Applied Mathematics and Mechanics (English Edition, Vol. 20, No. 5, May 1999)

Published by SU, Shanghai, China

A CATENARY ELEMENT FOR THE ANALYSIS

OF CABLE STRUCTURF~

Peng Wei (~'fl TJ_), Sun Bingnan (~ Jv )~) , Tang Jinchun ( )~ i ~)

Department of Civil Engineering, Zhejiang University,Hangzhou 310027, P R China

(Commtmicated by Chen Shanlin; Received Feb 28, 1998, Revised Jan 16, 1999)

Abstract: Based on analytical equations, a catenary element is presented for the finite element analysis of cable structures. Compared with usually used element (3- node element, 5-node element), a program with the proposed element is of less computer time and better accuracy.

Key words: cable structures; catenary elements; tangent stiffness matrix

Introduction

Cable structures arc nonlinear systems with large displacements, they should be analysed by nonlinear elastic theory. At present, cable structures are usually analysed by finite element method with bar element or multi-node curved element, these elements have approximation to certain degree. An alternative approach proposed by this paper is to use step-by-step methods based on approximate analytical equations of the elastic catenary. In contrast to the multi-element techniques, the cable may be represented by a single element. The potential savings in computer time make the method attractive for static response calculations.

1 Statics of the Elastic Catenary

1.1 Basic equations The cable as shown in Fig. 1 is suspened between two fixed points A and B. The span of the

cable is l = Ltt, the unstrained and strained length of the cable is L0 and L, respectively. A point

on the cable has Lagrangian coordinate s in the unstrained profile. Under self-weight of IV ( = mgL0) this point moves to occupy its new position in the strained profile described by Cartesian coordinates x and z and Lagrangian coordinate p.

A(0,0)

V

B(Lu, Lv)

W,/Lo r

Fig. 1 Cable system

The geometric constraint to be satisfied is:

Fig. 2 Forces on a segment

532

Abstract

The one-dimensional problem of the motion of a rigid flying plate under explosive attack has an analytic solution only when the polytropic index of detonation products equals to three. In general, a numerical analysis is required. In this paper, however, by utilizing the "weak" shock behavior of the reflection shock in the explosive products, and applying the small parameter pur- terbation method, an analytic, first-order approximate solution is obtained for the problem of flying plate driven by various high explosives with polytropic indices other than but nearly equal to three. Final velocities of flying plate obtained agree very well with numerical results by computers. Thus an analytic formula with two parameters of high explosive (i.e. detonation velocity and polytropic index) for estimation of the velocity of flying plate is established.

1. Introduction

Explosive driven flying-plate technique ffmds its important use in the study of behavior of materials under intense impulsive loading, shock synthesis of diamonds, and explosive welding and cladding of metals. The method of estimation of flyor velocity and the way of raising it are questions of common interest.

Under the assumptions of one-dimensional plane detonation and rigid flying plate, the normal approach of solving the problem of motion of flyor is to solve the following system of equations governing the flow field of detonation products behind the flyor (Fig. I):

ap +u_~_xp + au --ff =o,

au au 1 y =0,

aS as a--T =o,

p =p(p, s),

(i.0

293

where p, p, S, u are pressure, density, specific entropy and particle velocity of detonation products respectively, with the trajectory R of reflected shock of detonation wave D as a boundary and the trajectory F of flyor as another boundary. Both are unknown; the position of R and the state para- meters on it are governed by the flow field I of central rarefaction wave behind the detonation wave D and by initial stage of motion of flyor also; the position of F and the state parameters of products

A Catenary Element of Cable Structures 533

(dx/dp) 2 + (dz/dp) 2 = 1. (1)

With reference to Fig. 2, the balancing of horizontal and vertical forces yields

T . dx/dp = H, T 'dz /dp = V- W" s/Lo. (2)

A constitutive relation of Hooke' s law is:

T = EA(dp/ds - 1), (3)

where E is Young' s modulus, A is the uniform cross-sectional area in the unstrained profile. The end conditions at the cable supports A and B are

x =0, z=0, p =0 at A(s =0) , (4)

x = Ln , z = Lv , p = L at B (s = L0).

1.2 Parametr ic solutions Based on F_qs. ( 1 ) ~ (4), x, z and T can be expressed as functions of the indenpendent

variable s

T = T(s ) = H 2 + V- W. , (5)

H, _~[s inh_ l (V ) s inh_ l (V - W_ 's /Lo) ] , (6) = = +

z = z (s )=E~(~-2Zo)+f f -~{[ I+(~ _ + W~ }

(7) By using the end conditions s = Lo, x = L/~, z = Lv, we can obtain two equations in H

and V:

1 V Ln- EA + W t

2

rZto( V 1) Lv = EA x -2

Catenary E lement

+ -~-~{[ 1 + (V)2,] 1/2 - [ 1 + ( -~)2] 1/2},

(8)

(9)

For the purpose of developing procedure, it is best to replace the catenary element in Fig. 1 by that in Fig. 3, then

