Comparative study on thin and thick walled cylinder models subjected to thermo-mechanical loading

Composite Structures 134 (2015) 142–146

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Comparative study on thin and thick walled cylinder models subjected to thermo-mechanical loading Grzegorz Juszkiewicz ⇑, Tomasz Nowak ABB Corporate Research, Starowislna 13A Street, Cracow, Poland

a r t i c l e

i n f o

Article history: Available online 24 August 2015 Keywords: Thick-walled cylinder Thin-walled cylinder model Orthotropic material CLT

a b s t r a c t This paper gives a theoretical background and compares two analytical approaches, thin- and thickwalled models, analyzing composite cylindrical tubes under thermo-mechanical loadings. First, a theoretical background is introduced, and a lamination theory and an elasticity theory for thick-wall tubes are recalled. A systematic parametric study for various geometrical, material and load settings was performed to find out the difference between analyzed calculation approaches. It was generally observed, that the Classical Lamination Theory can be successfully applied for pressure loads, however this planestress assumption may generate remarkable errors if thermal loads are introduced. It is especially the case for highly orthotropic cylinders. The generalization of the achieved results allowed to recommend a new criterion for the selection of an appropriate calculation model. The proposed measure incorporates simple forms of tubes’ geometrical parameters (D/t) and material factor (C22/C33). Thanks to the applied approach the importance of through-thickness stresses can be quickly assessed. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction The mechanical behavior of pressurized cylinders made of an isotropic material is very well elaborated. The literature delivers a thick walled model based on Lamé theory, which incorporates principle stresses in all three directions, and a thin wall model, which neglects radial stresses [1–7]. Normally, it is assumed that if tube’s diameter-to-thickness ratio, D/t is more than 20, the radial stresses are an order of magnitude smaller than the other stress components. In this case a simplified, thin-wall assumption could be successfully applied. This approach may not work however, if a laminated cylinder is considered. In this case stresses within the tube are not related only to D/t ratio, but may be dependent on material properties (different for various plies), and lay-up design (layer thicknesses and orientations). For this reason it is frequently accepted that diameter-to-thickness ratio cannot be treated as the only factor allowing to decide if a plane-stress model may be used for a particular design case. The fundamental theoretical background for analysis of the anisotropic bodies was provided by Lakhnitskii [8], and his work has been referenced later in the large number of textbooks dealing with composites [9–12]. The application of the orthotropic material model into cylindrical structures was given by Scherrer [13], Pagano [14], Wilson and Orgill [15], and Pindera [16]. The solid ⇑ Corresponding author. Tel.: +48 222238439. E-mail address: [email protected] (G. Juszkiewicz). http://dx.doi.org/10.1016/j.compstruct.2015.08.085 0263-8223/Ó 2015 Elsevier Ltd. All rights reserved.

description of the analytical solution for the laminated circular tube subjected to the mechanical loading was provided by Herakovitch [17]. Also hybrid structures, like Fiber Reinforced Metal (FRM), or Fiber Metal Laminates (FML) are of interest, [18–20]. Today, with growing popularity of numerical methods, theoretical investigations are strongly supported by FEM analyses [21–23]. However, even if the literature covering the theory and practice of composite cylinders is quite reach, the systematic studies comparing two different calculation models, thick- and thin-walled, are not common. The work described in this paper compares both above approaches. The basic information about a classical lamination concept and an elasticity theory for thick-walled orthotropic tubes is given in Section 2, providing the insight into the applied material models and constitutive relations. Next, the numerical example is presented, and calculations managed for different loads, diameter-to-thickness ratios and material properties are described. The outcome of the study delivers a proposal of new criterion allowing to assess if the plane-stress assumption could be safely applied to the particular design case. The concluding remarks are given in the last section of the article. 2. Theoretical investigation 2.1. Isotropic cylinder In order to investigate a difference between thin- and thick walled theories for pressurized cylinders, the analysis of an

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isotropic structure will be recalled. The well known Lamé theory states that the hoop and radial stresses in a pressurized cylinder can be described as:

R21

rU;r ¼ p

R22  R21

1

R22 r2

!

