-
1
1. 1.1 1.1.1 (Normal stress) 1 , p , p . , . , , p. , . ,
n p = , , , .
Fig. 1.1 Isolated material subjected to hydrostatic pressure
-
2
1.1.2 (Shearing Stress) 2 , W , F .
WF = , . , . . 2 , , .
FA
=
Fig. 1.2 Isolated materials subjected to shearing force
1.2 1 A , A . A . , A . ,
-
3
Fn A .
0lim nn A
FA
=
, Ft , .
0l im t
A
FA
=
, . , . , . , . , . 1.2.1 1.3 p , n
pn = , 0= . , pn = , 0=. ( A) . ? . , 4 BC, DA 0x AB, CD
0y BC, DA 0xx = 0=xy AB, CD 0yy = , 0=yx .
x y x, y x, y . , xy x,y x=const. y
-
4
. 0xo , 00 =y , E
0xx = , 0=y , 0=xy , .
Fig. 1.3 Fig. 1.4 1.2.2
1.4 , , .
0=x , 0=y , 0xyxy = at BC, DA
0=x , 0=y , 0yxyx = at AB, CD
? , AB 0yx x , CD x . , x , x
-
5
. BC, DA . ABCD . , xy , z , z 0 . A z . ( ) ( ) 000 = DACDABBC
yxyx , CD=AB, DA=BC .
00 yxxy =
. , . 1.5 .
Fig. 1.5 Fig. 1.6
-
6
1.3 2 1.6 ,
.
0xx = , 0yy = , 0xyxy =
X-Y . , . , X-Y - , , . 1.6 ABC
1.7 ,
0xx = , 0xyxy = at AC, 0yy = , 0xyxy = at AB
, 1.7 AC AB - . BC 1 , X-Y - . (cosine), .
-
7
Fig. 1.6 Fig. 1.7
1.7 ABC . ( ) 11 = ( ) 11 llx + ( ) 11 mmy + ( ) 11 mlxy + ( )
11 lmxy ( ) 11 = ( ) 21 llx + ( ) 21 mmy + ( ) 21 mlxy + ( ) 21
lmxy
( ) , ( ) , . . , .
= 21lx + 21my +2 11mlxy = 22lx + 22my +2 22mlxy = 21llx + 21mmy
+ ( )1221 mlmlxy + 1.1
1.7 .
-
8
= 2cosx + 2siny +2 sincos xy = 2sinx + 2cosy -2 sincos xy = ( )
sincos xy + ( ) 22 sincos xy 1.2
, , . 1.4 3
Fig. 1.8 Fig. 1.9
1.8 , . () , 3 6 .
x , y , z , xy , yz , zx 6 , 1.9
-
9
- - , .
, 2
. , 1.1 1.2 , . , 2 3 , 2 z - , x-y , n1=n2=0, l3=m3=0 . 3 . = x
21l + y 21m + z 21n +2( 11mlxy + 11nmyz + 11lnzx ) = x 1l 2l + y 1m
2m + z 1n 2n + xy ( 1l 2m + 2l 1m )
+ yz ( 1m 2n + 2m 1n )+ zx ( 1n 2l + 1l 2n ) = x 1l 3l + y 1m 3m
+ z 1n 3n + xy ( 1l 3m + 3l 1m )
+ yz ( 1m 3n + 3m 1n )+ zx ( 1n 3l + 31l 2n ) , 1.9 3
, 4
x y z 1l 1m 1n
2l 2m 2n 3l 3m 3n
x y z 1l 1m 0
2l 2m 0 0 0 1
-
10
. , 3 , ABC 1 , OBC=l1, OCA=m1, OAB=n1 .
-
11
2. (Strain) 2.1 2 . 2.1 , ABCD . 2 (x,y) (u,v) . . x y . x,y .
2.1 , ABCD . ( ) , ( ) . . .
4 ABCD 2.2 , A x x (x+u) , B x .
dxxudxux
+++
Fig. 2.1 Fig. 2.2
-
12
, AB dx dxxudx )(+ .
x .
xu
dx
dxdxxudx
x =
+
==
, y yv
y = .
, . ABCD . , ABCD . , 2.3 .
, A,B , DC D', C' . . . 2.3 (b) . .
xv
yuBABDADxy
+=+= ''
2.3 a b . z .
(a) AD AB , . , AD AB ((b) ).
-
13
Fig. 2.3
, x
. .
xv
xv
yu
z =+
+ )(
21 or
yu
xv
yu
z =
+ )(
21
z .
)(21
yu
xv
z
= 2.2 3
3 , 1 2 . 2 x-y , y-z, z-x , x,y,z u,v,w 6 3 .
-
14
xu
x = ,
yv
y = ,
zw
z =
xv
yu
xy +
= , yw
zv
yz +
= , zu
xw
xz +
=
)(21
yu
xvwz
= , )(
21
zv
zvwz
= , )(
21
xw
zuwy
=
6 3 . 2.3 ( ) 2.3.1 2
Fig. 2.4
, x-y xyyx ,,
. 2.4 2.5
-
15
. , 2.4 C C' , OC, OC' 1 . 2.4 2,5 , O C yx , .
121 mxx += A 111 myy += A 2.8
, 11,mA . OC ,
11 myx += A
2.9
yx .
Fig. 2.5
2.8 2.9 ,
112
12
111212
12
1 )( mmmm xyyxyx AAAA ++=+++= 2.10 C' O yx
-
16
222 mlxx += , 221 ml yy += 2.11 .
222
22
2 mm xyyx AA ++= 2.12 2.8 2.11 2,5(b) , .
)()( 112221 mm yxyx+++=+= AA
= )(22 12212121 mmmm xyyx AAAA +++ 2.13 sin,cos ,
sincossincos 22 xyyx ++= sincoscossin 22 xyyx +=
)sin(cossincos)(2 22 += xyxy 2.14
. . , , , /2 . , 2 , . 1/2 .
-
17
2.3.2 3
x, y, z xyzyx ,,, , , 2 2.10 ~ 2.13 . 2 3 . .
1111112
12
12
1 AAA nnmmnm zxyzxyzyx +++++= 222222
22
22
22 AAA nnmmnm zxyzxyzyx +++++=
)()(2 2121212121 AAAA mmnmmm xyzyx ++++= )()( 12211221 AA nnnmnm
zxyz ++++
2.4 0
. , . , 2.8 z . ,
1211211121 )(21)(
21 mmm xxx ++=+= AA
1211121111 )(21)(
21 AAA +++=+= mm yyy
.
-
18
+
+
+=
1
1
21
21
1
1
21
21
)(21
)(21
)(21
)(21
mmy
x
y
x
y
x AA
+
=
1
1
1
1
00
21
21
mm zz
yxy
xyx AA
, 1 , 2
. , . , . . 2.4 C C' , . ,
0)(
21
21)(
=
yxy
xyx
2.18
2)()(
,22
21xyyxyx ++= 2.19
. 3 ,
-
19
.
0
)(21
21
21)(
21
21
21)(
=
yyzzx
yzyxy
zxxyx
2.20
3 321 ,, , .
0))()(( 321 = 2.21 .
3211 ++=++= zyxI
)(
)41
41
41(
133221
2222
++=++= zxyzxyxzzyyxI
321222
3 41
41
41
41 =+= xyzzxyyzxzxyzxyzyxI 2.22
, 321 ,, III . 1I . () , .
-
20
2.5
2.6 6 3 . , 6 3 . , 6 x,y,z 3 6 . . ( () ) .
. . 6 . .
),,(,, zyx DDDzyx
,
wDvDuD zzyyxx ===
uDwDwDvDvDuD zxzxyzyzxyxy +=+=+= 2.23
yx , u,v,w 3 . , 6 4 , 1 .
, xyzyx ,,, 4 , 1 .
-
21
xyzyx CCC =++ 321 , vDuDwDCvDCuDC xyzyx +=++ 321
,
0321 === CDDC
DD
Cy
x
x
y
,
xyyy
xx
x
y
DD
DD =+
xyyxyxxy DDDD =+ 22 ,
yxxyxyyx
=
+ 2
2
2
2
2
. 6 4
1546 =C . 15 6 .
)(222
2
2
2
2
zyxxzyyxxyxyzxyzxxyxx
+
+
=
=
+
)(222
2
2
2
2
zyxyxzzyyzxyzxyzyyzzy
+
=
=+
)(222
2
2
2
2
zyxxyxxzzxxyzxyzzzxxz
+
=
=+
2.24
6 .
-
22
2.24, 2.6 2.23 , . 2.24, 2.6 wvu ,, .
-
23
3. 3.1 2
Fig. 3.1
3.1 , bc 2 ,
. 0 2 , 0 . .
