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Stress analysis for the orthotropic pressurized structure of the cylindrical shell and spherical head
Thin–Walled Structures 111 (2017) 29–37
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Stress analysis for the orthotropic pressurized structure of the cylindrical shell and spherical head ⁎
⁎
Chunxiao Li, Tianyu Yao, Xiaohua He , Changyu Zhou
School of Mechanical and Power Engineering, Nanjing Tech University, Nanjing 211816, China
A R T I C L E I N F O
A BS T RAC T
Keywords: Orthotropy Cylindrical shell Spherical head Stress
According to anisotropic shell theory, the pressurized structure of cylindrical shell and spherical head is analyzed, and the shear force, bending moment and stress solution at the juncture are obtained. The influence of anisotropic mechanical properties and geometric parameters on shear force, bending moment and stress distribution along the cylindrical shell is discussed, and the results are compared with the isotropic results. Additionally, Hill48 anisotropic yield criterion is also used to derive the equivalent stress intensity expression of the plane stress state, and the effect of thickness ratio and elastic modulus ratio on Hill48 equivalent stress at the juncture is discussed. Finally, finite element method is applied to verify the results of stress intensity calculation, which shows that the finite element solution and the analytical solution are in good agreement.
1. Introduction In the pressure vessels, film stress distributes evenly along the wall of the spherical head, equally, and uniformly. At the same time, spherical head is easy to manufacture, widely applied as the middle or low pressure vessel. But the wall thickness, load, temperature and mechanical properties along the axial direction of shell may be abrupt for the combined structure, and these factors can be expressed as the discontinuity of pressure vessel structure [1]. It is worth noting that the juncture between cylindrical shell and spherical head is not continuous, which may cause a large local stress. For accurately understand the stress distribution of spherical head, Fu [2] according to the nonmoment theory deduced the stress expression of spherical head, and the stress distribution law was studied by experiment method and ANSYS finite element analysis. Awrejcewicz [3] estimated and predicted a critical set of critical parameters responsible for buckling of spherical circle axially symmetric shells. The buckling phenomenon under static loading was investigated. Cui [4] solved the basic governing differential equations for conical shells by performing magnitude order analysis and neglecting the quantities with h/R magnitude order with a simple and accurate solution for conical shells derived by solving the second-order differential equation. Rilo [5] analyzed the solutions of displacements and stresses in spherical heads over rectangular areas based on spherical shallow shell equations, and represented the displacement and loading terms by use of double Fourier series expressions. Awrejcewicz [6] analyzed stress-strain state of the laminated shallow shells under static loading by using the R-functions
⁎
theory together with the spline-approximation, and the comparison of obtained results with those results by using ANSYS were also presented. For the problem of vibration of cylindrical shell, A.V. Krysko [7] solved strongly nonlinear partial differential equations by modeling the dynamics of closed circular shells using the Bubnov–Galerkin approach, V.A. Krysko [8] studied the complex vibrations of closed cylindrical shells of infinite length and circular cross-section subjected to transversal local load. At present, the stress analysis and strength evaluation of pressurized structure are usually carried out by using the theory of isotropic mechanics. However, material in the rolling process will appear a texture phenomenon and anisotropy such as titanium which is widely used in recent years. Therefore, it is necessary to carry out detailed stress analysis on the anisotropic pressurized structure to meet the requirements of strength and stiffness in different directions. For the anisotropic structures, Li [9] analyzed the influence of anisotropy on the internal forces and moments of the laminated conical shells. According to the principle of energy deformation and the constitutive relation of orthotropic materials, Zeng [10] derived the elastic-plastic stress expression of the axially loaded cylindrical shell under the condition of simply supported at both ends. Rotter [11] combined the generalized Hooke's law, the axial film stress resultant force and the axial bending moment, and developed the complete solution equation of the bending theory for cylindrical shell under different pressure and axial loads. Chandrashekhara [12] applied the theory of elasticity, the classical shell theory and the shear deformation
Corresponding authors. E-mail addresses: [email protected] (X. He), [email protected] (C. Zhou).
http://dx.doi.org/10.1016/j.tws.2016.11.003 Received 5 March 2016; Received in revised form 3 November 2016; Accepted 8 November 2016 0263-8231/ © 2016 Elsevier Ltd. All rights reserved.
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theory to study the problem of infinite transversely isotropic cylindrical shell under the action of axial symmetrical radial load. The influence of anisotropy and the diameter thickness ratio of shell on the stress and displacement distribution was shown through some typical results. Kirichenko [13] established the mathematical modeling of evolutionary states of non-homogeneous multi-layer shallow shells with orthotropic initial imperfections. There are still many scholars to study the stress analysis at the juncture between cylindrical shell and head. The discontinuous stresses of the anisotropic cylindrical shell and the connection of the different head were analyzed by Paliwal [14], the corresponding discontinuous stress and displacement expressions are obtained. Mechanical analysis of the connection structure of the pressure cylindrical shell and the flat plate was made by Gao [15], the stress expression at the dangerous point of the structure were obtained, and the stress distribution of the joint was similar to that of the isotropic one, but the peak stress difference was higher which was associated with the Ex / Eθ . Through the analysis of the stress on the small end of orthotropic conical shell, Yao [16] obtained shear force, bending moment and stress solution at the small end of conical shell. In present paper, by analyzing the force and bending moment at juncture of the orthotropic cylindrical shell and the spherical head, the stress expression is obtained, and the impact of elastic modulus ratio and thickness ratio on the stress distribution is discussed. According to Hill48 anisotropic yield criterion, the equivalent stress intensity expression of the plane biaxial stress state is obtained, and the effect of thickness ratio and elastic modulus ratio on Hill48 equivalent stress at the juncture is discussed at the same time. The calculation results of the stress intensity are verified by finite element results which provide a reference for the design of the orthogonal anisotropic pressurized structure. 2. Mechanics analysis of the connection between orthotropic cylindrical shell and spherical head
Fig. 1. Load at the junction of cylindrical shell and spherical head.
d 4w + 4β14 w = 0 dx 4 where β1 =
4
3Eθ (1 − νθ νx ) Ex R2t12
Qx |x=0 = −D1 (
w=
e−β1x [β1 M0 (sin β1 x − cos β1 x ) − Q0 cos β1 x ] 2β13D1
(w )x=0 = −
(ϕ)x =0 = (
.
cos β1 x + C4 sin β1 x )
)x =0 = Q0
(5)
(6)
where D1 = 12(1 − v v ) is the flexural modulus. θ x The maximum displacement and rotation angle at x = 0 are：
1 1 M0 − Q0 2β12D1 2β13D1
dw 1 1 )x =0 = M0 + Q0 dx β1 D1 2β12D1
(7)
(8)
For cylindrical shell, the displacement and rotation angle caused by shear force and bending moment are as below. (2)
In the formula, C1, C2, C3 and C4 are integral constants, which are determined by the boundary conditions of cylindrical shell. When the cylindrical shell is long enough, with the increase of x, the bending deformation gradually decay and disappear, so the e β1x in formula (2) is close to 0, that is, C1=C2=0, so the formula (2) can be written as:
w=
dx 3
(4)
Ex t13
The general solution for the homogeneous equation:
e−β1x (C3
d 3w
Substituting the boundary conditions into the differential equation:
(1)
w = e β1x (C1 cos β1 x + C2 sin β1 x ) + e−β1x (C3 cos β1 x + C4 sin β1 x )
d 2w )x =0 = M0 dx 2
Mx |x=0 = −D1 (
Under the influence of internal pressure, the stress of spherical head (2) and cylindrical shell (1) is shown in Fig. 1. The inner pressure is p, the radius of cylindrical shell and spherical head is R, the thickness of cylindrical shell and spherical head are respectively t1 and t2. To solve edge load shear force and bending moment, the deformation caused by the load at the edge of its positive and negative direction are shown in Fig. 1: Radial displacement pointing to the rotation shaft is as the positive direction. The left layout of Fig. 1 is used for determining rotation direction and the counter clockwise is positive. On the edge of the cylindrical shell, shear force Q0 and bending moment M0 are applied along the circumference direction. The basic differential equation of the orthotropic cylindrical shell by moment theory [17] is:
(3)
At the juncture of cylindrical shell and spherical head, that is x = 0 , boundary conditions are as below:
Δ1M0 = −
1 M0 2β12D1
(9)
Δ1Q0 = −
1 Q0 2β13D1
(10)
φ1M0 =
1 M0 β1 D1
(11)
φ1Q0 =
1 Q0 2β12D1
(12)
The basic differential equation of orthotropic spherical head by 30
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moment theory [18] is:
d 4w + 4β24 w = 0 dx 4
(13)
According to the moment theory [19], the displacement and rotation angle of spherical head caused by shear force and bending moment are:
Δ2M0 = − Δ2Q0 =
(14)
1 Q0 2β2 3D2
(15)
1 M0 β2 D 2
(16)
1 Q0 2β 2 2 D 2
(17)
φ2M0 = − φ2Q0 =
1 M0 2β 2 2 D 2
where β2 =
4
Fig. 2. Shear force and bending moment at the juncture of cylindrical shell.
