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力学专业外文翻译

力学专业外文翻译
力学专业外文翻译

附录:外文翻译

5.1Introduction

Cylindrical shells are used innuclear,fossil and petrochemical industries. They are also used in heat exchangers of the shell and tube type.Generally.These vessels are easy to fabricate and install and economical to maintain. The design procedures in pressure vessel codes for cylindrical shells are mostly based on linear elastic assumption,occasionally allowing for limited inelastic behavior over a localized region.The shell thickness is the major design parameter and is usually controlledby internal pressure and sometimes by external pressure which can produce buckling.Applied loads are also important in controlling thickness and so are the disconti-nuity and thermal stresses.The basic thicknesses of cylindrical shells are Based on simpli?ed stress analysis and allowable stress for the material of construction.There are some variations of the basic equations in various design codes.Some of the equations are based on thick-wall Lame equations.In this chapter such equations will be discussed.Also we shall discuss the case of cylindrical shells under external pressure where there is a propensity of buckling or collapse.

5.2 Thin-shell equations

A shell is a curved plate-type structure.We shall limit our discussion to Shells of revolutions.Referring to Figure5.1 this is denoted by anangle ?,The meridional radius r1 and the conical radius r2,from the center line.The horizontal radius when the axis is vertical is r. If the shell thickness is t,with z being the coordinate across the thickness,following the convention of Flugge, We have the following stress resultants:

?-+

=

2

2

1 1) (

t

t

dz r

z

r

N

θ

θ

σ(5.1)

?-+

=

2

2

2 2) (

t

t

dz r

z

r

N

φ

φ

σ(5.2)

?-+

=

2

2

2 2) (

t

t

dz r

z

r

N

θφ

θφ

σ(5.3)

Figure 5.1 Thin shell of revolution .

?-+=2

21

1)(

t

t

dz r z r N θφφθσ (5.4) These stress resultants are assumed to be due only to an internal pressure, p,acting in the direction of r. For membrane shells where the Effects of bending can be ignored,all the moments are zero and further development leads to

θφφθN N =

The following equations result from considering force equilibrium along with the additional requirement of rotational symmetry:

0cos )

(1=-φφθφN r d rN d (5.6)

φφN r r pr N 122-

= (5.7) Noting that φsin 2r r = ,we have,by solving Eqs.(5.6)and(5.7),

2

2pr N =φ (5.8) )2(2122r r pr N -=

φ (5.9) The above two equations are the results for a general shell of revolution. Two speci?c cases result:

1. For a spherical shellof radius R, r1= r2 =R,which gives

2

pR N N ==θφ (5.10) 2. For a cylindrical pressure vessel of radius R,we have r1 =∞; r2 = R,which gives

2

pR N =φ (5.11) pR N =θ (5.12) This gives the hoop stress

t

pR t N hoop ==

=θφσσ (5.13) and the longitudinal stress t

pR t N long 2===θφσσ (5.14) These results will be shown to be identical to the results that follow. Let us consider a long thin cylindrical shell of radius R and thickness t, subject to an internal pressure p.By thin shell we mean the ones having the ratio R/t typically greater than about10.If the ends of the cylindrical shell are closed,there will be stresses in the hoop as well as the axial (longitudinal) directions.

A section of such a shell is shown in Figure5.2. The hoop(circumfer-ential)stress, hoop σand the longitudinal stress, long σ are indicated in the ?gure.The shell is assumed to be long and thin resulting in hoop σand long σto be uniform through the thickness.Therefore in this case hoop σand long σ are also referred to as membrane stress(there are no bending stresses associated with this type of loading).

