The application of line-solution techniques to the solution of plane-stress problems

h d . J . Mech. Sci. l ' e r g a m o n Press Ltd. 1964. VoL 6, ])P. 391-399. P r i n t e d in Great Britain

THE APPLICATION OF LINE-SOLUTION TECHNIQUES TO THE SOLUTION OF PLANE-STRESS PROBLEMS S. hi. A. KAZIMr and A. COULL Department of Civil Engineering, University of Southampton (Receiced 13 December 1963)

Summary--The paper presents the solution of a typical plane-stress problenl with mixed boundary conditions by tile use of the line-solution technique, in which the partial differential equation of plate theory is rendered uni-direetional by replacing derivatives in one direction by their finite difference equivalents, producing a set of ordinary linear equations which may be solved by means of the matrix progression nmthod. A solution has been obtained for the particular case of a rectangular shear wall, rigidly clamped along the base, and free along the other two edges, carrying a uniformly distributed load along one vertical edge. Reasonable agreement was reached with the stresses obtained from a photoelastic investigation. NOTATION

0(x, y) orthogonal co-ordinate system ~,~: non-dimensional co-ordinates L width of plate H height of plato h strip width r a

L/H hlH

Yolmg's modulus for plate material Poisson's ratio for plate material direct and shear stresses ~','0 displacements in x- and y-direction r non-dimensional displacements P applied load intensity P density of plate material .F wanted ftmctions A, B, G, L, K matrices nsed in analysis E

1. I N T R O D U C T I O N FOR plane-stress problems subject to mixed b o u n d a r y conditions, closed m a t h e m a t i c a l solutions to the governing partial differential e q u a t i o n are usually impossible to achieve. One m e t h o d of resolving the difficulty is b y the use of "line-solution" techniques, in which the governing e q u a t i o n is rendered uni-directional b y replacing the partial derivatives in one direction b y t h e i r finite difference equivalents; the problem is t h e r e b y reduced to the solution o f a set o f o r d i n a r y linear differential equations. A l t h o u g h this approach was originally suggested b y H a r t r e e a n d W o m e r s l e y 1 in 1937 as a more accurate m e t h o d of solution t h a n b y the sole use o f finite difference networks, it does n o t a p p e a r to h a v e been utilized b y engineers, due p r e s u m a b l y to the difficulty of solving the resulting set o f simultaneous differential equations. The a d v e n t o f the high-speed electronic 391

392

S. 5I. A. KAzL~tI and A. CoU[.L

digital computer has largely obviated this difficulty, and, with the aid of such elegant techniques as Tottenham's matrix progression method, ~ the solution of such sets of equations can be tackled successfully. As far as is known, the method has previously been used only for certain forms of shell structures 3.4 with comparatively simple boundary conditions; the present paper describes its use in the solution of the plane-stress problem of the laterally loaded shear wall (or, essentially the same problem, the deep cantilever beam). 2. T H E

LINE-SOLUTION

AND MATRIX-PROGRESSION

METHOD

The governing partial differential equation for any two-dimensional surface structure has the general form % ~--::_. + ~ + . . . + c , , - ~ - = .

= p

(1)

where 2'(x, y) is the wanted flmction, p(x, y) is the applied load, and coefficients c, a r e generally constants, depending on the form and material properties of the structure. Equation (1) reduces to the biharmonic form for plate structures, and to an eighth-order equation for shell structures.

line /'}'I - I

k~'l k k-I

m

o

0

Flc.. 1. Surface structure divided into strips. I f the structure is divided into a number of strips by a series of lines, m in number, parallel to the x-axis, the above equation (1) holds for all points along the strip boundaries. Tile governing equation m a y be rendered uni-directional by replacing the y-wise derivatives with their finite difference equivalents along the bolmdary lines. Typical transformations along the kth line, at y = Yk (Fig. 1), take the form, using central differences,

~y =

(F~+l(x)-F~_~(x)}

0y~

{r~+, - 2i~i + F~_,}

bx~y

~ l ~x

~x I

/

etc., where h is tile line spacing, assumed uniform for convenience. Tile transformations (2) must be modified where necessary at the boundaries, k = 0, m - - 1, to take account of the prescribed edge conditions, and derivatives m a y have to be expressed in alternative terms of backward or forward differences.

