2d Exercise 1
13
MEMBER ab LOCAL - ELE Notation for Node 1 a Notation for Node 2 b ua va Fxa = 1275.3041 0 AREA AND GEOMETRIC PROPERTIES Fya 0 5.5839586 Length (in) 268.3281573 L Mza 0 749.16667 Area (in^2) 11.8 A Fxb -1275.304 0 Iy (in^4) 0 Iy Fyb 0 -5.583959 Iz (in^4) 310 Iz Mzb 0 749.16667 J (in^4) 1 J E (ksi) 29,000 E TRANS G (ksi) 11153.846153846 G v 0.3 v Fxa Fya Fx'a = 0.447214 0.894427 Local : x', y', z' Global: x, y, z Fy'a -0.894427 0.447214 x to x' 63.43494882 Mz'a 0.000000 0.000000 y to x' 333.43494882 Fx'b 0.000000 0.000000 x to y' 153.43494882 Fy'b 0.000000 0.000000 y to y' 63.43494882 Mz'b 0.000000 0.000000 ua va Fxa = 570.33333 -4.994444 Fya 1140.6667 2.4972222 Mza 0 749.16667 Fxb -570.3333 4.9944444 Fyb -1140.667 -2.497222 Mzb 0 749.16667 ua va Fxa 259.52799 507.88806 Fya 507.88806 1021.3601 Mza -670.075 335.03752 Fxb -259.528 -507.8881 Fyb -507.8881 -1021.36 Mzb -670.075 335.03752 αx βx αy βy [K]
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Transcript of 2d Exercise 1
MEMBER ab LOCAL - ELEMENT STIFFNESS MATRIX [K']Notation for Node 1 a
Notation for Node 2 b ua va θza
Fxa
=
1275.3041 0 0
AREA AND GEOMETRIC PROPERTIES Fya 0 5.58395864 749.166667
Length (in) 268.3281573 L Mza 0 749.166667 134015.007
Area (in^2) 11.8 A Fxb -1275.3041 0 0
Iy (in^4) 0 Iy Fyb 0 -5.58395864 -749.166667
Iz (in^4) 310 Iz Mzb 0 749.166667 67007.5037
J (in^4) 1 J
E (ksi) 29,000 E TRANSFORMATION MATRIX [Γ]G (ksi) 11153.8461538462 G
v 0.3 v Fxa Fya Mza
Fx'a
=
0.447214 0.894427 0.000000
Local : x', y', z' Global: x, y, z Fy'a -0.894427 0.447214 0.000000
x to x' 63.43494882 Mz'a 0.000000 0.000000 1.000000
y to x' 333.43494882 Fx'b 0.000000 0.000000 0.000000
x to y' 153.43494882 Fy'b 0.000000 0.000000 0.000000
y to y' 63.43494882 Mz'b 0.000000 0.000000 0.000000
ua va θza
Fxa
=
570.333333 -4.99444444 -670.075037
Fya 1140.66667 2.49722222 335.037519
Mza 0 749.166667 134015.007
Fxb -570.333333 4.99444444 670.075037
Fyb -1140.66667 -2.49722222 -335.037519
Mzb 0 749.166667 67007.5037
[K] - FOR GLOBAL
ua va θza
Fxa 259.527988 507.888058 -670.075037
Fya 507.888058 1021.36007 335.037519
Mza -670.075037 335.037519 134015.007
Fxb -259.527988 -507.888058 670.075037
Fyb -507.888058 -1021.36007 -335.037519
Mzb -670.075037 335.037519 67007.5037
αx
βx
αy
βy
[Γ]T * [K']
[K] = [Γ]T * [K'] * [Γ]
LOCAL - ELEMENT STIFFNESS MATRIX [K']
ub vb θzb
-1275.3041 0 0 ua
0 -5.58395864 749.166667 va
0 -749.166667 67007.5037 θza
1275.3041 0 0 ub
0 5.58395864 -749.166667 vb
0 -749.166667 134015.007 θzb
TRANSFORMATION MATRIX [Γ]
Fxb Fyb Mzb
0.000000 0.000000 0.000000 Fxa
0.000000 0.000000 0.000000 Fya
0.000000 0.000000 0.000000 Mza
0.447214 0.894427 0.000000 Fxb
-0.894427 0.447214 0.000000 Fyb
0.000000 0.000000 1.000000 Mzb
ub vb θzb
-570.333333 4.99444444 -670.075037 ua
-1140.66667 -2.49722222 335.037519 va
0 -749.166667 67007.5037 θza
570.333333 -4.99444444 670.075037 ub
1140.66667 2.49722222 -335.037519 vb
0 -749.166667 134015.007 θzb
[K] - FOR GLOBAL
ub vb θzb
-259.527988 -507.888058 -670.075037 ua
-507.888058 -1021.36007 335.037519 va
670.075037 -335.037519 67007.5037 θza
259.527988 507.888058 670.075037 ub
507.888058 1021.36007 -335.037519 vb
670.075037 -335.037519 134015.007 θzb
[Γ]T * [K']
[K] = [Γ]T * [K'] * [Γ]
MEMBER bc LOCAL - ELEMENT STIFFNESS MATRIX [K']Notation for Node 1 b