Considering equilibrium across the cut section,we have,

tL R pL hoop σ2)2(=

Figure 5.2 Thin cylindrical shell. which gives

t

pR hoop =σ (5.15) Considering a cross-section of the shell perpendicular to its axis,we have

)2(2Rt R p long πσπ= Which gives

t pR hoop 2=

σ (5.16 5.3 Thick-shell equations

For R/t ratios typically less than 10,Eqs.(5.15) and (5.16) tend not to be accurate,and thick-shell equations have to be used. Consider a thick cylindrical shell of inside radius Ri and outside radius Ro subjected to an internal pressure p as shown in Figure5.3 .The stress function for this case(refer to AppendixI)is given as a Function of radius r as

2ln Br r A +=Φ (5.17)

Figure 5.3 Thick cylindrical shell .

with A and B to be determined by the boundary conditions. If we indicate the radial stress as rad o and the hoop and longitudinal Stress as indicated previously by hoop σand long σ,we have

B r A dr d r o rad 212+=Φ=

(5.18) B r

A dr d r o hoop 212+-=Φ= (5.19) The constants A and

B are determined from the following boundary conditions:

p rad -=σ at i R r =

0=rad σ at o R r = (5.20) Substituting(5.20)into(5.18)and(5.19),we have

)(22

22i o o

i R R R R A --= )(2222i o i R R p

R B -= (5.21)

Denoting the ratio of the outside to inside radii as m,so that m =Ro/Ri, We obtain theradial and hoop stresses

]1[122

2r

R m p o rad --=σ (5.22) ]1[1222r R m p o hoop

+-=σ (5.23) Figure5.4 shows the radial and hoop stress distributions.

The longitudinal stress,long σ is determined by considering the Equilibrium of forces across a plane normal to the axis of the shell,which gives

)(2

22i o long i R R R p -=πσπ (5.24)

This is of course based on the assumption that the longitudinal stress is a Form of membrane stress in that there is no variation across the thickness of The shell.Thus we have

Figure 5.4 Hoop and radial stress distribution .

)

1()(2222-=--=m p R R pR i o i

long σ (5.25) It should be noted however that the solutions indicated by Eqs.(5.22), (5.23),and(5.25) are valid for regions remote from discontinuities.

5.4 Approximate equations

For a moderately thick shell employing thin-shell theory and using the Mean radius Rm we get the expression of the hoop stress, hoop σ,as

t

t R p t pR i m hoop )2/(+=-=σ (5.26) Equating the hoop stress, hoop σ,to the code-allowable design stress, m S , We have

p

S pR t m i 5.0-= (5.27) Rewriting Eq.(5.27) in terms of the outside radius, o R we have,

t

t R p t pR o m hoop )2/(-=-=σ (5.28) Once again equating the hoop stress to the code-allowable design stress, S,we have

p

S pR t m 5.00+= (5.29) The equations in the ASME Boiler and Pressure Vessel Code are based On equating the maximum membrane stress to the allowable stress Corrected for weld joint ef?ciency.The allowable stress, m S ,is replaced by The term SE (to be explained later).In ASME Boiler and Pressure Vessel

Code,SectionVIII Division1, The Eqs .(5.27) and (5.29) are modi?ed as:

p

SE pR t 5.0-= (5.30) p

SE pR t 5.0+= (5.31) In ASME Boiler and Pressure Vessel Code SectionIII,Division1, the equationsusedare

p

SE pR t 6.0-= (5.32) p SE pR t 4.0+=

(5.33) In Eqs .(5.30)and (5.32), R stands for the inside radius, i R ,whereas in Eqs.(5.31) and (5.33) it stands for the outside radius, o R .In both of the above equations, S is the allowable stress and E is the joint ef?ciency.This joint ef?ciency is employed because cylindrical shells are often fabricated by welding.The values of E depend on the type of radio graphic examination performed at various welded seams of the shell.

5.5Buckling of cylindrical shells

Consider a long,thin cylindrical shell of mean diameter D and wall thickness t subjected to an external pressure P.The cylinder is in a stable con?guration as long as it remains circular in shape.If there is an initial ellipticity,the cylinder will be in an unstable condition and will eventually buckle.

