N 119865

1198773 N 119865

1198774 N 119865

1198775 N


Δ119870 1261 minus2448 4027 minus4829 7875

B

Detail B

X

Y

Z














Δ119860

= (

119865Ri (APPROX)


Δ119870

= (

119865Ri (RIGID)


(9)


4 Conclusion













Acknowledgment


References





































[36] ASME B311 1995



RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










N12N12N21 N21N1

N1N2 N2

120572212057221205721 1205721

AmAp

Am equivAp



M2

M1M12M21

120572212057221205721 1205721

DmDp

DmequivDp

M2M21 M1

M12




1 1205762 and 120576


1

1205812 and 120581




11986011

=

1198641198981

sdot ℎ

1 minus ]11989812

sdot ]11989821

11986012

=

1198641198982


1 minus ]11989812

sdot ]11989821

11986022

=

1198641198982

sdot ℎ

1 minus ]11989812

sdot ]11989821

11986033

= 11986611989812

sdot ℎ

(3)

where1198641198981





2(the axis



1 ]11989821


2 and 119866




11986311

=

1198641198871

sdot ℎ

3

12 sdot (1 minus ]11988712

sdot ]11988721

)

11986312

=

1198641198872

sdot ℎ

3sdot ]11988712

12 sdot (1 minus ]11988712

sdot ]11988721

)

11986322

=

1198641198872

sdot ℎ

3

12 sdot (1 minus ]11988712

sdot ]11988721

)

11986333

=

11986611988712

sdot ℎ

3

12

(4)


ℎ =radic

3 sdot 120587

16 sdot 119863 sdot 119896

sdot (119863


4) +

(119896 minus 119863) sdot 119905

3

119863 sdot 119896

(5)


h

k

120601D times 120575

1205722

1205722

1205723

1205723

t

b2



membrane wall


1

2

3

4

5

6

7

8

9

10

11

25

24

23

22

21

20

19

18

17

16

15

14

13

12

X

Y

Z



850

2500

2100

1925

1030

1080

4915

1790

890

11701440

X

Y

Z

1980



1198641198981

=

119864

119896 sdot ℎ

[

120587

4

sdot (119863


2) + (119896 minus 119863) sdot 119905]

1198641198982

=

119896 sdot 119864

ℎ sdot [3 sdot (119863120575 minus 1)

3sdot (1205878 minus 1120587) + 119887119905]

11986611989812

asymp

radic1198641198981

sdot 1198641198982

2 sdot (1 +radic]11989812

sdot ]11989821

)

]11989821

= ]11989812

sdot

1198641198982

1198641198981

(6)






1198641198871

=

119864

119896 sdot ℎ

3sdot [

3 sdot 120587

16

sdot (119863


4) + (119896 minus 119863) sdot 119905

3]

1198641198872

= 119864 sdot (

119905

ℎ

)

3

sdot

(119899 sdot 119896)

4

sum

119899

119894=1(119894 sdot 119896)

4minus (119894 sdot 119896 minus 119887)

4

11986611988712

asymp

radic1198641198871

sdot 1198641198872

2 sdot (1 +radic]11988712

sdot ]11988721

)

]11988721

= ]11988712

sdot

1198641198872

1198641198871

(7)


=



k = int

119860

B119879DB 119889119860 (8)





360

100

2120

100

2120

100

100

1035118013851315

1 2 3 4 5

1740

Superheater 1

Superheater 2

Superheater 3

(SH 1)

(SH 2)

(SH 3)

Seco

nd p

ass

Firs

t pas

s

Third

pas

s

A

A

A

A

Evaporator(EV)













119899 = 12


pass


119899 = 23



119899 = 23



119899 = 15



119899 = 21



119899 = 23



119899 = 23







119899 = 23










119899 = 34









ℎmm

1198641198981

GPa

1198641198982

GPa

11986611989812

GPa ]

11989821

1198641198871

GPa1198641198872

GPa11986611988712

GPa ]

11988721

5 7 17 19 22 2348 76492 1367 4915 0005361 454159 8064 29095 00053276 2348 76492 1367 4915 0005361 454159 8023 29025 000530013 2348 76492 1367 4915 0005361 454159 8203 29336 000541916 2348 76492 1367 4915 0005361 454159 7690 28439 000508018 2040 88592 3880 8722 0013138 442956 9583 31199 0006490