3.1, P bc 2p = 2 + 2 . bc 1 , , AB=m1, AC=l1 . ABC , BC x,y
.
=xp +1lx 1mxy =yp +1lxy 1my
0 , BC x,y .
=xp 1l , =yp 1m
-
24
0= .
+1lx =1mxy 1l 1lxy =+ 1my 1m ( ) x 1l 01 =+ mxy 1lxy ( ) + y 01
=m
1l 01 == m 1,1 ml 2 0 . , l1=m1=0 . ( )
( )
yxy
xyx =0
2 1 , 2 ,
.
1 , 2 = ( ) ( )2
4 22 xyyxyx ++ .
, . , . . , . , B .
-
25
=1
1
lm =tan
xy
y
1tan =
xy
y
0/ = dd ,
0 . 3.2 3 2 , 3 1 , 2 , 3 . , ( )
( )( )
zyzzx
yzyxy
zxxyX
=0
, ( ) ( ) 22223 zxyzxyxzzyyxzyx +++++ ( ) 02 222 =+
xyzzxyyzxzxyzxyzyx 3.1
3 , .
x , y , z 3 . , 2 . 3 1 , 2 , 3 . ( )( )( ) 0321 = 3.2
-
26
0322
13 = JJJ 3.3
=1J 1 + 2 + 3 =2J - ( )133221 ++
= ( ) ( ) ( ) ( )[ ]2321213232221 261 ++++ =3J 1 2 3
1J , 2J , 2J , 1, 2, 3 . 3.1 3.3 , , ( )zyx ++ , ( )+yx . , ,
x-y-z - - .
=1J 1 + 2 + 3 = zyx ++ = ++ =2J ( )133221 ++ = ( )222
zxyzxyxzzyyx ++
= ( )222 ++ =3J 1 2 3 = 2
222 xyzzxyyzxzxyzxyzyx + = 2222 +
3 3.1 , . 1J (hydrostatI c) (Hooke )
-
27
. 3J 3J 1J , 2J . 3.3 ( Hooke )
Hook . (Isotropic) . , . (anisotropic) . . , .
(Homogeneous), . , , . , . .
.......),,( xyyxx f =
0...... ==== zyx 0=x x , 2 .
-
28
zxyzxyzyxx CCCCCC 161514131211 +++++= 1611 ~ CC . , .
=y =z * * =xy * * =yz * *
xzx c 61= * * * * zxc 66 .
* * * * * *
* * * zxa 66 3.4 ija .
3.4 1 , xy x . . 014 >a , 3.1(a) xy , 0>x . , x . , (b) ,
x . , (a) (b)
0a ,
zxyzxyzyxx cccccc 161514121211 +++++=
zxyzxyzyxyx aaaaaa 161514131211 +++++=
=zx
-
29
014
-
30
)(21 yxzz aa ++= 3.7 xyxy a 3= yzyz a 3= zxzx a 3= 3.8
3.8, 0 . .
)( 322111 ++= aa )( 132212 ++= aa )( 212313 ++= aa 3.9
(123) (x,y,z) xy 1 2 3 xy 1 2 3
)(2 211212211 nnmmxy ++= AA 3.10 213212211 nnmmxy ++= AA
3.11
3.8 1 3.9~11 3a
1a 2a . , 3.8 3.10, 3,11
)()(2 2132122113211212211 nnmmannmm ++=++ AAAA 3.9
))(()([2 21212132122132122111 nnmmannmma +++++++ AAAA )])((
213212211212 nnmmaaa ++ AA
))((2 21321221121 nnmmaa ++= AA
-
31
)(2 213 aaa = 2 .
Ea /11 = 12 aa = Ea /)1(23 +=
, Hooke
)]([1 zyxx E +=
)]([1 xzyy E +=
)]([1 yxzz E +=
Gxyxy / = Gyzyz / = Gzxzx / = E: Youngs Modulus =Poissons Ratio
G: Shear Modulus
)1(2/ += EG 3.4 6 , 3
9 . x-y-z - - , .
-
32
=
333
222
111
nmlnmlnml
zyx
3.12
3.12 , .
=
333
222
111
nmlnmlnml
zyzxz
zyyxy
zxyxx
1
1
1
nml
.
=
333
222
111
nmlnmlnml
zyzxz
zyyxy
zxyxx
T
nmlnmlnml
333
222
111
3.12 ,
(tensor) . , , .
-
33
4. Cartesian 4.1 ?
(physical process) (coordinate system) . (quality) ?
(tensor analysis)' . ( 4 ) , Cartesian 3 . (tangent) .
. (order rank)' 0 , 1 . 3 Cartesian 1 (=30) , 3 (=31) . 2 9
(=32) n 3n( 6n) .
. (linear operator) . 2 . 3 2
-
34
33 . 31 u 33 T 31 v , .
Tu = v u, v 2 T . vuT = (coordianate-free invariant validity)' .
. . . Cartesian u (u1, u2, u3) , ),,( 321 uuu (transformation
equation) . 1 u u u . (3 ) 3 1 : 1 . () .
3 33 9 2 9 .
-
35
2 . . (an objective physical reality) . . Cartesian . Cartesian
, Cartesian . 4.2
2 e- - (Einstein ) (summation convention)' . ( ) ( 3 Cartesian
i=13) . . . .
-
36
(3 ) . (i) aixi a1x1 + a2x2 + a3x3 (ii) aijbjk ai1b1k + ai2b2k +
ai3b3k (iii) aijbjkck ai1b1k + ai2b2k + ai3b3k (iv)
(v) (dummy index)' (free index)' . , . , aifbjkckl aijbjjcjl
ailblkckl , aimbmkckl aimbmncnl . ( .) Kronecker delta"ij .
ij .
ikkjijjkij aaa == 4.1
3
3
2
2
1
1
xv
xv
xv
xv
i
i
+
+
32
2
22
2
12
22
xxxxx ii +
+
==
) 2() (1
jiij
-
37
(j) Kronecker delta (k) . (5.1) AI = A , A aij , U A .
iijj bb =
Kronecker delta .
) ( jkkjjiijkijkij bababa =
Kronecker delta (substitution operator)' . 3
Levei-Civita" (alternating) (permutation) eijk .
ij ( ij = ji) eijk . eijk eijk 3 .
+
=)(0
) 3,2,1 ,,(1) 3,2,1 ,,(1
kjikji
eijk
-
38
33 A eijk .
ijknkmjlilmn eAAAeA = 4.2 3 Laplace , NN . .
ij eijk (contracted) . , Cartesian , a = b c ai = eijkbjck i .
Tij = vicj 2 (outer product)' (dyadic product)'
T = bc .
e- - e-(identity)" .
jlimjmilklmijk ee = 4.3 ) 2 4 ( 6 ) . 4.3 . +1 (i=l j=mi)
4.4
-
39
-1 (i=m j=li) 4.5 0 ( I,jj,l,m ) 4.6 k 1klmijk ee ( k ), 0 k
I,l,j,m . 3 (1,2,3) , 0 i=l j=m . 4 . ( i=j l=m e 0 .) 4.3 4.4, 4.5
4.6 . 4.4 4.5 4.3 . (1) i=l, j=m, klmlmkijk eee == ijke +1 -1 , +1
, (2) i=m, j=l, klmlmkijk eee == , klmijk ee (n.s.) -1 . 4.3 .
4.3
.
.
4.7
krkqkp
jrjqjp
iriqip
lmnijk ee
=
-
40
4.3 . 4.7 4.3 .
kljmkmjlijmijk ee = 4.8
e ( i) , j, k, l, m . 4.8 j = l , kk = 3 .
kmkmkmijmijk ee 23 == 4.9 , k = m , .
6=ijkijk ee 4.10 4.3 Cartesian 4.3.1
3 Euclid Cartesian (base) ( ) , (orthonormal triad)' . {x, y, z}
{y1 ,y2 ,y3},
-
41
{i, j, k} {i1, i2, i3} . [ .] 4.2 e- - 4.11 4.12 .
pqqp ii = 4.11 mpqmqp ieii = 4.12
Cartesian i1, i2, i3 u
ui , i1,i2,i3 Cartesian ui . u 2 {i1, i2, i3} {i1, i2, i3} .
Fig 4.1 Cartesian
{ 321 ,, iii } { '',,' 321 iii }
-
42
O Oy1y2y3 Oy1y2y3 . Oy1y2y3 (direction cosine) . ljp j p . ij
lj1,lj2,lj3 . .
232211' ililili jjjj ++= (j=1,2,3) ( p )
pjpj ili =' 4.13
),'cos(' pjpjjp yyiil == 4.14 i1, i2, i3 .
''' 332211 ililili pppp ++= ljp ip ij .
'jjpp ili = 4.15 4.13 4.15 .
