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7/30/2019 Stability of structures - Solved examples
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0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
0 = 0
0 > 0
= P/Pkr = P L/(4k).
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L/2
F = 4k/L
22222222
dd dd '
Pk
0
cF
cos 0L/2 cos 0L/2
(; 0, ) =
1
2 k[2( 0)]2
P L(cos 0 cos ) 1
2 F L(sin 0 sin )
=d
d =
4k( 0) P L sin + 1
2F L cos
= 0 = 0
P = 4kL 0 +
12
cos
sin
= 20 L
P = 0
= 20
d
d= ( 0) P L sin + 4k0 cos
= 4k P L sin + 4k0(cos 1) = 0
= 0 PI,
P = 4
k
L
+ 0(cos 1)
sin PII
2 =d2
d2 ()2 = (4k P L cos 4k0 sin )()2 = 0
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EI
k
k
pP =1
48
F L3
EI
pq =5
384
qL4
EI.
jP =1
8
F L2
kand jq =
1
16
qL3
k.
jP = pP
kP = 6EIL
jq = pq
kq = 12EIL
Pcr =4k
L PPkr =
24EI
L2and Pqcr =
48EI
L2.
k
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sin = 0 (k1 4k2)L2 cos + 4k2L2 2k2Lu = 0
k2 = k k1 = k
P = kL 4 + ( 32
4)cos (, P)
cos = 1 +P
2kL 3
2
u
L,
P = kL4
2 + (8 3)
u
L ,(u, P)
= 0
u =P
3k
P = [4 + (32
4)cos ]kL
P =
kL
4 2 + (8 3) uL = 1 = 5
= 1
P = (4 52
cos )kL
P =1
3kL
2 + 5
u
L
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
-0.3 -0.2 -0.1 0 0.1 0.2 0.3
PkL
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0
0.5
1
1.5
2
2.5
3
3.5
4
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
PkL
u/L
0.333*(2+5*x)3*x
(Pkr =32
kL
= 5
P = (4 + 72
cos )kL
P =
kL 10
7u
L
0
1
2
3
4
5
6
7
8
-0.3 -0.2 -0.1 0 0.1 0.2 0.3
PkL
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0
1
2
3
4
5
6
7
8
0 0.5 1 1.5 2 2.5 3
PkL
u/L
-10+7*x3*x
k1 = 5k2
0
U =1
2k1L
2(sin sin 0)2 + k2u2 + 12
[u 2L(cos 0 cos )]2
= k1L2(sin
sin 0)cos + k2[u
2L(cos 0
cos )](
2L sin ) = 0
u= 2k2u + k2[u 2L(cos 0 cos )] P = 0
u
u =k1
2k2
sin sin 0tan
L 2L(cos cos 0)
P = 3k2u 2k2L((cos cos 0) = 3k12
sin sin 0tan
L 8k2L((cos cos 0)
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= (1, 2) =1
2k21(1, 2) +
1
2k22(1, 2) P(1, 2)
= 0
1= k1
11
+ k221
P1
= 0
2= k1
12
+ k222
P2
= 0
11
= 2 + 21 +1
222 12
2
1 = 1 1
2
2
1 1
2
2
2 + 12
1= L
21 2 + 31
3
2212 +
3
21
22
1
232
12
= 1 12
21 1
222 + 12
22
= 2 + 22 12 +1
221
2= L
22 1 + 32
1
231 +
3
2212
3
21
22
1 = 2 = = 0 1 = 2 = 0
2 =2
21[(1)]
2 + 22
12(1)(2) +
2
22[(2)]
2
= ((1) (2))
22
1
212
212
22
1
(1)
(2)
,
K = [2/ij ]
2
> 0 K
1 = 2 = 0
2
21= k1
2121
+ k
11
2+ k2
2221
+ k
21
2 P
2
21
2
21= 5k 2P L
2
12= 4k + P L
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2
22= 5k 2P L
P = k/L K K
Kx = x (K I)x = 0 det(K I) = 0
det
5 2 4
4 5 2
= 0
1 = 1 ja 2 = 9 3 = 1 = 3
< 1 K
= 1
3 33 3
1
2
=
0
0
1 = 2
Pkr = k/L
2222222222
= 3 1 11 1
12
=
00
1 = 2Pkr = 3k/L
2222222222 22222222
22 = 1
1 = 2 = 1 = 21
= 0 k + 16
3
2 + 12
2 1 P L 1
23
= 0
k 1 + 16
2
1 + 12
2 P L 1 + 1
22
= 0
= 0 tai PII =
1 + 16
2
kL
= 3 1 = 2 = 2 =1
1
= 0 k 3+ 32
3
2 + 52
2 (1 22) P L 3+ 92
3
= 0
3k 1 + 122 3 + 922 P L 3 +
92
2 = 0 = 0 tai PII I = 1 + 122 3kL
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0
0.5
1
1.5
2
2.5
3
3.5
4
-0.4 -0.2 0 0.2 0.4
PIIPII I
PII PII I
PII 1 = 2
2
21
PII
= k
+
1
63
+ k
2 +
1
222
+ k
+
1
63
0 + k(1)2 k
1 +1
62
2 +3
22
= k
3 +7
62 +
1
64
> 0
PII I 1 = 2, 1 = 2
2
21
PIII
= k
3+
3
23
3+ k
2 +
5
222
k
3+3
23
(2) + k(1 22)2
3k
1 +1
22
2 +15
22
= k 1 +7
2
2 +3
2
4 < 0, PII PII I
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cv
E' a ' Eac
P
T
c
L
(v, ) =1
2k2A +
1
2k2B PD
= k(v2 + a2 sin2 ) P[v + L(1 cos )]= ka2(u2 + sin2 ) P
u +
1
(1 cos )
,
v = au a = L P = ka
ka2
= = u2 + sin2
u + 1
(1 cos )
u= 2u = 0 u =
2
= 2 sin cos 1
sin = 0
= sin
2cos 1
= 0
sin = 0 = 0 = 2 cos
u = /2 = 0
2
u2= 2,
2
u= 0,
2
2= 2cos 2
cos
u = /2 = 0
K = 2 0
0 2
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x
x x
x
da = dx E x, u
cy, v
dadv
u u + du
sin = dvda
= dvdx
= v
= arcsin v
= 1/R =
1
R= =
v1 (v)2 ,
d arcsin x
dx=
11 x2
da E x, u
cy, v
dadv
dx
da = dv2 + dx2 = dx(v)2 + 1
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tan =dv
dx= v
= arctan v
=1
R=
a
not
x!!