Detail A

A

X

Y

Z





11986011

Nmm11986012

Nmm11986022

Nmm11986033

Nmm5 6 7 13 16 1719 22 1799282 9646 32152 115438

18 1814749 23843 79476 177966



11986311

Nsdotmm11986312

Nsdotmm11986322

Nsdotmm11986333

Nsdotmm5 7 17 19 22 491000497 2615341 8717804 314048896 490996567 2602240 8674134 3132927213 491014120 2660750 8869166 3166540416 490964096 2494004 8313348 3069652418 314156438 2038906 6796354 22084218



1198771 N 119865

1198772 N 119865

1198773 N 119865

1198774 N 119865

1198775 N







1198771 N 119865

1198772 N 119865

1198773 N 119865

1198774 N 119865

1198775 N


Δ119870 1261 minus2448 4027 minus4829 7875

B

Detail B

X

Y

Z














Δ119860

= (

119865Ri (APPROX)


Δ119870

= (

119865Ri (RIGID)


(9)


4 Conclusion













Acknowledgment


References





































[36] ASME B311 1995



RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










h

k

120601D times 120575

1205722

1205722

1205723

1205723

t

b2



membrane wall


1

2

3

4

5

6

7

8

9

10

11

25

24

23

22

21

20

19

18

17

16

15

14

13

12

X

Y

Z



850

2500

2100

1925

1030

1080

4915

1790

890

11701440

X

Y

Z

1980



1198641198981

=

119864

119896 sdot ℎ

[

120587

4

sdot (119863


2) + (119896 minus 119863) sdot 119905]

1198641198982

=

119896 sdot 119864

ℎ sdot [3 sdot (119863120575 minus 1)

3sdot (1205878 minus 1120587) + 119887119905]

11986611989812

asymp

radic1198641198981

sdot 1198641198982

2 sdot (1 +radic]11989812

sdot ]11989821

)

]11989821

= ]11989812

sdot

1198641198982

1198641198981

(6)






1198641198871

=

119864

119896 sdot ℎ

3sdot [

3 sdot 120587

16

sdot (119863


4) + (119896 minus 119863) sdot 119905

3]

1198641198872

= 119864 sdot (

119905

ℎ

)

3

sdot

(119899 sdot 119896)

4

sum

119899

119894=1(119894 sdot 119896)

4minus (119894 sdot 119896 minus 119887)

4

11986611988712

asymp

radic1198641198871

sdot 1198641198872

2 sdot (1 +radic]11988712

sdot ]11988721

)

]11988721

= ]11988712

sdot

1198641198872

1198641198871

(7)


=



k = int

119860

B119879DB 119889119860 (8)





360

100

2120

100

2120

100

100

1035118013851315

1 2 3 4 5

1740

Superheater 1

Superheater 2

Superheater 3

(SH 1)

(SH 2)

(SH 3)

Seco

nd p

ass

Firs

t pas

s

Third

pas

s

A

A

A

A

Evaporator(EV)













119899 = 12


pass


119899 = 23



119899 = 23



119899 = 15



119899 = 21



119899 = 23



119899 = 23







119899 = 23










119899 = 34









ℎmm

1198641198981

GPa

1198641198982

GPa

11986611989812

GPa ]

11989821

1198641198871

GPa1198641198872

GPa11986611988712

GPa ]

11988721

5 7 17 19 22 2348 76492 1367 4915 0005361 454159 8064 29095 00053276 2348 76492 1367 4915 0005361 454159 8023 29025 000530013 2348 76492 1367 4915 0005361 454159 8203 29336 000541916 2348 76492 1367 4915 0005361 454159 7690 28439 000508018 2040 88592 3880 8722 0013138 442956 9583 31199 0006490

Detail A

A

X

Y

Z





11986011

Nmm11986012

Nmm11986022

Nmm11986033

Nmm5 6 7 13 16 1719 22 1799282 9646 32152 115438

18 1814749 23843 79476 177966



11986311

Nsdotmm11986312

Nsdotmm11986322

Nsdotmm11986333

Nsdotmm5 7 17 19 22 491000497 2615341 8717804 314048896 490996567 2602240 8674134 3132927213 491014120 2660750 8869166 3166540416 490964096 2494004 8313348 3069652418 314156438 2038906 6796354 22084218