3Eθ (1 − νθ νx ) Ex R2t22
,D 2 =
Ex t 23 . 12(1 − vθ vx )
According to the non-moment theory [20] of orthotropic thin shell, radial displacement and rotation angle caused by pressure p are as below.
Δ1p =
φ1p
−pR2 (2 − νθ ) 2Eθ t1
=0
(18) (19)
For spherical head, it is assumed that wall thickness t2 = nt1:
Δ2p =
−pR2 (1 − νθ ) 2Eθ t2
φ2p = 0
(20) (21)
At the juncture, cylindrical shell and spherical head satisfy the deformation coordination relationship, the displacement and rotation angle should be consistent, therefore the deformation of cylindrical and spherical head should meet the following conditions:
Δ1p + Δ1M0 + Δ1Q0 = Δ2p + Δ2M0 + Δ2Q0
(22)
ϕ1p + ϕ1M0 + ϕ1Q0 = ϕ2p + ϕ2M0 + ϕ2Q0
(23)
Substituting the item (9)–(21) into formula (22) and (23):
−pR2 −pR2 1 1 1 (2 − νθ ) − M0 − Q0 = (1 − νθ ) − M0 2Eθ t1 2β12D1 2β13D1 2Eθ t2 2β 2 2 D 2 1 + Q0 2β2 3D2 (24) 1 1 1 1 M0 + Q0 = − M0 + Q0 β1 D1 2β12D1 β2 D 2 2β2 2D2
(25) Fig. 3. Effect of Ex /Eθ on the distribution of axial stress on the outer surface of cylindrical shell: (a) n=1; (b) n=0.5.
where t2 = nt1,D2 = n3D1, β2 = 1 β1. n The shear force and bending moment can be written as:
M0 =
pR2β12D1 n (1 − n2 )[(1 − υθ ) − n (2 − υθ )] Eθ t1 [n4 + 2(n2 + n3/2 + n5/2 ) + 1]
(26)
Q0 =
2pR2β13D1 n1/2 (1 + n5/2 )[(1 − υθ ) − n (2 − υθ )] Eθ t1 [n4 + 2(n2 + n3/2 + n5/2 ) + 1]
(27)
Mx1 = −D1
Mθ1 = −νx D1
So the internal force of cylindrical shell on the cross section are as follows:
Tx1 = 0 Tθ1 = −Ex t1
w + νx Tx1 = 2β1 R (β1 M0 f4 + Q0 f2 ) R
d 2w 1 = (β M0 f3 + Q0 f1 ) dx 2 β1 1 d 2w = νx Mx1 dx 2
(30)
(31)
(28)
Local stress for spherical head, due to the large curvature radius of the juncture, the spherical head can be equivalent to a cylindrical shell with a radius of R near the juncture.
(29)
Tx2 = 0 31
(32)
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Fig. 5. Effect of Ex /Eθ on the distribution of circumferential stress on the outer surface of cylindrical shell: (a) n=1; (b) n=0.5.
Fig. 4. Effect of Ex /Eθ on the distribution of axial stress on the inner surface of cylindrical shell: (a) n=1; (b) n=0.5.
σθ1 =
⎞ β ⎛ β Tθ2 = 2R 1 ⎜ 1 M0 ψ4 + Q0 ψ2⎟ ⎠ n⎝ n
(33)
n β1 ( M0 ψ3 + Q0 ψ1) Mx2 = β1 n
(34)
Mθ 2 = νx Mx 2
(35)
pR T 6Mθ1 + θ1 ± t1 t1 t12
(37)
At the juncture, the stress of spherical head can be written as:
where
σx 2 =
pR T 6Mx 2 + x2 ± 2nt1 nt1 (nt1)2
(38)
σθ 2 =
pR T 6Mθ 2 + θ2 ± 2nt1 nt1 (nt1)2
(39)
The formulae from(36)–(39) are a form of stress classification. In the above four equations, the first one is the membrane stress caused by pressure, the second is the edge effect of the membrane stress, the third are the edge effects caused by bending stress.
f1 = e−β1x sin(β1 x ) f2 = e−β1x cos(β1 x ) f3 = e−β1 x [cos(β1 x ) + sin(β1 x )] f4 = e−β1x [cos(β1 x ) − sin(β1 x )]
ψ1 = e− ψ2 = e−
β1x n
sin(
β1x n
cos(
β1 n
3. Analysis of shear force and bending moment at the juncture of the orthotropic cylindrical shell and spherical head
x)
β1 n
x)
⎡ ⎤ β1 β1 ψ3 = ⎢cos( n x ) + sin( n x ) ⎥ ⎣ ⎦ β1x ⎡ ⎤ β β ψ4 = e− n ⎢cos( n1 x ) − sin( n1 x ) ⎥ ⎣ ⎦
It is known by the stress calculation formula of the stress components for orthotropic cylindrical shell and spherical head that the influence factors of stress components include Ex / Eθ , νθ , R/t etc. In the analysis of the impact of Ex / Eθ , the impact of νθ can be neglected, and the νθ will be assumed as common metal alloy value νθ =0.3 (for metal alloy pressure vessel, Poisson's ratio is between 0.2–0.5). Here p=1 Mpa, t1=10 mm, t2 = nt1, R=500 mm, and n = 1 and n = 0.5 are discussed respectively. In the case of n = 0.5, n = 1, the ratio of bending moment and shear force at the juncture between anisotropic model and isotropic model with the change of elastic modulus ratio is shown in Fig. 2. As shown in
β1x e− n
The internal force of any point at the juncture of cylindrical shell is composed of the general membrane stress, local membrane stress and bending stress [21].
σx1 =
pR T 6M + x1 ± 2x1 2t1 t1 t1
(36) 32
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Fig. 6. Effect of Ex /Eθ on the distribution of circumferential stress on the inner surface of cylindrical shell: (a) n=1; (b) n=0.5.
Fig. 7. Effect of R/t on the distribution of axial stress on the outer surface of cylindrical shell: (a) n=1; (b) n=0.5.
Fig. 2, when thickness ratio n = 1, shear ratio Q0 / Q0(Ex / Eθ )=1 increases gradually with elastic modulus ratio Ex / Eθ . At this moment, there is M0 Ex / Eθ =1 = 0 , bending moment curve does not exist. When thickness ratio n = 0.5, bending moment ratio M0 / M0(Ex / Eθ =1) decreases with the ratio of elastic modulus Ex / Eθ , while shear ratio Q0 / Q0(Ex / Eθ )=1 increases gradually with elastic modulus ratio Ex / Eθ which is similar to curve under n=1. As can be seen from Fig. 2, the effect of thickness ratio on shear ratio is smaller than that of elastic modulus ratio Ex / Eθ .
thickness ratio n has significant influence on the axial stress σx on the outer surface of cylindrical shell. 4.2. Effect of Ex / Eθ on the distribution of axial stress on the inner surface of cylindrical shell Fig. 4(a) and (b) are drawn in the case of n = 0.5, n = 1for the variation of the axial stress on the inner surface of cylindrical shell along x direction with different Ex / Eθ values. When n = 1, the axial stress σx decays rapidly along the x direction, rises after reaching the minimum value, and then levels off. When x=0, the axial stresses are basically the same with σx /(pR / t ) = 0.5. The minimum stress of σx /(pR / t ) is among 0.33–0.39, and the variation range of stress fluctuation is approximately3 Rt . The minimum axial stress decays with Ex / Eθ . When n = 0.5, the variation trend of axial stress σx /(pR / t ) is the same as that with n = 1. The minimum stress of σx /(pR / t ) is 0.45–0.47, the variation range of stress fluctuation is roughly 2.5 Rt , and the variation range is smaller than that with n = 1. The results above show that the thickness ratio n has significant influence on the axial stress σx on the inner surface of cylindrical shell.