=1
1 + v2
a
dv
dx
where
d arctan x
dx=
1
1 + x2
=1
1 + v2d2v
dx21
1 + v2(where da = dx
1 + v2)
=v
(1 + v2)
1 + v2=
v
(1 + v2)3/2
= v
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Pcr
e ee eE'
v(4)1 + k2v1 = 0 k
2 = P/2EI
v(4)2 = 0
BC : v1L
2
= v1
L2
= v2
L2
= v2
L2
= 0
v1(0) = v2(0)
v1(0) = v2(0)
M1(0) = M2(0)
Q1(0) = Q2(0) + P v2(0)
Pz
M110
E
Q1#
P'
Q2
M2()'
v1 = C1 sin kx + C2 cos kx + C3x + C4
v1 = C1k cos kx C2k sin kx + C3v1 = C1k2 sin kx C2k2 cos kxv1 = C1k3 cos kx + C2k3 sin kxv2 = C5x
3 + C6x2 + C7x + C8
v2 = 3C5x2 + 2C6x + C7
v2 = 6C5x + 2C6
v2 = 6C5
Q1(0) = Q2(0) + P v2(0)
2EI v1 (0) = EI v2 (0) + P v2(0)2C1k
3 = 6C5 + 2k2C7C5 =
1
3
k3C1 +1
3
k2C7
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Pcr = 2, = 1
dd
dddd
dd
E'
= 1 (EA)1 = (EA)2 P1 = P2 = P/2v(4)1 + k
21v
1 = 0 k
21 =
P/22EI
= P4EI
v(4)2
k22v
2 = 0 k
22 =
P/2EI
= P2EI
k22 = 2k21 k2 = 2k1
BC : v1L
2
= v1
L2
= v2
L2
= v2
L2
= 0
v1(0) = v2(0)
v1(0) = v2(0)
M1(0) = M2(0)
Q1(0) = Q2(0) + P v2(0)
P2z
M110
E
Q1#
P'
Q2
M2()'
P2
z
v1 = C1 sin kx + C2 cos kx + C3x + C4
v1 = C1k cos kx C2k sin kx + C3v1 = C1k2 sin kx C2k2 cos kxv1 = C1k3 cos kx + C2k3 sin kxv2 = C5 sinh kx + C6 cosh kx + C7x + C8
v2 = C5k cosh kx C6k sinh kx + C7v2 = C5k2 sinh kx C6k2 cosh kxv2 = C5k3 cosh kx + C6k3 sinh kx
Q1(0) = Q2(0) + P v2(0)
2EI v1 (0) = EI v2 (0) + P v1(0)2C1k
31 = C5k32 + 4k21(C1k1 + C3)
C5 = 1
k32
2k31C1 + 4k21C3
=1
2C1 +2
2k1C3
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'
(v) =1
2
L0
EI(v)2 P(v)2 dx,
P
dx + du
dv
= ,vv =
L0
(EI vv P vv )dx = 0,
v
v(0) = v(0) = 0v
=
L
0
EI vv L0
(EI v)v dx L
0
P vv +
L0
vdx
=
L
0
EI vv L
0
(EI v)v L
0
P vv +
L0
[(EI v) + (P v)] vdx
v(0) = v (0) = 0
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M = EI v Q = (EI v)
= M(L)v (L) + [Q(L) P v(L)]v(L) +L0
[(EI v) + (P v)] vdx = 0,
v v(0) = v(0) = 0
(EI v) + (P v) = 0 x (0, L)
M(L) = 0
Q(L) P v(L) = 0
v(0) = 0
v(0) = 0
EI P
EI v(4) + P v = 0
EI v + P v = Cx + D, (C, D constants) v = A sin kx + B cos kx + Cx + D, k =
P
EI
v = Ak cos kx Bk sin kx + Cv = Ak2 sin kx Bk2 cos kxv = Ak3 cos kx + Bk3 sin kx
v(0) = 0 B + D = 0v(0) = 0 Ak + C = 0
v(L) = 0
A sin kL + B cos kL = 0
EI v(L)P v(L) = 0 EI(Ak3 cos kL+Bk3 sin kL)P(Ak cos kLBk sin kL+C) = 0
P = EI
L2 k =
L
A = 0 C = 0 B cos kL = 0 B = 0 cos kL = 0B = 0 v 0
cos kL = 0 kL = 2
+ n, n = 0, 1, 2,...
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n =
2+ n
2,
0 = 22 Pcr =
2
4
EI
L2
n
vn = B(cos knx 1), kn = 1L
2
+ n
||vn||2E =L
0
EI(vn)2dx.
L0
EI vnvmdx = 0, kun n = m.
vn ||vn||E = E1 E1[E1] =
Nm
vn = Bk2n cos knx
E21 = EI B2k4nL0
cos2 knxdx
y = knx, dx =1
kndy rajat
x = 0 y = 0x = L y =
2+ n
E21 = EI B2k3n
2
+n0
cos2 ydy = EI B2k3n1
2
2
+ n
B2 = 2E21
EI k3n2
+ n = 2E21
EI4
16(1 + 2n)4
B = 4
2L3/2E1
2(1 + 2n)2
EI
vn(x) = bn(cos knx 1), Bn =4
2
2(1 + 2n)2E1L3/2
EI
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L0
EI vnvmdx = 0, kun n = m.
L0
vnvmdx =
L0
cos
2(1 + 2n)
x
L
cos
2
(1 + 2m)x
L
merk. y =
2
x
L, dx =
2L
dy
=2L
L0
cos[(1 + 2n)y] cos[(1 + 2m)y]dy = 0, kun n = m.
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x = L
2
Mt =
qL2
2tan2
1
(x = 0) Mk =qL2
2sin2
1
P/Pkr Mt/qL2 Mk/qL
2 v(0)/ qL4
EI
22
32
3.62
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
MqL2
P/Pcr
MtMk
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P/PE
PE
e e
e eE
'c
v(4)1 + k
2v1 = 0 k2 = P
2EI
v(4)2 = 0
BC : v1L
2
= v1
L2
= 0
v2L2
= v2
L2
= 0
v1(0) = v2(0)
v1(0) = v2(0)M1(0) = M2(0)
Q1(0) = Q2(0) + P v2(0) + F
Pz
M110
E
Q1#
F
cP'Q2
M2(
)
'
v1 = C1 sin kx + C2 cos kx + C3x + C4
v1 = C1k cos kx C2k sin kx + C3v1 = C1k2 sin kx C2k2 cos kxv1
=
C1k3 cos kx + C2k
3 sin kx
v2 = C5x3 + C6x
2 + C7x + C8
v2 = 3C5x2 + 2C6x + C7
v2 = 6C5x + 2C6
v2 = 6C5
Q1(0) = Q2(0) + P v2(0)
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2EI v1 (0) = EI v2 (0) + P v2(0) + F2C1k
3 = 6C5 + 2k2C7 + FEI
C5 = 13
k3C1 +1
3k2C7 +
F
6EI
M1(0) = M2(0)
2EI v1(0) = EI v2(0)2C2k
2 = 2C6 C6 = k2C2
v1(0) = v2(0)
C1k + C3 = C7 C5 = 13
k3C1 +1
3k2(C1k + C3) +
F
6EI=
1
3k2C3 +
F
6EI
v1(0) = v2(0)C2 + C4 = C8
v1
L
2
= 0 C4 = C1 sin kL
2 C2 cos kL
2+ C3
L
2
v1
L
2
= 0 C3 = k(C1 cos kL
2+ C2 sin
kL
2)
v2
L
2
= 0
1
3k2C3 +
F
6EI
L
2
3 k2C2
L
2
2+ (C1k + C3)
L
2+ C2 + C4 = 0