Application of line-solution techniques

393

Substitution of equations (2) into (1) yields a set o f m ordinary differential equations of the n t h order, which may: in t u r n be reduced to a set of m n flrst-order equations b y the use of a now Set of Variables

dFk't=.F~,j§ 1 ( k = 0 , 1 , 2 . . . . . m - l , dx

j=.0;1,2 ..... n--2)

(3)

I n m a t r i x form, equation (1) finally becomes d F = _~F+B dx

(4)

where A is a square m a t r i x of order ran, containing the coefficients of the variables/'~,~+1 in equation (1), whilst F and B are column matrices of order (ran • 1), F containing the functions F,.t§ 1 a n d B the applied load, terms, evaluated a t th e s t r i p boundaries. I f matrices A and B contain c o n s t a n t coefficients only, and if the.boundaries of the structure lie a t x = 0, L (Fig. 1), the solution of.(4) becomes

r(x) = G(x) 17(0)-L(x) where

(5)

F(0) -~ value of functions F a t x ~- 0 (](x) -- exp.(As)- .=.I+Ax-I-~-v.x2_+.... L(x)

{I -- G(X)} A -1 B

where I is the unit matrix. I n practice, 9A is often ill-conditioned, and it is preferable for computational purposes to expand L(x).as AB _" A~B

/

_

I n practical problems, half the functions will be "known at each boundary, either explicitly or as a linear combination of the other half. iTherefore, it will always be possible to write 17(0) in the f o ~ a 17(0) = K(0) FC0) (6) where F(0) is a column m a t r i x containing the unkiiowri flmetions only, and K(0) is a conformable transformation matrix, generally of the partitioned form

K,0,_Similarly, at x .= L, we m a y write K(L) 17(L) = p

(7)

where K(L) is a transformation m a t r i x used to elimin.ato the unknown terms of F(L), and p is a eoluin a m a t r i x containing t h 0 ' k n o x ~ values of t h e functions~ K(L) will thus be of the general form, conformable with 17(L) K(L) = [I !0] On substituting (6) a n d (7) in (5), the solution becomes 17(x) = G(x) K(0) [K(L) G(L) K(0)] -~ [p + K(L) LCL)]--L(x)

(8)

enabling 17 to be computed directly a t a n y point.. I n s t e a d of using a single composite governing equation (1), it is often found more convenient with this method to work instead 'in terms of the component equations (eqfiilibrium, stress-strain relations, etc.) in order t o deal with the b o u n d a r y conditions encountered in a n y particular problem. The solution follows in exactly the sam~ w a y as outlined above. 27

394

S. 1~I. A. l~.zi)iz and A. C o u ~ 3. P L A N E - S T R E S S

PROBLEMS

Plane-stress problems (such as shear walls, deep beams, dam buttresses, etc.) m a y be formulated in terms of either stresses or displacements. I n terms of stresses, the equilibrium and compatibility relationships are, 5 in the orthogonal system considered (Fig. 2), -~x

-~y = 0

(9a)

~

(9b)

= p

(~7 x ~ + ~ - ~ / ( ~ x + ~ ) 2

~2

= o

(9c)

where a~ and a~ are the direct stresses in the x- and y-direction, respectively, ~-=~is the shear stress, and p is the density of the material. I t is assumed that no body forces other t h a n those due to gravity are acting on the structure.