4. Analysis on the stress at the juncture of orthotropic cylindrical shell and spherical head 4.1. Effect of Ex / Eθ on the distribution of axial stress on the outer surface of cylindrical shell Fig. 3(a) and (b) are drawn in the case of n = 0.5, n = 1 for the variation of axial stress on the outer surface of the cylindrical shell along x direction with different Ex / Eθ values. When n = 1, the axial stress σx increases rapidly along the x direction, drops after reaching the maximum value, and then levels off. The maximum axial stress does not occur at the juncture (x=0), and when x=0, that is the juncture of cylindrical shell and spherical head, the axial stresses are basically the same with σx /(pR / t ) = 0.5. The maximum stress of σx /(pR / t ) is among 0.62–0.68, and the variation range of stress fluctuation is approximately 3 Rt . The maximum axial stress increases with Ex / Eθ . When n = 0.5, the variation trend of axial stress σx /(pR / t ) is the same as that with n = 1. The maximum stress of σx /(pR / t ) is 0.53–0.55, the variation range of stress fluctuation is roughly 2.5 Rt , and the variation range is smaller than that with n = 1. The results above show that the
4.3. Effect of Ex / Eθ on the distribution of circumferential stress on the outer surface of cylindrical shell Fig. 5(a) and (b) are drawn in the case of n = 0.5, n = 1 for the variation of circumferential stress on the outer surface of cylindrical shell along x direction with different Ex / Eθ values. The minimum circumferential stress occurs at the juncture (x=0). When n = 1, the circumferential stress σθ increases rapidly along x direction, after 33
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Fig. 9. Effect of R/t on the distribution of circumferential stress on the outer surface of cylindrical shell: (a) n=1; (b) n=0.5.
Fig. 8. Effect of R/t on the distribution of axial stress on the inner surface of cylindrical shell: (a) n=1; (b) n=0.5.
4.5. Effect of R/t on the distribution of circumferential stress on the outer surface of cylindrical shell
reaching the maximum value levels off. The maximum stress of σθ /(pR / t ) is among 1.01–1.04, and the variation range of stress fluctuation is approximately 3 Rt . The maximum circumferential stress increases with Ex / Eθ . When n = 0.5, the variation trend of circumferential stress σθ /(pR / t ) is the same as that with n = 1. The maximum stress of σθ /(pR / t ) is 1.00–1.01, and the variation range of stress fluctuation is approximately 2.5 Rt , and the variation range is smaller than that with n = 1. The results above show that thickness ratio n has significant influence on the circumferential stress σθ on the outer surface of cylindrical shell.
Fig. 7(a) and (b) are drawn in the case of n = 0.5, n = 1 for the variation of axial stress on the outer surface of cylindrical shell along x direction with different R/t values. When n = 1, the axial stress σx increases rapidly along x direction, drops after reaching the maximum value, and then levels off. The maximum axial stress does not occur at the juncture (x=0), and when x=0, the axial stresses are basically the same as σx /(pR / t ) = 0.5. The maximum stress of σx /(pR / t ) is among 0.55–0.63. When n = 0.5, the variation trend of axial stress σx /(pR / t ) is the same as that with n = 1. The maximum stress of σx /(pR / t ) is 0.5– 0.54, and the variation range is smaller than that with n = 1. The results above show that the thickness ratio n has significant influence on the axial stress σx on the outer surface of cylindrical shell.
4.4. Effect of Ex / Eθ on the distribution of circumferential stress on the inner surface of cylindrical shell
4.6. Effect of R/t on the distribution of axial stress on the inner surface of cylindrical shell
Fig. 6(a) and (b) are drawn in the case of n = 0.5, n = 1or the variation of circumferential stress on the inner surface of cylindrical shell along x direction with different Ex / Eθ values. The minimum circumferential stress occurs at the juncture (x=0). When n = 1, the circumferential stress σθ increases rapidly along x direction, levels off after reaching the maximum value. The variation range of stress fluctuation is approximately 3 Rt . The maximum circumferential stress decays with Ex / Eθ . When n = 0.5, the variation trend of circumferential stress σθ /(pR / t ) is the same as that with n = 1. The variation range of stress fluctuation is approximately 2.5 Rt , and the variation range is smaller than that with n = 1. The results above show that thickness ratio n has significant influence on the circumferential stress σθ on the inner surface of cylindrical shell.
Fig. 8(a) and (b) are drawn in the case of n = 0.5, n = 1 for the variation of axial stress on the inner surface of cylindrical shell along x direction with different R/t values. When n = 1, the axial stress σx decays rapidly along x direction, rises after reaching the minimum value, and then levels off. When x=0, the axial stresses are basically the same as σx /(pR / t ) = 0.5. The minimum stress of σx /(pR / t ) is among 0.37–0.44. When n = 0.5, the variation trend of axial stress σx /(pR / t ) is the same as that with n = 1. The minimum stress of σx /(pR / t ) is 0.46– 0.48, and the variation range is smaller than that with n = 1. The results above show that the thickness ratio n has significant influence 34
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Fig. 11. The relationship between Hill48 equivalent stress and thickness radio n at the juncture of cylindrical shell under different Ex /Eθ : (a)σHill48 ; (b) σHill 48 /σHill 48(EX / Eθ =1) . Fig. 10. Effect of R/t on the distribution of circumferential stress on the inner surface of cylindrical shell: (a) n=1; (b) n=0.5.
5. Effect of thickness ratio and elastic modulus ratio on the Hill48 equivalent stress at the juncture of structure
on the axial stress σx of the inner surface of cylindrical shell. Mises or Tresca yield criterion is generally adopted in the design of isotropic structure. Due to the difference between the orthotropic and isotropic materials, the VonMises or Tresca yield criterion is no longer applicable for orthotropic one. Hill48 yield criterion is more suitable to describe the anisotropy of orthotropic material. For the structure of thin-walled cylindrical shell and spherical head, the equivalent stress intensity expression of the plane structure under biaxial stress state is [22]:
4.7. Effect of R/t on the distribution of circumferential stress on the outer surface of cylindrical shell Fig. 9(a) and (b) were drawn in the case of n = 0.5, n = 1 for the variation of circumferential stress on the outer surface of cylindrical shell along x direction with different R/t values. The minimum circumferential stress occurs at the juncture (x=0). When n = 1, the circumferential stress σθ increases rapidly along x direction, after reaching the maximum value levels off. When n = 0.5, the variation trend of circumferential stress σθ /(pR / t ) is the same as that with n = 1. The variation range is smaller than that with n = 1. The results above show that thickness ratio n has significant influence on the circumferential stress σθ on the outer surface of cylindrical shell.
σHill 48 =
3 Fσθ2 + Gσx2 + H (σx − σθ )2 ⋅ 2 F+G+H
(39)
where F, G and H are the coefficients related to the yield stress of the three direction X,Y and Z.
2F = 4.8. Effect of R/t on the distribution of circumferential stress on the inner surface of cylindrical shell
2G = 2H =
Fig. 10(a) and (b) were drawn in the case of n = 0.5, n = 1 for the variation of circumferential stress on the inner surface of cylindrical shell along x direction with different R/t values. The minimum circumferential stress occurs at the juncture (x=0). When n = 1, the circumferential stress σθ increases rapidly along x direction, after reaching the maximum value levels off. When n = 0.5, the variation trend of circumferential stress σθ /(pR / t ) is the same as that with n = 1. The change range is smaller than that with n = 1. The results above show that thickness ratio n has significant influence on the circumferential stress σθ on the inner surface of cylindrical shell.
1 Y2 1 Z2 1 X2
+ + +
1 Z2 1 X2 1 Y2
1
− − −
X2 1 Y2 1 Z2
(40)
In which X, Y and Z are the yield stresses in axial, circumferential and radial(thickness) directions. According reference [23] for TA2 alloy, Y=1.18X and Z=1.09X. σHill48 is the strength of the equivalent stress under Hill48 yield criterion, and its expression can be written as:
σHill 48 = 0.3 ×
9.345σθ2 + 13.059σx2 − 11.434σx σθ
(41)
According to axial and circumferential stress, Hill48 equivalent stress at the juncture of structure can be obtained. And the variation 35
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Fig. 13. Linearization path: (a) n=1; (b) n=0.5. Fig. 12. The relationship between Hill48 equivalent stress and thickness radio n at the juncture. spherical head under different Ex /Eθ : (a) σHill48 ; (b) σHill 48 /σHill 48(EX / Eθ =1) .
Table 2 Finite element solution and analytical solution of stress in cylindrical shell and its error.
Table 1 Material parameters of TA2. Ex GPa
Ey(Eθ) GPa
RP0.2(x) MPa
RP0.2(θ) MPa
vxy(vx)
vxy(vθ)
110
134
277
327
0.337
0.431
Primary stress plus secondary stress Error
tendency of Hill48 equivalent stress under different elastic modulus at the juncture of structure is drawn, as shown in Figs. 11 and 12.
FEM Analytical solution
σxs (MPa)
σθs (MPa)
n=1
n=0.5
n=1
n=0.5
24.86 25
12.93 12.50
36.98 37.5
23.28 23.12
0.56%
3.44%
1.38%
0.69%
When n > 0.41, Hill48 equivalent stress ratio σHill 48 / σHill 48(Ex / Eθ =1) decreases with the ratio of elastic modulus Ex / Eθ . Hill48 equivalent stress ratio of isotropic material is a straight lever line which is equal to 1. When σHill 48 / σHill 48(Ex / Eθ =1) > 1, the equivalent stress intensity of anisotropic material is greater than that of isotropic one, and is easy to reach yield limit; When σHill 48 / σHill 48(Ex / Eθ =1) < 1, equivalent stress of orthotropic material is less than that of isotropic one and difficult to reach yield limit. Hill48 equivalent stress at the juncture of cylindrical shell is controlled by the ratio of elastic modulus Ex / Eθ and thickness ration , and the maximum Hill48 equivalent stress in different thickness ration corresponds to different elastic modulus ratio Ex / Eθ .
