Integrated thermal-structure finite strip analysis

Integrated thermal-structure finite strip analysis Z. M A l q K O

Institute of C'TvilEngineering, Wroclaw Technical University, Iqroclaw, Poland

INTRODUCTION In order to calculate the temperature induced internal structural forces, one must have the knowledge of the thermal conduction, for which it is necessary to determine the temperature distribution in different points of the analysed structure. The thermal forces are calculated then on the basis of the plotted temperature distribution profiles. The available analytical methods, 1'2 however, usually limit themselves to very simple geometries and boundary conditions. The problem of heat conduction and distribution occurs, for instance, when several elements are welded together and the welds are executed at a specified speed as linear or point ones, as well as in the analyses of plates, shields, bridge box-sections, etc. which are exposed to temperature or a source of heat. Due to the complexity of these structures, various numerical methods such as the finite difference method and the finite element method have been found to play a significant role in both thermal and structural analyses. With the wide acceptance of the ffmite element method in structures a and its rapid growth in thermal analysis, it has been found to be particularly well suited for such analyses. At present, several thermalstructural programs exist for finite element thermal and structural analyses. Yet often a lack of compatibility between the thermal and structural analyses exists because the finite element thermal model may require a different discretisations than the Finite element structural model. Furthermore, improvement of t'mite element thermal methods is needed to reduce model size and high costs for analysis of complex structures. Dechaumphai et al.4 presented an approach called integrated thermal-structural analysis which improves the capability and efficiency of the finite element method. The unique feature of the approach is the use of nodeless variable interpolation functions for element temperatures. Generally, in the finite element method, convergence is achieved through increasing discretisation, but increasing discretisation leads to increasing computer costs. Therefore, a search for alternative, more economical methods for this kind of analyses becomes advisable. The finite strip method (FSM), s-a which is particularly suitable for thermal conduction analysis, the plotting of temperature propagation diagrams and the calculation of thermal forces is worthy of notice here. Its advantages become clearly visible in the case of orthogonal structures with identical boundary conditions at their ends 9 and when one takes into account the fact that it can be applied to the analysis of triangular plates. One must say, however, that in some cases, the advantages of the method become slightly less distant. The aim of the present work is to widen the

FSM so that it would cover mixed boundary conditions and more complicated structures. GENERAL OUTLINE OF THERMAL PROCEDURE The goals of the integrated approach to thermal-structural analysis are to: (I) provide thermal strips which accurately predict detailed temperature variations; (2) maintain the same discretisation for both thermal and structural models, and (3) provide accurate thermal loads for the structural analysis in order to improve the accuracy of displacements and stresses. In two-dimensional thermal analysis, element temperature can be written as

T(x,y, t)= [NT(X,y)] {T(t))

where [NT] denotes the strip temperature interpolation functions, and {T) denotes a vector of nodal line temperatures. 6,7 For a rectangular structure with two of its opposite ends held either insulated or maintained at a constant temperature, the plate may be imagined to be separated into strips, each having a defined constant thickness. The temperature function T may then be expressed as a finite Fourier series consisting of the product of a shape function and trigonometric series of the form r

X

in which i and/refer to the two sides of the strip, b = width of the strip (Fig. 1) and Tim and Tlm= generalised temperatures along the strip lines i and], respectively. The value of the basic function Ym defines a trigonometric series and is given by Yra = sinaraY/l for ends held at constant temperature, and Ym = c°S°tm2y/l for ends held insulated. Finally, r = positive integer representing the limit of summation. For such a temperature function, the temperature along boundary line x = 0 or x = b is given by

These two edges kept y/v Heat Input e~ther at constant ,,/__ temperature or J -[ - - / / j -,nsutoted / 0 , /"

/

/

/

0141-1195/86/030159--07 $2.00 © 1986 Computational Mechanics Publications

----7

F~ctdtous

Heat Inpot_ _ -]J-1 4

+3Q

,Oo_L_

I I

One S~de I Insulated I Other SLde I Kept at Con- I stont Tempe-I roture I h

I

3'

a) Typical Strip Accepted September 1985. Discussion closes September 1986.

(1)

Figure 1.

b) Division of Prate

Finite strip model

Adv. Eng. Software, 1986, Vol. 8, No. 3

159

r

(3)

T= Y rtmYm m=O

Such a representation of temperature now automatically describes the boundary conditions and reduces the twodimensional problem to a one-dimensional one.