! 4

T

1

11

f h - - - - 12

~, ~" 0

--

H

0

%=P "C=,~. o

a__,J: J~.,, V'" J ~ " ~ 0

lllo0

9

,I

Fie. 2. Shear wall under uniform lateral load. I n terms of displacements, the governing equations are 2

52 u

~2 v

2

02v

~2u

O1V u+-~x-~+~-~y = 0

(10a)

= 02~

(10b)

where u and v are the displacements in the x- and y-direction, V2 is the Laplacian operator, and C1 and O2 are functions of Poisson's ratio v for the material. ~rhen using the present method, neither set of equations alone is satisfactory for the solution of problems involving mixed boundary conditions, a n d a combined stressdisplacement approach must be adopted. This produces the simplest specification of b o u n d a r y conditions along each edge. I n this case, it is preferable to replace the compatibility condition (9c) with the three corresponding stress-strain relationships

E

[~v

~u\ | (9d)

E

I~u

!

~v\ |

Application of line-solution techniques

395

3.1. A p p l i c a t i o n to shear wall structures The shear wall considered is shown in Fig. 2, in which, for convenience, the boundary conditions are specified along each edge. The structure has been divided into four strips r u n n i n g parallel to the x-axis. I n a n y particular problem, the direction in which thestrips are chosen to lie is decided b y a consideration of the ease of satisfying the boundary conditions. The n u m b e r of strips used shotfld be as large as possible to ensure greatest accuracy, b u t in practice the m a x i m u m t h a t can be dealt with is a ftmction of the characteristics of the digital computer available. For computational purposes it is convenient to organize the problem in nondimensional form. Accordingly, if a new set of co-ordinates and variables is defined as x

v=Z,

y

~=~'

u

r

v

,

~=~'

c

~=~x,

c

~v' = ~ v

(11)

the equations of equilibritun and compatibility become

~: = p ~ , c

(12b)

@-2(l+v)U~, = o

(12e)

O~1 ~-vr

~+r ~r

~ +re

e~ (1-,,~)

~T/'rv 0~:

c

a" = 0

(12d)

where r = L / H . The arbitrary constant "c" in equations (11) is introduced in order to improve the conditioning of the final set of equations (15), and can be given a discrete value for the particular problem considered. The stress a~ has been eliminated from the five independent equations (9) to produce the simpler set of four governing equations (12); since these are of the first order in the x-direction, no further variables need be introduced (as in equations (3)) and integration can be performed directly. Along each line, the equations (12) m a y be rendered uni-direetional by converting the derivatives with respect to ~: into finite difference form. The four equations will yield respectively typical equations of the form, for the kth line, t

9

dr/ I- 2--a(%~Vk+x--%:Yk-l)= 0 dCv~.

pL

d~ d'--~- =

vr

(13a)

cr

2(1+~) r , _ ~ (r162 v

~'~uk

03~

2a

rick (1-v~) ' --L(~bk+l--r d'~--c ~ 2a

(13d)

where a = h / H . 3.2. T r e a t m e n t o f boundary conditions The finite difference expressions in equations (13) hold only for lines which are not affected b y the edge conditions at ~ = 0, 1, and special treatment must be given to the surrounding lines affected. The clamped condition along the lower edge, ~ =-0 (Fig. 2), is specified b y the satisfaction of the three conditions

r = ~o = ~r ~q = o

(l~a)

396

S. 1~. A. Ir

and A. CounT,

B y using forward instead of central differences for the finite difference approximation for ~ b ] ~ in equation (12d), the third condition becomes 9 ~'c (4~bx_r a~0 = 2a(l_v2)

(14b)

Along the upper stress-free edge, ~: = 1, the b o u n d a r y conditions to be satisfied are 9

a~, =

p ~rXV t

= 0

(14e}

The condition a~, = 0 must be replaced b y its equivalent stress-displacement con. dition, from equations (11) and (9d) o

which becomes, b y virtue of equations (12d} dr

1 ,

d'-~- = c a x '

(14d)

On substituting equatious (14) into (13), four variables (a~0, ~'~v,, ~o and r m a y be eliminated, so t h a t a set of 16 first-order differential equations is finally produccd, of the form dF dx = AF+B 05) ~vhere A is a square matrix, of order 16 • 16, embodying the coefficients of the variables in equations (13), modified where necessary b y the edge conditions, F is a column vector containing the 16 unknown functions, and B is the gravitational loading matrix, hlatrix A contains expressions for the derivatives in the y-direction, evaluated at; the u p p e r and lower surfaces, and these m u s t be expressed in their respective backward a n d forward difference forms 0F4.= ~h ' (F~ - 4F~ + 3F,) ~y