)'(' kkpjppjpj illili == 4.16
-
43
[ 4.15 j k . j 4.16 .] 4.16 .
''' kjpjkkpjp iiill == ik .
jkkpjpll = 4.17 4.13 4.15 .
qpqqjqjpjjpp iillili === '
Fig. 4.2 321 ,, iii ',',' 321 iii 3i
pqjqjpll = 4.18
-
44
Cartesian ( ) () 4.17 4.18 4.14 lij () R (orthogonal) . (
RT=T-1) i1, i2 i3 i1, i2, i3 ( 4.2 ). 4.13
3211 0sincos' iiii ++= 3212 0cossin' iiii ++= 4.19
3213 100' iiii ++= 4.17 4.18 . 4.3.2 u i1, i2, i3
.
ppiuiuiuiuU =++= 333211 4.20 . 4.13 .
'''''''' 333211 jj iuiuiuiuU =++= 4.20 .
jpjp luu '= 4.21
-
45
. 4.15
'')'( jjjjpppp iuiluiuU ===
pjpjppj ulluu ==' 4.32 (1 ) . 4.4 Cartesian 4.4.1
Cartesian . Cartesian ( ) . (crysual lattice)' . 3 ( ) 3 . . 3
p1, p2, p3 . [p1, p2, p3] 0 . 3 (basis)' ,
-
46
v . v1, v2, v3 {p1, p2, p3} v ()
v1p1, v2p2, v3p3 v . v () . v2 v3 v1 (power) .( (v1)2 (v1)3 .)
{p1, p2, p3} (reciprocal)' {p1, p2, p3} .
ji
ji pp = 4.34
i j 1 3 . p1 p2, p3 . p1p1 = 1 . )),cos(/(1 1111 pppp = cos(p1,
p1) p1 p1 . p1 p1 (+) cos(p1,p1) , p1 p1 . p2 p2 p3 p3 . 5.3 . {pi}
{pi} .
-
47
4.4.2 () Cartesian {p1, p2,
p3} , {p1, p2, p3} 4.33 p1, p2, p3 v . v p1, p2, p3 .
ii pvpvpvpvv =++= 332211 4.27
{v1, v2, v3} v (covariant)' . . {v1, v2, v3} v (contravariant)'
. 4.34 . . (non-Cartesian) . pj (5.33) .
iji
iji
ii vvppvpv ===
jji
iji
ij vvppvpv === v v .
ii pvv = , ii pvv = 4.38
-
48
4.3 2 {p1, p2} {p1, p2} v . Cartesian (self-reciprocal)' . .
.
Fig. 4.4 2 },{ 21 pp },{ 21 pp v
-
49
4.4.3 Cartesian Cartesian Cartesian pi Cartesian ij . () L11, ,
L33 .
33
122
111
11 iLiLiLP ++= 3
322
221
122 iLiLiLP ++=
33
322
311
33 iLiLiLP ++= ( ,
) .
jj
ii iLP = 4.39 () Lij Cartesian ( ai = aej) .
jij ipL = 4.40
pi 4.39 () Mij .
jj
ii iMp = 4.41 j ( , ) Cartesian ij = ij . Mij 4.40 .
-
50
j
ij
i ipM = 4.42 4.34 4.39 4.41 .
)()( li
jk
ki
ji
ji iLiMpp ==
kl
ljk
il
kljk
i LMiiLM == )(
jik
jki LM = 4.43
.
IML = 4.44 [ M(=Mij) () , L(=Lij) () .] Cartesian Cartesian . r
.
jjii iypxr == 4.45
(y1, y2, y3) Cartesian (x1, x2, x3) Cartesian . {yi} {xi} 4.45
pi 4.39 .
jjjj
ii
ii iyiLxex ==
-
51
yi = Lijxi , yj .
iji
j xLy = 4.46
Mkj yj xi , j , 4.34 .
kii
kijij
kjj
k xxxLMyM ===
jj
ii yMx = 4.47 4.46 4.47 . Cartesian . 4.5 1 0 () Cartesian {i1,
i2, i3} Cartesian . u u u1, u2, u3 .
jj iuu = (j=1,2,3) 4.48 u =ujij 4.11 .
-
52
{i1, i2, i3} Cartesian u u1, u2, u3 . 4.32 . uj = ljpup ljp ij
ip . 1 () Cartesian ( CT1 ) 4.32 . CT1 4.32 . {i1, i2, i3} {u1, u2,
u3} .
pjpj ulu =' pjjp iil = ' 4.49
4.3 R(=lij) 4.49 up = ljpuj( 4.31) . CT1 . [ (coordinate
invariance), () .] Cartesian Cartesian . u (column) .
Tuuu ),,( 321
-
53
4.32 .
Ruu =' 4.50 u (u) 9 u = ujij = ujij), R(=lij) ( 4.14 ). 4.3 R ,
RRT = RTR = I R .
ijjkik ll = ijkjkill = 4.51 Cartesian , ui = lijuj ui = mijuj ,
ML .
kikkjkijjiji uMLulmumu )(''' === 1 ( ) . , m Cartesian ( mx1,
mx2, mx3) . ( (m) ) ( ) () (x1, x2, x3) . 1 . , m L = rp = m(rr) 1
. L
-
54
(quotient law) eijk . 1 , . 0 . 0 ( 0 ). , r2 = y12 +y22 + y32 .
r2 = x12+x22+x32 y12 + y22 + y32 . uv ,
1 () 0 (). ) 2 Cartesian u v vu . ) }{ iy iivu , }'{ iy .
jjkkjkkjikijkikjijii vuvuvullvlulvu ==== '' 4.51 . , .
-
55
0 . , [ e , E Fdr, eEdr] . . , . 2 1 0 , 1 . , grad CT1 . 4.6 2
() Cartesian 4.6.1 2 Cartesian (CT2) ( ) CT1 . 3 CT2 9 Tij . I j 1
3 . 33 . 4.58
-
56
{i1, i2, i3} . {i1, i2, i3} Tij . .
pqiqipij TllT =' 4.59 lip ( 4.57) . CT2 CT1 ( 4.50) . CT2 2 T .
(Cartesian) T T .
TRTRT =' 4.60 (i) Tij=lipTpqljq (4.59) . (ii) ljp 2 RT q-j .
(block form)' 3 (indicinal form)' . CT2 CT1 (dyadic product)' .
Cartesian ui vi u v .
-
57
jiij vuT = T= (uv) .
4.59 CT2 . ) CT1 4.49 .
pipi ulu =' qjqj vlv ='
.
))((''' qjqpipjiij vlulvuT == qjqpip vlul=
pqjqip Tll 4.59 . )( uvT = .
TuvT =
u 3x1 Tv 1x3 . 4.50 .
Ruu =' TTT Rvv ='
-
58
4.60 4.50 .
))((''' TTT RvRuvuT == TTT RTRRuvR == )(
CT2 4.60 . 4.59 ijT T ijT . (linearity) ( CT2 ) . S T Cartesian
ijS ijT CT2 ijij TS + CT2 . (S+T ). T CT2 ijT . T . 2
. 4.60 .
TTTTTTTTT RRTRTRRTRT === )()()'( 4.61 TT
4.60 CT2 . vuuvT TT == )( , TT vuT = . () T
() T 4.61 . ( TT T = ')'( TT T =
-
59
.) . T (symmetric) CT2 , TT =' . CT2 S ijjiT SSSS == , (skew
symmetric) CT2 , . CT2 .
)(2/1)(2/1 TT TTTTT ++= 4.62 )(2/1 TTT +
)(2/1 TTT . CT2 . 1) 2 T , 3 3 2 T(3) . 2) T () ( T = Tijeiej )
T = (tij) . CT2 T v (mapping) Tv = , TV = .
-
60
3) U(m)v(n) ( U B m n ) (a) () m+n-2 (b) () m+n-1 (c) m+n .
4.6.2 2 CT2 2 . . 2 {i1,i2} u {u1,u2} 2 T 22 .
, {i1,i2,i3} 3 Cartesian i3 0 u u3 T T13,T23,T31,T32,T33 2 ( 4.2
). i3 i1 i2 ( )
22 . ( 4.19)).
-
61
4.60 .
T11+T22 = T11+T22 .
22 (trace) (invariant)'. . T12-T21=T12-T21 . i2 i1 i1 i2 . (i3 )
2 , 3 i1=i2, i2=i1, i3=-i3 i3 () . (T12-T21) (T12=T21)2 . 2 .
detR=1 detT=detT . .
2112221121122211 '''' TTTTTTTT =
4.64 .
-
62
2 ( T12=T21) . 4.65 T . ( T12=T21) 0 . Cartesian 2 T Tij y3 (i3)
( 4.2 ) Tij 4.65 Mohr ( 5.5) . Mohr . (a) 2 CT2 (T11, T12) (T22,
-T12) . (b) a . (c) 2 . (T11,T12) (T22, -T12). (d) (b) i1-i2
Cartesian T () ( i1-i2) . T (1,2) Mohr .
