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METODA PREMIKOV
izr. prof. dr. Vojko KILARasist. dr. David Koren
marec, 2012
GRADBENA MEHANIKA:
Fakulteta za arhitekturo
Univerza v Ljubljani
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Okvirne konstrukcije• SAP2000: 3-etažen okvir: Lx = 2 x 6 m, Het = 3 m, HEA 300 (stebri), IPE 240 (grede), jeklo S235
• obtežba vozlišča i: M = 100 kNm
• zasuk vozlišča i [10-3 rad]:
okvir 1 okvir 2 okvir 3
2,06 1,18 1,14
vozlišče i
M
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Splošna togostna matrika elementa• OBOJESTRANSKO VPETI NOSILEC
ui wi i uk wk k
Ni L
A E 0 0
L
A E 0 0
Qi 0 12 E I
L3 6 E I
L2 0 3L
I E 12
6 E I
L2
Mi 0 6 E I
L2 4 E I
L 0
2L
I E 6
2 E I
L
Nk L
A E 0 0
L
A E 0 0
Qk 0 3L
I E 12
2L
I E 6 0 12 E I
L3 2L
I E 6
Mk 0 2L
I E 6
2 E I
L 0
2L
I E 6
4 E I
L
k i k
i k
wi
iMi
Qi
Qk
i kMi
Qi Qk
wk
i k
i
Mi
Qi Qk
i k
k
i
i
kwi
iMi
Qi
Qk
i
i
Mi
Qi
k
k
Mk
wkMi Mk
Qi
i
Qk
k
k
Qk
Mk
iMi
Mk
Qi
Qk
kk
zasuk členka ne povzročanotranjih sil
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Splošna togostna matrika elementa• ENOSTRANSKO VPETI NOSILEC
k i k
i k
wi
iMi
Qi
Qk
i kMi
Qi Qk
wk
i k
i
Mi
Qi Qk
i k
k
i
i
kwi
iMi
Qi
Qk
i
i
Mi
Qi
k
k
Mk
wkMi Mk
Qi
i
Qk
k
k
Qk
Mk
iMi
Mk
Qi
Qk
kk
zasuk členka ne povzročanotranjih sil
ui wi i uk wk k
Ni L
A E 0 0
L
A E 0 0
Qi 0 3 E I
L3 3 E I
L2 0 3L
I E 3 0
Mi 0 3 E I
L2 3 E I
L 0
2L
I E 3 0
Nk L
A E 0 0
L
A E 0 0
Qk 0 3L
I E 3
2L
I E 3 0 3 E I
L3 0
Mk 0 0 0 0 0 0
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Obojestransko vpeti nosilec• SAP2000: L = 1 m, EI = 1, faktor za A in As >> 1
• obtežba desnega vozlišča: φ = 1 radφ
[M] kNm
[Q] kN
deformacije
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Obo
jest
rans
ko v
peti
nosi
lec
Togostna matrika:
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Obo
jest
rans
ko v
peti
nosi
lec
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Enostransko vpeti nosilec• SAP2000: L = 1 m, EI = 1, faktor za A in As >> 1
• obtežba vpetega (desnega) vozlišča: φ = 1 radφ
[M] kNm
[Q] kN
deformacije
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Enos
tran
sko
vpeti
nos
ilec
0 00 3
Togostna matrika:
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Enos
tran
sko
vpeti
nos
ilec
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Vpliv
zun
anje
obt
ežbe
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Primer 1
l/2
Pφ1
32
l
1
φ2 φ3
l/2
Podatki:
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Primer 1
l/2
Pφ1
32
l
1
φ2 φ3
l/2
Podatki:
1 2 2 3
Togostni matriki elementov
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Primer 1
1 2 2 3
Togostni matriki elementov
Togostna matrika konstrukcije:
=
=
=1
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Primer 1=0
=0
l/2
Pφ1
2
l
1
φ2 φ3
l/2
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Primer 1 – upogibni momenti [ ]P
21
φ2
2φ2 =0,29
4φ2 =0,57
3φ2 =0,43
+- -
φ2φ2
[Mφ2]
1,0
1,0+- [Mobt.]
0,14
- 1,0
1,29
1,14+
- [M]-0,43
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Primer 1 – prečne sile [ ]P
21
φ2
6/l·φ2 = 0,86 + +
φ2φ2
[Qφ2]
4,0+
[Qobt.]-
4,0
4,86+
[Q]+0,43
3/l·φ2 = 0,43
-3,14
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Primer 1 – reakcije [ ]P
21
φ2
4,86+
[Q]+0,43
-3,14
3
V1 = 4,86
M1 = 1,29H1 = 0
V2 = 3,57 V3 = 0,43
+QR
Smeri:
+Q R
[R]
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Primer 2
l1
421
3
l2
l1
q
Podatki:
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Primer 2
421
3
q Podatki:
1 2 2 4
2
3
Togostne matrike elementov
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Primer 21 2 2 4
2
3
Togostna matrika konstrukcije:
=
i = 1 … „moder“ in „zelen“ element
i = 2 … „rdeč“ element
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Primer 2
Ob predpostavki l1 = l2 velja:
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Primer 2Upogibni momenti [kNm] in deformirana lega
Program SAP2000
-predpostavka l1 = l2
-q = 10 kN/m
-l = 1 m, EI = 1
-faktor za A in As >> 1
1,136
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Primer 2a421
3
q Podatki:
1 2 2 4
2
3
Togostne matrike elementov
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Primer 2a
Togostna matrika konstrukcije:
=
i = 1 … „moder“ in „zelen“ element
i = 2 … „rdeč“ element
1 2 2
2
3
Ob predpostavki l1 = l2 velja:
4
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Primer 2aUpogibni momenti [kNm] in deformirana lega
Program SAP2000
-predpostavka l1 = l2
-q = 10 kN/m
-l = 1 m, EI = 1
-faktor za A in As >> 1
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Primer 3
l
52
6
l
q
Podatki:
43
l
1
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Primer 3
l
52
6
l
q
43
l
1
Togostne matrike elementov
Togostna matrika konstrukcije
φ1 φ2 φ3 φ4 φ5 φ6
0 0 0 0 0 0
0 11 2 0 2 0
0 2 8 2 0 0
0 0 2 8 2 0
0 2 0 2 11 2
0 0 0 0 2 4
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Primer 3φ1 φ2 φ3 φ4 φ5 φ6
0 0 0 0 0 0
0 11 2 0 2 0
0 2 8 2 0 0
0 0 2 8 2 0
0 2 0 2 11 2
0 0 0 0 2 4
=
sistem 5 enačb s 5 neznankami
(φ2, φ3, φ4, φ5, M6)
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Primer 3Upogibni momenti [kNm] in deformirana lega
Program SAP2000
-q = 10 kN/m
-l = 1 m, EI = 1
-faktor za A in As >> 1
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Togostne matrike konstrukcijk11 k12 . . . k1n
k21 k22 . . . k2n
.
.
.
.
.
.
.
.
.
kn1 kn2 . . . knn
=
Za togostno matriko konstrukcije [K] in za togostne matrike elementov velja, da so simetrične. Togostna matrika stabilne konstrukcije je pozitivno definitna (ne more biti singularna in jo lahko invertiramo dobimo podajnostno matriko konstrukcije).Diagonalizacija matrike problem lastnih vrednosti (λ):
xMxK
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