1198771 N 119865

1198772 N 119865

1198773 N 119865

1198774 N 119865

1198775 N







1198771 N 119865

1198772 N 119865

1198773 N 119865

1198774 N 119865

1198775 N


Δ119870 1261 minus2448 4027 minus4829 7875

B

Detail B

X

Y

Z














Δ119860

= (

119865Ri (APPROX)


Δ119870

= (

119865Ri (RIGID)


(9)


4 Conclusion













Acknowledgment


References





































[36] ASME B311 1995



RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










850

2500

2100

1925

1030

1080

4915

1790

890

11701440

X

Y

Z

1980



1198641198981

=

119864

119896 sdot ℎ

[

120587

4

sdot (119863


2) + (119896 minus 119863) sdot 119905]

1198641198982

=

119896 sdot 119864

ℎ sdot [3 sdot (119863120575 minus 1)

3sdot (1205878 minus 1120587) + 119887119905]

11986611989812

asymp

radic1198641198981

sdot 1198641198982

2 sdot (1 +radic]11989812

sdot ]11989821

)

]11989821

= ]11989812

sdot

1198641198982

1198641198981

(6)






1198641198871

=

119864

119896 sdot ℎ

3sdot [

3 sdot 120587

16

sdot (119863


4) + (119896 minus 119863) sdot 119905

3]

1198641198872

= 119864 sdot (

119905

ℎ

)

3

sdot

(119899 sdot 119896)

4

sum

119899

119894=1(119894 sdot 119896)

4minus (119894 sdot 119896 minus 119887)

4

11986611988712

asymp

radic1198641198871

sdot 1198641198872

2 sdot (1 +radic]11988712

sdot ]11988721

)

]11988721

= ]11988712

sdot

1198641198872

1198641198871

(7)


=



k = int

119860

B119879DB 119889119860 (8)





360

100

2120

100

2120

100

100

1035118013851315

1 2 3 4 5

1740

Superheater 1

Superheater 2

Superheater 3

(SH 1)

(SH 2)

(SH 3)

Seco

nd p

ass

Firs

t pas

s

Third

pas

s

A

A

A

A

Evaporator(EV)













119899 = 12


pass


119899 = 23



119899 = 23



119899 = 15



119899 = 21



119899 = 23



119899 = 23







119899 = 23










119899 = 34









ℎmm

1198641198981

GPa

1198641198982

GPa

11986611989812

GPa ]

11989821

1198641198871

GPa1198641198872

GPa11986611988712

GPa ]

11988721

5 7 17 19 22 2348 76492 1367 4915 0005361 454159 8064 29095 00053276 2348 76492 1367 4915 0005361 454159 8023 29025 000530013 2348 76492 1367 4915 0005361 454159 8203 29336 000541916 2348 76492 1367 4915 0005361 454159 7690 28439 000508018 2040 88592 3880 8722 0013138 442956 9583 31199 0006490

Detail A

A

X

Y

Z





11986011

Nmm11986012

Nmm11986022

Nmm11986033

Nmm5 6 7 13 16 1719 22 1799282 9646 32152 115438

18 1814749 23843 79476 177966



11986311

Nsdotmm11986312

Nsdotmm11986322

Nsdotmm11986333

Nsdotmm5 7 17 19 22 491000497 2615341 8717804 314048896 490996567 2602240 8674134 3132927213 491014120 2660750 8869166 3166540416 490964096 2494004 8313348 3069652418 314156438 2038906 6796354 22084218



1198771 N 119865

1198772 N 119865

1198773 N 119865

1198774 N 119865

1198775 N







1198771 N 119865

1198772 N 119865

1198773 N 119865

1198774 N 119865

1198775 N


Δ119870 1261 minus2448 4027 minus4829 7875

B

Detail B

X

Y

Z














Δ119860

= (

119865Ri (APPROX)


Δ119870

= (

119865Ri (RIGID)


(9)


4 Conclusion













Acknowledgment


References





































[36] ASME B311 1995



RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












1198641198871

=

119864

119896 sdot ℎ

3sdot [

3 sdot 120587

16

sdot (119863


4) + (119896 minus 119863) sdot 119905

3]

1198641198872

= 119864 sdot (

119905

ℎ

)

3

sdot

(119899 sdot 119896)

4

sum

119899

119894=1(119894 sdot 119896)

4minus (119894 sdot 119896 minus 119887)