5.1. Hill48 equivalent stress and thickness radio n at the juncture of cylindrical shell under different elastic modulus ratio Fig. 11(a) shows under different Ex / Eθ , the relationship between Hill48 equivalent stress σHill48 and thickness ratio at the juncture of cylindrical shell. Hill48 equivalent stress at the juncture of cylindrical shell increases first and then decreases with thickness ratio. The stress fluctuation range of σHill48 under different elastic modulus ratio is little, which indicates that elastic modulus ratio has less influence on Hill48 equivalent stress at the juncture of cylindrical shell than that of thickness ratio. The variation trend of equivalent stress ratio σHill 48 / σHill 48(Ex / Eθ =1) is shown in Fig. 11(b). With increase of thickness ratio, when elastic Ex / Eθ < 1, modulus ratio Hill48 equivalent stress ratio σHill 48 / σHill 48(Ex / Eθ =1) at the juncture of cylindrical shell decreases first, then increase and finally decreases again. When the elastic modulus ratio Ex / Eθ > 1, Hill48 equivalent stress ratio σHill 48 / σHill 48(Ex / Eθ =1) at the juncture of cylindrical shell increases first, then decrease and finally increases. When n < 0.41, Hill48 equivalent stress ratio σHill 48 / σHill 48(Ex / Eθ =1) increases with the ratio of elastic modulus Ex / Eθ .
5.2. Hill48 equivalent stress and thickness radio n at the juncture spherical head under different elastic modulus ratio Fig. 12(a) shows under different Ex / Eθ , the relationship between Hill48 equivalent stress σHill48 and thickness ratio at the juncture of spherical head. Hill48 equivalent stress at the juncture of spherical head decreases with thickness ratio. The stress fluctuation range of 36
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are consistent.
σHill48 under different elastic modulus ratio is little, which indicates that elastic modulus ratio has less influence on Hill48 equivalent stress at the juncture of spherical head than that of thickness ratio. The variation trend of equivalent stress ratio σHill 48 / σHill 48(Ex / Eθ =1) is shown in Fig. 12(b). With increase of thickness ratio, when elastic Ex / Eθ < 1, modulus ratio Hill48 equivalent stress ratio σHill 48 / σHill 48(Ex / Eθ =1) at the juncture of spherical head increases first, then decrease. When elastic modulus ratio Ex / Eθ > 1, Hill48 equivalent stress ratio σHill 48 / σHill 48(Ex / Eθ =1) at the juncture of spherical head decreases first, then increase. When n < 0.41, Hill48 equivalent stress ratio σHill 48 / σHill 48(Ex / Eθ =1) increases with the ratio of elastic modulus Ex / Eθ . When n > 0.41, Hill48 equivalent stress ratio σHill 48 / σHill 48(Ex / Eθ =1) decreases with the ratio of elastic modulus Ex / Eθ .
Acknowledgements The authors gratefully acknowledge the financial support by SixTalent Peak Project of Jiangsu Province(2014-ZBZZ-012) and the National Natural Science Foundation of China(51475223, 51675260). References [1] Z.W.Wang, R.L.Cai, Design of chemical pressure vessels, 2005. [2] L. Fu, Y.R. Luo, L. Fu, Study on stress distribution of spherical head, Mach. Des. Manuf. 10 (2013) 209–212. [3] J. Awrejcewicz, A.V. Krysko, I.V. Papkova, et al., On the methods of critical load estimation of spherical circle axially symmetrical shells, Thin-Walled Struct. 94 (2015) 293–301. [4] W.C. Cui, J.H. Pei, W. Zhang, A simple and accurate solution for calculating stresses in conical shells, Comput. Struct. 79 (3) (2001) 265–279. [5] N.F. Rilo, J.F.D.S. Gomes, J. Cirne, et al., Stresses from radial loads and external moments in spherical pressure vessels, Proc. Inst. Mech. Eng. Part E J. Process Mech. Eng. 215 (2) (2001) 99–109. [6] J. Awrejcewicz, L. Kurpa, A. Osetrov, Investigation of stress-strain state of the laminated shallow shells by R-functions method combined with spline-approximation, ZAMM J. Appl. Math. Mech.: Z. für Angew. Math. und Mech. 91 (91) (2001) 458–467. [7] A.V. Krysko, J. Awrejcewicz, E.S. Kuznetsova, et al., Chaotic vibrations of closed cylindrical shells in a temperature field, Int. J. Bifurc. Chaos 18 (5) (2005) 1515–1529. [8] V.A. Krysko, J. Awrejcewicz, N.E. Saveleva, Stability, bifurcation and chaos of closed flexible cylindrical shells, Int. J. Mech. Sci. 50 (2008) 247–274. [9] Z.Q. Li, Edge bending calculation and analysis of laminated shell, Fiber Reinf. Plast./Compos. 4 (1992) 17–21. [10] J.J. Zeng, Y.M. Fu, Elasto-plastic buckling analysis for orthotropic circular cylindrical shells under axial compression, Eng. Mech. 23 (10) (2006) 25–29. [11] J.M. Rotter, A.J. Sadowski, Cylindrical shell bending theory for orthotropic shells under general axisymmetric pressure distributions, Eng. Struct. 42 (2012) 258–265. [12] K. Chandrashekhara, P. Gopalakrishnan, Transversely isotropic infinite cylindrical shell subjected to a radial axisymmetric line load, Fibre Sci. Technol. 16 (4) (1982) 275–293. [13] V.F. Kirichenko, J. Awrejcewicz, A.V. Kirichenko, et al., On the non-classical mathematical models of coupled problems of thermo-elasticity for multi-layer shallow shells with initial imperfections, Int. J. Non-Linear Mech. 74 (2015) 51–72. [14] D.N. Paliwal, P. Kumar, I.K. Bhat, et al., Discontinuity stresses in orthotropic pressure vessels, Int. J. Vessels Pip. 72 (1997) 63–72. [15] T.T. Gao, C.Y. Zhou, X.H. He, et al., Discontinuity stresses at the junction of orthotropic pressurized cylindrical shell with a flat head, Chinese, J. Appl. Mech. 32 (1) (2015) 28–33. [16] T.Y. Yao, X.H. He, F.S. Kong, et al., Design by analysis for orthotropic pressurized structure with small end of conical shell and cylindrical shell based on Hill48 yield criterion, Thin-Walled Struct. 96 (2015) 220–226. [17] W.Flügge, Stresses in Shells, 1960. [18] Lloyd E. Brownell, Edwin H. Young, Process Equip. Des. (1959) 120–140. [19] S.T. Sun, R.G. Jiang, R.G. Liu, Anisotropic plates Shells Theory (1993). [20] K.Z.Huang, Z.X.Xia, M.D.Xue, et al., Plates and shells theory, 1987. [21] JB 4732-1995, Steel pressure vessels-design by analysis, Peking, China, pp. 1995. [22] R. Hill, Math. Theory Plast. (1956). [23] Y.H. Guo, Stress Analysis and Design of Orthotropic Pressurized Components Made of Titanium, Nanjing Tech University, Jiangsu, China, 2014.
6. Comparison and verification between analytical solution and finite element solution In order to verify the correctness of stress solution at the juncture of structure, finite element method is used to simulate the discontinuous stress value at the juncture. ANSYS is used to build models where R=500 mm, t2=10 mm, p=1 MPa, thickness ratios are n=1 and n=0.5, respectively. The material properties of finite element calculation and analytical solution refer to Table 1. Fig. 13 shows the linear path of spherical head and cylindrical shell. The axial and circumferential stress of cylindrical shell are extracted from path A-A and path B-B. According to the VIII-2 ASME stress classification method, the stress is obtained by the FEM method and analytical solution respectively, the results are shown in Table 2. From Table 2, the maximum error of analytical solution and finite element solution of spherical head and cylindrical shell is 3.44%, which shows that analytical solution and finite element solution are consistent, and the accuracy of analytical solution is verified by finite element method. 7. Conclusion (1) For the structure of orthotropic cylindrical shell and spherical head, the effect of thickness ratio on shear ratio at the juncture is smaller than that of elastic modulus ratio Ex / Eθ . (2) For the structure of cylindrical shell, thickness ratio n has significant influence on axial stress σx and circumferential stress σθ on the surface of cylindrical shell than the Elastic modulus ratio Ex / Eθ and R/t. (3) At the juncture of cylindrical shell and spherical head, Hill48 equivalent stress changes differently with elastic modulus ratio and thickness ratio. (4) By calculation, the maximum error of analytical solution and finite element solution of spherical head and cylindrical shell is 3.44%, which shows that analytical solution and finite element solution
37
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Full length article
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Stress analysis for the orthotropic pressurized structure of the cylindrical shell and spherical head ⁎
⁎
Chunxiao Li, Tianyu Yao, Xiaohua He , Changyu Zhou
School of Mechanical and Power Engineering, Nanjing Tech University, Nanjing 211816, China
A R T I C L E I N F O
A BS T RAC T
Keywords: Orthotropy Cylindrical shell Spherical head Stress
According to anisotropic shell theory, the pressurized structure of cylindrical shell and spherical head is analyzed, and the shear force, bending moment and stress solution at the juncture are obtained. The influence of anisotropic mechanical properties and geometric parameters on shear force, bending moment and stress distribution along the cylindrical shell is discussed, and the results are compared with the isotropic results. Additionally, Hill48 anisotropic yield criterion is also used to derive the equivalent stress intensity expression of the plane stress state, and the effect of thickness ratio and elastic modulus ratio on Hill48 equivalent stress at the juncture is discussed. Finally, finite element method is applied to verify the results of stress intensity calculation, which shows that the finite element solution and the analytical solution are in good agreement.
