finite difference equations for the analysis of thin rectangular
TRANSCRIPT
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Finite Bifierence Equations lot the Analysis of Thin Rectangular Plates With Coinblnations of FitMVaM. ^©e>':E«^es '. : '.- (Hon©:);
Eärtohj M* V, '•: University of Texas, Defense Research L*afo., Austin (Same)
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finite difference and differential e one edge fixed and'-three- edges free swept ^ings and fins*, Emphasis is for the determination öf deflections combinations of fixed and free •edge a square plate fixed along One edge
quätiöns are developed for the analysis of thin rectangular plates with „ . , as a preliminary' study of the structural characteristics of thia placed on the finite difference equations, and egressions are presented ... shears, moments, and reactions for a rectangular plate with various conditions and transverse loadings» The deflections and stresses for and carrying ä load at the center of the other edge was evaluated.
These quantities are- compared with experimental results. Good agreement is obtained between theoretical and experimental results,"
Copies of this -report obtainable from, 'CÄÜÖ«
Structures ft) Theory and; Analysis (2);
Pia' Wo - -f&& messes Equations, Differential Wings i Swept. * Stress analysis
DOCUMENTS DIVISION, t-2
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LiE FENS E RESEARCH L,A BOBATGRY THE UlN.iWERSITY OF ^TEWAS
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finite Difference Equations-, for, the Analysts- of TMn Rectaaagülar/
Plates with Combinations of Biased..and, Er.se idggg.. '
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DEFENSE RESEARCH LABORATORY THE UNIVERSITY OF TE^AS
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Finite, Difference,.i&uations for,the,Analysis öi1 .Thin Be.cta^ular ~; ' Plates with\ComlftnaTffona "ofFixed' .aM/FreCEdges-»""
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~ - M. ¥'i Barton
Abstract-
As a preliminary study of the structural characteristics of thin swept wings and fins,; the equations for the analysis of thin rectangular plates with one edge fixed are. developed in differential and finite difference form. Emphasis is: placed oh the finite difference equations and expressions are given for the determination of deflections, shearsj moments, and reactions for a rectangular plate with various combinations of fixed and free - edge conditions and transverse loadings.
A numerical evaluation is made of the deflections and stresses for a square plate fixed along one edge and carrying a load at the center of the other edge. These quantities are compared with experimental results. Beasonahle correlations (within about 5 percent:) between theoretical and experimental results are obtained.
Introduction
The present work is the beginning of ä more comprehensive project to determine the elastic and dynamic properties of skew plates which are similar to the sveptback fins used on supersonic guided missiles. The information to be obtained ultimately is expected to consist of elastic characteristics, such as the deflections., stress distributions, ©leietic axis positions and so forth for thin plates of low asp.ect ratio, with one edge clamped and three edges free.,, for various load conditions, as well as dynamic characteristics suchas modes of vibrations, natural frequencies^ and so forth. Analytical and experimental, procedures: are to he used. Such information should he of considerable, value in faciliteting the structural design of wings and fins and in providing much, needed information on. the structural phase of aerö- elastic problems.
In order to "prove in" methods and techniques, the relatively simple problem of the rectangular plate is first studied. This report consists of .an account of the plate equations ^or the static structural analysis p
rectangular thin plates with one e je fixed and three edges free, in terms öf
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the differential as well äs finite difference equations t in addition, ä numerical evaluation is presented for the deflections of a square rectangular cantilever plate with, a concentrated load applied at the center of the out- febkrd edge." Analytical results are compared with, test values.
It is expected that later reports will deal with the ©ore complete solutions of rectangular plateB with a Yäriety of loading and Will discuss the relative merits of methods of analysis such as the relaxation procedure and the solution of simultaneous equations, along with the effect of network size on the accuracy, of solutions of the. finite difference equations and the effect of Eoisson's ratio. Experimental results will also "be indicated. "A similar procedure will "be followed In presenting the. results of skew plate analyses.