Generation of element matrices Using the method of variational calculus, it can be shown 9'10 that the final matrix equation to be solved is given by

[Khl](r} + [C]

[[Kc] +

(i/') = (RQ} +

(Rq} + (Rh) (4)

in which [K] is the total heat potential matrix given by the assembly of each individual strip element matrix, where [Kc] and [K~] are element conductance matrices corresponding to conduction and convection, respectively, [C] is the specific heat capacity matrix given by the assembly of each individual strip element matrix. The above matrices are expressed in the form of integrals over volume V and surface A of a strip as follows: [C] =

fo e v (NT} [NT] dV

(5)

V

[ o1 =f [BT]T[k] [BT] dV

(6)

V

[K~] = f h "INr } [NT] dA

(7)

A

where [BT] denotes the temperature gradient interpolation matrix and [k] denotes the material conductivity matrix. In the above equations, p is the density, ev is the specific heat capacity and h is the convection coefficient. Finally, (J'} represents differentiation of temperature with respect to time. The right-hand side of equation (4) contains heat load vectors due to internal heat generation, specified heating and surface convection, etc. Generally, this side represents the combined heat input column matrix due to heat generation within the body and the heat input at the surfaces. These vectors are defined by

{Re}

=f Q {NT} dV

(8)

V

(Rq} = f

q(NT} dA

(9)

A

(Rn) = f hr {NT} dA

r

[(~ + m2rr2b)(

l+m2zr2b)-]

~/z~--b 24l-// [K] :m~__lkl~l+ 2 : l b ) ( i +m~lb) j

[CI = ~

cvPlb

(ll)

(12)

m=l

In Case 2, the corresponding elements of matrices [K] and [C] assume an almost identical form as in Case 1, with the difference that instead of l, one should substitute 0.5l, which leads to a change of some coefficients, and sum for m = 0. 7,9

Boundary conditions The boundary conditions in a simplified finite strip program may be classified under two categories as: 1. Inherent boundary conditions parallel to the generalised x-axis and already incorporated in equation (1) for the generalised strip element, namely, insulated or constant temperature condition. 2. External boundary conditions parallel to the generalised y-axis of each strip element, four types of these conditions are possible along these edges in engineering analysis. For transient analyses, the unknown temperatures are solved by numerical integration. This is a general method of solution and may be used for almost all types of boundary conditions. An explicit numerical integration procedure was adopted in the analysis of demonstration problems presented in this work. The third type is the boundary subjected to linear convective heat transfer and the fourth, heat input specified at the boundary or due to internal heat generation, which will be discussed below.

Convective boundary conditions. The general equation for convectional heat transfer over an area A is given by qini = a(T-- Tot) which qi = the heat

(13)

lot = f 2 (T--Tot)2 dA

(14)

in flux vector, n i = the outward unit normal on the surface, a = the surface heat transfer coefficient, and Tot = the ambient temperature. The variational integral for convective heat transfer is given by Visser: 1°

A

(10)

A

where Q is the internal heat generation rate per unit volume, q is the surface heating rate per unit area, and T is the convective medium temperature. The heat conduction problem is, therefore, reduced to a first-order differential equation which may now be solved numerically for various boundary conditions after evaluating of the above matrices. Two cases are considered to illustrate the validity of the procedure: a rectangular structure with two opposite ends held at a constant temperature (Case 1) and a rectangular

160 Adv. Eng. Software, 1986, Vol. 8, No. 3

structure with two opposite ends held insulated (Case 2). For Case 1 with boundary conditions x = 0 or x = b and for y = 0 or y = l, the initial conditions are T = 0 and the element matrices [K] and [C] are then given by

The input in the finite strip program having an element strip equation given by equation (2), Tot, may be expressed as a half-sine series or half-cosine series such that Tot =

Z

(15)

ZotmYm

re=O,1

Then, it may be shown that 1

Iot = Z ? (T, mTotm) m=0 n=0

--1

l J [ Totn

Ymrn da 0

(16)

3000

-Heat Input 669 kcat/m in 30sec.

T

[*K] 2000

1T 310

D',. oD

t.~

for 30 s ~ c f

A 11 I l l l l l

I

-

Illll

~IIIIIIB

.

1000- e * 900 800700600-

.

(',,I : o

C

o

•

i

..