ð1Þ

where p in an internal pressure acting on the inner surface, and R1 and R2 are the inner and outer radii, respectively. Focusing only on the highly stressed internal surface (r = R1), and expressing the diameter-to-thickness ratio as K, one may reformulate Eq. (1) to:

"

K2 K2 ¼p  1 4ðK þ 1Þ 4ðK þ 1Þ

rU;r

#

ð2Þ

where:

D1 2R1 ¼ t R2  R1

K¼

ð3Þ

where D1 is an internal diameter and t is the wall thickness of the cylinder. Based on Eq. (2) it is possible to state, that for large diameter-tothickness ratio (K  10) the hoop stress could be quite well estimated by Eq. (4), as proposed also by a thin-wall model:

"

rU ¼ p

#

2

K K pD1 ~U þ1 !p ¼ ¼r 2 2ðK þ 1Þ 2t

ð4Þ

The relative error between Lamé approach and thin-wall model in the case of an isotropic, pressurized cylinder depends only on the diameter-to-thickness ratio, and may be calculated as:

d¼

rU  r~ U K þ2 1 ¼ 2  ½100% K rU K þ 2K þ 2

ð5Þ

With the help of Eq. (5) one can simply estimate that the application of the thin-wall model into a pressurized cylinder, having K = 20, can introduce an error in stress calculation at the level of about 5%. It should be also noted, that even smaller error may be achieved, if a mean diameter (not D1) is used in Eq. (4). Lamé theory states also, that radial stresses vary across the wall thickness – from the value equal to p, at the inner surface, to zero – at the outer surface. While the thin-wall theory totally neglects stresses across the thickness direction. If an unconstraint isotropic cylinder subjected to the temperature load is considered, both models similarly predict that the thermal strains will not generate stresses. It is not the case for a laminated tube, where the thermal stresses will be generated. However, the comparison between these two theories is not so straightforward, if an anisotropic material, or orthotropic composite should be investigated. In this case the difference is not only affected by geometrical parameters of the cylinder, but also the material properties, which are different in principle directions. 2.2. Classical Lamination Theory The thin-wall composite tube may be analyzed with the use of well-established Classical Lamination Theory, CLT. It basically considers plane-stress state, assuming that radial stresses in thin cylinders are significantly smaller than the other stress components, thus may be ignored. Such assumption simplifies the calculation process, therefore it is popular in industrial applications. In this classical approach [17] the stress–strain relation is characterized by an equivalent generalized force (N, M) – generalized strain (e0,j) system:



N M

(

 þ

NT M

T

)

 ¼

A

B

B

D

(

e0 j

)

ð6Þ

where N, M are vectors of forces {Nx, NU, NxU} and moments {Mx, MU, MxU}, respectively; NT, MT refer to vectors of thermal forces and thermal moments, respectively; e0, j are vectors of strains due to in-plane forces and strains due to moments (curvatures), respectively; A, D and B are called tension stiffness, bending stiffness and coupling stiffness matrices respectively. They are calculated as follows:

Z ½A; B; D ¼

t=2

t=2

  ð1; z; z2 Þdz ½Q k

ð7Þ

where [Q]k is the reduced stiffness matrix of a single kth lamina, which is spaced from the neutral plane of the laminate by distance z, and t is the total thickness of the laminate. The reduced stiffness matrix [Q]k defines the relation between stresses and strains for a single lamina, as:

2

3

2

 11 rx k Q 6 7 6 4 rU 5 ¼ 4 Q 12 sxU 0

 12 Q  22 Q 0

3k 2

3

ex k 76 7 0 5 4 eU 5  cxU Q 66 0

ð8Þ

The vector of strains in a single lamina is an algebraic sum of mid-plane strains, curvatures and thermal strains:

ek ¼ e0 þ zj  eT

ð9Þ

where thermal strain vector eT reads:

2

3

2

3

ax k eTx k 6 T 7 6 7 4 eU 5 ¼ DT 4 aU 5 T axU cxU

ð10Þ

DT is the temperature change, and aj are thermal expansion coefficients in respective directions. It should be underlined, that the matrix B plays an important role in the lamination theory, since it causes complex interaction between the in-plane loads and bending effects (out-of-plane strains). However, composite structures are typically designed in such a way that all components of B matrix are zero, therefore generated stresses are only the result of in-plane strains, e, driven by in-plane forces, N. 2.3. Thick-walled composite cylinder The thick-walled analytical model used to study a composite cylinder under thermo-mechanical load assumes a general orthotropic laminate. In the most universal case, there are 15 unknowns (3 displacements, 6 strains and 6 stresses) to be derived from the equilibrium, constitutive, and continuity equations. Using these relations, and neglecting the radial shears, one can prove [17,23], that the radial, axial and tangential displacements can be calculated as:

wðrÞ ¼ Ar k þ Br k þ Ce0x r þ Xc0 r 2 þ Wr DT ð11Þ

uðx; rÞ ¼ xe0x

v ðx; rÞ ¼ xrc0 where: A, B, ex0, c0 are constants of integration – to be determined from the boundary conditions, and: k, C, X, W are material coefficients, defined as:

sffiffiffiffiffiffiffi C 22 k¼ ; C 33

C¼

C 12  C 23 C 33  C 22

;

X¼

C 26  2C 36 4C 33  C 22

;

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W¼

ðC 13  C 12 Þax þ ðC 23  C 22 ÞaU þ ðC 33  C 23 Þar þ ðC 36  C 26 ÞaxU C 33  C 22 ð12Þ

If constants of integrations are known, the strains may be written in matrix notation as:

2 6 6 6 4

3

32

2

3

2

3

ex A 0 0 1 0 0 76 B 7 6 W 7 6 r k1 k1 eU 7 r C X r 76 7 6 7 7 6 76 7 þ 6 7DT 7¼6 er 5 4 krk1 krk1 Cr 2Xr 54 e0x 5 4 W 5 cxU 0 0 0 r c0 0

ð13Þ

The stresses can be described by generalized Hooke’s Law, or by directly using the constants of integration as:

2

rx

6r 6 U 6 4 rr

3

2

C 11 7 6C 7 6 12 7¼6 5 4 C 13

sxU

C 12

C 13

C 22

C 23

C 23

C 33

C 16

3

C 26 7 7 7 C 36 5

0

2

rx C 11 6 rTR 7 6 C 6 U 7 6 12 6 TR 7 ¼ 6 4 rr 5 4 C 13

0

sTR xU

C 16

C 16

32

31

2

3

rTR x C 7 7 7 6 6 Xr 76 B 7C 6 rTR U 7 76 0 7C  6 TR 7 2Xr 54 ex 5A 4 rr 5 r c0 sTR xU 0

A

32

ax

3

C 12

C 13

C 22 C 23

C 23 C 33

6 7 C 26 7 76 aU  W 7 76 7DT C 36 54 ar  W 5

C 26

C 36

C 66

ð15Þ

axU

8 rr ðrin Þ ¼ pin > > > > < rr ðr out Þ ¼ pout R rout > > Px ¼ 2p rin rx rdr > > : T ¼ 2p R rout s r 2 dr x xU rin

ð16Þ

pin

3

3

rTR r



TR rTR r i1  rr i

rki1 1 rki1 1 Ci1  Ci ðXi1  Xi Þr rki 1 ari1 rki1 1 bri1 rki1 1 cri1  cri ðdri1  dri Þr ari r ki 1