kL2 kL cos kL2 1 + 124 (kL)2 + sin kL2 C1+
1 1
4(kL)2 cos kL
2 kL sin kL
2
1 +
1
24(kL)2
C2 = F L
3
48EI
v2
L
2
= 0
k2C3 +
F
2EI
L
2
2 2k2C2L
2+ C1k + C3 = 0
1
1 +1
4(kL)2
cos
kL
2
kC1 +
kL
1 +
1
4(kL)2
sin
kL
2
kC2 = F L
2
8EI
M1(x) = 2EI k
2(C1 sin kx + C2 cos kx) when L2 x 0M2(x) = EI(6C5x + 2C6) when 0 < x L2
C1 C2 [] C1 + [] C2 = FL348EI
[] kC1 + [] kC2 = FL28EI
C5 C6 C1 C2
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v0(x) = v0sin(x/L)
P y E
v0 = L/1000
f
@@@@4
44
dd dd'
&%'$
M(x) + P[v(x) + v0(x)] = 0
v(x) + k2v(x) = k2v0(x), where k2 = PEI
vy(x) = A sinx
L
2L2
+ k2
A sin xL
= k2v0 sin xL
A = k2v0
k2 2L2
v(x) = C1 sin kx + C2 cos kx + C3x + C4 k2v0
k2 2L2
sinx
L
v(0) = C2 + C4 = 0
v(0) = k2C2 = 0 C2 = 0 C4 = 0v(L) = k2C1 sin kL = 0 C1 = 0()
v(L) = C3L = 0
() kL = n k
v(x) =
k2v0
k2
2
L2
sinx
L
,
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M(x) = EI v(x) = EI k2v0
2
k2L2 2 sinx
L
(k2
= P/EI)
M
L
2
= P v0
2
PL2
EI 2
PE = 2EI/L2
M
L
2
= P v0
PPE
1M(L/2) P
PE
= PA
MW
= P
1
A 1
PPE
1v0W
A = (502 452) = 1492mm2I =
4(504 454) = 1.688 106mm4
W =1
50mm2 I = 33760mm
3
k2 = 1.41 103 , when P = 50 kN
= 33.5 15.4 MPaP
y
y = P1A
+ 1PPE
1v0W
PPE
1
y PA v0
WP = 0
P2 mA + PE + PEAv0W P + yPEA = 0
P
2
509.5P + 45945 = 0 P1 = 117.1 kN
P2 = 392.4 kN
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n =117.1
50= 2.34
v0
= 0.49
i =
I
A= 33.6 mm
k =Lci
fyE
= 1.53
=
1 + (k
0.2) + 2k
22k = 0.852
fck = (
2 1/2k)fy = 66.99 MPaNRc = fck
A
m= 66.99 1492 = 99.97 kN
NRc
f =99.97
50= 2.0
= 0.21
f =118
50= 2.36
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N
dd dd
Mt'' MtEE
NE
Ny Nz
Mz
xy
My xz
EI y = Ny + MzEI z = Nz My
y = C1erx, z = C2e
rx
EI r2C1e
rx = NC1erx
MC2re
rx
EI r2C2erx = NC2erx + MC1rerx
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(45)1 C2(EI r2 N) = MC1r(45)2 EI r2C1 NC1 + Mr MC1r
EI r2 N = 0
(EI r2 N)2 + M2r2 = 02 = r2
EI 2 N = M 2 + MEI
+ NEI
= 0
EI 2 + N = M 2 MEI
+ NEI
= 0
1, 2 > 0
1,2 = MEI MEI2 4 NEI
2
y = A1 sin 1x + B1 cos 1x + C1 sin 2x + D1 cos 2x
z = A2 sin 1x + B2 cos 1x + C2 sin 2x + D2 cos 2x
441
EI z = EI A2(21)sin 1x EI B221 cos 1x EI C222 sin 2x EI D222 cos 2xNz = NA2 sin 1x NB2 cos 1x NC2 sin 2x ND2 cos 2x
My = MB11 sin 1x MA11 cos 1x + MD12 sin 2x MC12 cos 2x
EI z Nz My = 0 x
(EI 21 N)A2 + MB11 = 0(EI 21 N)B2 MA11 = 0(EI 22 N)C2 + MD12 = 0(EI 22 N)D2 MC12 = 0
A2 = MB11EI21+N = MB11M1 = B1
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B2 =MA11EI2
1+N
= MA11M1 = A1
C2 =MD12EI2
2+N
= MD12M2 = D1D2 =
MC12EI2
2+N
= MC12M2 = C1
z = B1 sin 1x + A1 cos 1x D1 sin 2x + C1 cos 2x
y(0) = 0 B1 + D1 = 0z(0) = 0 A1 + C1 = 0y(L) = 0 A1 sin 1L + B1 cos 1L + C1 sin 2L + D1 cos 2L = 0z(L) = 0 B1 sin 1L + A1 cos 1L D1 sin 2L + C1 cos 2L = 0
2 2(cos 1L cos 2L + sin 1L sin 2L) = 2 2 cos(1L 2L) = 0
(1 2)L = n2
1 2 = 2
M
2EI2
NEI
1 = 2
N = M = 0
y = z = 0
2 1 = 0 (2 1)L = 2
2
M
2EI2
NEI
L = 2
M2EI
2 N
EI=
L
2
Mcr = 2EI
N
EI+
L
2
N > 0 Mcr
N < 0 Mcr N = 2EI/L2
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L
Pcr Ibolt = Isleeve
Pcr
sleeve : 12 = M12L
3EI11 M21 L
6EI11
= M0 L3EI1
(1 +1
21)
bolt : 12 = M12L
3EI22
M21
L
6EI22
= M0L
3EI2(2 +
1
22)
12,sleeve = 12,bolt
M0 L3EI1
(1 +1
21) + M0
L
3EI2(2 +
1
22) = 0
M0 = 0 I1I2
(2 +1
22) + (1 +
1
21) = 0
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I1I2
3
(kL)2
1
tanh(kL)2 1
(kL)2+
1
(kL)2 1
sinh(kL)2
+
3
(kL)1 1
(kL)1 1
tan(kL)1 +
1
sin(kL)1 1
(kL)1
= 0
I1I2
(kL)1(kL)2
1
tanh(kL)2 1
sinh(kL)2
+
1
sin(kL)1 1
tan(kL)1
= 0
Pcr = EI1k21
I1 = I2 (kL)1 = (kL)2 = kL
cosh kL 1sinh kL
=cos kL 1
sin kL kL 4.73 Pcr = 22.4EI
L2
2 = 2 = 1
I1I2
3
2+ (1 +
1
21) = 0
3(kL)1
1
sin(kL)1 1
tan(kL)1
= I1
I2
3
2
I1/I2 = 1
kL
4.057
Pcr = 16.5
EI
L2v1 = v2
M1 = EI1v1 = P v1 M0M2 = EI2v2 = P v2 + M0
EI1v1 + EI2v2 = 0 v1 = 0 v1 = Ax + B
v1
0 (since v1(0) = v1(L) = 0)
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L/h = 20
dd P'
L
hc
T
EI
EI v(4) + P v = 0
v
EI
L
0
v(4)vdx + P
L
0
vvdx = 0
EIL0
v(3)vdx PL0
vvdx = 0
(M = EI v)
L
0
Mv + EI
L0
vvdx PL0
vvdx = 0
M(L)v(L) M(0)v(0) + EIL
0
vvdx PL
0
vvdx = 0
M(0) + P Rv(0) = 0
x = L
M(L) P Rv(L) = 0v(x) = v0 sin
xL
, v(x) = sin xL
P R[v(L)v(L) + v(0)v(0)] + EIL0
vvdx PL0
vvdx = 0
EI v0
L
4 L2
+ 2P Rv0
L
2 P v0
L
2 L2
= 0
v0
L
2 L2