+++++
=
2sin2cos)(
21)(
212cos2sin)(
21
2cos2sin)(212cos)(
21)(
21
1211222211121122
12112222112211
TTTTTTTT
TTTTTTTT
-
63
Fig. 4.5 CT2 T ),,( 221211 TTT ()
)',','( 221211 TTT Mohr )
)/tan )....(cos(sincos 22 abbaba =+=+ )/tan )....(sin(sincos 22
abbaba =+= 4.66
4.65 .
)22cos()(2/1' 221111 ++= aTTT )22sin('12 = aT
)22cos()(2/1' 221122 += aTTT 2/1
1222
2211 ])(4/1[ TTTa += )/()2(2tan 221112 TTT = 4.67
a Mohr , T () ( ) .( 2/1== a
)]/()2[(tan 1211121 TTT 'ijT .
4.67 ijT
-
64
'ijT 4.59 Mohr . Mohr 2 , . 3 CT2 Mohr .
-
65
5.
Fig. 5.1 Fig. 5.2
,
. X Y Z . , 5.1 X 5.2 . x , .
dzdxdzdxdyy
dydzdydzdxx xy
xyxyx
xx
++ )()(
0)( =+++ Xdxdydzdxdydxdydz
z zxzx
zx 5.1
-
66
0)( =++
+
dxdydzXzyxzxxyx 5.2
0=++
+
Xzyxzxxyx 5.3
y,z . ,
0=++
+ X
zyxzxxyx
0=++
+
Yzyxyzyxy
0=++
+ Z
zyxzyzzx 5.4
, , , . 3 3 6. 2 5.4 z , .
0=++
Xyxxyx
0=++
Y
yxyxy 5.5
2 , 2 3. 2 3 . , 5.5 ),( r
-
67
.
01 =+++
r
rrr Frrr
021 =+++
F
rrrrr 5.6
FFr r .
-
68
6. Saint-Venant
Fig. 6.1
, 6.1 . . 6.1(b), 6.1(a)
. (a) A-A, B-B, C-C , (c) . (c) , P
-
69
. A-A C-C
, C-C . 6.1(b) , L=2W, C-C . 6.2, .
Fig. 6.2 , A-A 5.1 A-A
. C-C 5.1 . , 6.1 P 6.2 P 2 , C-C .
-
70
. Saint-Venant . 7.
, . . . . 7.1
. 0 , . 7.1 , 2 const= , . , no , nto , , . x, y ,X Y .
Xmxyx =+ A Ymyxy =+ A
),( mA x, y .
-
71
Fig 7.1
, ,
. , () . , Saint-Venant . , . 7.2 .
0 . . ,
-
72
- .
-
73
8. 2
2 , , CLOSED FORM SOLUTION , . CLOSED FORM SOLUTION . 2 , CLOSED
FORM SOLUTION , . CLOSED FORM SOLUTION 2 , , . 8.1 8.1.1 (Plane
Stress Problem) x-y , z
),,( xyyx 0 . . , . Hooke .
)(1 yxx E = )(1 xyy E =
-
74
)( yxz E += G
xy
xy
= 0= z 0= yz 0= zx 8.1 8.1.2 (Plane Strain Problem)
( z) . , , . , . .
, y . Poisson . , . , . ,
. Hooke .
)}({1 zyxx E += )}({1 xzyy E +=
-
75
0)}({1 =+= yxzz E )( yxz += G
xy
xy
= 0= yz 0= zx 8.2 8.2 , .
)(1 ** yxx E = )(1 *
* xyy E = Gxy
xy * = 8.3
,
2*
1 =EE )1(
*
= GEEG =+=+= )1(2)1(2 *
**
8.4 xyyx ,, , yx , , Young's modulus E Poisson 8.4 yx , . ,
Young's modulus E
)1(1
2 ,
3.0 , 10% . 8.2
)}({1 zxxx E += )( yxz +=
-
76
)1
()1(
})1()1{(1
))}(({1
2
2
yx
yx
yxyxx
E
E
E
=
+=
++=
)1
()1( 2 xyy E =
8.2 ( )
Fig 8.1
. , . , . , .
-
77
, 8.1 . .
0=x 0= x 0= xy ax = 0=x )(2 0 ayxy = 0=y 0= y 0= xy 8.5 ay= 0=y
)(2 0 axxy =
.
20
)(ax
x = 20 )( ayy = )( 202 axyxy = 8.6 , .
0=+
yx
xyx 0=+
yx
yxy 8.7 8.6 8.7 , x, y 8.6 . 8.6 8.1 . Hooke , .
ayx
EE yxx 2
22
0)(1 ==
-
78
axy
EE xyy 2
22
0)(1 ==
axy
GGxy
xy 202 == 8.8 yv
xu
yx =
= , , , )()
3(1 2
3
20 yfxy
Eu x
a+= 8.9
)()3
(1 23
20 xgyx
Ev
ya
+= 8.10 , )(yf y , )(xg x
.
xxg
yyfxy
Eaxv
yu
xy +
+=+
= )()(4 2 0 8.11
, xy 7.8 xy . , .
yxxyxyyx
=+
22
2
2
2
8.12
,
. , , 8.8 xy 8.11 xy
-
79
, , . 8.3
2 (, , ) .
8.7 , 2 3. , (x, y) .
),( yxfx = ),( yxgy = ),( yxhxy = 8.13
, , 8.7 . 3 2, 8.13 1 . , f(x, y) , xy 8.7 .
)(),( xGdyx
yxfxy +
= 8.14 )(xG x . , 8.13 , ,
2
2
yx = . x
, 8.7 xy y . , ,
-
80
2
2
yx = 2
2
xy =
yxxy =
2
8.15
x , y , xy , Airy's stress function ( ) . 8.15 , 8.12 ,
Hooke
yxExyExyxyyx
+=
+
22
2
2
2 )1(2)()(1 8.16
8.15 , .
02 44
22
4
4
4
=+
+
yyxx 8.17
Laplacian 22
2
22
yx +
= ,
0))(( 422222
2
22
2
2
2
2
===+
+
yxyx
8.18
. . 8.18 E, . E .
xyyx ,, 8.3
-
81
. 8.18 , 21, )( 21 + , 1 2 . (Principle of superposition) . , ,
. 8.4
. , ),( r . , 8.18 . ,
0)11)(11 22
22
2
2
2
22
2
=+
+
+
+
rrrrrrrr 8.19 , .
2
2
2
11
+=
rrrr 2
2
r= )1(
=rrr
8.20
, r , 0= r r r 8.19 .
0)1)(1( 22
2
2
=++drd
rdrd
drd
rdrd 8.21
,
-
82
0112 32
23
3
4
4
=+++drd
rdrd
rdrd
rdrd 8.22 , )log( rtorer t == ,
0)2( 22 = DD 8.23 ,
dtdD = .
8.23
eCCCC ttt 2'4'3'2'1 )( +++= 8.24 r ,
DCrrBrrA +++= 22 loglog 8.25 ABCD .
-
83
Fig 8.3
8.20 .
)log21(21 2 rBrAC
drd
rr+++==
)log23(2 222
rBrAC
drd ++== 8.26
8.26 8.3(a)-(c) . A, B, C 8.3(a)-(c) . 3 . 8.3(a) , 3 . : (i) r
= a 1 =r
(ii) r = b 2 =r : u, v r . , u , v=0 . .
drdu
r = 8.27
ru= 8.28
8.28 u , 8.27
drdu
-
84
. . ,r Hooke , 8.26 , A, B, C . , . 8.27 8.28
04)()(1)(1)( ===EB
drd
drd
Er
EEdrrd r
rrr
8.29 B=0 8.30 , 8.3(a) .
22 rACr += 221 r
CCr +=
22 rAC = 221 r
CC = 8.31 8.31, . 8.3(b) , v
0B . ) a, b iP , 0P ? (Hint> r = a, R = b (8.31) .)
-
85
8.4.1
Fig 8.8
, . (stress concentration) . , , .
, 8.8 , x 0 =x , a
. , , , , . 8.8 .
, 8.8 04 = ( 8.18) . Saint-venant , , . 1) 0 =x 0=y 0=xy at Edge
(Cart.) 2) 0=r 0= r at )( ar = (Polar)
-
86
Fig. 8.5
, 1), 2) . , 2) , 1) . , 1) . 1)
22cos
2cos 0020
+==r 22sin0 =r
1 ) 2) , 8.4 8.5 . 8.5(b) . 8.5(c) .
