NUMERICAL-ANALYTICAL METHOD OF CALCULATING

OF THE SHALLOW SHELL RECTANGULAR IN PLANE

REINFORCED BY RIBS OF VARIABLE STIFFNESS

D.P. Goloskokov

Admiral Makarov State University of Maritime and Inland Shipping,

Dvinskaya St. 5/7, St. Petersburg, 198035, Russia

e-mail: dpg1954@mail.ru

Abstract. The analytical solution of a boundary value problem for a differential equation of

deforming shallow shell rectangular in plane reinforced by ribs of variable stiffness is

received. The decision is presented in the form of a series from combinations of regular and

special discontinuous functions. This decision leads to quickly converging series and simple

computing algorithm. The example of calculation of a reinforced cylindrical panel is given.

1. Introduction

The main method of calculation of the stress and deformed status (SDS) of composite

constructions, for example, of the shells reinforced by stiffness ribs is the finite-element

method (FEM). With development of ADP equipment and creation of systems of analytical

computation, for example Maple, Mathematica, etc., there is an opportunity of the analysis

enough complex challenges of mathematical physics by analytical methods [1-4]. In this

article the mathematical model based on analytical solutions of boundary value problems of

the theory of ribbed shallow shells is offered.

2. Mathematical model

The shallow shell having a rectangular form in the plan is considered: 0 x a , 0 y b .

The shell is reinforced by stiffness ribs parallel to x axes and y on lines y = yi (i = 1,2,...,K1), x

= xj (j = 1,2,...,K2). The accounting of ribs is carried out with the help δ-function Dirac and its

derivatives to the second order. It is considered that ribs have variable stiffness.

Determination of the stress status and deformed status for a ribbed shells is consolidated to

solution of a boundary value problem for the equation of the fourth order to complex function 2( , ) w( , ) nF( , ) , n 12(1 ) , 1x y x y i x y Eh h i , where w ,x y — function of

normal relocation of points of a median surface of a shell (deflection), F ,x y — function of

efforts, E — Young’s module, h — shell's thickness, ν — Poisson coefficient [5].

1 2 24 2 1 1

12 21

1n ( ) ( )

n n

K

n k kk k i k

k

q f i g ii y y g y y

D D x x

22 2

2 2

22 21

1( ) ( )

n n

Kj j

j j j

j

f gi ix x g x x

D y y

. (1)

where

Materials Physics and Mechanics 26 (2016) 66-69 Received: September 28, 2015

© 2016, Institute of Problems of Mechanical Engineering

2 2 2

1 11 1 12 2 2

F F w, i i

i i i

E Sf x y E J

Eh y x x

,

2 2 22 2

2 2 22 2 2

F F w,

j j

j j j

E Sf x y E J

Eh x y y

,

2 2 2

1 11 1 12 2 2

F F w, i i

i i i

E Fg x y E S

Eh y x x

,

2 2 22 2

2 2 22 2 2

F F w,

j j

j j j

E Fg x y E S

Eh x y y

,

1 1 1 1 1, , , , 1,2,...,i i i iE F S J i K — Young’s modules (elastic modulus), the areas of transverse

sections, static and axial inertia moments of transverse sections of the ribs located along x-

axis; 2 2 2 2 2, , , , 1,2,...,j j j jE F S J j K — similarly for the ribs located along y-axis.

We will consider formally the following series

1 1

w , w sin w sinmy m nx n

m n

x y y x x y

, 1 1

F , F sin F sinmy m nx n

m n

x y y x x y

,

1, 1,

1, 1,

1 1

, sin , , sini i

i my m i my m

m m

f x y f y x g x y g y x

,

2, 2,

2, 2,

1 1

, sin , , sinj j

j nx n j nx n

n n

f x y f x y g x y g x y

.

The common decision of the equation (1) can be presented in the form

4

1 1

( , ) ( , ) sin( ) ( )mq m mk k

m k

x y x y x A Z y

1

2 1 1 1

1

1( ) ( ) ( ) ( ) ( )

n nm m

Kk k k

m my k my k k my k k

k

i if y g y y g y y

D

4

1 1

sin( ) ( )nn nk k

m k

y B Z x

2

2 2 2 2

1

1( ) ( ) ( ) ( ) ( )

n nn n

Kj j j

n nx j nx j j nx j j

j

i if x g x x g x x

D

. (2)

Here q(x,y) –– any private solution of the equation 4 2n nk

qi

D . This decision can

be taken in the form of the decision of Navier of a double trigonometric series.

