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Nelinearna statika stapnih konstrukcija
Geometrijske imperfekcije
M 1(x) + Hw 1(x) − V(x) = 0
w(x) = wimp(x) + wel(x)
wimp.
wel.
w
M 1(x) + Hw 1(x) − V(x) = 0
∣
∣
∣
∫ x
0
w(x) = wimp(x) + wel(x)
M(x) + Hw(x) −
∫ x
0
V(ξ) dξ = 0
M 1(x) + Hw 1(x) − V(x) = 0
∣
∣
∣
∫ x
0
w(x) = wimp(x) + wel(x)
M(x) + H[
wimp(x) + wel(x)]
−
∫ x
0
V(ξ) dξ = 0
M 1(x) + Hw 1(x) − V(x) = 0
∣
∣
∣
∫ x
0
w(x) = wimp(x) + wel(x)
M(x) + Hwel(x) = −Hwimp(x) +
∫ x
0
V(ξ) dξ
M(x) + Hwel(x) = −Hwimp(x) +
∫ x
0
V(ξ) dξ
M(x) = −EIw2el(x)
−EIw2el(x) + Hwel(x) = −Hwimp(x) +
∫ x
0
V(ξ) dξ
M(x) + Hwel(x) = −Hwimp(x) +
∫ x
0
V(ξ) dξ
M(x) = −EIw2el(x)
EIw2el(x) − Hwel(x) = Hwimp(x) −
∫ x
0
V(ξ) dξ
EIw2el(x) − Hwel(x) = Hwimp(x) −
∫ x
0
V(ξ) dξ
EIw2el(x) − Hwel(x) = Hwimp(x) −
∫ x
0
V(ξ) dξ
EIw2el(x) − Hwel(x) = Hwimp(x)
w2el(x) −
H
EIwel(x) =
H
EIwimp(x)
w2el(x) −
H
EIwel(x) =
H
EIwimp(x)
wel(x) = wel,h(x) + wel,p(x)
w2el(x) −
H
EIwel(x) = 0
w2el(x) =
H
EIwel(x)
wel,h(x) = eµx
w2el,h(x) = µ2 eµx
w2el(x) −
H
EIwel(x) = 0
H = −Pt , Pt ą 0
w2el(x) +
Pt
EIwel(x) = 0
h = ℓ
c|H|
EI= ℓ
cPt
EI
w2el(x) +
h2
ℓ2wel(x) = 0
w2el(x) +
h2
ℓ2wel(x) = 0
wel,h(x) = eµx
w2el,h(x) = µ2 eµx
µ2 eµx +h2
ℓ2eµx = 0
(
µ2 +h2
ℓ2
)
eµx = 0
eµx ą 0
µ2 +h2
ℓ2= 0, µ2 = −
h2
ℓ2
µ1,2 = ¯ ih
ℓ
wel.,h,1(x) = e−ixh{ℓ, e−ix = cos x − i sin x
wel.,h,1(x) = cos
(
h
ℓx
)
− i sin
(
h
ℓx
)
wel.,h,2(x) = eixh{ℓ, eix = cos x + i sin x
wel.,h,2(x) = cos
(
h
ℓx
)
+ i sin
(
h
ℓx
)
wel.,h(x) = b1wel.,h,1(x) + b2wel.,h,2(x)
= b1
[
cos
(
h
ℓx
)
− i sin
(
h
ℓx
)]
+ b2
[
cos
(
h
ℓx
)
+ i sin
(
h
ℓx
)]
= (−i b1 + i b2) sin
(
h
ℓx
)
+ (b1 + b2) cos
(
h
ℓx
)
wel.,h(x) = (−i b1 + i b2) sin
(
h
ℓx
)
+ (b1 + b2) cos
(
h
ℓx
)
a1 = −i b1 + i b2
a2 = b1 + b2
wel.,h(x) = a1 sin
(
h
ℓx
)
+ a2 cos
(
h
ℓx
)
w2el(x) −
H
EIwel(x) = 0
H = Pv , Pv ą 0
w2el(x) −
Pv
EIwel(x) = 0
h = ℓ
c|H|
EI= ℓ
cPv
EI
w2el(x) −
h2
ℓ2wel(x) = 0
w2el(x) −
h2
ℓ2wel(x) = 0
wel,h(x) = eµx
w2el,h(x) = µ2 eµx
µ2 eµx −h2
ℓ2eµx = 0
(
µ2 −h2
ℓ2
)
eµx = 0
eµx ą 0
µ2 −h2
ℓ2= 0, µ2 =
h2
ℓ2
µ1,2 = ¯h
ℓ
wel.,h,1(x) = e−xh{ℓ, wel.,h,2(x) = exh{ℓ