=
021 =r
abb >>)(
(a) + 2cos21
0=r
2sin21
0=r(c) b >> a
-
87
,04 = a) ar = 0=r 0= r b) =r
22cos0 =r 2
2sin0 =r , 8.19 (a), (b) . , abr >>= .
2cos
2111
02
2
2 =+
=rrrr
2sin2
1)( 0=
=
rrr 8.33
8.33 , .
2cos)( = rf 8.34 , f(r) r . 8.34 8.19 , 2cos .
0)41)(41( 222
22
2
=++r
fdrdf
rdrfd
rdrd
rdrd 8.35
,
0992 322
23
3
4
4
=++drdf
rdrfd
rdrfd
rdrfd 8.36
8.22 , 423
42
21)( Cr
CrCrCrf +++= 7.37
-
88
2cos)( 4234221 CrCrCrC +++= 8.38
2cos)462(11 244312
2
2 rC
rCC
rrrr++=
+= 8.39
2cos)6122( 4322122
rCrCC
r++=
= 8.40
2sin)2662()1( 2443221 r
CrCrCC
rrr+=
= 8.41
(a), (b) 41 ~CC ,
01 41=C 02 =C 0
4
3 4aC = 0
2
4 2aC = 8.42
8.39~8.41 , 8.5(c)
. 8.5(b) , 8.5(a) .
2cos)431(2
)1(2 2
2
4
40
2
20
ra
ra
ra
r ++=
2cos)31(2)1(2 44
02
20
ra
ra ++=
2sin)231(2 22
4
40
ra
ra
r += 8.43
8.43 . .
ar = 0=r 0= r 8.44 2cos2 00 = 8.45
-
89
8.44 , 8.45
, 2 = , ,0=
.
2 = 0max 3 = ,0= 0min =
8.5(a) ,
3 , Kt=3 . , 8.6 . y
x .
.
Fig 8.6
-
90
8.4.2 8.7 , , , x y .
+
++
++= 22
222
2 )()(3
21
)1(1
11
babba
baba
baba
baa
oy
+ ba
aba
bba
b 232
2
)1(4 8.46
,
2222
, bacc
cxx =+= 8.47 A(x=a) y . maxy
oy ba
+= 21 8.48
x x
y
0A
B
aa
t t
b
0
0
-
91
t t
0
0
+t21K t
Fig. 8.7
tK
t
baKt 21
21 +=+= 8.49 A ab /2= , at = . B x b/a .
oxB = 8.50 8.47 8.50 . , .
0
+t21K t
0
Fig. 8.8 Fig. 8.9
-
92
8.8 , , 8.9 . 8.9 , . . . 8.4.3 (Edge)
x x
y y
P Q
Fig. 8.10 Fig. 8.11
8.10 8.11 ,
, . .
, 04 = . 8.10 , .
(1) 2 = 0, = r
-
93
(2) r yx FF , r PFF yx == ,0 .
8.20 ( ) , .
2 = . const
r=
. 1 constr
= 8.51
.
)( fr = 8.52 8.50 04 =
)11)(11(2
22
2
2
2
22
2
+
+
+
+
rrrrrrrr
0)2(1 22
4
4
3 =++= fdfd
dfd
r
02 22
4
4
=++ fd
fdd
fd 8.53
A-D .
sincossincos DCBAf +++= 8.54 , 8.52 8.54 ,
)sincossincos( DCBAr +++= 8.55 8.20 ,
)cos2sin2(1 DCrr
+= 0= 0= r 8.56
-
94
8.10 r C=0 .
cos2rD
r = 8.57 (2) ,
=22
cos Prdr 8.58
=22
0sin rdr 8.59
8.57 8.58
PD = 8.60
, 8.10 .
sin= rP 8.61
rP
r
cos2 = 0= 0= r 8.62
r 0 , simple
radial distribution . 8.11 , .
cos= rQ 8.63
rQ
r
sin2 = 0= 0= r 8.64
-
95
, simple radial distribution , 8.10 8.11 simple radial
distribution . 8.4.4
x
y
= ++ o
3rP
ord
(a) (b) (c) (a)
or
1r1 r1 2r
r2
2
Fig. 8.12
8.12 ,
( P) , , . 8.12(a) , (b),(c) (d) . (b) (c) 8.10 , d r1, r2 .
1
11
cos2r
Pr
= 8.65
2
22
cos2r
Pr
= 8.66
221 =+ 11 cosdr = 22 cosdr =
-
96
1r 2r . dP
r 2
1 = dP
r 2
2 =
21 rr = ),( ),( 2211 rr , (b)(c) ,
dP
2 .
, (d) dP
r 2
3 = . , (b),(c),(d) (a) y x
dP
x 2= 8.67
x y
])4(
41[223cos4 2224
xdd
dP
dP
rP
y +=+=
.
-
97
8.5
. . , . 8.5.1
O
y
xl
T x
y
rdr
d
z
z z
Fig. 8.13
. 8.13 zz = , z . d T r dr dT .
-
98
drrdT z 22= 8.68
2/dr = , z max ,
maxmax 2/2 d
rdr
z == 8.69
=== max32/0 max3 164 ddrrddTTd
8.70
PZT
dT == 3max 16 8.71
0 1
Fig. 8.14
o ,
8.14
PP GIT
dGZT
dGd====
)2/()2/(2/max
0 8.72
32
4dI P= 8.73
-
99
PGITll == 0 8.74
8.71 8.74 2 . 8.5.2 (Thin-walled)
(a) (b)
Fig. 8.15
8.15 T . () A,B 8.15(b) . 8.15(b) . A,B ,
11 = BBAA hh 8.75
BA hh , A,B . A,B 8.75
-
100
.
.consth= 8.76 8.76 .
Fig. 8.16
T
8.16 ds O z ,
rhdsdT = 8.77 (9) , == dsrhdsrhT 8.78
, dsr OCD 2 dsr
A 2 . ,
hAT 2= 8.79
T o T .
-
101
= 02 212 ThdsG 8.80 (12) (13)
= hdsGAT20 4 8.81
0h 0s .
02
00 4 GhA
Ts= 8.82 8.5.3 Saint-Venant
Saint-Venant . 3 .
(i) z=0 1zz = , 2zz = 21,
2121 // zz= 8.83 (ii) z . (iii) . (iii)
-
102
. 0 . (i) (ii) .
T
z
O x
y
Fig. 8.17
(x + u, y + v)
0r(x , y)
0
O x
y
Fig. 8.18
Saint-Venant
. 8.17 . z x-y
-
103
0 . x,y,z u,v,w , Saint-Venant .
===
),(00
0
yxwzxv
zyu
8.84
o , ),( yx
0),( =yx . Navier O . Saint-Venant 8.18 ,, . 8.84 .
+=
+=
=
+=
====
)(
)(
0
0
0
xyz
vyw
yxz
uxw
yz
zx
xyzyx
8.85
+=
=
====
)(
)(
0
0
0
xy
G
yx
G
yz
zx
xyzyx
8.86
-
104
8.85 , 8.86 ),( yx . , . , 8.84 , . 8.5.4
, , . . ()
0=++
+ Z
zyxzyzzx 8.87
(x,y) . ,
0,0 == Zz
0=+
yxyzzx 8.88
),(),,( yxgyxf yzzx == 2
8.88 , yzzx , . ),( yxf
yzx =
xyz = 8.89
. .
-
105
() () 8.18,
0)( =+
yxxzxyz 8.90
0)( =
yxyzxyz 8.91
0)( =+
yxzzxyz 8.92
. zxyz , z
8.92 . 8.90 8.91
'Cyxzxyz =
+ 8.93
Cyxzxyz =
+ 8.94
8.93 8.94
Saint-Venant 8.84 8.85 8.86 . 8.89 8.94
Cyx
=+
2
2
2
2 8.95 , 8.86 8.94 ,
02 GC = 8.96
02
2
2
2
2 Gyx
=+
8.97
-
106
8.97 . ()
x
y
z
A
B A
B
x
y
l
m
1
A
B A
B
O1 m
l1
O2
yz
xz x
y
dy
dx
A
Bds
Fig. 8.19
8.19
. AA BB . ,
21 'OAAO z BBOO '21 z 0 . , z
011 =+ ml yzzx 8.98
l,m x,y s .
dsdxm
dsdyl == , 8.99
-
107
, 8.99 8.98 0=
dsdx
dsdy
yzzx 8.100 8.92
0==
+
sdsdx
xdsdy
y 8.101
, C ,
C= () 8.102
C , C=0 . , 8.97 8.102 T . 8.20 , 0= . , +By By y ,
+Bx
Bx .