3 Ai1 6B 7 i1 7 #6 6 0 7 e 7 r ki 1 6 6 x 7 6 0 bri rki 1 6 c 7 7 7 6 4 Ai 5 Bi 2

3. Analytical example In order to compare the results of the analytical models presented shortly in Sections 2.2 and 2.3, the numerical studies will be provided. In the analyzed example, a three-ply laminated tube has been investigated. The ply sequence was symmetrical and balanced: a/90/a, where the helical angle a varied from 0° (transversely isotropic layers – fibers in longitudinal direction) to 90° (orthotropic layers – fibers in hoop direction). The glass fiber reinforced epoxy resin was considered, with typical material properties and the fiber fraction equal to 55%. The overall thickness of cylinder, t was equal to 3 mm, and the internal diameter D1 varied from 60 to 165 mm, to evaluate different diameter-to-thickness settings. The diameter-to-thickness ratios, which were tested, allowed to assume that CLT should generate an acceptable error (below 5%). Two different load conditions have been tested during the study: internal pressure and temperature drop. Various load levels were evaluated. 3.1. Pressure load

After integration of Eq. (16), the unknown vector of integration constants can be finally derived from:

2

Wi DT i  Wi1 DT i1

ð19Þ

In order to determine the four unknown constants of integration: A, B, ex0, c0 it is necessary to solve the set of four equations composed of two radial stresses (pressures) acting on the inner and outer surfaces of the cylinder, the axial force (Px) equilibrium condition, and the torque (Tx) equilibrium condition:

2

 )

"

where the vector of temperature related stresses [rTR] reads: TR 3



¼

ð14Þ 2

wðrÞ

rr ðrÞ

C 16 C 26 C 36 C 66 02 0 0 1 B6 r k1 r k1 C B6  B6 k1 @4 kr krk1 Cr 0

Having a single-layer orthotropic cylinder analyzed as presented above, the multi-layer structure could be investigated. In this case, every additional ith layer introduced into the cylindrical structure provides four constants of integration: Ai, Bi, exi, ci. However, one can apply the compatibility requirement, which assumes that all the layers are perfectly bonded. It requires that all displacements, including axial and tangential ones, should be continues from layer to layer, what implies that: exi = ex0, and ci = c0. It finally drives to the conclusion, that every additional ith layer involves just two equations in order to specify unknown integration constants Ai and Bi. These two additional conditions may be derived using continuity of layers’ interface in terms of radial displacements and radial stresses as:

The comparison of hoop stress results for CLT and thick-walled models in case of the pressure load is presented in Fig. 1. It can be

7 6 7 6 p 7 6 rTR r 7 6 out 7 6 6 Px 7 þ 6 TR r2out r2in 7 7 4 2p 5 6 rx 2 5 4 Tx 2p

2

3

3

r out rin sTR xU 3

ar r k1 in 6 a r k1 6 r out 6 ¼ 6 rkþ1 r kþ1 6 ax outkþ1in 4 r kþ2 rkþ2 axU outkþ2in

k1 br r in k1 br r out

bx

kþ1 r kþ1 r in out

bxU

kþ1

rkþ2 r kþ2 out in kþ2

cr

dr rin

cr

dr r out

r 2 r2 cx out2 in

c xU

r 3out r 3in 3

dx

r 3out r3in 3

dxU

r4out r 4in 4

3

2 3 A 7 76 B 7 76 7 76 0 7 74 ex 5 5

ð17Þ

c0

where:

ak ¼ C j2 þ kC j3 bk ¼ C j2  kC j3 ck ¼ C j1 þ ðC j2 þ C j3 ÞC

ð18Þ

dk ¼ C j6 þ ðC j2 þ 2C j3 ÞX For k = (x, U, r, xU) and j = (1, 2, 3, 6), respectively.

Fig. 1. Relative error for the hoop stress results – the pressure load condition.