2
EI
L2 P
1 4 R
L
= 0
P = 2
1 4RL
EI
L2
if R =L
40 P = 10
92
EI
L2
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dd P'
L/2 L/2
2EI EI
v2, 2, 3
dd
P'
L/2 L/2
j j
'c
c
P = EI/L2 Kx =
Sx K S
K
(e)
=
EI
L
12L2
6L
12L2
6L
4
6L
2
12L2 6L
symm. 4
S(e) = N(e)
65L
110
65L
110
2L15
110
L30
65L
110
symm. 2L15
v1, v1, v2, v
2 N
(1) = N(2) = P =
EI/L2
1 = v(1)2 = v
(2)1 , 2 =
(1)2 =
(2)1 , 3 = (2)2
K11 = K(1)33 + K
(2)11 =
8 12L3
2EI +8 12
L3EI = 288
EI
L3
K12 = K(1)34 + K
(2)12 = 24
EI
L2
K13 = K(2)14 = 24
EI
L2
K22 = K(1)44 + K(2)22 = 24 EIL
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K23 = K(2)24 = 4
EI
L
K33 = K(2)44 = 8
EI
L
S11 = S(1)33 + S
(2)11 = P
24
5L = 24EI
5L3
S12 = S(1)34 + S
(2)12 = 0
S13 = S(2)14 = P
1
10=
EI
10L2
S22 = S(1)44 + S
(2)22 = P
2L
15=
2EI
15L
S23 = S(2)24 = P
L
60= EI
60L
S33 = S(2)44 = P
L
15=
EI
15L
EI
L
288 24 2424 24 424 4 8
1/L
2
3
= EIL
245
0 110
0 215
160
110
160
115
1/L
2
3
,
1 1/L Li
det
288 245 24 24 110
24 24 215
4 + 160
24 110
4 + 160
8 115
= 0 kr = 26.32
Chk C
h k
1 = (h1) 2 = (h2)
1 = ex + Chk1
2 = ex + Chk2
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C = (1 ex)hk1
2 = ex + (1 ex)h2h1
k
ex =2 1
h2h1
k1
h2h1
k
= 25.18
k = 4 h1 = L/2, h2 = L/10
h2/h1 = 0.2
1ex = 25.18
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EA =
e e
c c
e ej1c
jv2'u2T2
c jv3'u3T
3c
j4c
EA =
u2 = u3 = 0
v2 = v3
1 = 4
2 = 3
1, v2, 2
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1 4
dd
dd
dd
dd
'
(EA)iu = 0
u1 = C1x + C2u2 = C3x + C4
Ni = (EA)iu =
EAC1 L2 x < 0EAC3 0 < x L2
u1
L
2
= u2
L
2
= 0 C1L
2= C2, C3
L
2= C4
u1(0) = u2(0) C2 = C4 C1 = C3
N1 + P = N2
EAC1 + P = EAC3
C3 = C1 + PEA
C1 = 11 +
P
EA= C3
N1 = EAC1 = 1 +
P (compression)
N2 = EAC3 = 11 +
P (tension)
= 1
N1 =
N2 =
P/2
N1
P, N2
0
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-3000
-2000
-1000
0
1000
2000
3000
1 1.5 2 2.5 3 3.5 4
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e e
e eE
'
v1(0) v1(0)
K11 = K(1)33 + K
(2)11 =
122EI(L/2)3
+ 12EI(L/2)3
=288EI
L3
K12 = K(1)34 + K
(2)12 = 62EI(L/2)2 + 6EI(L/2)2 =
24EI
L2
K22 = K(1)44 + K
(2)22 =
42EIL/2
+ 4EIL/2
=24EI
L
N1 = P, N2 = 0 P = EIL2
S11 = S(1)33 =
3630(L/2)
P = 12EI
5L3
S12 = S(1)34 = 110P =
EI
10L2
S22 = S(1)44 =
215
L2
P = EI
15L
EI
L2 288 1L 2424 24L
v1
v1
=
EI
L2 125L
110
110
L15
v1v1
EI
L2
288 12
5 110
24 110
24 24 115
v1L
v1
=
0
0
2 480 + 42240 = 0 = 116, 1 = 363.9
= 114.04
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Ln
= A2
+ B + C 0 = 392MPa , 0 =0, 002 Et()
Pcr = 20A(
K+ 1)/K K = (0L2nA/
2I)2
dd P'L = 9000D = 410
t = 10jdd dd L
a
L/a = 20
= () = A2 + B + C
= 0 = 0 C = 0 = 0 = 0 = 0, dd = 0
d
d
=0
= 2A0 + B = 0 B = 2A0
0 = A20 + B0 = A20 A =
020
= 0
20
2
+ 2
0
0 d
d = Et = 2
0
0
1
0
d
d
=0
= 200
= E Et = E
1 0
= ()
0=
0
2+ 2
0
0= 1
1
0
Et = E1
0
, when < 0
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Pcr =2EtI
L2n, kr =
PcrA
, merk. =2I
L2n
Pkr = 2E2
20A 4E4
420A2 + 2E2 =
2E2
20A
1 +420A
2
2E2 1
=20A
K
1 + K 1
, K =
0L
2nA
2I
2
A (D t)t = 1.257 104 mm2, I /8(D t)3t = 2.513 108 mm4, L = 9.0 m Ln = 0.699L = 6.291 m K = 0.1608 Pcr = 4.742 kN
A = a2, I = 1/12 a4, L = 20a K = 0.947 cr = Pcr/A = 327
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cr = Ln/i
Ln i =
I/A
Et
d
d = Et = Ey
y c ,
y c
(kr/y) (kr/y) [0, 1], [0, 200] c = 0, 9 E/y = 500
E/y = 200
Pcr =2EtI
L2n,
Et = Ey
y c cr =2Ey
ycI
L2nA
i =
I/A, = Ln/i
cr =2Ei2
L2n
y cry
ccr
2cr = 2E y cry ccr
2cry
(1 ccry
) = 2E
y(1 cr
y)
2c
cry
2
2 + 2E
y
cry
+ 2
E
y= 0
cry
=1
22c
2 + 2 E
y
2 + 2E
y
2 42c2 E
y
c = 0.9 E/y
Pcr =2EI
L2n
cry
=
E
y
2
2
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0
0.2
0.4
0.6
0.8
1
0 50 100 150 200
cry
tang. mod. E/y = 200tang. mod. E/y = 500
elastic E/y=200elastic E/y=500
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v = PE1 + E2
+E2 E1E1 + E2
a
L (P ) = E2 E1E1 + E2
aP +4E1E2
E1 + E2a2
(P ) = E2 E1E1 + E2
aL
P + 4E1E2E1 + E2
a2L
=E2 E1E1 + E2
a
L, P1 = 4
a2
L
E1E2E1 + E2
(P ) = P + P1integrating
tt0
P P00 = (P P0) + P1( 0)
0 = 0
P = (P P0) + P1 P = P0 P1
E1 = E2 = 0E1 = E2 = E P = P1 = 2Ea2L = PEE1 = E2 = ET P = P1 = 2ETa
2
L = PT i < 0 = 0 i > 0 Ei = E
E1 = E, E2 = ET P1 = PR = 2ER a2L ER = 2(EET)/(E+ ET)P0 (PT, PR)
P0 (PT, PE)