Many investigators have solved a great variety öf plate problems. However, exact solution, by which is meant an analytical solution to the governing differential equations, have "been largely confined to rectangular or circular plates with the boundary ed^ss fixed, simply supported, or a combination of fixed, free and simply supported. An example of this type of solution is that of the "rectangular plate with two opposite edges simply supported, the third edge free and the fourth edge built in or Bimply supported" given by Timoshenko^' on page 215 of "Theory of Plates and Shells"» The problems of the skew plate or the rectangular plate with more complicated
the great Complexity of obtaining an "exact" solution or the tremendous labor involved in obtaining approximate numerical solutions by means of finite difference equations or other methods. Some work? however, has been done on . these problems by means of finite difference equations. For example, Jensen*2' solves a number of skew plate, problems for a. variety of lateral loads With two opposite edges simply supported and the other two edges free or elastically supported. HallOi has solved fche problem of ä cantilever rectangular plate for which the fixed edge is four times the cantilever length and which carries a concentrated load at the center of the outboard edge.
differential ^Equations
The equations of equilibrium of an element of thin plate in terms of rectangular coordinates save well known (see reference 1) and will only be cited here. The directions -corresponding to positive quantities are shown in Figure 1.
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DEFENSE .RESEARCH LABORATORY THE UNIVERSITY OF TEXAS
August 6, 19)48 *3-
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Moments:
Shears:
Edge forces
Concentrated corner force:
M = 2C
daröyc
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= M = D{l-u} iLi
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p =: intensity of distriMtsd load: fpö.i)
D :=
CF-iöG5 , BBL-lfl
Eh?
12(1-^)
E.» modulus of elasticity
(i » Polsson's ratio
& 0 plate thickness (in)
v t= deflection of plat® in Z direction, (in)
Moments, shears, axti reaction are for a unit length of edge»
The equations are limited "by the following assumptions.
1. £h© thickness of the plate is small in comparison •with its other
dimensions.
2. Deformations are snail so that the curvature is apprcximted "by
the second derivative of the deflection with respect to coordinates
la las© plane of the plate«
3. Stretching of the middle surface of the plat® is not considered.
Th.3. laBt assuHptioa has as a coroHary| the fact that the deflection is eipail coshered vith the thickness of the plat© for certain 'boun&ary conditions. If this assumption is not made, then the so-called large deflection theory must "foe used which takes into account the effect of the msabrans forces in which case the relation "between load and deflection is not linear. Fortunately,, our experiments show that for the "boundary conditions used corresponding to the cantilever plat*©* the load and deflections are essentially linear up to a deflection of at least 20 times the plate thickness. Hence, the use of the linear theory se@ss justified in this particular profit».
Finite Difference Equations •iwnnwi» 11 iTiiimniiiiinnwi-niiwiiirniiniwiiBiiinii nnwrtiin.ii n iw
For the square network shown in Figure 2, the approximate mlues of the derivatives in terms of finite differences of the values of the deflections at
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the imxitms polsts can "be shown to tie e Se© SoutJraölX*'.
• (öL^(w* -**"** 7) Vu 1 r-jl=9rj- (veE - 2vt + 2vw - *w )
AV OT/o " -5? ^
(vPc - liwft + 6vc "-"l»ww + Vww )' (2)
"Sus i 1 V>y/o 2X,
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(»S ~ WN )
lOT";a**v*,rs ***" "Ww"v
9*v U"?^! = b LW-+V+w^^w-^s +ws ^e *« )^o
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The equations for load, tsoaiönt, shear and reaction in terms of finite differences thus "beefs©
load: 20w0 - 8(vN + we + vs •+ vw )* 2{vN= 4- wSE + *,„, 4- wHW )
•WK T "BS. T "SS
Moments:
D_ r "I = - K2 |_-(2+2|i)vo + vM + ws + H(WW 4- wE )j fa
p(l-u) <May)o "^J^^** " w^
Shears:
cy. D r
Edge forces:
Corner force: (R)& = -^T^ _(v&6 -»H6 »v^ «wKW ) 2X'
Boundary Conditions
The conditions at the edge of the plate to "be considered here correspond to a fixed edge and a free edge along which the tsotuent is zero although it BJ&y
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carry a distributed applied force or a concentrated force at the comer. These conditions are-specified formally as follows.