400-

..ooeOO-**..o...o~

*I

300. .*'0 * $ * S • "

Q)

"

~

B ~-

i

T

[OK] 310

I

I

b

I

I

0

I

I

0,25 0,50 0,75 y/t 1,0 Distance orang axis y , x = b / 2

o

- Rnite strip m=7 Anotyticet series m=l •

m=l ~..* Point C / 2 ' o . * * 256 o **

4

* Anatyticot series m=7 •

o

Y,

o

]

Point B Point A

,,

o

310,92

290

//'

* Finite strip-21 nodal lines * Finite element-*63node points , o.** ** o

* Finite strip m =1

.

//

--~-~"

Temperature profite (at the ends for the voriabte wide prate time =184 seconds

Time in hours

330

./'~

-

Finite strip m=l

i

/

*'*'Point C ~

0.01 0,02 0,03 0,04 0.05

0

/

I-x /

/

2s4~ /

. Finite strip m=7 * Finite etement-82nodes

200

r-31j -:3

/ ~:o.8~ ,/

b = 1,0[

Point A *

0

I '

b=o,6~

Ol..

~0

OI'-L -u

152,4

%

500-

uJ

Va riabl.e

----4111111

I

c3

-29o I~ ~ !

[I --&llllll

-4"

E ,41" I.O

Ill[ill 0o

q Heatlnput / 4 -~ I " - / 6 6 9 k c a t / m / I

o

310.92

** °O CO

o

t.r3

270

X

255,37

c)

I,

*"

..~,

d)

I

I

I

I

o

o

O

O n

Selected distribution

of temperatures for

The integral p. YmYn dA consists of trigonometric orthogonal functions and its value is zero for all values of m not equal to n and has a defmite value for all values o f m equal to n . 2"-4'7'8

Heat input boundary or internal heat generation. In cases where the analysis requires the consideration of a specific

L 255,3?

255 ,

~ Illllll

I

I

0 0,2 0,4 0,6 0,8 y/l 1,0 0,001 0,01 Distance along cent ert i ne- a xis y lqme in

Figure 2.

~A ~

(3

O 0

o x exo x • xO~ • x

250

,~B~"

** ~"

lr

Itttll[

I

t I IllllL

x

[

I

111111

0,1 1,0 10,0 hours (tog score)

analysed cases of rectangular plate heat input along a boundary line or a specified heat generation within the body, the heat input q can be resolved into basic function series in the Y direction similar to the inherent temperature function such that r

q= ~

qmYm

(17)

m=O

Adv. Eng. Software, 1986, Vol. 8, No. 3

161

in which o2

f qm--

qYmdy

¢1

1

(18)

A free-supported square plate temperature loaded at fi)ur of its edges

f y2dy o

for distributed heat input from y -- cl to y = c2, and

qrn-

QoYm(c)

(19)

1

f y2 dy o

for a point heat source Qo at y = c (Fig. 1). APPLICATIONS To demonstrate the capabilities and efficiency of the finite strips in thermal-structural analyses the following examples are analysed.

A rectangularplate with heat input The problem is first formulated for a plate 254 mm long and 152.4 mm wide. Three sides of the plate are kept at 255.37 K while a heat input of magnitude of 688.98 kcal/m occurs during the first 30 s. The heat input is then taken off and the system is allowed to cool at an ambient temperature of 310.82 K. That was done to check the correctness of the algorithm and then the results were compared with the Chakrabarti solution. 9 The plate was divided into 20 strips and 1 and 4 harmonics were assumed. Figure 2a shows the temperature history obtained at three points along the centre line of the plate during the first 180 s. The preceding problem can now be extended by varying the plate widths and assuming the ends of the plates as being insulated and the temperature rise at the insulated faces can be calculated. The width at which the temperature rise at the boundary is reasonably small may then be used as the design width and the temperatures evaluated using that width will represent the true temperature distribution around the heated zone. In the demonstration problem considered herein, the results of analyses assuming 152.4 mm (model I), 203.2 m (model II) and 2 5 4 m m (model III) widths are shown in Fig. 2b. Therefore, a width slightly greater than in model III should be used to compute the temperature distribution around the heated zone, since the temperature of the edges is already relatively low. The advantage in the finite strip problem in such a case is that changing of the widths does not require any additional computer time because only the specification for width needs to be changed. Moreover, the accuracy of the finite strip program was analysed using as an example the plate shown in Fig. 2a which had three of its edges kept at a temperature of 255.37 K and the assigned temperature of 310.92 K. The steady-state temperature profile along the centre line of the plate along with its finite strip model and the comparison of results with the standard analytical series solution are shown in Fig. 2c. The transient analysis solution capability of the finite strip method is demonstrated in Fig. 2d which shows the transient temperature distribution versus time at points A, B, C. The statement of

162

Adv. Eng. Software, 1986, Vol. 8, No. 3

the problem is almost the same as that given above, except that the third side suddenly acquires the temperature of 310.92 K. The accuracy of the FSM considering only one term of the series shows a convergence level superior to that of the fine mesh FEM.