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EI kGA(v ) = 0
kGA(v ) P v = 0
Pcr =PE
1 + PE,
PE PE = 2EI /(4L2) = 1/(kGA) k = 1/
P'
M Q = 0
Q P v = 0
v
M = EI Q = kGA(v ) (49)
EI, kGA P
kGA
v + EI = 0(1 P)v = 0
v(0) = 0
Q(0) = 0
M(L) = EI (L) = 0Q(L) = P v(L)
rrrQ
! P'
v
v = Aerx, = Berx
[Ar + B(EIr2
1)] erx = 0
[(1 P)Ar2 Br] erx = 0
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r EIr2 1(1 P)r2 r
A
B
=
0
0
A = 0 = B
det = 0
r2 1 + (1 P)(EIr2 1) = 0 r1,2 = 0 tai 1 + (1 P)(EIr2 1) = 0
r2 = P(1 P)EI
P > 0 (1 P) = 1 P/kGA P = EI/L2
P = EI
kGAL2 = 2(1 + )k1 I
AL2 (inserting G =
E
2(1 + ) , k1
= )
= 2(1 + )I
AL2, 0 1
2, 1, I/A
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A1, B1, B3, B4
v(x) = A1 + B1x +
1 + EI2
B4 sin x 1 + EI2
B3 cos x(x) = B1 + B3 sin x + B4 cos x
1) v(0) = 0 A1 1+EI2 B3 = 02) (0) = 0 B1 + B4 = 03) (L) = 0 B3 cos L B4 sin L = 04) Q(L) = P v(L)
kGA[v(L) (L)] P v(L) = 0 (1 P)v(L) (L) = 0 (1 P) [(1 + (1 + EI2)cos L) B4 + (1 + EI2)B3 sin L]
(1 + cos L)B4 B3 sin L = 0 (1 P)B4 + cos LB4 (1 + cos L)B4 = 0 P B4 = 0 B4 = 0 B1 = 0
B3
cos L = 0 B3
= 0
cos L = 0 L = 2
P1 P =
2EI
4L2= PE
P = (1 P)PE (1 + PE)P = PE P = PE
1 + PE
v(x) = 1+EI2 B3(1 cos x)(x) = B3 sin x
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Pcr
dd
dd
dd
ddd
P E
2Pc
2Pc
L
L L
j
j j
j j
M21 + M23 = 0
M32 + M34 + M35 = 0
21 = 23 = 2
32 = 34 = 35 = 3
M21 = A212
M23 = A232 + B323
M32 = A323 + B232
M34 = A343
M35 = A353
Aij, Bij
Aij
=2((kL)ij)
42 ((kL)ij) 2 ((kL)ij)6(EI)ij
Lij, B
ij=
((kL)ij)
42 ((kL)ij) 2 ((kL)ij)6(EI)ij
Lij
kL
(kL)221 =2P
2EIL2 =
P L2
EI= k2L2
(kL)223 =P
2EIL2 =
P L2
2EI= 1
2k2L2
(kL)234 =2P P
3EIL2 =
P L2
3EI= 1
3k2L2
(kL)235 =
2P
EI (2L)2
=
2
2P L2
EI = 22k2
L2
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A21 + A23 B32
B23 A32 + A34 + A35
2
3
=
0
0
K
(A21 + A23)(A32 + A34 + A35) B232 = 0 kL = 3.564 Pkr = 12.705 EI
L2
g(x) = 0
x0g x0
g(x) g(x0) + g(x0)(x x0) + ...
xk+1 = xk g(xk)g(xk)
x0 x
g
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(kL)
kL = 0
kL
det(K)
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P
HEP
c Pcq
c c c c c c c c
L
2L
j
j j
j
EI
EI
EI
q = 180 kN/m2
H = 20 kN
L = 6 m
A = 36 103 mm2I = 650 106 mm4
W = 3, 25 106 mm3E = 210 GPa
m = 220 MPa
M23, M34, 12
M21 + M23 = 0 M21 = M23M32 + M34 = 0 M32 = M34
21 = 23
21 =
2L3EI
(k12L)M21 + 12
23 =L3EI
M23 L6EIM32 + 023
LEI
2
3(k12L) +
1
3M23 +L
6EIM34 12 + 023 = 0
32 = 34
32 = L6EIM23 + L3EIM32 + 03234 =
2L3EI
(k22L)M34 + 12
LEI
2
3(k22L) +
1
3
M34 +
L
6EIM23 + 12 032 = 0
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Q21 + Q34 = H
Q21 = Q021
M212L
N1212
Q34 = Q
0
34 M34
2L N3412
HE
Q21'
Q34'
M23 M34 2L(N12 + N34)12 = 2LM N23 M34 2L(qL + 2P)12 = 2LM
L
EI
23
(k12L) +13
16
11
6
2
3(k
22L) + 1
31
1 1 2L(qL + 2P) LEI
M23
M34
EIL
12
=
0230322HL2
EI
023 =qL3
24EI, 032 =
qL3
24EI, =
H
qL, q =
EI
L3, P =
EI
L2,
23
(k12L) +13
16
116 23(k22L) + 13 1
1 1 2L(qL + 2P) LEI
M23
M34EIL
12
= 124
1242
qL2
k1 =
N12EI
N12 = Q23 + P = Q023
M23 + M32L
+ P = P +qL
2+
1
L(M34 M23)
k2 = N34EI
N34 = P Q32 = P + qL2
+ 1L
(M23 M34)
M23 =M23qL2
, M34 =M34qL2
, 12 =12EI
qL3
M23 = M23qL2 = M23EIL
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M34 = M34qL2 = M34
EI
L
12 = 12qL3
EI= 12
N12 =
+
1
2+ M34 M23
EI
L2
N34 =
+
1
2+ M23 M34
EI
L2
2k1L = 2
+
1
2+ M34 M23
2k2L = 2 + 12 + M23 M34
M23E
Q23
T
qc c c c c c c c M(x)
'
xE
M(x) = Q23x + M23 12
qx2
Q(x) = Q23
qx
Q(x) = 0 x = Q23q
=1
2L +
M34 M23qL
H E
Q21'
N23'
N23 = H Q21= H Q021 +
M212L
+ N1212
= H M232L
+
12
+ M34 M23 + EIL2
12
P = 0 M23, M34 = 0
< m P = EI/L
= 0, 01
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m = 220 =
0, 22 = 0, 23 max = 0, 22 Pmax = 0, 834
Pallowable = Pmaxh
=0, 834
1, 5MN = 0, 56MN
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d b
2h L Ep
En In
dd
dd
dd
P'
dAA
b
P = PE1 + PE
, PE = 2EI
L2
= Q
=QS
bI
S =
h0
bydy =1
2bh2
I =1
12b(2h)3 =
2
3bh3
= Q
=S
bI=
S
bI
=
3
4bh
db
db = kn = knh
kn
= dbknh = 3d4knh2
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kn
EEEEE
h2
T
c
v
1
2Epv
h
2= Q Q = knv
kn = 14
Eph
1 = 3d4Eph3
In
ffdd
ff v
v(0)
c
Qc
Ep
En, In
v(0) =2
EpQ, =
Ep
4EnIn
1/4
kn = Ep2
2 = 32
dEph2
= 32
h dEph3
2 > 132
h > 3 h > 2En = 210
Ep 8 13
EpEn
120
h h(80In)1/4
, In =D4n
64
h hDn
,