1. Fixed .Edge
Deflection zero: w = 0
2w at x = 0
Slope zero: = 0 »x
<*>
2. Free edge (y = const)
Moment zero: M = 0 = —x + H ; 7 "äy^ dy"
Vat. y = constant
Edge forces: E = -D 3? 9y3 ay3xc
All (5)
3. Free corner
Moment zero: M - M =0: ^-5 = il3 = n
' ~ * i? 2y2 ° y
Edge force: E- -D [ 2* + (2^) -4ÜÜ
Ey=-D
32V Corner force: B = 2D ^^j _£_!
i at corner 1 (6)
In terms of the finite difference equation, the boundary conditions may he expressed as
1. Fixed edge w0 = 0
w£ - ww =0 (7)
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2. Free edge (y = const.)
-(2 + 2^) v0 *vH + vs +V (vv + w& )
(6-2n)(vk -^)+(2-n)(wsE+^w-^|w.wK1E)rWwK +vss -
3. Free corner
= 0
2V3VTJ 1
D
(ßi
v£ - 2wQ + ww =0
ws - 2w0 + vN =0
(6-2ii) (vw -we)+(2-^i)(wut+w3f= -wWw-wSw)-w«,w +wr, 2\3(Ex)c
Ee
(6-2|i)(wN -ws)+(2-j»)(wia +w5vy -vNw-vNe)-wNK +v
D
2X ss <Vo
(9)
, 2X2(5J0
)(i-n)
Since w„ in each "boundary condition is for a point on the edge of the plate, it is apparent that some of the deflections indicated are fictitious ...«90s lying outside the plate. Take for example the conditions far a free edge along a line y = constant. From Figure 3 it is noticeable that there are deflections at four points off of the plate from the condition for the edge force.
•t U\.«J
MM
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I
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I Jj
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DEFENSE RESEARCH LABORATORY THE UNIVERSITY OF TEXAS
M73:f.s August 6, 19k8
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The deflections are w<,s . w3_, vSv, and wSH . These deflections can he evaluated in terms of the deflections on the plate so that the conditions for a free edge can he determined in terms of deflections at points on the plate. Since four
quantities are to he eliminated.,, five conditions are specified along the free „edges,.. .They are. . ...... ._.
load: 20wo - 8(vN+ wE* ws+ ^) + 2(trHw 4- wu6 + wSP + w,w )
1» + V^ + WN* + VF- + Ws=, Po*-
edge force: (6-2n)(vK -ws) + (2~u)(wSF +v^-vNw -wN£ ) -wWK +vbs = - 2X ^P'y^° D
moment at: -(24-2^)^ + wM + v5 4= n{ww +we ) = 0 o:
moment at -(2+2n)v..v + vHw + w^ 4- n(vWvV + v0) = 0 w:
moment at -(2+2nW 4- w.._ + w,._ 4- n(w__ + w„) = 0
Eliminating wss , w?w t v,^ , "ws from these equations gives
(l6-8|i-6n )w0 4- (-124^u)wN 4- (-ö+kinJ*H2Kww 4- we )
(10)
For an edge -with no loading, p =(B ) = 0.
The equation for the edge condition may he schematically represented as indicated in Figure k.
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0-4
.rr~.
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DEFENSE RESEARCH LABORATORY THE UNIVERSITY OF TEXAS
MVB:fs August 6, 19kB
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Other conditions have "been determined in a similar manner and the results are tabulated, Table I gives the relations between the deflections and loads for various points on the plate. Tables II and III schematically indicate the same information for the special case in which [x=0 and ji=0'."3 respectively.
By means of the relations in Table I, a set of simultaneous equations can be determined representing the relationship between loads and deflections for a rectangular plate with various boundary conditions. Thase_ equations are then solved to determine the deflections for specified loadings. An application to a specific problem is given in a later section.