1. Introduction In the pressure vessels, film stress distributes evenly along the wall of the spherical head, equally, and uniformly. At the same time, spherical head is easy to manufacture, widely applied as the middle or low pressure vessel. But the wall thickness, load, temperature and mechanical properties along the axial direction of shell may be abrupt for the combined structure, and these factors can be expressed as the discontinuity of pressure vessel structure [1]. It is worth noting that the juncture between cylindrical shell and spherical head is not continuous, which may cause a large local stress. For accurately understand the stress distribution of spherical head, Fu [2] according to the nonmoment theory deduced the stress expression of spherical head, and the stress distribution law was studied by experiment method and ANSYS finite element analysis. Awrejcewicz [3] estimated and predicted a critical set of critical parameters responsible for buckling of spherical circle axially symmetric shells. The buckling phenomenon under static loading was investigated. Cui [4] solved the basic governing differential equations for conical shells by performing magnitude order analysis and neglecting the quantities with h/R magnitude order with a simple and accurate solution for conical shells derived by solving the second-order differential equation. Rilo [5] analyzed the solutions of displacements and stresses in spherical heads over rectangular areas based on spherical shallow shell equations, and represented the displacement and loading terms by use of double Fourier series expressions. Awrejcewicz [6] analyzed stress-strain state of the laminated shallow shells under static loading by using the R-functions
⁎
theory together with the spline-approximation, and the comparison of obtained results with those results by using ANSYS were also presented. For the problem of vibration of cylindrical shell, A.V. Krysko [7] solved strongly nonlinear partial differential equations by modeling the dynamics of closed circular shells using the Bubnov–Galerkin approach, V.A. Krysko [8] studied the complex vibrations of closed cylindrical shells of infinite length and circular cross-section subjected to transversal local load. At present, the stress analysis and strength evaluation of pressurized structure are usually carried out by using the theory of isotropic mechanics. However, material in the rolling process will appear a texture phenomenon and anisotropy such as titanium which is widely used in recent years. Therefore, it is necessary to carry out detailed stress analysis on the anisotropic pressurized structure to meet the requirements of strength and stiffness in different directions. For the anisotropic structures, Li [9] analyzed the influence of anisotropy on the internal forces and moments of the laminated conical shells. According to the principle of energy deformation and the constitutive relation of orthotropic materials, Zeng [10] derived the elastic-plastic stress expression of the axially loaded cylindrical shell under the condition of simply supported at both ends. Rotter [11] combined the generalized Hooke's law, the axial film stress resultant force and the axial bending moment, and developed the complete solution equation of the bending theory for cylindrical shell under different pressure and axial loads. Chandrashekhara [12] applied the theory of elasticity, the classical shell theory and the shear deformation
Corresponding authors. E-mail addresses: [email protected] (X. He), [email protected] (C. Zhou).
http://dx.doi.org/10.1016/j.tws.2016.11.003 Received 5 March 2016; Received in revised form 3 November 2016; Accepted 8 November 2016 0263-8231/ © 2016 Elsevier Ltd. All rights reserved.
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theory to study the problem of infinite transversely isotropic cylindrical shell under the action of axial symmetrical radial load. The influence of anisotropy and the diameter thickness ratio of shell on the stress and displacement distribution was shown through some typical results. Kirichenko [13] established the mathematical modeling of evolutionary states of non-homogeneous multi-layer shallow shells with orthotropic initial imperfections. There are still many scholars to study the stress analysis at the juncture between cylindrical shell and head. The discontinuous stresses of the anisotropic cylindrical shell and the connection of the different head were analyzed by Paliwal [14], the corresponding discontinuous stress and displacement expressions are obtained. Mechanical analysis of the connection structure of the pressure cylindrical shell and the flat plate was made by Gao [15], the stress expression at the dangerous point of the structure were obtained, and the stress distribution of the joint was similar to that of the isotropic one, but the peak stress difference was higher which was associated with the Ex / Eθ . Through the analysis of the stress on the small end of orthotropic conical shell, Yao [16] obtained shear force, bending moment and stress solution at the small end of conical shell. In present paper, by analyzing the force and bending moment at juncture of the orthotropic cylindrical shell and the spherical head, the stress expression is obtained, and the impact of elastic modulus ratio and thickness ratio on the stress distribution is discussed. According to Hill48 anisotropic yield criterion, the equivalent stress intensity expression of the plane biaxial stress state is obtained, and the effect of thickness ratio and elastic modulus ratio on Hill48 equivalent stress at the juncture is discussed at the same time. The calculation results of the stress intensity are verified by finite element results which provide a reference for the design of the orthogonal anisotropic pressurized structure. 2. Mechanics analysis of the connection between orthotropic cylindrical shell and spherical head
Fig. 1. Load at the junction of cylindrical shell and spherical head.
d 4w + 4β14 w = 0 dx 4 where β1 =
4
3Eθ (1 − νθ νx ) Ex R2t12
Qx |x=0 = −D1 (
w=
e−β1x [β1 M0 (sin β1 x − cos β1 x ) − Q0 cos β1 x ] 2β13D1
(w )x=0 = −
(ϕ)x =0 = (
.
cos β1 x + C4 sin β1 x )
)x =0 = Q0
(5)
(6)
where D1 = 12(1 − v v ) is the flexural modulus. θ x The maximum displacement and rotation angle at x = 0 are：
1 1 M0 − Q0 2β12D1 2β13D1
dw 1 1 )x =0 = M0 + Q0 dx β1 D1 2β12D1
(7)
(8)
For cylindrical shell, the displacement and rotation angle caused by shear force and bending moment are as below. (2)
In the formula, C1, C2, C3 and C4 are integral constants, which are determined by the boundary conditions of cylindrical shell. When the cylindrical shell is long enough, with the increase of x, the bending deformation gradually decay and disappear, so the e β1x in formula (2) is close to 0, that is, C1=C2=0, so the formula (2) can be written as:
w=
dx 3
(4)
Ex t13
The general solution for the homogeneous equation:
e−β1x (C3
d 3w
Substituting the boundary conditions into the differential equation:
(1)
w = e β1x (C1 cos β1 x + C2 sin β1 x ) + e−β1x (C3 cos β1 x + C4 sin β1 x )
d 2w )x =0 = M0 dx 2
Mx |x=0 = −D1 (
Under the influence of internal pressure, the stress of spherical head (2) and cylindrical shell (1) is shown in Fig. 1. The inner pressure is p, the radius of cylindrical shell and spherical head is R, the thickness of cylindrical shell and spherical head are respectively t1 and t2. To solve edge load shear force and bending moment, the deformation caused by the load at the edge of its positive and negative direction are shown in Fig. 1: Radial displacement pointing to the rotation shaft is as the positive direction. The left layout of Fig. 1 is used for determining rotation direction and the counter clockwise is positive. On the edge of the cylindrical shell, shear force Q0 and bending moment M0 are applied along the circumference direction. The basic differential equation of the orthotropic cylindrical shell by moment theory [17] is:
(3)
At the juncture of cylindrical shell and spherical head, that is x = 0 , boundary conditions are as below:
Δ1M0 = −
1 M0 2β12D1
(9)
Δ1Q0 = −
1 Q0 2β13D1
(10)
φ1M0 =
1 M0 β1 D1
(11)
φ1Q0 =
1 Q0 2β12D1
(12)
The basic differential equation of orthotropic spherical head by 30
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moment theory [18] is:
d 4w + 4β24 w = 0 dx 4
(13)
According to the moment theory [19], the displacement and rotation angle of spherical head caused by shear force and bending moment are:
Δ2M0 = − Δ2Q0 =
(14)
1 Q0 2β2 3D2
(15)
1 M0 β2 D 2
(16)
1 Q0 2β 2 2 D 2
(17)
φ2M0 = − φ2Q0 =
1 M0 2β 2 2 D 2
where β2 =
4
Fig. 2. Shear force and bending moment at the juncture of cylindrical shell.