==
+
=+=
+=
+
+
dydx
dydxxdxdyy
dydxx
xy
y
dydxxy
dydxxdydxyT
B
B
B
B
xx
yy
yzzx
yzzx
2
}]{[}]{[
)(
)(
)(
8.103
-
108
,2= A dAT 8.104
x
y
+By
+BxBx
By
O
dy
dxT
Fig. 8.20 () ( 8.21)
01// 2222 =+ byax 8.102 C=0 .
y
xO a
b
a > b
Fig. 8.21
-
109
)1( 22
2
2
0 += by
axC 8.105
0C 8.97 8.104
Cba
baC += )(2 2222
0 8.106
33
22
0)(22
babaTGC
+== 8.107 8.5.5 (Membrane analogy)
L. Prandtl(1903) , .
O
x
y
xq
O
dy
s n
dx
dy
dx
S S
xdx
z
(a) (b) Fig. 8.22
-
110
q , 8.22 . , dxdy
2
2
2
2 1,1y
zx
z
yx =
= 8.108 , z . ,
xx
dxd = , yydyd = 8.109
, S
0=+ dyqdxSdxdySdydxyx 8.110
8.108 8.110
Sq
yz
xz =
+
2
2
2
2
8.111
8.97 . , . z
0= () 0=z 02 G Sq / = dxdyT 2 zdxdy2 ( 2
)
-
111
n
( ) nz
( ) .
Sqz
G /2 0=
8.112
Sqnz
Gn
//
2/
0
= 8.113
Sq
Gnz =
02 8.114
,
= qASdsnz 8.115 , 8.114 . = AGds 02 8.116 , memebrane analogy .
8.5.6
const.= 02= const.
0const.1 ==)a( )( b
Fig. 8.23
-
112
8.23(a) . 8.23(b) . , membrane analogy . ()
8.24 membrane analogy , q . .
Sq
yz
xz =
+
2
2
2
2
8.117
, 22 / yz 0/ 22 = yz ,
Fig. 8.24
Sq
dxzd =2
2
8.118
,
1CxSq
dxdz += 8.119 x=0 0/ =dxdz 01 =C . ,
22
2Cx
Sqz += 8.120
-
113
2/hx = z=0 SqhC 8/22 = . ,
Sqhx
Sqz
82
22 += 8.121
, 2/hx =
Sqh
dxdz
2max= 8.122
V ,
= 203
122
h
SqbhbzdxV 8.123
oG2 Sq / ,
0
3
32 GbhVT == 8.124
xG 02 = 8.125 0max hG= 8.126
GbhT30
3= 8.127
2max3bh
T= 8.128 8.5.7
8.81 8.82 A
-
114
. . ()
)a( )b(
h
b b
h
Fig. 8.25
8.25 (a) (b) 01 02 .
31
013
GbhT= 8.129
hbGT
GhAsT
32
202
024
4 == 8.130 21 TT =
1)(4
343 2
3
3
02
01 >>==hb
bhhb
8.131 0201 =
-
115
2
2
1 )(34
bh
TT
= 8.132 1.0/ bh 321 104/ TT . 8.25(b) (a) .
-
116
9.
, . . 9.1 (Strain energy) , , . Cauchy . Cauchy(1868) . Cauchy ,
.
9.1~9.4 Cauchy . 9.1 zyx PPP ,, 9.1 zyx ( nm,,A ) .
-
117
Fig. 9.1
nmP zxyxxx ++= A nmP zyyxyy ++= A nmP zyzzxz ++= A 9.1
yxxy = xzzx = zyyz = 9.2
0=++
+ X
zyxzxyxx
0=++
+
Yzyxzyyxy
0=++
+ Z
zyxzyzzx 9.3
xu
x =
yv
y =
zu
xw
zx +
= 9.4
-
118
Clapeyron . , 0E . , , 1E . , 01, EE 01 EEE = , 0>E . 0>E
0>E () . .
9.2 , .
zYXzxyxzxyx ,,,,,,
Fig. 9.2 Fig. 9.3
Xxx ,, 9.3
udydzdxudydzdxx xx
xx +
+ 21)()(
21
-
119
dxdydzx
udxdydz xxx +
21
21 dxdydz
xu xxx )(2
1+= 9.5
XudxdydzdxdydzdxuX x 21)
21(
21 + 9.6 xz 9.4
wdydzdzdxxwwdydzdx
x zxxx
zx +
+ 21)()(
21
dxdydzx
wxw xz
zx )(21
+
9.7
Fig 9.4
dxdydzy
v yyy )(21
+ dxdydz
zw zxz )(2
1+
dxdydzx
vxv xy
xy )(21
+
dxdydzy
uyu xy
xy )(21
+
dxdydzy
wyw yz
yz )(21
+
dxdydzz
vzv yz
yz )(21
+
dxdydzz
uzu zx
zx )(21
+
-
120
Y,Z
Yvdxdydz21 , Zwdxdydz
21
dU
dxdydzZzyx
w
Yyyy
vXzyx
u
dU
zyzzx
zyyxyzxyxx
zxzxyzyzxyxyzzyyxx
)](
)()(
)[(21
++
++
++
+++
++
+
+++++=
9.8 [ ] 2,3,4 0. ,
dxdydzdU zxzxxx )(21 ++= 9.9
U0 dU=dxdydz
)(21
0 zxzxyzyzxyxyxyxyzzyyxxU ++++++= 9.10 0U . Hooke 0U , .
-
121
)(21
)}(2){(21
222
2220
zxyxxy
xzzyyxzyx
G
EU
+++
++++= 9.11
)(21
)}{()()21)(1(2
222
22220
zxyzxy
zyxzyx GEU
+++
++++++= 9.12
0U . Poisson . 9.2 . , A , 1, 2 . 1: zxxzyx ,, 2: zxxzyx ,, 1, 2
, , , 1 ,
0=++
+ X
zyxzxxyx
-
122
0=++
+
Yzyxyzyxy
0=++
+
Zzyx
zyzxz 9.13 x,y,z
zyx PPP , ( nm,,A )
xzxyxx pnm =++ A yzyyxy pnm =++ A zzyzxz pnm =++ A 9.14
, 2 9.14 9.15 . 9.15 9.16 1 2 = xxx
0=+
+
zyxzxxyx 0=
++
zyxyzyxy
0=+
+
zyxzyzzx 9.17
0=++ nm zxyxx A
yzyyxy nm 0=++ A zzyzxz nm 0=++ A 9.18
9.17 9.18 1 2 ,
-
123
. , 0, 0U 9.8 9.9
0==== zyx == ,, yyxx , . , . , , , , () . 9.3 (Principle of
virtual work) ,
( ) 0 . .
, , 0 . . . ( .) S: V:S Su
S :()
-
124
P: S )( zyx PPP F:(X,Y,Z)
uS:wvu sU
dSwPvPuPU
s zyxs ++= )( 9.19 +++++= s yzyxyzxxyxs vnmunmU )()[( AA
dSwnm zyzzx ])( +++ A +++++= s yzyxyzxxyx mwvuwvu )()[( A
dsnwvu zyzzx ])( +++ 9.20 Gauss (Divergence theory) ,
+++++= v yzyxyzxxyxs wvuywvuxU )()([ dVwvu
z zyzzx)]( ++
+
vzyx
uzyx
yzyxyzxv
xyx )()[( +
++
++
= zw
yv
xuw
zyx zyxzyzzx
+
++
++
+ )(
dVxw
zu
zv
yw
yu
xv
zxyzxy )]()()( +
++
++
+ 9.21 ,
+++++ Vs zyx dVwZvYuXdSwPvPuP )()( dVzxzxV yzyzxyxyzzyyxx )(
+++++= 9.22
-
125
. . , , . , . .
Fig 9.5
9.5 2 ,
, .
-
126
sincos yx PP + += cos)sinsin( RQ ++ sin)coscos( RQ 9.4
9.22 . 9.10 , +++V
zxzxyyxx dV)(
=++= V Vzxzxxx dVUdVUU
000 )( 9.23
8.22
)2
cos()2
cos(
sincos
++=+
RQ
PP yx
coscossinsin
RQPRQP
y
x
+=+=
-
127
++++=V
zyS
xV
dVwZvYuXdSwPvPuPdVU )()(0
9.24 ( zyx PPP ) (X,Y,Z)
++++=V
zyS
xV
dVZwYvXudSwPvPuPdVU )()(0
9.25 ,
++++=V
zyS
xV
dVZwYvXudSwPvPuPdVU )()(0
9.26 (potential energy of the system) . 9.26 1 , 3 , 2 .
, 9.10 , . , , ,
0= .