Moments, Shears and Reactions
After the simultaneous equations have been solved for the deflections, the moments, shears and reactions if any, can be determined at each point by means of equations (jj) providing the deflections specified in the equations are on the plate. In the case that the moment, shear^ or reaction expression is in terms of deflections off of the plate, then the expressions must be modified by the application of the boundary conditions so that the points external to the plate are eliminated. This procedure is similar to that used before so that only the results will be indicated. Table I? gives the expressions for moments, shears and reactions for points on or near various edges.
Principal Moments and Maximum Stress
In order to determine the maximum stress at a point, the maximum value of the moment should be known. The maximum and minimum values of the moments at a point are called the principal moments and they can be determined by combining the bending moments and twisting moments in the arbitrarily chosen x and y directions according to the following relation.
max:
mm:
*L
Mo = KMX+My) t JjOVMy)] + **7 (11)
The angle between the moment in the x direction and the maximum principal moment is given by
tan 20 = Ms-My
(12)
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DEFENSE RESEARCH LABORATORY THE UNIVERSITY OF TEXAS
MVBsfs August' 6, 19^8
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The value of the maximum normal stress corresponding to the maximum principal moment is
. . h ... -- (12)
vhere the stress is at the fiber in the surface of the plate.
Analysis of a Square CantileTered Plate:., (^0)
Consider a square plate with side length L fixed on one side and supporting a transverse load P at the center of the outboard side as shown in Figure |. Since the plate is symmetrical about a longitudinal axis through
/ id
Fin. 5.
the load, the conditions for only half of the plate need he considered. Thus the deflection at point is equal to the deflection at point 2 and so forth. A square network is imposed on the half plate so that there are 12 abodes or grid points to he considered in addition to those along the fixed edge for which the deflections are zero*
condi By means of the equations of Table I or the patterns of Table II, the
tions relating the leads and deflections for ji=0 at each point are specified. It should be noted that since there is no applied normal plate pressure, p is
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zero at all points. Sitatlarly along the fre© ©dgea of the plat© there is no applied fore© except at point 12 vher© ä distributed force of taagnitude
S. is asstased to act. In this. ffi&3ffi©r the applied concentrated force is. approaamted oy a fore© distributed over 6ns grid spacing X n
The following 12 slmÄtsneous equations result from the applications, jösäf the relations .given in Table XL.
point
-9
9
17^ - 12w24- 2w^ - 8^ 4 1»^ * y „ « 0
-6^ 4- 21w2 - 8^ 4 2^ .8^ASW.A¥ »Ö
2«^ - l$Wg 4 21^_ 4- -»W5 - 8w, *VQSO
-8^ + l»v2 4 16V,. - 12^ 4. 2wg - 8\*„4 lmg 4- ^1Q » 0
Sw^ - 6^? 4- 2v_ - oVjj 4- 20w,_ - 6Vg 4- 2w, - 8w„ * 2w + v » 0
W2 -8w -4- 2WJ; —l&rtj-+-20vg;+^»wg—Oar—^-»r^-s-o- —- -
y^ - SWJ^ 4. k-w 4- 15v - 12«- 4- 2VQ - 6V10 4- toj^ o 0
v2 4 2Vij ~ 8w5 4 2w6 - 6V 4 19*g - 8w 4 2KL. - 6V 4 2sr12 = 0
v3 4 ^ - 8w 4 2w - 16Vg 4 19*^0 + ^w_ - 6V « 0
10) 2^ - 12w„ 4 8vg 4 12ir10 - 12Wy, " ^ia * °
IJ/ 2Vfe 4 W~ - 12wg 4 Wg - 6V, „ 4 16VTT = 8*r,~ e s10 1ÄW11 = 8"12 " °
lg) 2w6 4 8vg - 12yg.4 2w1Q - 16» 4 16V.