3Eθ (1 − νθ νx ) Ex R2t22
,D 2 =
Ex t 23 . 12(1 − vθ vx )
According to the non-moment theory [20] of orthotropic thin shell, radial displacement and rotation angle caused by pressure p are as below.
Δ1p =
φ1p
−pR2 (2 − νθ ) 2Eθ t1
=0
(18) (19)
For spherical head, it is assumed that wall thickness t2 = nt1:
Δ2p =
−pR2 (1 − νθ ) 2Eθ t2
φ2p = 0
(20) (21)
At the juncture, cylindrical shell and spherical head satisfy the deformation coordination relationship, the displacement and rotation angle should be consistent, therefore the deformation of cylindrical and spherical head should meet the following conditions:
Δ1p + Δ1M0 + Δ1Q0 = Δ2p + Δ2M0 + Δ2Q0
(22)
ϕ1p + ϕ1M0 + ϕ1Q0 = ϕ2p + ϕ2M0 + ϕ2Q0
(23)
Substituting the item (9)–(21) into formula (22) and (23):
−pR2 −pR2 1 1 1 (2 − νθ ) − M0 − Q0 = (1 − νθ ) − M0 2Eθ t1 2β12D1 2β13D1 2Eθ t2 2β 2 2 D 2 1 + Q0 2β2 3D2 (24) 1 1 1 1 M0 + Q0 = − M0 + Q0 β1 D1 2β12D1 β2 D 2 2β2 2D2
(25) Fig. 3. Effect of Ex /Eθ on the distribution of axial stress on the outer surface of cylindrical shell: (a) n=1; (b) n=0.5.
where t2 = nt1,D2 = n3D1, β2 = 1 β1. n The shear force and bending moment can be written as:
M0 =
pR2β12D1 n (1 − n2 )[(1 − υθ ) − n (2 − υθ )] Eθ t1 [n4 + 2(n2 + n3/2 + n5/2 ) + 1]
(26)
Q0 =
2pR2β13D1 n1/2 (1 + n5/2 )[(1 − υθ ) − n (2 − υθ )] Eθ t1 [n4 + 2(n2 + n3/2 + n5/2 ) + 1]
(27)
Mx1 = −D1
Mθ1 = −νx D1
So the internal force of cylindrical shell on the cross section are as follows:
Tx1 = 0 Tθ1 = −Ex t1
w + νx Tx1 = 2β1 R (β1 M0 f4 + Q0 f2 ) R
d 2w 1 = (β M0 f3 + Q0 f1 ) dx 2 β1 1 d 2w = νx Mx1 dx 2
(30)
(31)
(28)
Local stress for spherical head, due to the large curvature radius of the juncture, the spherical head can be equivalent to a cylindrical shell with a radius of R near the juncture.
(29)
Tx2 = 0 31
(32)
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Fig. 5. Effect of Ex /Eθ on the distribution of circumferential stress on the outer surface of cylindrical shell: (a) n=1; (b) n=0.5.
Fig. 4. Effect of Ex /Eθ on the distribution of axial stress on the inner surface of cylindrical shell: (a) n=1; (b) n=0.5.
σθ1 =
⎞ β ⎛ β Tθ2 = 2R 1 ⎜ 1 M0 ψ4 + Q0 ψ2⎟ ⎠ n⎝ n
(33)
n β1 ( M0 ψ3 + Q0 ψ1) Mx2 = β1 n
(34)
Mθ 2 = νx Mx 2
(35)
pR T 6Mθ1 + θ1 ± t1 t1 t12
(37)
At the juncture, the stress of spherical head can be written as:
where
σx 2 =
pR T 6Mx 2 + x2 ± 2nt1 nt1 (nt1)2
(38)
σθ 2 =
pR T 6Mθ 2 + θ2 ± 2nt1 nt1 (nt1)2
(39)
The formulae from(36)–(39) are a form of stress classification. In the above four equations, the first one is the membrane stress caused by pressure, the second is the edge effect of the membrane stress, the third are the edge effects caused by bending stress.
f1 = e−β1x sin(β1 x ) f2 = e−β1x cos(β1 x ) f3 = e−β1 x [cos(β1 x ) + sin(β1 x )] f4 = e−β1x [cos(β1 x ) − sin(β1 x )]
ψ1 = e− ψ2 = e−
β1x n
sin(
β1x n
cos(
β1 n
3. Analysis of shear force and bending moment at the juncture of the orthotropic cylindrical shell and spherical head
x)
β1 n
x)
⎡ ⎤ β1 β1 ψ3 = ⎢cos( n x ) + sin( n x ) ⎥ ⎣ ⎦ β1x ⎡ ⎤ β β ψ4 = e− n ⎢cos( n1 x ) − sin( n1 x ) ⎥ ⎣ ⎦
It is known by the stress calculation formula of the stress components for orthotropic cylindrical shell and spherical head that the influence factors of stress components include Ex / Eθ , νθ , R/t etc. In the analysis of the impact of Ex / Eθ , the impact of νθ can be neglected, and the νθ will be assumed as common metal alloy value νθ =0.3 (for metal alloy pressure vessel, Poisson's ratio is between 0.2–0.5). Here p=1 Mpa, t1=10 mm, t2 = nt1, R=500 mm, and n = 1 and n = 0.5 are discussed respectively. In the case of n = 0.5, n = 1, the ratio of bending moment and shear force at the juncture between anisotropic model and isotropic model with the change of elastic modulus ratio is shown in Fig. 2. As shown in
β1x e− n
The internal force of any point at the juncture of cylindrical shell is composed of the general membrane stress, local membrane stress and bending stress [21].
σx1 =
pR T 6M + x1 ± 2x1 2t1 t1 t1
(36) 32
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Fig. 6. Effect of Ex /Eθ on the distribution of circumferential stress on the inner surface of cylindrical shell: (a) n=1; (b) n=0.5.
Fig. 7. Effect of R/t on the distribution of axial stress on the outer surface of cylindrical shell: (a) n=1; (b) n=0.5.
Fig. 2, when thickness ratio n = 1, shear ratio Q0 / Q0(Ex / Eθ )=1 increases gradually with elastic modulus ratio Ex / Eθ . At this moment, there is M0 Ex / Eθ =1 = 0 , bending moment curve does not exist. When thickness ratio n = 0.5, bending moment ratio M0 / M0(Ex / Eθ =1) decreases with the ratio of elastic modulus Ex / Eθ , while shear ratio Q0 / Q0(Ex / Eθ )=1 increases gradually with elastic modulus ratio Ex / Eθ which is similar to curve under n=1. As can be seen from Fig. 2, the effect of thickness ratio on shear ratio is smaller than that of elastic modulus ratio Ex / Eθ .
thickness ratio n has significant influence on the axial stress σx on the outer surface of cylindrical shell. 4.2. Effect of Ex / Eθ on the distribution of axial stress on the inner surface of cylindrical shell Fig. 4(a) and (b) are drawn in the case of n = 0.5, n = 1for the variation of the axial stress on the inner surface of cylindrical shell along x direction with different Ex / Eθ values. When n = 1, the axial stress σx decays rapidly along the x direction, rises after reaching the minimum value, and then levels off. When x=0, the axial stresses are basically the same with σx /(pR / t ) = 0.5. The minimum stress of σx /(pR / t ) is among 0.33–0.39, and the variation range of stress fluctuation is approximately3 Rt . The minimum axial stress decays with Ex / Eθ . When n = 0.5, the variation trend of axial stress σx /(pR / t ) is the same as that with n = 1. The minimum stress of σx /(pR / t ) is 0.45–0.47, the variation range of stress fluctuation is roughly 2.5 Rt , and the variation range is smaller than that with n = 1. The results above show that the thickness ratio n has significant influence on the axial stress σx on the inner surface of cylindrical shell.
4. Analysis on the stress at the juncture of orthotropic cylindrical shell and spherical head 4.1. Effect of Ex / Eθ on the distribution of axial stress on the outer surface of cylindrical shell Fig. 3(a) and (b) are drawn in the case of n = 0.5, n = 1 for the variation of axial stress on the outer surface of the cylindrical shell along x direction with different Ex / Eθ values. When n = 1, the axial stress σx increases rapidly along the x direction, drops after reaching the maximum value, and then levels off. The maximum axial stress does not occur at the juncture (x=0), and when x=0, that is the juncture of cylindrical shell and spherical head, the axial stresses are basically the same with σx /(pR / t ) = 0.5. The maximum stress of σx /(pR / t ) is among 0.62–0.68, and the variation range of stress fluctuation is approximately 3 Rt . The maximum axial stress increases with Ex / Eθ . When n = 0.5, the variation trend of axial stress σx /(pR / t ) is the same as that with n = 1. The maximum stress of σx /(pR / t ) is 0.53–0.55, the variation range of stress fluctuation is roughly 2.5 Rt , and the variation range is smaller than that with n = 1. The results above show that the
4.3. Effect of Ex / Eθ on the distribution of circumferential stress on the outer surface of cylindrical shell Fig. 5(a) and (b) are drawn in the case of n = 0.5, n = 1 for the variation of circumferential stress on the outer surface of cylindrical shell along x direction with different Ex / Eθ values. The minimum circumferential stress occurs at the juncture (x=0). When n = 1, the circumferential stress σθ increases rapidly along x direction, after 33
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Fig. 9. Effect of R/t on the distribution of circumferential stress on the outer surface of cylindrical shell: (a) n=1; (b) n=0.5.