The algorithm given by Chakrabarti 9 was extended so that it would cover other support conditions using the functions given by Cheung 6 and the examples of plate shown in Fig. 3a,b. The difference in temperature between the top and the bottom face was AT = 20 K, the temperature was changing linearly in the two directions along the plate thickness. The calculations were made for two divisions of the plate into strips differing only in the strip width, each time assuming 10 strips in the direction of axis y. In the first division, all the strips had identical width and in the second division the extreme strips were narrower. Moreover, a similar plate was analysed but its thickness was changed starting from the half of its width (Fig. 3b). The obtained results are presented in Table 1 and some of them have been compared with the results of Maulbetsch 2 and Szilaqyi. 1~ Table 1 shows that a relatively good convergence of results depending on the number of harmonics m and a good agreement with the other solutions have been obtained. It is worth noticing that a better convergence of results at a smaller number of harmonics than in the case of Szilaqyi n (m = 40-60) has been obtained, although a somewhat great number of them is required in this type of analyses than in the case of external loads. 6-8 Moreover, the solution with narrower extreme strips is more accurate at identical number of harmonics, e.g. m = 40. In the case

aT=20 ° K Cl)

~-

T1 PLATE 1 L ] -~ - 4 - - -~ - - e - -e - -¢ - --q - --e - -4,~-_~=0,30

Dtv,s,on

(~)

"r2

(~)

PLATE 2

b) + + + + ÷ Ii-Number

C)

~ 1

o 3

2

,y

o 4

o 5

(~-Number

il

-+ ÷

of nodQi.

o 6

Lines

o 7

o 8

of s t r o p s -

- 11

o 9

o~ 1011

9=0,167 C£t= 0, 000012

10 D = 6 4 M m = E ( 3 / 1 2 ( 1 2 - ~ 2)

®®

(!) @ ® ® @

C

@

g

B,D4

%

® m

e 91

O O ,'

th c) c) u5

"6 o

g

ff

b'3

d) ~ 1 , 0 0 ,, L__

~,_

4 "1,93662

_ ~,225352

10,00 m

Figure 3. Discretisation of two types of plate

Table 1.

A comparison o f the obtained remits. Units w in (1 O-" ra) mx, my or mxy in (I0-" MN)