If the cylinder is suf?ciently long,the end effects may be neglected and The problem may be considered as two-dimensional. Summing up the forces in the radial direction,we have

0)(22)(=--++++Pds V ds ds

dv V d N d ds dS dN N θθ or

0=-+Pds ds ds

dv Nd θ or

)2/(,0D R PRd ds ds

dv Nd ==-+θθ 0=--R

N dS dV P Summation of forces in the tangential direction(see Figure5.5) gives -02

2)()(=+++++θθd v d ds ds dv V N ds dS dN N 0=+-θvd ds ds

dN 0=-ds

dN R V

Figure 5.5 Equilibrium of a shell element.

With ds dM

V =,we have

ds dM

R ds dN 1

= Assuming deviation from circular shape to be small,we have

C R M

N +=

Where C is a constant

When

2D

R =

0==V M

2D

P N = Therefore

2D

P R M

N +=

022=--R N

s d M d P

01

2222=---R D P R M s d M d P

0)21(22

22=-++D R D P R M s d M d For a curved shell:

0)21()1(1223=--=D

R v Eh M 0))1(61(3

2222=-++Eh PD v R M s d M d with 2D

R ?

0))1(61(32222=-++M Eh PD

v R s d M

d With ))

1(64(32

22Eh v PD D k -+=

0222=+M k s d M

d

ks C ks C M cos sin 21+=

ks k C ks k C ds dM

cos sin 21+=

0=ds dM

at 0=s

0=ds dM at 4D

s π=

0)4sin(=D

k π

ππn D

k =)4(

n kD 4=

))

1(64(3222Eh v PD D k -+=

)14(4)

1(62232-=-n D Eh v D P cr With a minimum value of n=1,we get

32)()1(2d t

v E

P cr -=

(5.34)

For cylinder with shorter lengths,where the ends are free to expand Axially and rotate with the restriction of expanding radially,thecritical Pressure is given by

??????

??????+-+????????????+--+--=)41)(1(1)(24112)1()()1(3222222222222320D L n n D t E D L n v n n d t v E P cr

ππ When D

l is large: )1()()1(322320--=n d

t v E P cr (5.35) So that it becomes identical to the buckling pressure inEq.(5.34)for n= 2. In the ASME code,the critical pressure is calculated for two situations, Involving the ratio of the outside diameter to the thickness(Do/t) 1.10≥t

D o 2.

10

5.6 Discontinuity stresses in pressure vessels

Let us take the special case of discontinuity at a juncture between a cylindrical vessel and ahemispherical head subjected to internal pressure p. For simplicity let us assume the spherical head and the cylindrical shell are Of the same thickness.If the mean radius and thethickness of the shell are Denoted by Rm and t respectively,then the hoop and the longitudinal Stresses in the cylindrica lshell are given by:

t

pR m hoop c =σ (5.36)

t

pR m long c 2=σ (5.37) The hoop and the longitudinal stresses in the spherical shell are given by:

t

pR m hoop s 2=σ (5.38) t pR m long s 2=

σ (5.39) The radial growth or dilation of the cylindrical shell under internal pressure p is given by

)2(22

v Et

pR m r c -=δ (5.40) That of the spherical region is given by

)1(22

v Et pR m r s -=δ (5.41) Where v is the Poisson’s ratio.

If the spherical and the cylindrical portions were separated,the Difference in the radial growth would be

Et

pR m r s r c r 22

=-=δδδ (5.42) In the actual vessel the hemispherical head and the cylindrical shell are Kept in place by shear force, V and moment M per unit circumference. These discontinuity forces produce local bending stresses in the adjacent Portions of the vessel.The de?ection and the slope induced at the edges of

The cylindrical and spherical portions by the force V are equal.The Continuity at the juncture will bes atis?ed if M equals zero and V is such That it produces a de?ection of δ/2. Applying the results from semi-in?nite beam on an elastic foundation Due to M and V , And substituting the spring rate of the foundation by 2

m R Et/

))sin()(cos(2)cos(2222x x e Et R M x e Et R V x m x m βββββδββ--=-- (5.43) where β is the attenuation factor,given by