4

11986611988712

asymp

radic1198641198871

sdot 1198641198872

2 sdot (1 +radic]11988712

sdot ]11988721

)

]11988721

= ]11988712

sdot

1198641198872

1198641198871

(7)


=



k = int

119860

B119879DB 119889119860 (8)





360

100

2120

100

2120

100

100

1035118013851315

1 2 3 4 5

1740

Superheater 1

Superheater 2

Superheater 3

(SH 1)

(SH 2)

(SH 3)

Seco

nd p

ass

Firs

t pas

s

Third

pas

s

A

A

A

A

Evaporator(EV)













119899 = 12


pass


119899 = 23



119899 = 23



119899 = 15



119899 = 21



119899 = 23



119899 = 23







119899 = 23










119899 = 34









ℎmm

1198641198981

GPa

1198641198982

GPa

11986611989812

GPa ]

11989821

1198641198871

GPa1198641198872

GPa11986611988712

GPa ]

11988721

5 7 17 19 22 2348 76492 1367 4915 0005361 454159 8064 29095 00053276 2348 76492 1367 4915 0005361 454159 8023 29025 000530013 2348 76492 1367 4915 0005361 454159 8203 29336 000541916 2348 76492 1367 4915 0005361 454159 7690 28439 000508018 2040 88592 3880 8722 0013138 442956 9583 31199 0006490

Detail A

A

X

Y

Z





11986011

Nmm11986012

Nmm11986022

Nmm11986033

Nmm5 6 7 13 16 1719 22 1799282 9646 32152 115438

18 1814749 23843 79476 177966



11986311

Nsdotmm11986312

Nsdotmm11986322

Nsdotmm11986333

Nsdotmm5 7 17 19 22 491000497 2615341 8717804 314048896 490996567 2602240 8674134 3132927213 491014120 2660750 8869166 3166540416 490964096 2494004 8313348 3069652418 314156438 2038906 6796354 22084218



1198771 N 119865

1198772 N 119865

1198773 N 119865

1198774 N 119865

1198775 N







1198771 N 119865

1198772 N 119865

1198773 N 119865

1198774 N 119865

1198775 N


Δ119870 1261 minus2448 4027 minus4829 7875

B

Detail B

X

Y

Z














Δ119860

= (

119865Ri (APPROX)


Δ119870

= (

119865Ri (RIGID)


(9)


4 Conclusion













Acknowledgment


References





































[36] ASME B311 1995



RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










360

100

2120

100

2120

100

100

1035118013851315

1 2 3 4 5

1740

Superheater 1

Superheater 2

Superheater 3

(SH 1)

(SH 2)

(SH 3)

Seco

nd p

ass

Firs

t pas

s

Third

pas

s

A

A

A

A

Evaporator(EV)













119899 = 12


pass


119899 = 23



119899 = 23



119899 = 15



119899 = 21



119899 = 23



119899 = 23







119899 = 23










119899 = 34









ℎmm

1198641198981

GPa

1198641198982

GPa

11986611989812

GPa ]

11989821

1198641198871

GPa1198641198872

GPa11986611988712

GPa ]

11988721

5 7 17 19 22 2348 76492 1367 4915 0005361 454159 8064 29095 00053276 2348 76492 1367 4915 0005361 454159 8023 29025 000530013 2348 76492 1367 4915 0005361 454159 8203 29336 000541916 2348 76492 1367 4915 0005361 454159 7690 28439 000508018 2040 88592 3880 8722 0013138 442956 9583 31199 0006490

Detail A

A

X

Y

Z





11986011

Nmm11986012

Nmm11986022

Nmm11986033

Nmm5 6 7 13 16 1719 22 1799282 9646 32152 115438

18 1814749 23843 79476 177966



11986311

Nsdotmm11986312

Nsdotmm11986322

Nsdotmm11986333

Nsdotmm5 7 17 19 22 491000497 2615341 8717804 314048896 490996567 2602240 8674134 3132927213 491014120 2660750 8869166 3166540416 490964096 2494004 8313348 3069652418 314156438 2038906 6796354 22084218



1198771 N 119865

1198772 N 119865

1198773 N 119865

1198774 N 119865

1198775 N







1198771 N 119865

1198772 N 119865

1198773 N 119865

1198774 N 119865

1198775 N


Δ119870 1261 minus2448 4027 minus4829 7875

B

Detail B

X

Y

Z














Δ119860

= (

119865Ri (APPROX)