wel.,h(x) = b1wel.,h,1(x) + b2wel.,h,2(x)
= b1 e−xh{ℓ + b2 e
xh{ℓ
sh (x) =ex − e−x
2, ch (x) =
ex + e−x
2
wel.,h(x) = a1 sh
(
h
ℓx
)
+ a2 ch
(
h
ℓx
)
w2el(x) −
H
EIwel(x) =
H
EIwimp(x)
wimp(x) = w0 (ax2 + bx + c) = w0 swimp(x)
wimp(x) = w0 (ax2 + bx + c) = w0 swimp(x),
swimp(0) = 0 ⇒ c = 0
swimp(ℓ{2) = 1 :ℓ2
4a +
ℓ
2b = 1
swimp(ℓ) = 0 : ℓ2 a + ℓ b = 0
}a = −
4
ℓ2
b =4
ℓ
wimp(x) = w0
$’%4
ℓx −
4
ℓ2x2
,/-
w2el(x) −
H
EIwel(x) =
H
EIwimp(x)
w2el(x) +
h2
ℓ2wel(x) = −
h2
ℓ2wimp(x)
w2el(x) +
h2
ℓ2wel(x) = −
h2
ℓ2wimp(x)
wel,p(x) = c0 + c1 x + c2 x2, w2
el,p(x) = 2 c2
wimp(x) = w0
$’%4
ℓx −
4
ℓ2x2
,/-
2 c2 +h2
ℓ2
$%c0 + c1 x+ c2 x
2,- = −
h2
ℓ2w0
$’%4
ℓx −
4
ℓ2x2
,/-
$’’%2 c2 +
h2
ℓ2c0
,//- +
$’’%h2
ℓ2c1 +
4w0 h2
ℓ3
,//-x
+
$’’%h2
ℓ2c2 −
4w0 h2
ℓ4
,//-x2 = 0
h2
ℓ2c2 −
4w0 h2
ℓ4= 0 ⇒ c2 =
4w0
ℓ2
h2
ℓ2c1 +
4w0 h2
ℓ3= 0 ⇒ c1 = −
4w0
ℓ
2 c2 +h2
ℓ2c0 = 0 ⇒ c0 = −
8w0
h2
wel,p(x) = w0
$’%−
8
h2−
4
ℓx +
4
ℓ2x2
,/- = −
8w0
h2− wimp(x)
wel(x) = wel,h(x) + wel,p(x)
= a1 sin
(
h
ℓx
)
+ a2 cos
(
h
ℓx
)
+ w0
$’%−
8
h2−
4
ℓx +
4
ℓ2x2
,/-
slobodno oslonjena greda:
wel(0) = 0 & wel(ℓ) = 0
a2 −8w0
h2= 0
a1 sinh + a2 cosh −8w0
h2= 0
a2 =8w0
h2
a1 =8w0
h2
1− cosh
sinh
wel(x) =8w0
h2tg
h
2sin
(
h
ℓx
)
+8w0
h2cos
(
h
ℓx
)
+ w0
$’%−
8
h2−
4
ℓx +
4
ℓ2x2
,/-
w2el(x) −
H
EIwel(x) =
H
EIwimp(x)
w2el(x) −
h2
ℓ2wel(x) =
h2
ℓ2wimp(x)
w2el(x) −
h2
ℓ2wel(x) =
h2
ℓ2wimp(x)
wel,p(x) = c0 + c1 x + c2 x2, w2
el,p(x) = 2 c2
wimp(x) = w0
$’%4
ℓx −
4
ℓ2x2
,/-
2 c2 −h2
ℓ2
$%c0 + c1 x+ c2 x
2,- =
h2
ℓ2w0
$’%4
ℓx −
4
ℓ2x2
,/-
$’’%2 c2 −
h2
ℓ2c0
,//- +
$’’%−
h2
ℓ2c1 −
4w0 h2
ℓ3
,//-x
+
$’’%−
h2
ℓ2c2 +
4w0 h2
ℓ4
,//-x2 = 0
−h2
ℓ2c2 +
4w0 h2
ℓ4= 0 ⇒ c2 =
4w0
ℓ2
−h2
ℓ2c1 −
4w0 h2
ℓ3= 0 ⇒ c1 = −
4w0
ℓ
2 c2 −h2
ℓ2c0 = 0 ⇒ c0 =
8w0
h2
wel,p(x) = w0
$’% 8
h2−
4
ℓx +
4
ℓ2x2
,/- =
8w0
h2− wimp(x)
wel(x) = a1 sh
(
h
ℓx
)
+ a2 ch
(
h
ℓx
)
+ w0
$’% 8
h2−
4
ℓx +
4
ℓ2x2
,/-
slobodno oslonjena greda:
wel(0) = 0 & wel(ℓ) = 0
a2 +8w0
h2= 0
a1 shh + a2 chh +8w0
h2= 0
a2 = −8w0
h2
a1 = −8w0
h2
1− chh
shh
wel(x) =8w0
h2th
h
2sh
(
h
ℓx
)
−8w0
h2ch
(
h
ℓx
)
+ w0
$’% 8
h2−
4
ℓx +
4
ℓ2x2
,/-
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