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145

generally seen, that there is rather a small difference between both models – just a few per-cents. However, this value is not constant and varies with the angle of fiber orientation. Interesting to note, that for fiber angles between 0° and 30° the Classical Lamination Theory slightly overestimates the hoop stress results, while starting from 40° CLT shows lower results than the thick-walled model. As one could expect, the relative error in hoop stress calculations decreases if diameter-to-thickness ratio is elevated. It should be also underlined that the applied load level (higher pressure) does not increase the discrepancy between both theories. Thus, it may be concluded, that plane stress model works well for the pressurized thin-walled composite structures. The relative error is primarily independent on the applied load, and its slight variation with the fiber angle is reduced for higher diameter-tothickness ratios. 3.2. Temperature load When analyzing results of the temperature load condition, Fig. 2, it is clearly visible that discrepancy between the models is much bigger. Generally, in this case the thick-walled model generates higher results than CLT, and the difference grows significantly when increasing the material orthotropy of the homogenized composite (fiber angles closer to 90°). If one realizes that the ply setup ‘‘0/0/0” reflects an ideal transversely isotropic material, while the composite marked as ‘‘90/90/90” describes an ideal orthotropic structure, then the variation of the relative error, as presented in Fig. 2, may be attributed to the increased degree of the material orthotropy. The reason for the difference between two theories in case of the thermal load is explained in Fig. 3. In this analysis a tube having one layer only was considered. According to the thick-walled theory the trialed temperature drop of 300 K introduces the notable hoop stresses at the external layer of the cylinder. The level of the stresses is significant for the fiber angles higher than 50°. It is important to note, that the hoop (and radial) stresses generated by the thermal load are not constant across the wall thickness. In contrast, the mechanism of stress distribution across the wall for a single-layer cylinder cannot be captured properly by the CLT model. In this case coupling stiffness matrix B is zero, thus also the curvatures j, needed for through-thickness strains, are zero. It basically explains the behavior shown in Fig. 2. If the set-up of the composite forms the structure being more homogenous and orthotropic, then the temperature-driven stresses vary more significantly over the wall thickness. This behavior cannot be accurately described by the plane-stress approach.

Fig. 2. Relative error for the hoop stresses – the temperature load condition.

Fig. 3. Hoop stress distribution across the thickness of a single-layer tube for different fiber angles. (According to the thick-walled theory. DT = 300 K, t = 3 mm.)

It should be noted, that the effect of material orthotropy can be easily measured by a coefficient k, as proposed by Eq. (12). Lambda generally ‘describes the root of the ratio between C22 and C33 stiffness constants in a global co-ordinate system, thus it covers, at the same time, material properties (Young modules and Poisson ratios) and material orientation (fiber directions). Since the coefficient k can be treated as an universal measure of the material orthotropy, therefore in some cases it would be more convenient to present results using lambda (instead of fiber orientation angle, a). The exemplary results for the external layers of the analyzed composite are presented in Fig. 4. It is evident, that higher material orthotropy (driven by the orientation of fiber angle and the difference in material properties in principle directions) generates higher discrepancy between CLT and the referenced thick-walled theory. In order keep the error of the CLT model at the acceptable level, the material of thin-wall tube (D/t > 20) subjected to the thermal load should not exceed the lambda level of 1.5. Within this range the relative error of the CLT model can be estimated by a simple, engineering rule as:

d¼

  2 k4 t C 22 ¼  33 ½100% for D=t > 20 and k < 1:5 D C K

ð20Þ

Fig. 4. Relative error in stress calculation with respect to the coefficient k, reflecting material orthotropy.