4

222)1(3t R v m -=β

(5.44) We have with 2/δδ= and M=0.at x=0

Et R V m 222βδ

=

(5.45) Substituting the value of δ from Eq.(5.45)we have

β8p

V =

(5.46)

The longitudinal stress and the hoop stress distribution in the cylindrical region is then given by

)sin(86222x e p t t pR x m long c ββσβ-±=

(5.47) )sin(43)cos(422x e t vp x e t pR t pR x x m m hoop c ββ

βσββ--±-= (5.48)

Using numerical values as, p= 2MPa, Rm = 1m, t = 25mm,and Poisson’sratio, v =0.3,we have from Eq.(5.44), β = 0.008127/mm and the Longitudinal and the hoop stresses become

)sin(3640x e x long c βσβ-±=Mpa (5.49)

)sin(9.10)cos(2080x e x e x x hoop c ββσββ--±-=Mpa (5.50)

In Eq .(5.49) the ?rst quantity –the membrane longitudinal stress –is a constant (equal to 40MPa) along the length of the vessel,while the second quantity –the bending stress –varies along the length.In Eq.(5.50) the membrane hoop stress (equal to 80MPa) stays constant along the length, while the direct compression stress due to shortening of the radius and the bending stress varies along the length of the vessel.

1.Find the thickness of a cylindrical shell 2m in diameter if it is required to contain an internal pressure of 7MPa.The allowable stress in the material is140MPa.

2.A thick cylindrical shell of 1.2m inside diameter and 1.5m outside diameter is

subjected to an internal pressure of 35MPa.Determine the following:

a.Magnitude and location of the maximum hoop stress

b.Magnitude of the maximum radial stress and its location

c.Average hoop stress

3.A thick cylinder has an inside diameter of 300mm and an outside Diameter of 450mm.If the allowable stress is175MPa,what is the maximum internal pressure that can be applied?

4.A cylinder has an inside radius of 1.8m and is subjected to an Internal pressure of 0.35 MPa.What is th required thickness if the Allowable stress is105 MPa?

5.For problem 4,what is the required thickness if thick cylinder equations were used?

https://www.doczj.com/doc/275309856.html,ing ASME Boiler and Pressure Vessel Code equations,determine the thickness of a cast-iron pressure vessel subjected to an internal pressure of 0.5 Mpa using a joint ef?ciency of 85 percent and a corrosion allowance of 1.5mm.The allowable stress (Sm)of the material is14MPa.

第五章

5.1简介

圆柱形容器主要用于核能,石油化工工业,它们也用于管壳式换热器。通常,这些容器制造、安装容易,维护经济。压力容器标准中圆柱形容器的设计程序主要是基于线性弹性假设,有时允许局部区域有限的非弹性行为。外壳厚度是主要设计参数,通常是由内部压力控制,有时是由产生屈曲的外部压力控制。添加载荷、不连续应力、热应力在控制厚度的过程中也很重要。圆柱形容器的基本厚度主要是基于简化应力分析和建筑物料的许用应力。在不同的设计标准中基本方程有一些变化。有一些方程是基于厚壳方程。在这一章节,这些理论将被讨论。另外,我们还将讨论圆柱形容器外压失稳在情形。

5.2薄壳方程

对于弯板型结构,我们只讨论旋转壳。参考图5.1,从中心,记为角?,经向半径记为r1和锥形半径记为r2当轴是垂直时水平半径为r。当沿Z坐标方向的壁厚为t,根据下面的草图,我们得到残余应力。

?-+

=

2

2

1 1) (

t

t

dz r

z

r

N

θ

θ

σ(5.1)

?-+=2

2

2

2)(

t

t

dz r z r N φφσ (5.2) ?-+=222

2)(

t

t

dz r z r N θφθφσ

(5.3)