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P E
y
Pcr = PE1 + PE
, where PE = 2
EIoL2
Io = 2
Izo + Aa
b
2
2= 281 106 mm4 PE = 5.84 MN
Ia, Aa
Ib, Ab b = 304
a = 985
Ia = Iz0 = 4.95 106 mm4Ib = 4.5 106 mm4
Ab = 2400 mm2
E = 210 GPa, G =E
2(1 + )= 80.8 GPa
=ab
12(EI)b+
a2
24(EI)a(1 ) +a
b
GA
b
= Pcr2
2
(EI)aa2
Pkr
= 0
Pcr
a1, a2, a3, 1, 2
(EI)b = a1(EI)o, (EI)a = a2(EI)o, (GA)b = a3 (EI)oL2
, ab
= 1, aL
= 2
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=
22
121a1+
2224a2(1 ) +
1
a3
L2
(EI)o=
L2
(EI)o
Pcr =2(EI)o
L2
1 + L2
(EI)o
2(EI)oL2
= 2(EI)o1 + 2
=PE
1 + 2
= 2(EI)o
2L2(1 + 2)22L
2
2a2(EI)o=
222a2(1 + 2)
Pcr = 3.77 PE
Io = 2Iyo = 1160.6 106 mm4
Pcr = 2
(EI)oL2
= 3.33 MN
Py = y2Aa = 2.82
Py
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Q y
= Q/
Q
d1, d2, d3, d4 D1, D2, D3, D4
d1 =b
3
2j b
2i + ak
d2 =b
3
2j +
b
2i + ak
d3 = b
3
2j +
b
2i + ak
d4 = b
3
2j b
2i + ak
d =a2 + b2
cos(d1, x) =d1 i|d1|
= b2d
= cos 2
cos(d2, x) =cos
2cos(d3, x) =
cos
2
cos(d4, x) = cos 2
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cos(d1, y) =d1 j|d1|
=b
3
2d=
3
2cos
cos(d2, y) =
3
2cos
cos(d3, y) =
3
2 cos
cos(d4, y) =
3
2cos
y Q
2(D1 + D2) j = Q D1 cos(d1, y) + D2 cos(d2, y) = Q2
2(
D1 +
D2) i = 0 D1 cos(
d1, x) + D2 cos(
d2, x) = 0
D1 = D2 = Q2
3cos
y Q
(D1 + D2 + D2 + D4) j = Q (D1 + D2 D3 D4)
3
2cos = Q
(D1 + D2 + D2 + D4)
i = 0
(
D1 + D2 + D3
D4)cos
2= 0
D3 = D4 = Q2
3cos
S
=
D1
D2
D3
D4
=Q
2
3cos
1
1
11
=
D1
D2
D3
D4
=
d1/2 EAd
d1/2 EAd
d1/2 EAd
d1/2 EAd
D1
D2
D3
D4
=
S
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aQ =
T
S
aQ =Q
23cos 1 1 1 1 Q
23cos
1
1
11
=Q2
4 3cos2 d
1/2 EAd
1 1 1 1
1
1
11
=
4dQ2
6EAd cos2
= 2dQ3EAda cos2
=2d3Q
3b2EAda
=
Q =2d3
3ab2EAd
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L
PE dd dd ddj j jL L
P v0 = L
dd
dd @@@@@
@@@@
ddv0PE
L L
M21 + M23 = 0
21 = 23 M21 + 21 = M23 + 23 2M21 = 2
21 = 23 = ja = L3EI
(kL)
P = EI
L2 kL = M21 = 3EI
L(kL) = 3EI
L2() v0 = 3EI
L()
T = Q23 Q21 = 2
M21L
+ P
=
2EI
L2
3
(
)
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TL2
EI
2
0 T 6EI/L2
P
6EI/L2 T
P = 8EI/L2 = 1/100
0.204EI/L2
P
0ij(kL)
dd 222dd 222ddv0 nv0PE
L L
v0 = L
M21 + M23 = 0
21 = 23 M21 + 021 = M23 + 023 2M21 = 023 021, 023 = 021 =
(kL)2
2 (kL)2
M21 = 1 + n2
(kL)2
2 (kL)23EI
L(kL)
kL =
, P = EI
L2
M21 = 3(1 + n)EI2L
(2
)(
)
T =2M21
L=
3(1 + n)EI
L2
(2 )(
)
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'T
Ec2qL(1
)2qL
qkr + qc c c c c c c c c c c c c c c c c c c c
qLT 2qL(1 )T qLT
2
2 1 =5
384
q(2L)4
EIv (1
)
48
q(2L)4
EIv
= (q)
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Pcr
tcT
b
EIzv(4) + P(v + zv) = 0
EIyw(4) + P(w yv) = 0
EI(4) GIt + P(zvv yvw + r2) = 0
r2 = y2v + z2v + (Iz + Iy)/A zv = yv = 0
EIzv(4) + P v = 0
EIyw(4) + P w = 0
EI(4) (GIt P r2) = 0
v, w
1 2
Pcr,z = Pcr,y = 2EIz
L2
v = A sinx
L, w = B sin
x
L
Iz = Iy, Iyz =
yzdA = 0
Iz = Iz cos2 + Iy sin2 2Izy sin cos = Iz = I
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3 I = 0
(4) +P r2 GIt
EI = 0
(4)
+ k
2
= 0 , where k2
=
P r2
GIt
EI
P > GItr2
= A + Bx + Ccos kx + D sin kx
(0) = 0 A + C = 0(L) = 0
A + BL + Ccos kL + D sin kL = 0
(0) = 0 C = 0 A = 0(L) = 0 Ccos kL + D sin kL = 0 D sin kL = 0 B = 0
A = B = C = 0
sin kL = 0 k = nL
P r2 GItEI
=n
L
2
Pcr, =GIt + EI
2
L2
r2
Iz = Iy =bt
12(b2 + t2)
It = 0.78bt3
I = 0
Pcr, =
It
Iz + Iy
GA = 4.67t2
b2
+ t2
GA = 4.67GA
1 + (b/t)2
I = 0
3 I = 0
(P r2 GIt) = 0 Pcr, = GIt/r2
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Pcr
b = 10t, = 0
tcT
b
b
EIzv(4) + P(v + zv) = 0
EIyw(4) + P(w yv) = 0
EI(4) GIt + P(zvv yvw + r2) = 0
b
c
T
b' E
yc
zEd =
b4Tc
zv = 0, EI = 0
yv = b/4Iy b
3t
12, Iz =
5
24b3t
It 23
t3b, r2 =IpA
+ y2v + z2v =
5
24b2
EIzv(4) + P v = 0
EIyw(4) + P[w yv] = 0
GIt + P[yvw + r2
] = 0
y
z y
Py = 42EIz
L2
w = B 1 cos2nx
L = C
1 cos2 nx
L
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P = GItr2
GIt
r2 yv
yv r2(1 ) B
C = 0
0
= 42(r/L)2EIy/GIt A, B
1
yvr
22 (1 + ) + = 0
Iy = I Iz =52
I It =225
I = 0 G = E/2 GIt =125
EI
(yv/r)2 = 3
10
2 107
(1 + ) + 107
= 0
= 1256
2(b/L)2
1 =57
(1 + )
1
1 14
5(1 + )2
1 1
Py = 42EIz
L2
= 2502 r
L2 GIt
r2
=625
12
2b
L2
GIt
r2
Pz,,1 = 1GIt
r2
z
Pz = 42EIy
L2=
GItr2
=125
62
b
L
2GItr2
=2
5Py > Pz,,1