-
128
Fig. 9.6
9.6 . . : 342321 xaxaxaaw +++= : ,0 0 10 === aw x 0 0 20 == = aw
x
34
23 xaxaw += 243 32 xaxaw += xaaw 43 62 +=
A A
0 0
22 )(21 )2/( dxwEIdxEIM
000
2)(21 MdxwEI = A
A== xw0 )32()62(
21)()(
21 2
4300
2430
0
2 AAA
A
AaaMdxxaaEIwMdxwEI x ++== =
)32()12124(21 2
43032
42
432
3 AAAAA aaMaaaaEI +++=
-
129
0=
03
=a 0
4
=a
02)128(21
042
3 =+ AAA MaaEI
03)2412(21 2
043
32 =+ AAA MaaEI
EIMa2
03 = 04 =a 202 xEI
Mw = , , Rayleigh-Ritz 9.5
Fig. 9.7 Fig. 9.8
9.7 , U, P
. P 9.8 . OBC CU (Complementary strain energy)
-
130
. , 9.7 .
PUU C =+ 9.27 9.8
= PU = UP 9.28
PUC = PUC= 9.29
, 9.7 , .
EAPA= 9.30
)2/()2/(21 22 AA EAEAPPUU C ==== 9.31
9.7 P M,
. 9.31 , PU / , 9.29 . 9.9 (a),(b) CUU P , PU / .
-
131
Fig. 9.9
,
, () . , .
0)()()( =++
++++
+ XXzyx
zxzxyyxx 9.32 , (Castigliano) .
dVUUUUUUU zxzx
Cyz
yz
Cxy
xy
C
Vz
z
Cx
y
Cx
x
CC ))(
++
++
+=
9.33
xx
CU = y
y
CU =
+++=V
zxzxyyxxC dVU )( 9.34
-
132
++++= V xyzyxC xv
yu
zu
yu
xuU )()[(
dVzu
xw
yw
zv
zxyz ])()( +
++
+
})()({})()(
{})()([{z
wz
wy
vy
vx
ux
u zzyyxxV
+
+
})()(
{})()(
{})()(
{z
vz
vx
vx
vy
uy
u yzyzxyxyxyxy
+
+
+
dVz
uz
ux
wx
wy
wy
w zxzxzxzxyzyz }])()(
{})()(
{})()(
{
+
+
+
9.35
9.32
+
++
++
=V
yzyxyzxxyxC z
vy
vx
vz
uy
ux
uU })()()(
{})()()([(
dVZuYuXuz
wy
wx
w zyzzx )](})()()({ +++
++
)()[( yzyxy
Szxxyx nvmvvnumuu +++++= AA
++++++V
zyzzx dVZwYvXudSnumww )()]( A
+++++=V
zyxS
dVZwYvXudSPwPvPu )()(
++++++++=VSu
zyxzyxS
dVZwYvXudSPwPvPudSPwPvPu )()()( 000000
0=== ZYX , 0000 === wvu (
-
133
) xx PdSP =0 , yy PdSP =0 , zz PdSP =0 ( )
=
++=n
izyxC iPwPvPuU
1)( 9.36
xi
Ci P
Uu =
yi
Ci P
Uv =
zi
Ci P
Uw = 9.37
xyz .
i
Ci P
U= 9.38
, , .
ii
UP = 9.39
-
134
10.
. , , . , , . , 0 .
-
135
h, l, b . 9.1(a) (b) .
O
yl
h x O
y
x
Fig. 10.1
10.1 oTyxT = ),(
9.1(a) oyTxT T == 0=xyT 0=== xyyx . 10.1(b), lTo = . 10.1(b) l
)1( oTl + , P l
EBH
PlTEbhTPl
o
o ++= 2)1(
)1( 10.1
,
EbhTP o= 10.2 , .
-
136
ETox = , 0=y , 0=xy 10.3 10.2
hyTyxT o2),( =
y , , 10.1(a)
, . 10.1(b) , x
hyToxT 2= x .
hyETox 2= 10.4
, 0x = lx = , x 0 , .
ETbhM o62
= 10.5 , M (). , 10.1(a) , 10.1(b)
10.5 10.1(b) . 10.4 , 10.1(a) . , .
-
137
10.3 22),(
=hyTyxT o
, 10.2 , 10.1(a)
. 10.1(b) , x
22
=hyToxT x
.
22
=hyETox 10.6
, 0x = lx =
, P .
ETbhbdyP oh
h x
32/
2/== 10.7
10.1(a) , 10.1(b)
10.7 10.1(b) . , .
ETbhp
ox 31' == 10.8
10.6 10.8 , .
2231
=
hyETET oox 10.9
-
138
, 0x = lx = 10.9 . 10.9 , x 0 , 10.9 Saint-Venant . , 10.1(a)
10.9 .
, Hooke . T , T ( ) () . Hooke . , . :
[ ])(1 zyxx ET ++= [ ])(1 xzyy ET ++= [ ])(1 yxzz ET ++=
10.10
Gxyxy / = , Gyzyz / = , Gzxzx / = 10.11 :
-
139
[ ])(1 zrr ET ++= [ ])(1 rzET ++= [ ])(1 ++= rzz ET 10.12
Grr / = , Gzz / = , Gzrzr / = 10.13
() , 10.10 10.12 2 . , , . 11.
, , closed form solution . , , , .
-
140
, , , , . 11.1 1 1) ,
Fig. 11.1 : kuF = Hooke , A
)211 ( AAAA uukF = , )( 122 AAAA uukF = 11.1 ,
=
2
1
2
1
A
A
A
A
AA
AA
FF
uu
kkkk 11.2
-
141
11.2 .
=
2
1
2
1
2221
1211
A
A
A
AAA
AA
FF
uu
kkkk 11.3
, 2 Ak11 , Ak12 , jiij kk = k . ijk . Ak11 , 2 1 11 =Au , 1 (
10.2). Ak12 , 1 2 12 =Au 1 ( 10.3).
Fig. 11.2 Fig. 11.3
.
AA kk =11 , AA kk =12 , AA kk =21 , AA kk =22
11.3 . [ ]{ } { }AAA fuk = 11.4 B , .
=
2
1
2
1
2221
1211
B
B
B
BBB
BB
FF
uu
kkkk
11.5
-
142
[ ]{ } { }BBB fuk = 11.6
BB kk =11 , BB kk =12 , BB kk =21 , BB kk =22
11.1 , 12 BA uu = , 11.3, 11.5 .
+=
+
2
12
1
3
2
1
2221
12112221
1211
0
0
B
BA
A
BB
BBAA
AA
FFF
F
uuu
kkkkkk
kk 11.7
11 Auu = , 122 BA uuu == , 23 Buu = , 11.1 .
=+=
=
0
0
12
02
1
BA
B
FFFF
u
Fig. 11.4
== ,, 1211 AAAA kkkk , 11.7
-
143
=
+
0
1
3
2 00
0
)(0
F
F
uu
kk
kkkkkk A
BB
BBAA
AA
11.8
11.8 132 ,, AFuu . ,
, . 11.8 , .
=+=+
=
032
32
12
0)(Fukukukukk
Fuk
BB
BBA
AA
11.9
32 ,uu , 11.9 2,3 ,
1 1 , , 1AF . .
AkFu 02 = ,
BA kF
kFu 003 += , 01 FFA = 11.10
. , , . , 11.7 () . , 11.8 11.9 1 . 2:
-
144
Fig. 11.5
A 1 11.1 .
=
2
1
2
1
2221
1211
A
A
A
AAA
AA
FF
uu
kkkk
11.11
A
AAA
lESk =11 ,
A
AAA
lESk =12 ,
A
AAA
lESk =21 ,
A
AAA
lESk =22 11.12
B
=
2
1
2
1
2221
1211
B
B
B
BBB
BB
FF
uu
kkkk
11.13
B
BBB
lESk =11 ,
B
BBB
lESk =12 ,
B
BBB
lESk =21 ,
B
BBB
lESk =22 11.14
11.11 11.13 , 1
+=
+
2
12
1
3
2
1
2221
12112221
1211
0)(
0
B
BA
A
BB
BBAA
AA
FFF
F
uuu
kkkkkk
kk 11.15
2312211 ,, BBAA uuuuuuu ==== .
-
145
11.5 ,
+=
+
2
12
1
3
2
1
0
)(
0
B
BA
A
B
BB
B
BB
B
BB
B
BB
A
AA
A
AA
A
AA
A
AA
FFF
F
uuu
lES
lES
lES
lES
lES
lES
lES
lES
11.16
11.16 . 11.2
1 . FEM 1 2 3 . , , . FEM , . 11.3 3
11.6 11.6(b) . c . , . 11.6 , , , b
-
146
. , 1 , . 11.7 3 6 . , 1, 2 . ,
Fig. 11.6(a) Fig. 11.6(b)
Fig. 11.7 [ ] [ ] { }fuk = 11.17
-
147
[ ]
=
666564636261
565554535251
464544434241
363534333231
262524232221
161514131211
kkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkk
k 11.18
[ ]
=
k
k
j
j
i
i
vuvuvu
u , [ ]
=
k
k
j
j
i
i
YXYXYX
f 11.19
2 , lmk .
lmk , . lmk . - - () , -
, , . , . , , . , .