Fig. 8. Effect of R/t on the distribution of axial stress on the inner surface of cylindrical shell: (a) n=1; (b) n=0.5.
4.5. Effect of R/t on the distribution of circumferential stress on the outer surface of cylindrical shell
reaching the maximum value levels off. The maximum stress of σθ /(pR / t ) is among 1.01–1.04, and the variation range of stress fluctuation is approximately 3 Rt . The maximum circumferential stress increases with Ex / Eθ . When n = 0.5, the variation trend of circumferential stress σθ /(pR / t ) is the same as that with n = 1. The maximum stress of σθ /(pR / t ) is 1.00–1.01, and the variation range of stress fluctuation is approximately 2.5 Rt , and the variation range is smaller than that with n = 1. The results above show that thickness ratio n has significant influence on the circumferential stress σθ on the outer surface of cylindrical shell.
Fig. 7(a) and (b) are drawn in the case of n = 0.5, n = 1 for the variation of axial stress on the outer surface of cylindrical shell along x direction with different R/t values. When n = 1, the axial stress σx increases rapidly along x direction, drops after reaching the maximum value, and then levels off. The maximum axial stress does not occur at the juncture (x=0), and when x=0, the axial stresses are basically the same as σx /(pR / t ) = 0.5. The maximum stress of σx /(pR / t ) is among 0.55–0.63. When n = 0.5, the variation trend of axial stress σx /(pR / t ) is the same as that with n = 1. The maximum stress of σx /(pR / t ) is 0.5– 0.54, and the variation range is smaller than that with n = 1. The results above show that the thickness ratio n has significant influence on the axial stress σx on the outer surface of cylindrical shell.
4.4. Effect of Ex / Eθ on the distribution of circumferential stress on the inner surface of cylindrical shell
4.6. Effect of R/t on the distribution of axial stress on the inner surface of cylindrical shell
Fig. 6(a) and (b) are drawn in the case of n = 0.5, n = 1or the variation of circumferential stress on the inner surface of cylindrical shell along x direction with different Ex / Eθ values. The minimum circumferential stress occurs at the juncture (x=0). When n = 1, the circumferential stress σθ increases rapidly along x direction, levels off after reaching the maximum value. The variation range of stress fluctuation is approximately 3 Rt . The maximum circumferential stress decays with Ex / Eθ . When n = 0.5, the variation trend of circumferential stress σθ /(pR / t ) is the same as that with n = 1. The variation range of stress fluctuation is approximately 2.5 Rt , and the variation range is smaller than that with n = 1. The results above show that thickness ratio n has significant influence on the circumferential stress σθ on the inner surface of cylindrical shell.
Fig. 8(a) and (b) are drawn in the case of n = 0.5, n = 1 for the variation of axial stress on the inner surface of cylindrical shell along x direction with different R/t values. When n = 1, the axial stress σx decays rapidly along x direction, rises after reaching the minimum value, and then levels off. When x=0, the axial stresses are basically the same as σx /(pR / t ) = 0.5. The minimum stress of σx /(pR / t ) is among 0.37–0.44. When n = 0.5, the variation trend of axial stress σx /(pR / t ) is the same as that with n = 1. The minimum stress of σx /(pR / t ) is 0.46– 0.48, and the variation range is smaller than that with n = 1. The results above show that the thickness ratio n has significant influence 34
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Fig. 11. The relationship between Hill48 equivalent stress and thickness radio n at the juncture of cylindrical shell under different Ex /Eθ : (a)σHill48 ; (b) σHill 48 /σHill 48(EX / Eθ =1) . Fig. 10. Effect of R/t on the distribution of circumferential stress on the inner surface of cylindrical shell: (a) n=1; (b) n=0.5.
5. Effect of thickness ratio and elastic modulus ratio on the Hill48 equivalent stress at the juncture of structure
on the axial stress σx of the inner surface of cylindrical shell. Mises or Tresca yield criterion is generally adopted in the design of isotropic structure. Due to the difference between the orthotropic and isotropic materials, the VonMises or Tresca yield criterion is no longer applicable for orthotropic one. Hill48 yield criterion is more suitable to describe the anisotropy of orthotropic material. For the structure of thin-walled cylindrical shell and spherical head, the equivalent stress intensity expression of the plane structure under biaxial stress state is [22]:
4.7. Effect of R/t on the distribution of circumferential stress on the outer surface of cylindrical shell Fig. 9(a) and (b) were drawn in the case of n = 0.5, n = 1 for the variation of circumferential stress on the outer surface of cylindrical shell along x direction with different R/t values. The minimum circumferential stress occurs at the juncture (x=0). When n = 1, the circumferential stress σθ increases rapidly along x direction, after reaching the maximum value levels off. When n = 0.5, the variation trend of circumferential stress σθ /(pR / t ) is the same as that with n = 1. The variation range is smaller than that with n = 1. The results above show that thickness ratio n has significant influence on the circumferential stress σθ on the outer surface of cylindrical shell.
σHill 48 =
3 Fσθ2 + Gσx2 + H (σx − σθ )2 ⋅ 2 F+G+H
(39)
where F, G and H are the coefficients related to the yield stress of the three direction X,Y and Z.
2F = 4.8. Effect of R/t on the distribution of circumferential stress on the inner surface of cylindrical shell
2G = 2H =
Fig. 10(a) and (b) were drawn in the case of n = 0.5, n = 1 for the variation of circumferential stress on the inner surface of cylindrical shell along x direction with different R/t values. The minimum circumferential stress occurs at the juncture (x=0). When n = 1, the circumferential stress σθ increases rapidly along x direction, after reaching the maximum value levels off. When n = 0.5, the variation trend of circumferential stress σθ /(pR / t ) is the same as that with n = 1. The change range is smaller than that with n = 1. The results above show that thickness ratio n has significant influence on the circumferential stress σθ on the inner surface of cylindrical shell.
1 Y2 1 Z2 1 X2
+ + +
1 Z2 1 X2 1 Y2
1
− − −
X2 1 Y2 1 Z2
(40)
In which X, Y and Z are the yield stresses in axial, circumferential and radial(thickness) directions. According reference [23] for TA2 alloy, Y=1.18X and Z=1.09X. σHill48 is the strength of the equivalent stress under Hill48 yield criterion, and its expression can be written as:
σHill 48 = 0.3 ×
9.345σθ2 + 13.059σx2 − 11.434σx σθ
(41)
According to axial and circumferential stress, Hill48 equivalent stress at the juncture of structure can be obtained. And the variation 35
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Fig. 13. Linearization path: (a) n=1; (b) n=0.5. Fig. 12. The relationship between Hill48 equivalent stress and thickness radio n at the juncture. spherical head under different Ex /Eθ : (a) σHill48 ; (b) σHill 48 /σHill 48(EX / Eθ =1) .
Table 2 Finite element solution and analytical solution of stress in cylindrical shell and its error.
Table 1 Material parameters of TA2. Ex GPa
Ey(Eθ) GPa
RP0.2(x) MPa
RP0.2(θ) MPa
vxy(vx)
vxy(vθ)
110
134
277
327
0.337
0.431
Primary stress plus secondary stress Error
tendency of Hill48 equivalent stress under different elastic modulus at the juncture of structure is drawn, as shown in Figs. 11 and 12.
FEM Analytical solution
σxs (MPa)
σθs (MPa)
n=1
n=0.5
n=1
n=0.5
24.86 25
12.93 12.50
36.98 37.5
23.28 23.12
0.56%
3.44%
1.38%
0.69%
When n > 0.41, Hill48 equivalent stress ratio σHill 48 / σHill 48(Ex / Eθ =1) decreases with the ratio of elastic modulus Ex / Eθ . Hill48 equivalent stress ratio of isotropic material is a straight lever line which is equal to 1. When σHill 48 / σHill 48(Ex / Eθ =1) > 1, the equivalent stress intensity of anisotropic material is greater than that of isotropic one, and is easy to reach yield limit; When σHill 48 / σHill 48(Ex / Eθ =1) < 1, equivalent stress of orthotropic material is less than that of isotropic one and difficult to reach yield limit. Hill48 equivalent stress at the juncture of cylindrical shell is controlled by the ratio of elastic modulus Ex / Eθ and thickness ration , and the maximum Hill48 equivalent stress in different thickness ration corresponds to different elastic modulus ratio Ex / Eθ .