图5.1:旋转壳

?-+=2

21

1)(

t

t

dz r z r N θφφθσ (5.4) 径向的残余应力由内压产生,对于膜壳,弯曲的影响可以忽略。所有的弯矩为零,进一步推得:

θφφθN N =

由旋转对称产生的力的平衡导出下面的方程:

0cos )(1=-φφ

θφN r d rN d (5.6) φφN r r pr N 1

22-= (5.7) 由于φsin 2r r =,我们解出(5.6)和(5.7)得:

2

2pr N =φ (5.8) )2(2122r r pr N -=

φ (5.9) 以上两个方程是普通壳的旋转的结果。两个具体的例子:

1.对于球壳,半径为R 则 r1= r2 =R 得出:

2

pR N N ==θφ (5.10) 2.对于圆柱形压力容器,半径为R , 则r1 =∞; r2 = R 得出:

2

pR N =φ (5.11) pR N =θ (5.12)

环向应力:

t

pR t N hoop ==

=θφσσ (5.13) 轴向应力 t

pR t N long 2===θφσσ (5.14) 这些结果与下面的结果一致。考虑一细长圆柱壳的半径为R ,厚度为t ,受内压为p 。薄壳是指壳的半径和厚度R/t 之比大于10。如果圆柱壳的两端是封闭的,还受环向应力和纵向应力。壳的局部图如图5.1所示。环向应力hoop σ和纵向应力long σ如图所示。外壳被假定为细长所以hoop σ和long σ沿厚度方向分布均匀。因此,这种情况下的hoop σ和long σ也称薄膜应力(这种类型载荷不存在弯曲应力)

考虑切节的平衡,得出:

tL R pL hoop σ2)2(=

图5.2:圆柱壳

得出:

t

pR hoop =σ (5.15) 考虑横截面的壳垂直于轴心,我们得出:

)2(2Rt R p long πσπ=

t

pR hoop 2=

σ (5.16) 5.3 厚壳理论

壳的半径和厚度R/t 之比小于10,式(5.15)和 式(5.16)将会不准确,这时将用到厚壳理论,厚壳圆柱形容器的内径为Ri ,外径为Ro ,受内压为p ,如图5.3这种情况下的应力函数(参阅附录I )由半径r 的函数给出: 2ln Br r A +=Φ (5.17)

图5.3厚壳圆柱形容器

A 和

B 由边界条件决定。如果用rad o 表示径向应力、用hoop σ表示环向应力和用long σ表示纵向应力,我们得出:

B r A dr d r o rad 212+=Φ=

(5.18) B r

A dr d r o hoop 212+-=Φ= (5.19) 常量A 和

B 由下面的边界条件决定:

p rad -=σ at i R r =

0=rad σ at o R r = (5.20)

将式 (5.20) 代入式(5.18)和式(5.19)得出:

)(22

22i o o

i R R R R A --= )(2222i o i R R p

R B -= (5.21)

外径和内径之比记为m ,则m =Ro/Ri 我们得出径向和周向应力

]1[1222r

R m p o rad --=σ (5.22) ]1[12

22r R m p o h o o p +-=σ (5.23) 图5.4显示出径向和周向应力的分布

纵向压力long σ垂直壳轴的面受力平衡得出:

)(222i o long i R R R p -=πσπ (5.24)

假定纵向应力是薄膜应力的一种,而且壳的厚度没有变化。因此,得出:

图5.4径向和周向应力的分布

)