The solution to these equations give
12 - D w
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DEFENSE RESEARCH LABORATORY THE UNIVERSITY OF TEXAS
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w, =
v„ = j
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2.
v- =
V. •-
w -
W^ -
Ö. 05002 &L. D
0*03143
0.03211
0.106.U
0.10978
0.11183
w* * 0.21250 IE 7 D
wg = 0.21923
VQ, = 0.221*05
w1(r 0.33331
wll= °*34370
W12= °-35430
2
These deflections have "been substituted into the moment equations given in Table U. The'maximum moment and stress at each point has been deter- mined "by means of equations (11) and (12). The vmxLwm stresses in the top surface of the plate are thus found to he
h£ «^ m 1) .0444 Eg. <T7 .=. 1.2649 £-
7 h2 ^a =• 5.7633
<s-2 . 2t.li8ii9 <3"g » 1.4429 CT-jj « 6.0351»
^5 « 4.5307 ^•9 - 1.4518 <rc = 6.1659
<3"2, = 2.6458 °io -° 0*5 a 2.9512 ^ll • 0.4328
<T6 - 3^0023 Cr^ - i.8516
The positive sign of these quantities denotes tension.
The values of the deflections and stresses have been carried out to more significant figures than the data mrrant^-. This ws done for comparison of results between different methods of solution.
Comparison between Analytical and Test Results
In order to have a check on analytical methods, a simple plate test
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set-up -was made consisting of a rigid clamp to hold the plate along one edge and a device for applying a static load at any point along the opposite edge, peflections -mre Bsaeured by meass of mAses dial gage supported from a rigid reference table. Th© st&ains were determined at four points "by means of SR--4 electric rosette gäges faslSÄed to the surface of tue plate.
The specimen used was a 2l»ST aluminum plate one foot square and 0„127 inches thick. She following data -Her© obtained for an applied edge load of 15*^35 lb* and a Essduius of elasticity 3 of 10.3 s 10® psi. lliecretical results for n=0 and ^»0.3 are give» for the sake of comparison. It should be emphasized that although the deflections are determined for the two values of fisO and p»0.3 the value of p»0.3 has been consistently used in the moment expressions.
point
1- ]
1 ' J jb - —•
%A 3
M 5 /
5
6
7
7
8 j
8.i
I
Deflections (in)
theoretical Escperimsntfil
recorded average
0.030
0,032
0.032
0.106
0.3,09
Ü.li2
0.213
0.220
0.025
0.033
0.031»
0.102
0.111$
0.118
0.213
0.228
0.023
0.029
0.032
0.029
0.033
0.100
0.101}
0.111
o,ni>
0.116
0.205
0.211*
0.219
0,225
0.026
0.031
0.033
0.102
0.113
0.116
0*210
0,222
percent difference for H«Q.3 values
-3.81»
6.1*5
-3.03
40.88
1.72
1.1)3
2.70
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9 0.221» 0.236 . _P«229
1.0 |_„. .
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.Q.33& _ o..-3i»o O.332
0,331
11 1 0.31)1* 0.30 0.351
0.350
12 0.555 0.375 0.362
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0.229 3.06
0.332 2.1}1
0.351
0.362
2.56
3.59
point
2
2
8
8'
Maximum Principal Stress (psi)
Theoretical Esperitaental
*x=Q }i=0.3 ' recorded average
Percent difference for
|i=äÖ.3
10,112
3,253
10,150
3,1J28
10,l|21
10,1» 02
3^96
3,6Hi
10,1*12
-3,555
>2.52
*5-57
The values of the minimum principal stresses determined "by es^eritoent did not check the calculated values nearly as closely as the corresponding values for the mxitsum stresses. These values are fairly emll so that sisall arror may mk© a considerahl© difference so that even the sign of the quantity say be effected.
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Keferences
GF-IÖ05 EKL-175 WEC-l-M-3
1. Y. £. Jens©», "Analysis of Skew Slabs", University of Illinois, Engineer lag Expfclisent St&tlea, Bulletin Series Ho. 332> Toi XH32 Ho. 3, Sept. 9, 19*11 •
-2-.- H. Itocus, "Die Theorie elastischer Gewepe vua. ihre Anvending auf die Berechnung T»eiegsas®r Platten", Julius Springer, Berlin 1932.