cr = 1 (L/b)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 20 40 60 80 100
cr
L/b
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E
m
h
t
tcT
h
h
EIzv(4) + P[v + zv)
] = 0
EIyw(4) + P[w yv)] = 0
EIz(4)
GIt
+ P(zvv
yvw
+ r2)) = 0
(yv, zv) r2 = y2v + z
2v + (Iz + Iy)/A
Iy = 13h3t Iz = 712h3t It = ht3 A = 3ht
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s = 0
s = h
s = h
s = 2hh2
B = B(s) =
s0
h(s)ds
h s h , B = 0h s 2h , B = h(h s)
yv = yB + Iz/Iy = 0 (symmetry)zv = zB + Iy/Iz
Iy
h/2h/2
h/2
y
Iy = y(s)B(s)t(s)ds = t2h
h
h
2
h(h
s)ds =
th4
4
zv = h3
1
4h4t
712
h3t
= 16
21h
r2 = y2v + z2v +Iz + Iy
A=
521
588h2
v
hhhhhhhhhhhh
37
h2 114
h2
12
h2
37
h
vv
s
v(s)t(s)ds = 0
C v
s
vtds +s
Ctds =
vtds + Cs
tds = 0
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C = s
vtds
A= 3
14h2
s y = 0 z =
h/3
hhhhhhhhhhhh
314
h2
314h2
27
h2
2
7h2
vTs
v(s) =
h28
(14s + 13h) , 32
h s h2
33
hs , h2
s h2
h28
(14s 13h) , h2
s 32
h
I =
A
2vdA = 2t
h/20
9
49h2s2ds +
3h/2h/2
1
282h2(14s 13h)
= 5h5t
84
Pc
xT
Lv(0) = w(0) = (0) = 0
v(0) = w(0) = (0) = 0
v(L) = w(L) = (L) = 0
v = C1
cos
x
2L 1
, w = C2
cos
x
2L 1
, = C3
cos
x
2L 1
2
4L2EIz P
C1 P zvC3 = 0
2
4L2EIy P
C2 = 0
P zvC1 +
2
4L2
EI + GIt
P r2C3 = 0
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Py =2
4L2EIz, Pz =
2
4L2EIy, P =
2
4L2EI + GIt
Py P 0 zvP
0 Pz P 0zvP 0 P P r2
C1
C2
C3
= 0
00
det[]
(Pz P)
(Py P)(P P r2) z2vP2
= 0
P1 = Pztai (r2 z2v)P2 (P + Pyr2)P + PyP = 0
P2,3 =(P + Pyr
2)
(P + Pyr2)2 4PyP(r2 z2v)2(r2
z2v
)
P1 = 1727 kN, P2 = 964 kN, P3 = 11454 kN
Pkr = P2 =
PkrA
= 321 MPa > m = 220 MPa
Pkr 660 kN
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M
Mcr
M'&E
TTTTTTTTTTTTTTTTTTTTTT
dd
dd
M$%'
L = 50b
E ' b
EIyw
(4) M0z = 0GIt M0z w = 0
M0z /EIy
(M0z )2
EIy + M0z w
(4) = 0
GIt(4) M0z w(4) = 0
(M0z )2
EIy GIt(4) = 0 trial = erx
(M0z )2
EIyr2erx GItr4erx = 0
r2 = (M0z )2
EIyGIt
k2 = r2
= A1 sin kx + A2 cos kx + A3x + A4
w = B1 sin kx + B2 cos kx + B3x + B4
B1 = A1GI
tM0z
, B2 = A2GI
tM0z
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z
w +5b
2 0 w
= 5b
2
w
B1A1
=B2A2
= 5b2
M0z = 2GIt5b
k =M0z
EIyGIt=
n
Ln halfwaves
M0z =n
L
EIyGIt =
2GIt5b
Iy =5b b3
12=
5
12b4
It =1
35b b3 = 5
3b4
L = 50b and if G = 0.4E
M0z =n
50bEb4
0.5 5/12 5/3 = 0.0331nEb3 = 0.267Eb3
n = 8.
2
GIt M0z
5b2
= 0
= 0 = 0 = Ax + BA = B = 0 0
GIt M0z
5b2
= 0 M0z =
2GIt5b
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Mcr
b hh
b
M'&
E
dd
dd
M$%'
EIyw
(4) M0z = 0GIt M0z w = 0
w(0) = 0, w(0) = 0, (0) = 0
ee
ee
ee
ee
ee
ee
yee
zB z, wE
y, vc
dd
M''
c
w(L) = h2
(L)
Mz%% Mye
eueeu
M'' zE
yc
zT
zE
$
$$$$
Mx$$X
MzgggMc
w(L)
M
Mz MMy = EIyw
(L)MMx =
w(L)M
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w(0) = 0 w(L) = h2
(L)
w(0) = 0 EIyw(L) = (L)M(0) = 0 GIt(L) = w(L)M
2 1
w(4) + k2w = 0, k2 =M2
EIyGIt w = A sin kx + B cos kx + Cx + D
w(0) = w(0) = 0 D = B = 0 w = A sin kx + Cx
2
= Esin kx + F x GItk2Esin kx Mk2A sin kx = 0 E = M
GItA
GIt(L) = w(L)M GIt(k M
GItA cos kx F) = (Ak cos kx + C)M
F = MGIt
C
w(L) = h2
(L) A sin kL + CL = h2
M
GIt(A sin kL + CL)
1 Mh2GIt
A sin kL +
1 Mh
2GIt
CL = 0
EIyw
(L) =
(L)M
EIyk
2A sin kL =M
GIt
(A sin kL + CL)
EIyk2 M
2
GIt
A sin kL + M
GItC = 0
k2 = M2/EIyGIt (M/GIt)C = 0 C = 0
1 Mh2GIt
sin kL
= 0
Mcr = min2GIt
h , EIyGIt
L
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w(x) = A sin kx
(x) = MGIt
A sin kx
Mcr =2GIth
kL = w(L), (L) = 0 Mcr = 2GIth kL =
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Mcr =
EIyGIt/L
= (k,h/L) k
k (1, 1) L/h = 20, = 0
M'&E
dd
dd
kM$%'
L b
h
=1
2
L0
GIt(
)2 + EIy(w)2 + 2(M0z )
w
dx
=1
2
L0
GIt(
)2 + EIy(w)2 + 2(M0z
+ M0z )w
dx
M0z = M(1 x/L) + kMx/L = M[1 + (k 1)x/L] M0z = (k 1)M/L = 0 sin x/L w = w0 sin x/L
L0
M0zwdx = 0
L0
x
Lcos2
x
Ldx =
L
4
= 12
GIt
L2
20L2 + EIy
L4
w20L2 + 2M
L2
0w0
L2 + (k 1) L4
0=
2GIt2L
0 + M
L
2w0
L
2+ (k 1) L
4
= 0
w0=
4EIy2L3
w0 + M
L
20
L
2+ (k 1) L
4
= 0
2EIy2L3
M2L
1 + k1
2
M2L1 + k1
2 GIt
2L
w0
0 =
0
0
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Mcr
b = 10t, L = 20b, = 1/3 M
M'&E
dd
dd
M$%'
L
d
dd
b
bdd
dds
It = 23
t3b, Iy = 13
tb3, Iz = 112
tb3, yv = 24
b, zv = 0
z =1
Iz
y(y2 + z2)dA 2yv,
y3dA = 0,
yz2dA = 2
bt
6
2
4b
1
2b2 =
2
24tb4 z =
2b
EIyw(4) M = 0
GIt Mw zM = 0 = MGIt + zM
w
w(4) + M2
EIy(GIt + zM)w = 0