Point A

Point C

Point B

Pomt D

Terms m

w

rnx

my

w

rnx

my

mx

rnxy

1 5 11 19 1 5 11 19

1.6007 1.6181 1.6243 1.6247 1.6006 1.6182 1.6242 1.6247

2.9351 2.7325 2.5207 2.4900 2.9320 2.7304 2.5199 2.4901

2.9535 2.8007 2.5008 2.3805 2.9482 2.7012 2.4989 2.4782

0.1125 0.1188 0.1192 0.1193 0.1126 0.1189 0.1193 0.1193

5.3281 5.0232 4.8354 4.7862 5.3234 4.9826 4.8013 4.7879

0.2007 0.1802 0.1531 0.1401 0.1934 0.1732 0.1692 0.1637

5.5828 5.3375 5.1492 4.9983 5.5329 5.3170 5.1283 4.9935

3.5218 3.4273 3.3312 3.2602 3.5175 3.4113 3.3195 3.2585

5.8525 5.3311 5.1438 5.1024 5.5326 5.3127 5.1235 5.0034

2.4925 2.4925 2.4940 2.4940 2.4889

2.3527 2.4316 2.4409 2.4673 2.4889

0.1180 0.1445 0.1870

4.6810 4.8240 4.8354 4.8796 4.9778

3.1151 3.1316 3.1334 3.1360 3.2300

4.8003 4.8855 4.8939 4.9200 4.9778

2.7737 2.5681 2.3582 2.3216 2.7709 2.5618 2.3498 2.3218

2.7943 2.5708 2.4209 2.3014 2.7903 2.5613 2.4089 2.3018

0.1842 0.1707 0.1609 0.1548

5.4123 5.1417 4.9739 4.8237 5.4039 5.1327 4.9684 4.8241

3.3578 3.2527 3.1639 3.0925 3.3428 3.2412 3.1583 3.0952

5.4182 5.1635 5.0027 4.8313 5.4098 5.1516 4.9582 4.8315

Plate 1 I division

II division

20 40 40 60

Ref. 9 I division II division Ref. 6 Plate 2 I division

1 5 11 19 1 5 11 19

II division

1.4237 1.4374 1.4408 1.4408 1.4239 1.4389 1.4408 1.4409

4.7848 4.7889 4.7908 0.0892 0.0908 0.0923 0.0923 0.0892 0.0908 0.0924 0.0924

of identical numbers of harmonics, slightly better results have been obtained only for some points of the plate, e.g. m x in points A and B as opposed to m y in point A or m x y in point C. Moreover, for the same plate of constant (plate 1) and variable thickness (plate 2) cases considered previously by Chakrabarti 9 were analysed. Case 1 was where

5.1537 4.8132 4.6305 4.5999

mx

the square plate has three of its edges kept at a temperature of 273.15 K and has an assigned temperature of 373.15 K. The steady-state temperature profiles along the centre line of the plates are shown in Fig. 4a. Calculations were made for 1 and 5 harmonics at two different divisions of the plate into 10 strips. A relatively good agreement and convergence

370

-.q 1

T

[°C

I

.

* Finite strip m=l

2 IY 373,15 i

350

330

• Fimte str~p m:5

LO .¢--

t.C)

r4 t'--

rfi t'~

C,4,

, ,'x,i

Point B' Point A ~

o

•

o

. ~, •

o

i

X

o

273,15 310

"-

Point B

o Finite strip m=l Ftntte strip m=5

?,

o o

293,15 i

290 o

•

~

°" ~t

Prate 1 Prate 2

270 9

I

@

g

First dlwslon

g

o •

3

Second division

t L I J 250 0 0,2 0,4 0,6 0,8 (a) DISTANCE ALONG CENTERLINE- AXIS Y

Y/I

1,0

0,01 TIME IN HOURS

Q1 (LOG SCALE)

1,0 (b)

Figure 4

Adv. Eng. Software, 1986, Vol. 8, No. 3

163

©@

@@

1 2

11

T2=373,15*K

@® lS

15 17 I

1 m°J

I I 12

3

14

16

5

®

125

@

~

r 2500x12 I

1_

-z
i

AT=80°K \P°int 9

Iii F--

t =20,00rn (~

/ /

t.lJ 330 Q.

250×30

,~o

//

25o 22.5,

3 × 250

T1=293,15"K ®®

P o i j 1_66

LIJ

1500×14

7

8 910

m=lg

Pr" 350 -I

®

370

~ Finite strip method

,o, I

200×16 6

I

o

310 strips

-

16

- Number

of

- Number

of nodal hnes -17

alDISCRETISATIONOFTHESPAN

290' 0

I

I

i

1

I

I

2

3

i

]

i

/.

TIME IN HOURS

b)

Figure 5

of results have been obtained for both models. The statement of Case 2 is almost the same as that of Case 1 except that the fourth edge suddenly acquires the temperature of 373.15 K, and the remaining edges have the temperature of 293.15 K. The transient temperature distributions versus time at the points A, B and B' are shown in Fig. 4b. The results are compared with respect to the results of a 1 and 5 terms analysis.

plates are cooled by surface convection to a medium temperature T = 293 K. Figure 6c shows the comparative temperature distributions at y = 1/2 for the Finite strip method for both models of the plates for each rib.

Case 2. The steel orthotropic deck plates with constant cross-sectional areas encased between two supports. Temperature at both ends are specified, and at time t = 0 the internal heat generation rate increases abruptly. Compara-

A two-beam bridge span loaded temperature A steel span of a railway bridge with two main girders was subjected from above to heating from the ambient temperature of T1 = 293.15 K up to the temperature of 7"2 = 373.15 K. Figure 5b shows diagrams of temperature distribution for two selected points at the half of the length depending on the time of heating. The calculations were made for the calculation model presented in Fig. 5a assuming 16 strips and 19 harmonics of expansion into Fourier series for a half of the span.