3... S... Titaoshenko, "Theory of Plates and Shells", McCSrav-Hlll Book Co. 19^0 •
k. B. Y« Southwell, "Pelasation Hsthoda in Theoretical Physics", Clarendon Press, O&ford, X$k6.
5. H. W. Emons, "The Stasric&l Solution of Partial U .ferential E^tlons", Qa&rterly of Applied Matheiaatics, Yol II Ho. 3, Oct. I9M.
6. D. L. Holl, "Cantilever Plate with Concentrated Edge Load'J, Journal ef Applied Ifechanios, Yol. kf lo, 1, March 1937»
7. F. L. Ehasz, "Structural Skew Plates", Am. Soc. for Civil Eisgineers, Transactions Paper So. 2287, Yol III, p. 1011, I9IJ6.
Table I ' '
" .^ *- '--••- -
Tabulation of Deflection Relations "for Esotangular Plates
\
N^NSW "
free?
1» Point adjacent to frssd edge.
Z114 * 8 I &N + U)e + UJS )• + t ( u)Me + UJSE) + UJ^ -v uüss v^e D
I
ga Point adjacent to fixed-free corner.
+ (2- |A)UJSE +• u)M* + we£ -« -^
I*
3« Point adjacent to fixed corner on free ©dgee
(n-8|i-5^)i4't (-1.2. + 4pO u)^-v(~8+4\i*4-f£) ti%
-0--1
4« Point on £r®@ edge*,
. t—1 ö—
5«j > .
'1
5« Point adj&eent to free edge.
II w0 - 8twN* uiw*u»e^ + C-6+2n)u^ + 2Cu)ME-v w^
6, Point oa free eoraer*
<>—
+
mi.. --*•« n *
, T,. i- _ '^m ^«••^p--j-'Sitf.n'' »"•>»"»• t^TjJ^ . "*t«-' u*- ; ,>• , ^*,;*f^1**^'->*.*!-'"-^'''-r i „ ,**&****->• inn jyro *r*>'i7' "y^ r^--s*^-^^^'?'^5Tl!r'^'?^?r T^B^*^W«W
7o Point inside fy©e»fre© corner0
-o
8, Point adjacent to £rae~frse cornea? on free edge.
2. W pax ^ ax
Wu> O -T !>(®-r)o
9« Inside point»
1
D
1 JSt^•r j|¥. ,*77". " • ~"7*T
table II
Deflection Relations for Rectangular Piates for n = 0
....... fixed —NNNVsNV
free —e i=T_
I« Point adjacent to fixed edge.
£ov
~~i"~
3» Point adjacent to fixed corner on free edge.
T~ * — (V,
Point adjacent to fixed-free corner.
1» *oX
D
4.» Point on free edge.
D D lVo
5« Point adjacent to free edge.
D
6. Point on free corner.
2xJr: T D ?x>o + CV3 J
.»«„•-' ^+ K
--. -y: »»i».*/«"» *-» ~ JS•'^*••^, „ryyt^m,
gaEEasg^gtgTflgjgiwffwagiMns wmtm&amm mem
7« Poiat insj.de free-free corner, 8« Point adjacent to .free-free - corner ba~ffee edge.
9. Inside polat«
^'H?* . -iJ^T-"- - Jrltf.... i .?^^,^*^^^L^^w,* •j» iwnwiwwa» u ij» jj?WWf^ WWBPtFy-«<S=i»'W.-j»—
^^W&ffSflhWrtWl^ • ,'-*^=.>e-,\:..^ . :*.. .^^ynj»--.»! „
SabUe III
Deflection Relations for Pöctangular Plates for ^ = 0.J
fixed free
N\W\\\
1. Point adjacent to fixed edge.
D
3, Point adjacent to fixed corssr on free edge»
iT* y'o D
2. Point adjacent to fixed-free co3,-»er.
?oX
P
h. Point on free edge.
