mathcad - análise matricial flambagem
TRANSCRIPT
MATRIZ DE RIGIDEZ GEOMÉTRICA - FLAMBAGEM DE PÓRTICOS
1 Dados
E 3 106
⋅:= C 8:= H 3.5:=
2 Seções transversais dos elementos
Bv 0.30:= Hv 0.80:= Av Bv Hv⋅ 0.24=:= Iv
Bv Hv3
⋅
120.013=:= Lv C:=
Lp H:=Bp 0.30:= Hp 0.30:= Ap Bp Hp⋅ 0.09=:= Ip
Bp Hp3
⋅
126.75 10
4−×=:=
3 Matrizes de rigidez (SLC)
ke kT kG+=
ke
E A⋅
L
0
0
E A⋅
L−
0
0
0
12 E⋅ I⋅
L3
6 E⋅ I⋅
L2
−
0
12 E⋅ I⋅
L3
−
6 E⋅ I⋅
L2
−
0
6 E⋅ I⋅
L2
−
4 E⋅ I⋅
L
0
6 E⋅ I⋅
L2
2 E⋅ I⋅
L
E A⋅
L−
0
0
E A⋅
L
0
0
0
12 E⋅ I⋅
L3
−
6 E⋅ I⋅
L2
0
12 E⋅ I⋅
L3
6 E⋅ I⋅
L2
0
6 E⋅ I⋅
L2
−
2 E⋅ I⋅
L
0
6 E⋅ I⋅
L2
4 E⋅ I⋅
L
N
1
L
0
0
1
L−
0
0
0
6
5 L⋅
1
10−
0
6
5 L⋅−
1
10−
0
1
10−
2L
15
0
1
10
L
30−
1
L−
0
0
1
L
0
0
0
6
5 L⋅−
1
10
0
6
5 L⋅
1
10
0
1
10−
L
30−
0
1
10
2 L⋅
15
⋅−=
Elementos 1 e 3:
kT1
E Ap⋅
Lp
0
0
E Ap⋅
Lp
−
0
0
0
12 E⋅ Ip⋅
Lp3
6 E⋅ Ip⋅
Lp2
−
0
12 E⋅ Ip⋅
Lp3
−
6 E⋅ Ip⋅
Lp2
−
0
6 E⋅ Ip⋅
Lp2
−
4 E⋅ Ip⋅
Lp
0
6 E⋅ Ip⋅
Lp2
2 E⋅ Ip⋅
Lp
E Ap⋅
Lp
−
0
0
E Ap⋅
Lp
0
0
0
12 E⋅ Ip⋅
Lp3
−
6 E⋅ Ip⋅
Lp2
0
12 E⋅ Ip⋅
Lp3
6 E⋅ Ip⋅
Lp2
0
6 E⋅ Ip⋅
Lp2
−
2 E⋅ Ip⋅
Lp
0
6 E⋅ Ip⋅
Lp2
4 E⋅ Ip⋅
Lp
:=
kT3 kT1:=
kG1
1
Lp
0
0
1
Lp
−
0
0
0
6
5 Lp⋅
1
10−
0
6
5 Lp⋅−
1
10−
0
1
10−
2Lp
15
0
1
10
Lp
30−
1
Lp
−
0
0
1
Lp
0
0
0
6
5 Lp⋅−
1
10
0
6
5 Lp⋅
1
10
0
1
10−
Lp
30−
0
1
10
2 Lp⋅
15
0.286
0
0
0.286−
0
0
0
0.343
0.1−
0
0.343−
0.1−
0
0.1−
0.467
0
0.1
0.117−
0.286−
0
0
0.286
0
0
0
0.343−
0.1
0
0.343
0.1
0
0.1−
0.117−
0
0.1
0.467
=:=
kG3 kG1:=
Elementos 2:
kT2
E Av⋅
Lv
0
0
E Av⋅
Lv
−
0
0
0
12 E⋅ Iv⋅
Lv3
6 E⋅ Iv⋅
Lv2
−
0
12 E⋅ Iv⋅
Lv3
−
6 E⋅ Iv⋅
Lv2
−
0
6 E⋅ Iv⋅
Lv2
−
4 E⋅ Iv⋅
Lv
0
6 E⋅ Iv⋅
Lv2
2 E⋅ Iv⋅
Lv
E Av⋅
Lv
−
0
0
E Av⋅
Lv
0
0
0
12 E⋅ Iv⋅
Lv3
−
6 E⋅ Iv⋅
Lv2
0
12 E⋅ Iv⋅
Lv3
6 E⋅ Iv⋅
Lv2
0
6 E⋅ Iv⋅
Lv2
−
2 E⋅ Iv⋅
Lv
0
6 E⋅ Iv⋅
Lv2
4 E⋅ Iv⋅
Lv
90000
0
0
90000−
0
0
0
900
3600−
0
900−
3600−
0
3600−
19200
0
3600
9600
90000−
0
0
90000
0
0
0
900−
3600
0
900
3600
0
3600−
9600
0
3600
19200
=:=
ke2 kT2:=
4 Matrizes de transformação (SLC -- SGC)
Elementos 1 e 3:
γc1π
2:=
T1
cos γc1( )
sin γc1( )−
0
0
0
0
sin γc1( )
cos γc1( )
0
0
0
0
0
0
1
0
0
0
0
0
0
cos γc1( )
sin γc1( )−
0
0
0
0
sin γc1( )
cos γc1( )
0
0
0
0
0
0
1
0
1−
0
0
0
0
1
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
1−
0
0
0
0
1
0
0
0
0
0
0
0
1
=:=
T3 T1:=
Elementos 2:
γc2 0:=
T2
cos γc2( )
sin γc2( )−
0
0
0
0
sin γc2( )
cos γc2( )
0
0
0
0
0
0
1
0
0
0
0
0
0
cos γc2( )
sin γc2( )−
0
0
0
0
sin γc2( )
cos γc2( )
0
0
0
0
0
0
1
1
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
0
0
0
0
0
1
=:=
5 Matrizes de incidência
Elemento 1:
L1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
1
0
0
0
1
0
0
0
0
0