A steel orthotropic deck plate with surface convection and with internal heat generation Two types of deck plates, i.e. a plate with open ribs (Model I) and a plate with closed ribs (Model II) are considered. Identical discretisation is used for the two plates, i.e. the same as that adopted in the statical analysis. 7'8 In Model I, 58 strips and 59 nodal lines are distinguished and 75 strips and 67 nodal lines are distinguished in Model II. In all the calculations, 19 harmonics are adopted each time. Two cases of thermal analysis, i.e. steel orthotropic plates with surface convection (Case 1 - Fig. 6a) and with internal heat generation (Case 2 - Fig. 6b) are considered.

Case 1. The steel orthotropic plates deck (Fig. 6a,b) have a specified temperature To along the boundaries. The

164

Adv. Eng. Software, 1986, Vol. 8, No. 3

G:58

1=59 I

I

I

I

tt[! I

I

I

I

I

I

T~ = 293OK I

I

I

l

2 3 /. b 6 7 8 g- Number of rlbs

a)

b) C

[1 @:

1=67

X_/ 1

75

k_/ 3

2

1,0 T X 7~ \"\\ \

O5-

Q }~

'

k._/ 5

4.

~ , .......

\

200 kW/m 3 I

k_/

I

k_/

1 xJ

i Q xJ _L d)

Case 2

=10hr 4

t60c /

\&

1

kJ

6 - N u m b e r of r~bs

closed r~bs t T open ribs |[°K

Case 1

1000 kW/m 3 't 1

M./

I LT °

-

-

! =O.qhr , ~ / ~ / ~

,

~ 3 o o

""~'4

0~

i

!~'~ 273,15

~ ~'

,200

t

~-----~

I

I

°-cl°sednbsl -open rlbs

~

: I

[

0.125 [225 0,375xlb 0,5 0 0.25 0,5 0.75 yll 1,0 Cam ~mtlve tempeoa~'ure d~stnbuhons Compamtwe fempem~rure dlstrlbuhons alone y=1t2

Figure 6.

Steel orthotropic deck plate

tive temperature distributions at time t = 0, 0.1 and 1.0 h are shown in Fig. 6d.

CONCLUSIONS The f'mite strip approach for improved thermal-structural analyses is presented. The finite strip method permits a standard computer program to solve a wide range of temperature problems. By use of this technique a two-dimensional problem is reduced to a pseudo-one-dimensional analysis with an accuracy which is believed to be greater than the accuracy obtainable from a comparable finite element analysis. Temperatures and stresses predicted by the finite strips are compared with refined conventional f'mite element models and with the nodeless variable elements and/or analytical solutions. Results demonstrate that the finite strips increase the accuracy and efficiency of the finite element thermal-structural analysis. The success of the finite strips in thermal-stress analysis suggests that the approach offers potential for more effective solutions to other coupled problems requiring different models. The relatively low calculation costs are also worthy of notice.

REFERENCES 1 Boley, B. A. and Weiner, J. H. Theory of Thermal Stresses, John Wiley & Sons Ltd, New York, 1960 2 Maulhetsch, F. L. Thermal stresses in plate, J. Appl. Mech. 1935, 2 (1), 141 3 Zienkiewicz, O. C. The Finite Element Method, 3rd Edn, McGraw-HiU, London, 1977 4 Dechaumphai, P. and Thornton, E. A. Nodeless variable finite elements for improved thermal-structural analysis, Proc. Int. Conf. on Finite Element Methods, Vol. I, Science Press, Beijing, pp. 139-144, 1982 5 Cheung, Y. K. Finite strip method analysis of elastic slabs, J. Eng. Mech. Div., ASCE 1968, 94 (6), 1365 6 Cheung, Y. K. Finite Strip Method in Structural Mechanics, Pergamon Press, London, Oxford, 1976 7 Mafiko, Z. Statical analysis of chosen steel bridge spans, Reports of the Cir. Eng. Inst., Wroclaw Technical University, Poland, Vol. 5, p. 242, 1975 8 Mafiko, Z. Statische Analyse yon Stahffahrbahnplatten, Der Stahlbau 1979,48 (6), 176 9 Chakrabarti, S. Heat conduction in plates by finite strip method,J. Eng. Mech. Div., ASCE 1980, 106 (2), 233 10 Visser, W. The Finite Element Method in Deformation and Heat Conduction Problems, Proefschnfft, Delft University of Technology, The Netherlands, 1968 11 Szilagyi,G. Quelques applications de la m~thode des bandes, IABSEPublications 1974, 34 (II), 149

Adv. Eng. Software, 1986, Vol. 8, No. 3

165