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Eq. (20) involves both the geometric parameter (D/t) and the measure of material orthotropy (k), thus is could serve as convenient evaluation for the applicability of the CLT model into a particular design case. 4. Conclusions This paper investigated and compared thin- and thick walled theories for the composite cylinders exposed to the axisymmetric thermo-mechanical load. The Classical Lamination Theory and the elasticity theory of thick cylindrical tubes were recalled, since they formed a basis of the analysis. The performed study showed that CLT model provides satisfied results for pressurized thin-walled cylinders. The relative error in stress calculations is not affected by the applied load level, but is slightly dependent on the fiber orientation. This effect is however reduced in composite tubes having higher diameter-to-thickness ratio. In the case of the temperature load condition, the difference between stress results produced by CLT and thick-walled models is much bigger. It was basically shown, that the discrepancy between both theories grows with the increased degree of the material orthotropy. For highly homogeneous orthotropic materials the thick-walled model exhibits a significant variation of the stresses across the thickness of the tube. In contrast, the CLT model is not able to reproduce this behavior. Thus, the relative error of the plane-stress theory grows with the increased material orthotropy, which is driven, for example, by higher fiber orientation angles (a > 50 deg). It was proposed that material coefficient lambda, defined as the root of the C22/C33 ratio, could be treated as a good indicator of the material orthotropy. It was suggested that, beside geometrical restriction (D/t > 20), the composite cylinders analyzed by CLT model should not exceed the value of lambda equal to 1.5. Within this range, the relative error of the CLT model for a particular layer could be estimated by very simple equation incorporating a dimensional factor (D/t) and a material coefficient (k).

References [1] Hill R. The mathematical theory of plasticity. Oxford: Clarendon Press; 1950. [2] Durban D, Kubi M. A general solution for the pressurized elastoplastic tubes. ASME J Appl Mech 1992;59:20–6. [3] Fryer DM, Harvey JF. High pressure vessels. London: Chapman & Hill; 1998. [4] Parker AP. Autofrettage of open end tubes – pressures, stresses, strains and code comparisons. ASME J Press Vessel Technol 2001;123:271–81. [5] Perry J, Aboudi J. Elasto-plastic stresses in thick walled cylinders. ASME J Press Vessel Technol 2003;125:248–52. [6] Moss D. Pressure vessel design manual. 3rd ed. California: Elsevier; 2004. [7] Annaratone D. Pressure vessel design. Berlin: Springer-Verlag; 2007. [8] Lekhnitskii SG. Theory of elasticity of an anisotropic body (english translation). Moscow: Mir Publishers; 1981. [9] Mallick PK. Composites engineering handbook. New York: CRC Press; 1997. [10] Jones R. Mechanics of composite materials. 2nd ed. New York: CRC Press; 1998. [11] Gay D, Hoa SV, Tsai SW. Composite materials: design and applications. New York: CRC Press; 2002. [12] Crawford RJ. Plastics engineering. New York: Elsevier; 2002. [13] Scherrer RE. Filament-wound cylinders with axial-symmetric loads. J Compos Mater 1967;1:344–55. [14] Pagano NJ. Stress gradients in laminated composite cylinders. J Compos Mater 1971;5:260–5. [15] Wilson JF, Orgill G. Linear analysis of uniformly stressed orthotropic cylindrical shells. J Appl Mech 1986;53:249–56. [16] Pindera MJ. Local/global stiffness matrix formulation for composite materials and structures. Compos Eng 1991;1:69–83. [17] Herakovitch C. Mechanics of fibrous composites. New York: John Wiley & Sons; 1998. [18] Lifshitz JM, Dayan HM. Filament-wound pressure vessel with thick metal liner. Compos Struct 1995;32:313–23. [19] Xia M, Kemmochi K, Takayanagi H. Analysis of filament-wound fiberreinforced sandwich pipe under combined internal pressure and thermomechanical loading. Compos Struct 2001;51:273–83. [20] Esfandiar H, Daneshmad S, Mondali M. Analysis of elastic–plastic behavior of fiber metal laminates subjected to in-plane tensile loading. Int J Adv Des Manuf Technol 2011;5(1):61–9. [21] Kabir M. Finite element analysis of composite pressure vessels with a load sharing metallic liner. Compos Struct 2000;49:247–55. [22] Widlak G. Radial return method applied in thick-walled cylinder analysis. J Theor App Mech 2010;48(2):381–95. [23] Nowak T, Schmidt J. Theoretical, numerical and experimental analysis of thick walled fiber metal laminate tube under axisymmetric loads. Compos Struct 2015;131:637–44.