, lmk ,
-
148
. 11.4
. )(
1 2 yxxE += , )(1 2 xyy
E += , 0=z
xyxyxyEG )1(2 2== , 0=yz , 0=zx 11.20
11.20 , { } [ ]{ } D= 11.21 ,
{ }
=xy
y
x
, { }
=xy
y
x
, [ ]
=
2/)1(000101
1 2
ED
11.22 ,
=
xy
y
x
xy
y
x E
2/)1(000101
1 2 11.23
11.5
11.6 , ()
-
149
, , . , , , . , , . , , , . , . , , . .
yxu 321 ++= , yxv 654 ++= 11.24
, 11.24 ,
, . , 11.8 11.9 . 11.8 11.9 , .
-
150
Fig. 11.8 Fig. 11.9
, 24
++=++=++=++=++=++=
kkkkkk
jjjjjj
iiiiii
yxvyxuyxvyxuyxvyxu
654321
654321
654321
,
,,
11.25
, .
11.25 , { } [ ]{ }Tu = 11.26 ,
{ }
=
=
=
6
5
4
3
2
1
}{,
10000001
10000001
10000001
][,
kk
kk
jj
ji
ii
ii
k
k
j
j
i
i
yxyx
yxyx
yxyx
T
vuvuvu
u
-
151
}{][}{ 1 uT = 11.27
=
21]T[ 1
ijkijk
jiikkj
ijjikiikjkkj
ijkijk
jiikkj
ijjikiikjkkj
xxxxxxyyyyyy
yxyxyxyxyxyxxxxxxxyyyyyy
yxyxyxyxyxyx
000000
000000
000
000
11.28
kk
jj
ii
yxyxyx
111
det2 = 11.29
.
xv
yu
yv
xu
xyyx +
==
= 11.30 11.24 11.30 .
5362 +=== xyyx 11.31 11.31 ,
-
152
)16()63(
010100100000000010
6
5
4
3
2
1
=
xy
y
x
11.32
, { } ]][[ B= 11.33 11.27 11.33 , { } ]][[][]][[ 1 uNuTB ==
11.34
1]][[][ = TBN
11.34 x,y , . , . , , , .
{ } { } Txyxyyyxxw =++=int 11.35 11.21 ,
{ } { } ][int Dw T= 11.34 ,
{ } { }uNDNuw TT ]][[][int = 11.36 1
-
153
{ } { }udxdydzNDNudvolwW TT
V
]]][[][[intint == 11.37 ,
}{][ uFW Text = 11.38 , F ,
{ }
=
k
k
j
j
i
i
YXYXYX
F 11.39
, { } { } { } { }udzdydxNDNuuF TTT ]]][[][[ = 11.40 , , { } { }
]]][[][[ = dzdydxNDNuF TTT { } { }udzdydxNDNF T ]]][[][[ = 11.41
11.41 [ ] . , [ ] = dzdydxNDNk T ]][[][ 11.42 to
-
154
[ ] = dydxNDNtk T ]][[][0 11.43 x,y [ ] ]][[][0 NDNtk T= 11.44 [
]k 11.18 66 , 11.28, 11.32 11.34 .
[ ] [ ][ ]
==
jiijikkikjjk
ijkijk
jiikkj
yyxxyyxxyyxxxxxxxx
yyyyyyTBN 000
000
211
11.45 , [ ]D 11.22 , [ ]k .
)1(4 20
=Etc
+= 2211 )(2
1)( jkkj xxyyck
+= )()(
21)()(12 kjjkjkkj yyxxxxyyck
+= )()(
21)()(13 kijkikkj xxxxyyyyck
+= )()(
21)()(14 ikjkkikj yyxxxxyyck
+= )()(
21)()(15 ijjkjikj xxxxyyyyck
-
155
+= )()(
21)()(16 jijkijkj yyxxxxyyck
1221 kk =
+= 2222 )(2
1)( kjjk yyxxck
+= )()(
21)()(23 kikjikjk xxyyyyxxck
+= )()(
21)()(24 ikkjkijk yyyyxxxxck
+= )()(
21)()(25 ijkjjijk xxyyyyxxck
+= )()(
21)()(26 jikjijjk yyyyxxxxck
,1331 kk = 2332 kk =
+= 2233 )(2
1)( kiik xxyyck
+= )()(
21)()(34 ikkikiik yyxxxxyyck
+= )()(
21)()(35 ijkijiik xxxxyyyyck
+= )()(
21)()(36 jikiijik yyxxxxyyck
,1441 kk = ,2442 kk = 3443 kk =
+= 2244 )(2
1)( ikki yyxxck
+= )()(
21)()(45 ijikjiki xxyyyyxxck
+= )()(
21)()(46 jiikijki yyyyxxxxck
,1551 kk = ,2552 kk = ,3553 kk = 4554 kk =
-
156
+= 2255 )(2
1)( ijji xxyyck
+= )()(
21)()(56 jiijijji yyxxxxyyck
,1661 kk = ,2662 kk = ,3663 kk = ,4664 kk = 5665 kk =
+= 2266 )(2
1)( jiij yyxxck
11.6
, . A,B .
-
157
3
4
Fb
Fa
2
1
1
2
Fig. 11.11
11.11, , 4
. , . , .
==
b
a
FyFxvyux
"
""
4
01 03,1
11.46
, 1 , 1
}{][}{ 111 kf = 11.47 , .
-
158
=
3
3
2
2
1
1
166
161
126
122
121
116
115
114
113
112
111
13
13
12
12
11
11
vuvuvu
kk
kkkkkkkkk
YXYXYX
"""""""######"""""
11.48
, mnX , mnY m , n . mk11 , mk12 , .
11.46 11.48 .
,
,
,
31562
1542
153
13
31362
1342
133
12
31162
1142
113
11
vkvkukXvkvkukXvkvkukX
++=++=++=
++=++=++=
31662
1642
163
13
31462
1442
143
12
31262
1242
123
11
vkvkukYvkvkukYvkvkukY
11 .49
2
=
4
4
2
2
3
3
266
261
226
222
221
216
215
214
213
212
211
24
24
22
22
23
23
vuvuvu
kk
kkkkkkkkk
YXYXYX
"""""""######"""""
11.50
, 1 11.49
10.46 .
-
159
42664
2652
2642
2633
262
24
42564
2552
2542
2533
252
24
42464
2452
2442
2433
242
22
42364
2352
2342
2333
232
22
42264
2252
2242
2233
222
23
42164
2152
2142
2133
212
23
vkukvkukvkY
vkukvkukvkx
vkukvkukvkY
vkukvkukvkx
vkukvkukvkY
vkukvkukvkx
++++=++++=++++=++++=++++=++++=
11.51 ,
24
24
23
13
22
12
22
12
,
0,0,0
YFXFYYYYXX
ba ==+=+=+=
11.52
, 11.49 11.51 .
)853.11(
)753.11(
)653.11()()()(
)553.11()()()(
)453.11()()()(
)353.11()()()(
)253.11(
)153.11(
42664
2653
2622
2642
263
24
42564
2553
2522
2542
253
24
42264
2253
222
1662
224
1642
223
163
23
13
42164
2153
212
1562
214
1542
213
153
23
13
42464
2453
242
1462
244
1442
243
143
22
12
42364
2353
232
1362
234
1342
233
133
22
12
31262
1242
123
11
31162
1142
113
11
++++=++++=
+++++++=++++++++=++++++++=++++++++=+
++=++=
vkukvkvkukYvkukvkvkukX
vkukvkkvkkukkYYvkukvkkvkkukkXXvkukvkkvkkukkYYvkukvkkvkkukkXX
vkvkukYvkvkukX
11.53 8 , 4
. 11.53, 53-3, 53-4, 53-6, 53-7, 53-8 5 11.52 , . 5, 5
-
160
. , 53-1,53-2,53-5 ( ) .
+++++++++
=
4
4
3
2
2
266
265
262
264
263
256
255
252
254
253
226
225
222
166
224
164
223
163
246
245
242
146
244
144
243
143
236
235
232
136
234
134
233
133
)()()(
)()()(
)()()(
0000
vuvvu
kkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkk
FF
b
a
11.54
11.54 1.
, 53-1, 53-2, 53-5 . 11.54 . { } [ ]{ }KF = 11.55
11.54 , 11.34 . 11.23 . , , . . 11.11 , 11.54 5 , . , 2 , 100
200 1 .
-
161
11.7
Fig. 11.12
Fig. 11.13
-
162
Fig. 11.14
, 11.12 .
A b . b . 11.13 a , . , . 11.14 . , .