~---~ = ~ 3.3. Laterally loaded 8hear wall The particular case has been considered of the shear wall loaded b y a uniform lateral load of intensity p along one vertical edge (Fig. 2). I n this case, the b o u n d a r y conditions are At

~/=0,

7~vk = o ~ = 0

At

~ = 1,

~',vk'= 0;

( k = 0,1 . . . . . 4)

a,x,= p

( k = 0,1 . . . . ,4)

I f gravitational stresses are ignored, the loading m a t r i x B vanishes. Simple transformation matrices m a y be used to eliminate the unknown terms, r and ~b, a n d introduce t h e known values of a~ a n d -r~, a t each end, as discussed earlier, and the solution finally reduces to ~'(~) = G(~) F(0) where F(0) = ~ ( o ) [K(1) 0(1) K(0)] -1 p

Computation of the functions 2' was performed a t eight sections across t h e plate, b y continued operation on F(0), i.e. r(~) = G(~) F(0) F(88 = G(}) [G(~) F(0)], etc. On forming the Final product F(1), a check is afforded on the accuracy of the computations, since the stress components of F(1) are tmown. W i t h the four strips used in

Application of line-solution techniques

397

the present analysis, convergence on the stress components of F(1) was obtained to within 0.5 per cent. 4. E X P E R I M E N T A L

INVESTIGATION

The theoretical stresses were checked b y a photoeIastic investigation on a 4 in. x 4 in. • ~ in. Araldite plate. The uniformly distributed lateral load was simulated b y a "tree arrangement" of rollers and beams, loaded b y a screw jack, the total applied

PROVING RING

r LOADIN( TREE

9 r

r*

9

9 1 6 2 1 6 2

MODEL

rBtw

SHEAR

WALL

Fro. 3. Diagrammatic representation of loading rig. load being measured by a proving ring (Fig. 3). Gluing the specimert into a 88in. deep slot in a steel block provided a satisfactory method of obtaining the clamped condition along the lower edge. 5. A G R E E M E N T

BETWEEN

THEORY

AND EXPERIMENT

A comparison between theoretical and experimental stresses is shown in Figs. 4, in which are also shown the stresses calculated from a simple analytical stress-function solution which exists for a cantilever beam subjected to a lmiform loading along one edge. ~ Although this solution does n o t satisfy all the b o u n d a r y conditions, a set of selfequilibrating forces existing along the free tip edge, it is felt that the stresses should be reasonably accurate in the interior of the plate, oven although the span : depth ratio is small. The direct stresses ax calculated from the present theory agree closely with experimental values throughout the upper half of the plate; it proved impossible to measure the direct stresses accurately near the base, owing to lack of experimental data, and only the results for lines 2, 3 and 4 are included. Agreement with the stress-function solution is good, particularly in the central region. Agreement between shear stresses is reasonable, discrepancies increasing towards the clamped edge, due presumably to the physical impossibility of satisfying completely the encastre condition specified. Agreement with the stress-function solution is again best in tho interior of tho plate. 6. T H E

LINE-SOLUTION

TECHNIQUE

Line-solution techniques have several advantages over conventional finite difference solutions for engineering problems. Derivatives in the governing partial differential equation are approximated by Finite differences in one direction only, the problem then being solved exactly as a set of ordinary differential equations in the other direction; the accuracy obtainabIe with a given n m n b e r of strips should thus be greater t h a n that from a two.dimensional finite difference approximation using a corresponding mesh. The continuous solution in one direction should yield a n y peak values of the ftmctions which, b y falling between two mesh nodes, would be missed b y a finite difference solution. No difficulties are encountered with such configurations as re-entrant corners, where a n overabundance of botmdary conditions m a y be obtained with finite difference networks involving fictitious nodes. By suitable formulation of the governing equations, both

398

S.M.A.

I ~ z I ~ I and A. COULL

~2

L

~

"6

,0

0.25

0'5

% (i).