Δ119870

= (

119865Ri (RIGID)


(9)


4 Conclusion













Acknowledgment


References





































[36] ASME B311 1995



RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation
















119899 = 12


pass


119899 = 23



119899 = 23



119899 = 15



119899 = 21



119899 = 23



119899 = 23







119899 = 23










119899 = 34









ℎmm

1198641198981

GPa

1198641198982

GPa

11986611989812

GPa ]

11989821

1198641198871

GPa1198641198872

GPa11986611988712

GPa ]

11988721

5 7 17 19 22 2348 76492 1367 4915 0005361 454159 8064 29095 00053276 2348 76492 1367 4915 0005361 454159 8023 29025 000530013 2348 76492 1367 4915 0005361 454159 8203 29336 000541916 2348 76492 1367 4915 0005361 454159 7690 28439 000508018 2040 88592 3880 8722 0013138 442956 9583 31199 0006490

Detail A

A

X

Y

Z





11986011

Nmm11986012

Nmm11986022

Nmm11986033

Nmm5 6 7 13 16 1719 22 1799282 9646 32152 115438

18 1814749 23843 79476 177966



11986311

Nsdotmm11986312

Nsdotmm11986322

Nsdotmm11986333

Nsdotmm5 7 17 19 22 491000497 2615341 8717804 314048896 490996567 2602240 8674134 3132927213 491014120 2660750 8869166 3166540416 490964096 2494004 8313348 3069652418 314156438 2038906 6796354 22084218



1198771 N 119865

1198772 N 119865

1198773 N 119865

1198774 N 119865

1198775 N







1198771 N 119865

1198772 N 119865

1198773 N 119865

1198774 N 119865

1198775 N


Δ119870 1261 minus2448 4027 minus4829 7875

B

Detail B

X

Y

Z














Δ119860

= (

119865Ri (APPROX)


Δ119870

= (

119865Ri (RIGID)


(9)


4 Conclusion













Acknowledgment


References





































[36] ASME B311 1995



RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation














ℎmm

1198641198981

GPa

1198641198982

GPa

11986611989812

GPa ]

11989821

1198641198871

GPa1198641198872

GPa11986611988712

GPa ]

11988721

5 7 17 19 22 2348 76492 1367 4915 0005361 454159 8064 29095 00053276 2348 76492 1367 4915 0005361 454159 8023 29025 000530013 2348 76492 1367 4915 0005361 454159 8203 29336 000541916 2348 76492 1367 4915 0005361 454159 7690 28439 000508018 2040 88592 3880 8722 0013138 442956 9583 31199 0006490

Detail A

A

X

Y

Z





11986011

Nmm11986012

Nmm11986022

Nmm11986033

Nmm5 6 7 13 16 1719 22 1799282 9646 32152 115438

18 1814749 23843 79476 177966



11986311

Nsdotmm11986312

Nsdotmm11986322

Nsdotmm11986333

Nsdotmm5 7 17 19 22 491000497 2615341 8717804 314048896 490996567 2602240 8674134 3132927213 491014120 2660750 8869166 3166540416 490964096 2494004 8313348 3069652418 314156438 2038906 6796354 22084218



1198771 N 119865

1198772 N 119865

1198773 N 119865

1198774 N 119865

1198775 N







1198771 N 119865

1198772 N 119865

1198773 N 119865

1198774 N 119865

1198775 N


Δ119870 1261 minus2448 4027 minus4829 7875

B

Detail B

X

Y

Z














Δ119860

= (

119865Ri (APPROX)


Δ119870

= (

119865Ri (RIGID)


(9)


4 Conclusion













Acknowledgment


References





































[36] ASME B311 1995



RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












1198771 N 119865

1198772 N 119865

1198773 N 119865

1198774 N 119865

1198775 N


Δ119870 1261 minus2448 4027 minus4829 7875

B

Detail B

X

Y

Z














Δ119860

= (

119865Ri (APPROX)


Δ119870

= (

119865Ri (RIGID)


(9)


4 Conclusion













Acknowledgment


References





































[36] ASME B311 1995



RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










4 Conclusion













Acknowledgment


References





































[36] ASME B311 1995



RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation

























[36] ASME B311 1995



RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









Research Article Calculation of Reaction Forces in the ...

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