F1 =- H, F 2 = V, F3 =- F1,

Fa =- F2 + W, T, = ( F~ + F~) 1/2

Tj = (F 2 + F2) 1/2,

with respect to

W = mgL0,

sinh-lx = ln[x + (1 + x2)1/2],

rewritting Eqs. (8) and (9) yields

FI

Fig. 3 Catenary dement

L

+ ___lln Tj + F4] mg T, - - f -22 '

(10)

Abstract

The one-dimensional problem of the motion of a rigid flying plate under explosive attack has an analytic solution only when the polytropic index of detonation products equals to three. In general, a numerical analysis is required. In this paper, however, by utilizing the "weak" shock behavior of the reflection shock in the explosive products, and applying the small parameter pur- terbation method, an analytic, first-order approximate solution is obtained for the problem of flying plate driven by various high explosives with polytropic indices other than but nearly equal to three. Final velocities of flying plate obtained agree very well with numerical results by computers. Thus an analytic formula with two parameters of high explosive (i.e. detonation velocity and polytropic index) for estimation of the velocity of flying plate is established.

1. Introduction

Explosive driven flying-plate technique ffmds its important use in the study of behavior of materials under intense impulsive loading, shock synthesis of diamonds, and explosive welding and cladding of metals. The method of estimation of flyor velocity and the way of raising it are questions of common interest.

Under the assumptions of one-dimensional plane detonation and rigid flying plate, the normal approach of solving the problem of motion of flyor is to solve the following system of equations governing the flow field of detonation products behind the flyor (Fig. I):

ap +u_~_xp + au --ff =o,

au au 1 y =0,

aS as a--T =o,

p =p(p, s),

(i.0

293

where p, p, S, u are pressure, density, specific entropy and particle velocity of detonation products respectively, with the trajectory R of reflected shock of detonation wave D as a boundary and the trajectory F of flyor as another boundary. Both are unknown; the position of R and the state para- meters on it are governed by the flow field I of central rarefaction wave behind the detonation wave D and by initial stage of motion of flyor also; the position of F and the state parameters of products

534 Peng Wei, Sun Bingnan and Tang Jinehun

1 Lr - 2EAmg

Eqs. (10) and ( 11 ) may be written as

where

r [K] = IF]-1 = [

"det[

kll =

km=

k21 --

k~ =

from Eqs. (10) and ( 11 ) we see that

Lo

i (

L~ = f (F1 ,F2) ,

4 = g(&,F2) ,

8FI1 fSL:,

kl, k,2] = [ f . k21 k99 J - f21

F] = Alf22 - A2f21,

f~ I (A , f= - A2f21 ) ,

- A2 / (A , f~ - A2A1) ,

- f21/(A1f22 - A2AI ),

A1/ (A l f~ - A2A, ) ,

(11)

1 in Tj + F4 " 1 {F2 A, =-~- zg T -& +~tT ,

F , (1 1) f ,~ = f~, = ~ - ~ ,

Lo 1 (F2 F4) s~ =-~-~T, +~

(12)

(13)

References

- fn ] 1 fn ] ~ ' (14)

(15)

(16)

+ , (17)

(18)

(19)

[ 1 ] Jayaraman H B, Knudson W C. A curved element for the analysis of cable structures[J]. Comput and Struct, 1981,3(4) :325 ~ 333

[2] Irvine H M. Cable Structures[M].Cambridge,Mass: MIT Press, 1981

Abstract

The one-dimensional problem of the motion of a rigid flying plate under explosive attack has an analytic solution only when the polytropic index of detonation products equals to three. In general, a numerical analysis is required. In this paper, however, by utilizing the "weak" shock behavior of the reflection shock in the explosive products, and applying the small parameter pur- terbation method, an analytic, first-order approximate solution is obtained for the problem of flying plate driven by various high explosives with polytropic indices other than but nearly equal to three. Final velocities of flying plate obtained agree very well with numerical results by computers. Thus an analytic formula with two parameters of high explosive (i.e. detonation velocity and polytropic index) for estimation of the velocity of flying plate is established.

1. Introduction

Explosive driven flying-plate technique ffmds its important use in the study of behavior of materials under intense impulsive loading, shock synthesis of diamonds, and explosive welding and cladding of metals. The method of estimation of flyor velocity and the way of raising it are questions of common interest.

Under the assumptions of one-dimensional plane detonation and rigid flying plate, the normal approach of solving the problem of motion of flyor is to solve the following system of equations governing the flow field of detonation products behind the flyor (Fig. I):

ap +u_~_xp + au --ff =o,

au au 1 y =0,

aS as a--T =o,

p =p(p, s),

(i.0

293

where p, p, S, u are pressure, density, specific entropy and particle velocity of detonation products respectively, with the trajectory R of reflected shock of detonation wave D as a boundary and the trajectory F of flyor as another boundary. Both are unknown; the position of R and the state para- meters on it are governed by the flow field I of central rarefaction wave behind the detonation wave D and by initial stage of motion of flyor also; the position of F and the state parameters of products