0-75

I'O

-

DIRECT

STRESS

LINE I

i

1

"."Z----- --" ....

0-25

(ii)

0"5

0"75

I'0

SHEAR STRESS

LEGEND THEORETICAL

.STRESS

( LINE

~'~

THEORETICAL

STRESS

(STRESS

SOLUTION ~)

....

EXPERIMENTAL STRESS

FUNCTION )

FIG. 4. Comparison between theoretical and experimental stress contributions.

Application of line-solution techniques

399

stresses a n d displacements m a y be computed directly, thus avoiding the difficulties associated with approximate solutions to, say, a stress function, where additional errors are introduced when differentiating twice to obtain corresponding stresses. The disadvantage of the method lies in the increased amotmt of eomputation neeessary. Tim m a t r i x progression technique enables a complete solution to be obtained b y forming a single matrL'~ product (equation (8)) involving both sets of botmdary conditions; the computations effectively proceed from one b o u n d a r y to another a n d back again, and hence should converge on known initial conditions, affording a eheck on the accuracy obtained. As the number of strips, and hence the order of the matrices, increase, the computed values become less accurate as each multiplication tends to produce further inaccuracies due to rounding-off errors. W i t h a larger computer, the increase in accuracy given b y the larger number of digits available will produee better convergence on the known initial conditions. W i t h the five lines used in the present analysis, convergence was obtained to within 0.5 per eent; however, on extending the analysis to six lines, the accuracy dropped to about 12 per eenb with the computer available. Since the calculations were performed in stages across the plate, as described earlier, only the last values obtained will be in error b y this amount, a n d other values in the interior of the plato should be much more a c c u r a t e . This problem could be 0vereome b y using double-length arithmetic when computing. I n an a t t e m p t to reduce as far as possible a n y ill-eonditioning of the matrices involved, an a r b i t r a r y constant "v" has been introduced in the non-dimensional stresses used (equation (11)). The precise value of "c" to be adopted Was arbitrarily ehosen to make the orders of the stress and displacement elements the same in m a t r i x A (equation (15)). I n the present formulation of the problem chosen for example, backward and forward finite differences have beeh utilized in the b o u n d a r y conditions to eliminate certain of the required ftmctions in order to reduce the size of the matrices. Other formulations of a n y particular problem m a y utilize central differences a t the botmdaries as well as in the interior of the plate. 7. C O N C L U S I O N S The results of this paper indicate that the line-solution tcchnique can be applied successfully to plane-stress problems. Although the application of the t e c h n i q u e h a s b e e n d e m o n s t r a t e d f o r a v e r y s i m p l e case, t h e m a t h e m a t i c a l formulation of more difficult practical problems, such as elastically supported w a l l s c o n t a i n i n g o p e n i n g s , is s t r a i g h t f o r w a r d . T h e m o s t i m p o r t a n t l i m i t a t i o n lies i n t h e a c c u r a c y a n d s t o r a g e c a p a c i t y o f t h e c o m p u t e r a v a i l a b l e , a s t h e matrices associated with practical structures are large and often ill-condi[ioned.

Acknowledgement---The help of Mr. H. T o t t e n h a m in this work is gratefully acknowledged. REFERENCES 1. D. R. tIA_rCTREE and J. R. ~Vo~aSLEY, Prec. Roy. Soc. A161, 353 (1937). 2. I t . To~'rENm~r, The Linear Analysis of Thi~ Walled Spatial Structures. Symposium on Use of Computers in Structural Engineering, Southampton University (1959). 3. S. M. K . CnETT~', Th'~ Analysis of.Hyperbolic Paraboloid Shells. Ph.D. Thesis, University of Southampton (1961). 4. J. R. DESAI and H. TOTrE,nHA~t, Approximate Solutions in the Shell Theory of Arch Dams. International Symposium on the Theory of Arch Dams, Southampton University, 1964 (to be published b y Pergamon Press, Oxford). 5. C. T. ~V~ro, Applied Elast&ity. McGraw-Hill, New York (1953).