Functions ( ), ( )m mi iy y

–– private decision and its second derivative of the equation

4 2nm m m mi k i ii y y

,

22 2

4 2 2 2

1 22 2,

m mm k m

d dk k

dy dy

. (3)

Functions mkZ y , k=1, 2, 3, 4 –– fundamental solutions of a homogeneous equation (3).

Functions ( ), ( )n nj jx x

, nkZ x turn out from functions ( ), ( )

m mi iy y

,

mkZ y by change y x, m n, i j, k1 k2.

The common decision (2) contains unknown constants: Amk, Bnk — arbitrary constants

of integration; 1, 1, 2, 2,, , , i i j j

my i my i nx j nx jf y g y f x g x — values of singular functionalities. We

67Numerical-analytical method of calculating of the shallow shell rectangular in plane...

apply standard procedure to determination of these constants — we calculate Fourier

coefficients of the received common decision and we use conditions of a continuity of

functions and boundary conditions of the task.

The deflection of a shell w( , )x y and function of efforts F( , )x y are determinate through

the real part Re ( , )x y and imaginary part Im ( , )x y of function ( , )x y ; then all

elements of the stress status and deformed status of a ribbed shell — efforts, moments and

stress are determined by known formulas.

3. Numerical example

Let's considered as a numerical example the cylindrical panel with sizes in plane of a = 8

meter, b = 6 meter, radius of shell R = 22,6 meter, and arrow of rise f = 0,2 meter. Elastic

modulus of material of a shell and stiffness ribs of E = 2×105 MPa, Poisson coefficient of

ν = 0,3.

The panel is supported with three ribs located along the lines x1 = 3 meters, x2 = 5

meter, x3 = 6,5 meters (variable stiffness) and two ribs located along the lines y1 = 2 meters, y2

= 4 meters (constant stiffness). Sizes of ribs (constant stiffness): tauris — width of a shelf of

0,35pb meter, thickness of a shelf of 0,03pt meter, wall height of 0,6sh meter, wall

thickness of 0,02st meter. Sizes of ribs (variable stiffness): rectangular section, variable on

height (geometrical characteristics were approximated by polynomials — an inertia moment a

polynomial of the 6th level, the static moment of the area — a polynomial of the 4th level, the

area of section — a polynomial of the 2nd level).

Shell thickness h = 0,012 meters. The shell is affected by a transverse load — hydrostatic

pressure of q(x, y) = ρgx (ρg 10000 N/m3, a total head of 8 meters) and is the free support

on a contour.

The results of calculation in the form of diagrams of deflections and stress in a shell are

given on Fig. 1 and Fig. 2. All dimensionalities are specified in system of SI (meters,

Newtons, Pascals). In series about 15 members (M = N = 15) on each variable were retained.

Fig. 1. Deflection of shell on lines y = 3 мeter and x = 6 мeter.

Fig. 2. Flexural stress for z = h/2: x (on lines y = 3 мeter) and y (on lines x = 4 мeter).

68 D.P. Goloskokov

4. Outputs

Implementation of many analytical methods on digital computers results in computing

instability of the majority of them that is connected to accumulation of the rounding-off errors

arising by operation on a set of the real numbers with limited number of significant figures in

a mantissa. Calculations for this operation were executed in system of analytical computation

of Maple [6] where this problem doesn't exist. The system to 600 algebraic equations decided;

in a mantissa of number about 45 signs were retained. Advantage of the offered approach —

is received the analytical solution of the task.

References

[1] D.P. Goloskokov, In: 2014 International conference on computer technologies in physical

and engineering applications (ICCTPEA), ed. by E.I. Veremey (IEEE, IEEE Catalog

number CFP14BDA-USB, 2014), p. 55.

[2] D.P. Goloskokov, In: IV-th International simulation seminar in systems of computer

mathematics KAZCAS-2014. International school on mathematical simulation in systems

of computer mathematics KAZCAS-SCHOOL-2014, ed. by Y.G. Ignatiev (Publishing

House of Kazan. University Press, Kazan, 2014), p. 33.

[3] D.P. Goloskokov // The GUMRF Bulletin of the Admiral S.O. Makarov 4 (2012) 22.

[4] D.P. Goloskokov, V.A. Daniluk // The GUMRF Bulletin of the Admiral S.O. Makarov 1

(2013) 8.

[5] D.P. Goloskokov, Numerical and analytical methods for calculating the elastic thin-

walled constructions irregular structure (Publishing House A. Kardakov, St. Petersburg,

2006). (In Russian).

[6] D.P. Goloskokov, Practical Course of mathematical physics in the Maple: manual for

higher education institutions (OOO «ParkKom», St. Petersburg, 2010). (In Russian).

69Numerical-analytical method of calculating of the shallow shell rectangular in plane...