:=
Elemento 2:
L2
0
0
0
0
1
0
0
0
1
0
0
0
1
0
0
0
0
0
0
0
0
0
0
1
0
0
0
1
0
0
0
1
0
0
0
0
:=
Elemento 3:
L3
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
1
0
0
0
1
0
0
0
0
:=
5 Matrizes de transformação e de incidência
A1 T1 L1T
⋅
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
1−
0
0
0
0
0
0
0
=:= A2 T2 L2T
⋅
0
0
1
0
0
0
0
0
0
0
0
1
0
1
0
0
0
0
0
0
0
0
1
0
1
0
0
0
0
0
0
0
0
1
0
0
=:= A3 T3 L3T
⋅
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
1−
0
=:=
6 Matrizes de rigidez transformadas
kpT2 A2T
ke2⋅ A2⋅
1.92 104
×
9.6 103
×
3.6− 103
×
3.6 103
×
0
0
9.6 103
×
1.92 104
×
3.6− 103
×
3.6 103
×
0
0
3.6− 103
×
3.6− 103
×
900
900−
0
0
3.6 103
×
3.6 103
×
900−
900
0
0
0
0
0
0
9 104
×
9− 104
×
0
0
0
0
9− 104
×
9 104
×
=:=
kpT1 A1T
kT1⋅ A1⋅
2.314 103
×
0
6.073 1014−
×
0
991.837−
0
0
0
0
0
0
0
6.073 1014−
×
0
7.714 104
×
0
4.689 1012−
×
0
0
0
0
0
0
0
991.837−
0
4.689 1012−
×
0
566.764
0
0
0
0
0
0
0
=:=
kpG1 A1T
kG1⋅ A1⋅
0.467
0
0
0
0.1−
0
0
0
0
0
0
0
0
0
0.286
0
0
0
0
0
0
0
0
0
0.1−
0
0
0
0.343
0
0
0
0
0
0
0
=:=
kpT3 A3T
kT3⋅ A3⋅
0
0
0
0
0
0
0
2.314 103
×
0
6.073 1014−
×
0
991.837−
0
0
0
0
0
0
0
6.073 1014−
×
0
7.714 104
×
0
4.689 1012−
×
0
0
0
0
0
0
0
991.837−
0
4.689 1012−
×
0
566.764
=:=
kpG3 A3T
kG3⋅ A3⋅
0
0
0
0
0
0
0
0.467
0
0
0
0.1−
0
0
0
0
0
0
0
0
0
0.286
0
0
0
0
0
0
0
0
0
0.1−
0
0
0
0.343
=:=
kpT kpT1 kpT2+ kpT3+
2.151 104
×
9.6 103
×
3.6− 103
×
3.6 103
×
991.837−
0
9.6 103
×
2.151 104
×
3.6− 103
×
3.6 103
×
0
991.837−
3.6− 103
×
3.6− 103
×
7.804 104
×
900−
4.689 1012−
×
0
3.6 103
×
3.6 103
×
900−
7.804 104
×
0
4.689 1012−
×
991.837−
0
4.689 1012−
×
0
9.057 104
×
9− 104
×
0
991.837−
0
4.689 1012−
×
9− 104
×
9.057 104
×
=:=
kpG kpG1 kpG3+
0.467
0
0
0
0.1−
0
0
0.467
0
0
0
0.1−
0
0
0.286
0
0
0
0
0
0
0.286
0
0
0.1−
0
0
0
0.343
0
0
0.1−
0
0
0
0.343
=:=
H kpG1−
kpT⋅
4.851 104
×
2.194 104
×
1.26− 104
×
1.26 104
×
1.126 104
×
6.4 103
×
2.194 104
×
4.851 104
×
1.26− 104
×
1.26 104
×
6.4 103
×
1.126 104
×
8.229− 103
×
8.229− 103
×
2.732 105
×
3.15− 103
×
2.4− 103
×
2.4− 103
×
8.229 103
×
8.229 103
×
3.15− 103
×
2.732 105
×
2.4 103
×
2.4 103
×
5.811 104
×
6− 104
×
1.861 1011−
×
2.143− 1012−
×
2.811 105
×
2.8− 105
×
6− 104
×
5.811 104
×
2.143− 1012−
×
1.861 1011−
×
2.8− 105
×
2.811 105
×
=:=
NCR eigenvals H( )
562173.137
67983.962
1584.894
25500.332
278290.327
270000.000
=:=
M eigenvecs H( )
0.152
0.152−
0
0
0.691
0.691−
0.68−
0.68−
0.082−
0.082
0.174−
0.174−
0.02
0.02
1.818 103−
×
1.818− 103−
×
0.707
0.707
0.707−
0.707
0
0
6.412 103−
×
6.412− 103−
×
0.056
0.056
0.705−
0.705
0.016
0.016
0
0
0.707−
0.707−
0
0
=:=