w = A sin kx + B cos kx + Cx + D k2 =M2
EIy(GIt + zM)
w(0) = 0 B + D = 0w(0) = 0 B = 0
w(L) = 0 A sin kL + CL = 0
w(L) = 0 Ak2 sin kL = 0 kL = n,
n = 1
M2 z 2
L2EIyM EIyGIt
2
L2= 0
M =
EIyGIt/L EIy =
2GIt
2 2 zL
2 = 0
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Nx10
2Da2
h
h = 4h
w(x, y)
h = a/40 a
Nx
EEEEEEEEEEE
an+1
nNx
'''''''''''
c ay
E
a
x
w(x, y) = w0 sinx
asin
y
a
= U + V = Uplate + Ubeams + Vplate + Vbeams
Uplate =D
2
A
(w)2dA
w = w,xx + w,yy, and w,xx = w02
a2 sin xa sin ya = w,yy
Uplate = D2
4
a2w20
Vplate = Nx2
A
w2,xdA = Nx2
2
4w20
Ubeams =n
i=1
EI
2
a0
w2,xxdx =EI
4
4
a3w20
sin2
i
n + 1
Vbeams
=
n
i=1
xhh
2
a
0
w2,x
dx, where x
h = Nx
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= Nx4
h
aw20
2
sin2i
n + 1
=
D
2
4
a2 Nx
2
2
4+
EI
4
4
a3
sin2
i
n + 1 Nx
4
h
a2
sin2i
n + 1
w20
Vpalkit Nx
Nx = 2Eh3
12a2, I =
3h4
12, D =
Eh3
12, when = 0
= 4 h = a/40
= Eh3
24
4
a2w20 1 + 3
h
2a
n
i=1
sin2i
n + 1
1
4+
h
2a
n
i=1
sin2i
n + 1 = 0 w0 = 0
2 = 0 2
w20= 0
= 1 + 3 h2a
sin2 i
n+114
+ h2a
sin2 i
n+1
10
= 4 h = a/40 n
n = 1
sin2i
n + 1= 1 = 1 +
45
14
+ 120
= 6
n = 2
sin2i
n + 1= 2 3
4=
3
2 = 1 +
4532
14
+ 12032
6.8
n = 5
sin2i
n + 1= 3 = 8.5
n = 9
sin2i
n + 1= 5 = 10
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cr
w(x, y) = A sin(y/b) sin[(x y)/s]s
D cr =5.352D/b2t t
E E E E E E E E E E
' ' ' ' ' ' ' ' ' '
b
w(x, y) = A siny
bsin
s(x y)
s x w = 0 x = y
x = y + s
y
s
b
x
=D
2
A
(w)2dA + Nxy
A
w,xw,ydA
y = 0, y = b, x = y x = y + s
w,x = As
sin yb
cos s
(x y)
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r
R
N =-P
Nr +
Nrr
dr
(r + dr)d Nrrd 2N d
2dr = 0
Nrr
+Nr N
r= 0
N0r N0
Dd4w
dr4 +2
r
d3w
dr3 1
r2d2w
dr2 1
r3dw
dr
= p
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p N0r N0
prdrd =
N0r
dw
dr
rd +
N0r
dw
dr+
r
N0r
dw
dr
dr
(r + dr)d
= N0rdw
dr
drd +
rN0r
dw
dr rdrd +
rN0r
dw
dr (dr)2d
p = N0r
r
dw
dr+
d
dr
N0r
dw
dr
d4w
dr4+
2
r
d3w
dr3 1
r2d2w
dr2 1
r3dw
dr=
1
D
N0rr
dw
dr+
d
dr
N0r
dw
dr
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+
R0
(rNr + N)rdr 2R0
N0r u +
d
dr(N0r u)r
dr
= D
R
0
(2r + 2 + 2r)rdr + 2
R
0
(N0r r + N0)rdr
2
R
0
N0r u + ddr
(N0r u)r dr +
r =du
dr+
1
2
dw
dr
2, =
u
r, r = d
2w
dr2, = 1
r
dw
dr
= D
R
0
(w)2 +
(w)2
r2+
2
rww
rdr
+2
R0
N0r
u +
1
2(w)2
(N0r u)
rdr
+2
R0
(N0 N0r )udr + R0
(rNr + N)rdr
N0r N0
Nr =E
12
(r + ), where r =du
dr
= u
N =E
12 ( + r) =ur
dNrdr
+Nr N
r= 0
u + r
u u 1r2
+1
r
u +
u
r u
r u
= 0
u + 1r
u 1r2
= 0
r2d2u
dr2 + r
du
dr u = 0
r = et t = lnrdu
dr=
du
dt
dt
dr=
1
r
du
dtd2u
dr2=
d
dr
1
r
du
dt
=
1
r2
d2u
dt2 du
dt
r2d2
udr2
+ r dudr
u = 0
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=
R0
D
(w)2 +
w
r
2+
2
rww
P(w)2
rdr
w
w = w0
1 r
2
R2
w = 2w0 r
R2 w = 2 w0
R2
= D 4r
R4+ 4
r
R4+ 8
r
R4 P4r3
R4 drw20
d2
dw20= 0 P = 4(1 + ) D
R2
w = w0
1 r
2
R2
2
w = 4w0
R
1 r2
R2 r
R
w = 4w0R2
1 3 r
2
R2
=
32D
3R2 2
3P
w20
P = 16 DR2
D/R2
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P
L
c
T
L
c
T
k1q1
k2
q2
P
V
Fi = Vqi
W = Fiqi = V
qi qi = V
W = PixBi
P = P sin q2i P cos q2jxB = (sin q1 + sin q2)Li + (cos q1 + cos q2)Lj
xB = (q1 cos q1 + q2 cos q2)Li (q1 sin q1 + q2 sin q2)Lj W = P L( cos q1 sin q2 sin q1 cos q2)q1 + 0 q2
Vq1
= P L sin(q1 + q2)Vq2
= 0 V = V(q1)
P
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K = Lq
(qe, 0) Qq
(qe, 0)
= k1 0
0 k2
2 1
1 1
P L, attheequilibriumpoint qe =0
0
M =J
q(qe, 0, 0) =
1
3
5 1
1 1
mL3
q = estx
K+ s2M
x = 0
k1 = k2 = k P = k/L r2 = s2mL3
4
9r4 + k
2 5
3
r2 + k2(1 3 + 2) = 0
r2 = k2 + 5
3
2 4
3 + 20
9
8/9
r = 0
2
3 + 1 = 0 =3
2
5
2
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x1
1 y1 = Sx1
k = 1, 2,... |(k+1 k)/k+1| < TOL
Kxk+1 = yk
yk = Sxk
(xk+1) =xTk+1yk
xTk+1yk+1
yk+1 =yk+1
(xTk+1yk+1)1/2
yT1 1 = 0
yk+1 S1, (xk+1) 1, when k
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K = S = aS + (1 a)S (K S) = S K = S
= +
K p1S
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x1
1 y1 = Sx1
k = 1, 2,... |(k+1 k)/k+1| < TOL
(K (xk)S)xk+1 = ykyk = Sxk
(xk+1) =xTk+1yk
xTk+1yk+1+ (xk)
yk+1 =
yk+1
(xTk+1yk+1)1/2
yT1 1 = 0
yk+1 S1, (xk+1) 1, when k
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1e-20
1e-10
1
1 2 3 4 5 6 7
ln(virhe)
iteraatiokierros
kaanteisiteraatio
shift ++
+
++
+
++
Rayleigh