5.1. Hill48 equivalent stress and thickness radio n at the juncture of cylindrical shell under different elastic modulus ratio Fig. 11(a) shows under different Ex / Eθ , the relationship between Hill48 equivalent stress σHill48 and thickness ratio at the juncture of cylindrical shell. Hill48 equivalent stress at the juncture of cylindrical shell increases first and then decreases with thickness ratio. The stress fluctuation range of σHill48 under different elastic modulus ratio is little, which indicates that elastic modulus ratio has less influence on Hill48 equivalent stress at the juncture of cylindrical shell than that of thickness ratio. The variation trend of equivalent stress ratio σHill 48 / σHill 48(Ex / Eθ =1) is shown in Fig. 11(b). With increase of thickness ratio, when elastic Ex / Eθ < 1, modulus ratio Hill48 equivalent stress ratio σHill 48 / σHill 48(Ex / Eθ =1) at the juncture of cylindrical shell decreases first, then increase and finally decreases again. When the elastic modulus ratio Ex / Eθ > 1, Hill48 equivalent stress ratio σHill 48 / σHill 48(Ex / Eθ =1) at the juncture of cylindrical shell increases first, then decrease and finally increases. When n < 0.41, Hill48 equivalent stress ratio σHill 48 / σHill 48(Ex / Eθ =1) increases with the ratio of elastic modulus Ex / Eθ .
5.2. Hill48 equivalent stress and thickness radio n at the juncture spherical head under different elastic modulus ratio Fig. 12(a) shows under different Ex / Eθ , the relationship between Hill48 equivalent stress σHill48 and thickness ratio at the juncture of spherical head. Hill48 equivalent stress at the juncture of spherical head decreases with thickness ratio. The stress fluctuation range of 36
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are consistent.
σHill48 under different elastic modulus ratio is little, which indicates that elastic modulus ratio has less influence on Hill48 equivalent stress at the juncture of spherical head than that of thickness ratio. The variation trend of equivalent stress ratio σHill 48 / σHill 48(Ex / Eθ =1) is shown in Fig. 12(b). With increase of thickness ratio, when elastic Ex / Eθ < 1, modulus ratio Hill48 equivalent stress ratio σHill 48 / σHill 48(Ex / Eθ =1) at the juncture of spherical head increases first, then decrease. When elastic modulus ratio Ex / Eθ > 1, Hill48 equivalent stress ratio σHill 48 / σHill 48(Ex / Eθ =1) at the juncture of spherical head decreases first, then increase. When n < 0.41, Hill48 equivalent stress ratio σHill 48 / σHill 48(Ex / Eθ =1) increases with the ratio of elastic modulus Ex / Eθ . When n > 0.41, Hill48 equivalent stress ratio σHill 48 / σHill 48(Ex / Eθ =1) decreases with the ratio of elastic modulus Ex / Eθ .
Acknowledgements The authors gratefully acknowledge the financial support by SixTalent Peak Project of Jiangsu Province(2014-ZBZZ-012) and the National Natural Science Foundation of China(51475223, 51675260). References [1] Z.W.Wang, R.L.Cai, Design of chemical pressure vessels, 2005. [2] L. Fu, Y.R. Luo, L. Fu, Study on stress distribution of spherical head, Mach. Des. Manuf. 10 (2013) 209–212. [3] J. Awrejcewicz, A.V. Krysko, I.V. Papkova, et al., On the methods of critical load estimation of spherical circle axially symmetrical shells, Thin-Walled Struct. 94 (2015) 293–301. [4] W.C. Cui, J.H. Pei, W. Zhang, A simple and accurate solution for calculating stresses in conical shells, Comput. Struct. 79 (3) (2001) 265–279. [5] N.F. Rilo, J.F.D.S. Gomes, J. Cirne, et al., Stresses from radial loads and external moments in spherical pressure vessels, Proc. Inst. Mech. Eng. Part E J. Process Mech. Eng. 215 (2) (2001) 99–109. [6] J. Awrejcewicz, L. Kurpa, A. Osetrov, Investigation of stress-strain state of the laminated shallow shells by R-functions method combined with spline-approximation, ZAMM J. Appl. Math. Mech.: Z. für Angew. Math. und Mech. 91 (91) (2001) 458–467. [7] A.V. Krysko, J. Awrejcewicz, E.S. Kuznetsova, et al., Chaotic vibrations of closed cylindrical shells in a temperature field, Int. J. Bifurc. Chaos 18 (5) (2005) 1515–1529. [8] V.A. Krysko, J. Awrejcewicz, N.E. Saveleva, Stability, bifurcation and chaos of closed flexible cylindrical shells, Int. J. Mech. Sci. 50 (2008) 247–274. [9] Z.Q. Li, Edge bending calculation and analysis of laminated shell, Fiber Reinf. Plast./Compos. 4 (1992) 17–21. [10] J.J. Zeng, Y.M. Fu, Elasto-plastic buckling analysis for orthotropic circular cylindrical shells under axial compression, Eng. Mech. 23 (10) (2006) 25–29. [11] J.M. Rotter, A.J. Sadowski, Cylindrical shell bending theory for orthotropic shells under general axisymmetric pressure distributions, Eng. Struct. 42 (2012) 258–265. [12] K. Chandrashekhara, P. Gopalakrishnan, Transversely isotropic infinite cylindrical shell subjected to a radial axisymmetric line load, Fibre Sci. Technol. 16 (4) (1982) 275–293. [13] V.F. Kirichenko, J. Awrejcewicz, A.V. Kirichenko, et al., On the non-classical mathematical models of coupled problems of thermo-elasticity for multi-layer shallow shells with initial imperfections, Int. J. Non-Linear Mech. 74 (2015) 51–72. [14] D.N. Paliwal, P. Kumar, I.K. Bhat, et al., Discontinuity stresses in orthotropic pressure vessels, Int. J. Vessels Pip. 72 (1997) 63–72. [15] T.T. Gao, C.Y. Zhou, X.H. He, et al., Discontinuity stresses at the junction of orthotropic pressurized cylindrical shell with a flat head, Chinese, J. Appl. Mech. 32 (1) (2015) 28–33. [16] T.Y. Yao, X.H. He, F.S. Kong, et al., Design by analysis for orthotropic pressurized structure with small end of conical shell and cylindrical shell based on Hill48 yield criterion, Thin-Walled Struct. 96 (2015) 220–226. [17] W.Flügge, Stresses in Shells, 1960. [18] Lloyd E. Brownell, Edwin H. Young, Process Equip. Des. (1959) 120–140. [19] S.T. Sun, R.G. Jiang, R.G. Liu, Anisotropic plates Shells Theory (1993). [20] K.Z.Huang, Z.X.Xia, M.D.Xue, et al., Plates and shells theory, 1987. [21] JB 4732-1995, Steel pressure vessels-design by analysis, Peking, China, pp. 1995. [22] R. Hill, Math. Theory Plast. (1956). [23] Y.H. Guo, Stress Analysis and Design of Orthotropic Pressurized Components Made of Titanium, Nanjing Tech University, Jiangsu, China, 2014.
6. Comparison and verification between analytical solution and finite element solution In order to verify the correctness of stress solution at the juncture of structure, finite element method is used to simulate the discontinuous stress value at the juncture. ANSYS is used to build models where R=500 mm, t2=10 mm, p=1 MPa, thickness ratios are n=1 and n=0.5, respectively. The material properties of finite element calculation and analytical solution refer to Table 1. Fig. 13 shows the linear path of spherical head and cylindrical shell. The axial and circumferential stress of cylindrical shell are extracted from path A-A and path B-B. According to the VIII-2 ASME stress classification method, the stress is obtained by the FEM method and analytical solution respectively, the results are shown in Table 2. From Table 2, the maximum error of analytical solution and finite element solution of spherical head and cylindrical shell is 3.44%, which shows that analytical solution and finite element solution are consistent, and the accuracy of analytical solution is verified by finite element method. 7. Conclusion (1) For the structure of orthotropic cylindrical shell and spherical head, the effect of thickness ratio on shear ratio at the juncture is smaller than that of elastic modulus ratio Ex / Eθ . (2) For the structure of cylindrical shell, thickness ratio n has significant influence on axial stress σx and circumferential stress σθ on the surface of cylindrical shell than the Elastic modulus ratio Ex / Eθ and R/t. (3) At the juncture of cylindrical shell and spherical head, Hill48 equivalent stress changes differently with elastic modulus ratio and thickness ratio. (4) By calculation, the maximum error of analytical solution and finite element solution of spherical head and cylindrical shell is 3.44%, which shows that analytical solution and finite element solution
37