T~ • IT *Vo "
5. Point adjacent to fro© edge,
1*
6o Point on free corner.
?o^
+
2X? D
D
Mo * < V
auixcssf£D*&ft"
7.. Point, inside free-free corner, 8. Poini/Jg^acent to free-free cornej "on.free edge.
v4°-+4mi
9. Inside point.
-%
• at • '"T~
rr« •!<--.. ,»- 7—^, -rr-rrrfr
Table IV
Expressions for Moments, Shears and Reactions*
Fixed;
Free
xxwww
1. Point on. fixed edge.
w * -1 <U* %* "«• § ••pvö'e-
M'^ a O D
x3 Q-Lj = -% (uJ--uJwe)
s
! -•
i ..- i
2* Point on fixed edge near fixed-free corner.
My = - 3^ H «1B
l Q
3» Point at fixed-free corner,
MK= - -^r u)E
ID
Mxy * 0
"C-LAT-'-J ' 5WBF- iitM^nTJ-yyy» ifwnJgf y n^-
!*Mye^^*iXS¥*saSt!tt>*Vt ^•ffv-ffy ^•^^rWiJfl^WTH'mifflfo-ir^ ^^••j^iiiajMmfcjinmiwmirtbMai
4-e Point adjacent, to fixed edge0
0
A'
ÜJ ME
Q* = " -^[-^e"4vx)^-t UJHe.-vuJs£T UJEE^
Qv* ' " U? 14(U)H" ^ ^ ^S " ^US ' ^NN -" ^&s]
V
>:
\ -A-
Point insid© fixed-free corner.
M(| a - - Sj. [-( £•* 2 j>j-uui0 -t uj^ -r UJ& •*• -|>tUE 1
Mxu= 2Ll^ (USE- LUHe>
6„ Point on free edge adjacent to fixed-free corner.
vKj - O
M M _ D(!-^
4-A2-
t(l+ I*) U»o "K"^"* 2. U) U)g * ( * ~ M)LUE-£ 3
J h
....-"*^.- , T._, ._..? %; ; - *~ ^ -,=< .-,-...-*.—r=•:
*«7yjl^iJi^^^ieiM^i|J^^u^u^U>^U4i.|X'^^W^Vy«^W^^^tJW»tUI
•• +i» Sx daw v fflfiliiiMr^-7-|,TVtr.MjnmMhn^Mi»»ii^^ Jjjj j..
7o Point on. free edge«
Mx= - -~L Ü- ^H~ &üJo ^ itf» + %y 1 MLJ - P.
--+•
.--o-
89 Point on free edge adjacent to free-free corner,
M^ - 0
9. Point on free-free corner.
Mx - 0 •-A
M«j = o
Mx^-- ^j[<.-6»-<H\t2V;UM-«- w*- jj0)i- fi~M7")(a>^w " w^+(4-4-^)uy«j,]
I»K> f.'c A'",K:. "."' rj~"tp •fi"i-???",^K^:^J"^£Vi
. li-
ft
10. Point inside free-free comer.
D
Mv fc.(:\ •*•!*)•
1 ~~ ~^s-I %« "* «^'M 6 ~ w-%^ "*. u-W 3
Vs-
11. Point adjacent to free edge.
D YA,K- ~ -5J. t"(2+ 2 H) ^O + w« H- wE + .JA-(U/„ f wä I
MM a EÜ^ ItiJsg - Awe- ^su) t «vl
2"
v-1-
<*r
V: ,^ ' -~t \~ U-V Z JA\ lijg -V ufo -*. vil^ •* y(.u^ * vx)^
12. Interior point.
Q»
Q^
.4-^
2>^
-1- 1 r- I
^4-tu^- Ui^.tui^t^-ui MUJ " ^ivO U) WvO W
rtf" «*W
'7 -_r . »\ , :AW J - . -_*.- -"~rj~ TT"
=-=. — -w-. ^ wwt^gl^^S^^.^jjiafäaaBJgyii- •iw» »•«
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