1()(2222-=--=m p R R pR i o i

long σ (5.25) 值得一提的是:由式(5.22)、 (5.23) 和(5.25)推出的结果在远离不连续性区域处有效。

对于中等厚壳,运用薄壳方程和平均半径Rm ,我们得出环向应力的方程。

t

t R p t pR i m hoop )2/(+=-=σ (5.26) 等值的环向应力hoop σ,允许设计应力Sm ,得出:

p

S pR t m i 5.0-= (5.27) 将外径代入(5.27)式得:

t

t R p t pR o m hoop )2/(-=-=σ (5.28) 又等同的环向应力的允许设计应力S 得:

p

S pR t m 5.00+= (5.29) 这些方程在ASME 锅炉及压力容器标准中主要是基于等同最大薄膜应力的容许应力,根据焊接接头效率校正。许用应力Sm 用SE (稍后解释)替代,在ASME VIII -1锅炉及压力容器标准中,.(5.27) 式和(5.29)式修改为:

p

SE pR t 5.0-= (5.30) p

SE pR t 5.0+= (5.31) 在ASME VIII -1锅炉及压力容器标准中,方程变为:

p

SE pR t 6.0-= (5.32) p SE pR t 4.0+=

(5.33) 在(5.30) 式和(5.32)式中,Ri 代表内径,然而在(5.31)式和 (5.33)式中Ro 代表外径。在上面两个式子中,S 为许用应力,E 是接头效率。用接头效率是因为圆柱壳容器通常是用焊接制造的。E 的值取决于壳上采用的焊缝的种类。

5.5圆柱壳的屈曲

细长圆柱壳容器的中径为D ,厚度为t ,受外压为p ,只要它仍然是圆形,圆柱体是一个稳定的配置,如果最初的椭圆,柱体将工作在不稳定工况,最终将屈曲。

如果圆柱体足够长,端面的因素可以忽略,问题可以认为在二维平面内。 总结径向方向上的受力,得出:

0)(22)(=--++++Pds V ds ds

dv V d N d ds dS dN N θθ 0=-+P d s ds ds

dv Nd θ )2/(,0D R PRd ds ds

dv Nd ==-+θθ 0=--R

N dS dV P 总结切向方向上的受力(见图5.5),得出: -02

2)()(=+++++θθd v d ds ds dv V N ds dS dN N 0=+-θvd ds ds

dN 0=-ds

dN R V

图5.5壳单元的平衡

代入 ds dM

V =,得

ds dM

R ds dN 1=

假定圆形变形偏差可以忽略不计,得出:

C R M

N +=

其中C 是一个常量 因为2D

R =,0==V M ,2D

P N =

所以:

2D

P R M

N +=

022=--R N

s d M d P

01

2222=---R D P R M s d M d P

)21(2222=-++D R D P R M

s d M

d 对于弯曲壳体:

0)2

1

()1(1223

=--=D R v Eh M

复合材料与工程专业毕业设计外文文献翻译

毕业设计外文资料翻译 题目POLISHING OF CERAMIC TILES 抛光瓷砖 学院材料科学与工程 专业复合材料与工程 班级复材0802 学生 学号20080103114 指导教师 二〇一二年三月二十八日

MATERIALS AND MANUFACTURING PROCESSES, 17(3), 401–413 (2002) POLISHING OF CERAMIC TILES C. Y. Wang,* X. Wei, and H. Yuan Institute of Manufacturing Technology, Guangdong University ofTechnology, Guangzhou 510090, P.R. China ABSTRACT Grinding and polishing are important steps in the production of decorative vitreous ceramic tiles. Different combinations of finishing wheels and polishing wheels are tested to optimize their selection. The results show that the surface glossiness depends not only on the surface quality before machining, but also on the characteristics of the ceramic tiles as well as the performance of grinding and polishing wheels. The performance of the polishing wheel is the key for a good final surface quality. The surface glossiness after finishing must be above 208 in order to get higher polishing quality because finishing will limit the maximum surface glossiness by polishing. The optimized combination of grinding and polishing wheels for all the steps will achieve shorter machining times and better surface quality. No obvious relationships are found between the hardness of ceramic tiles and surface quality or the wear of grinding wheels; therefore, the hardness of the ceramic tile cannot be used for evaluating its machinability. Key Words: Ceramic tiles; Grinding wheel; Polishing wheel

毕业论文外文翻译模版

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热电材料外文翻译

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外文翻译

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材料英文文献翻译

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