Notation for Node 2 c ub vb θzb
Fxb
=
1425.83333 0 0
AREA AND GEOMETRIC PROPERTIES Fyb 0 7.80381944 936.458333
Length (in) 240 L Mzb 0 936.458333 149833.333
Area (in^2) 11.8 A Fxc -1425.83333 0 0
Iy (in^4) 0 Iy Fyc 0 -7.80381944 -936.458333
Iz (in^4) 310 Iz Mzc 0 936.458333 74916.6667
J (in^4) 1 J
E (ksi) 29,000 E TRANSFORMATION MATRIX [Γ]G (ksi) 11153.8461538462 G
v 0.3 v Fxb Fyb Mzb
Fx'b
=
1.000000 0.000000 0.000000
Local : x', y', z' Global: x, y, z Fy'b 0.000000 1.000000 0.000000
x to x' 0 Mz'b 0.000000 0.000000 1.000000
y to x' 270 Fx'c 0.000000 0.000000 0.000000
x to y' 90 Fy'c 0.000000 0.000000 0.000000
y to y' 0 Mz'c 0.000000 0.000000 0.000000
ub vb θzb
Fxb
=
1425.83333 4.7804E-16 5.7365E-14
Fyb -2.6203E-13 7.80381944 936.458333
Mzb 0 936.458333 149833.333
Fxc -1425.83333 -4.7804E-16 -5.7365E-14
Fyc 2.6203E-13 -7.80381944 -936.458333
Mzc 0 936.458333 74916.6667
[K] - FOR GLOBAL
ub vb θzb
Fxb 1425.83333 -2.6155E-13 5.7365E-14
Fyb -2.6155E-13 7.80381944 936.458333
Mzb 5.7365E-14 936.458333 149833.333
Fxc -1425.83333 2.6155E-13 -5.7365E-14
Fyc 2.6155E-13 -7.80381944 -936.458333
Mzc 5.7365E-14 936.458333 74916.6667
αx
βx
αy
βy
[Γ]T * [K']
[K] = [Γ]T * [K'] * [Γ]
LOCAL - ELEMENT STIFFNESS MATRIX [K']
uc vc θzc
-1425.83333 0 0 ub
0 -7.80381944 936.458333 vb
0 -936.458333 74916.6667 θzb
1425.83333 0 0 uc
0 7.80381944 -936.458333 vc
0 -936.458333 149833.333 θzc
TRANSFORMATION MATRIX [Γ]
Fxc Fyc Mzc
0.000000 0.000000 0.000000 Fxb
0.000000 0.000000 0.000000 Fyb
0.000000 0.000000 0.000000 Mzb
1.000000 0.000000 0.000000 Fxc
0.000000 1.000000 0.000000 Fyc
0.000000 0.000000 1.000000 Mzc
uc vc θzc
-1425.83333 -4.7804E-16 5.7365E-14 ub
2.6203E-13 -7.80381944 936.458333 vb
0 -936.458333 74916.6667 θzb
1425.83333 4.7804E-16 -5.7365E-14 uc
-2.6203E-13 7.80381944 -936.458333 vc
0 -936.458333 149833.333 θzc
[K] - FOR GLOBAL
uc vc θzc
-1425.83333 2.6155E-13 5.7365E-14 ub
2.6155E-13 -7.80381944 936.458333 vb
-5.7365E-14 -936.458333 74916.6667 θzb
1425.83333 -2.6155E-13 -5.7365E-14 uc
-2.6155E-13 7.80381944 -936.458333 vc
-5.7365E-14 -936.458333 149833.333 θzc
[Γ]T * [K']
[K] = [Γ]T * [K'] * [Γ]
GLOBAL STIFFNESS MATRIXNODE NOTATIONS
Node 1 a
Node 2 b ua va θza ub vb
Node 3 c Fxa
=
260 508 -670 -260 -508
Node 4 Fya 508 1021 335 -508 -1021
Node 5 Mza -670 335 134015 670 -335
Node 6 Fxb -260 -508 670 1685 508
Node 7 Fyb -508 -1021 -335 508 1029
Node 8 Mzb -670 335 67008 670 601
Node 9 Fxc 0 0 0 -1426 0
Node 10 Fyc 0 0 0 0 -8
Mzc 0 0 0 0 936
Fx
Fy
Mz
Fx
Fy
Mz
(APPLY BOUNDARY AND COMPATIBILITY CONDITIONS)
ub vb θzb
Fxb
=0
=1685 508 670
Fyb 0 508 1029 601
Mzb 1500 670 601 283848
DISPLACEMENTS
40.2492236 -35.124612 2250
ub
=0.00070 -0.00034 0.00000 40.2492236
=vb -0.00034 0.00114 0.00000 -35.124612
θzb 0.00000 0.00000 0.00000 2250
SUPPORT REACTIONS
[ Kff ]
{ Δ } = [ Kff ] 1 ̶� * { F }
ub vb θzb
Fxa
=
-260 -508 -670 0.0380647
+
Fya -508 -1021 335 -0.057565
Mza 670 -335 67008 0.00795888
Fxc -1426 0 0
Fyc 0 -8 -936
Mzc 0 936 74917
GLOBAL STIFFNESS MATRIX
θzb uc vc θzc u v θz u v θz
-670 0 0 0
335 0 0 0
67008 0 0 0
670 -1426 0 0
601 0 -8 936
283848 0 -936 74917
0 1426 0 0
-936 0 8 -936
74917 0 -936 149833
(APPLY BOUNDARY AND COMPATIBILITY CONDITIONS)
FIXED-END FORCES
ub
+-40.24922359
vb 35.125θzb -750.000
DISPLACEMENTS
DISPLACEMENTS
0.03806 in-0.05757 in0.00796 rad
SUPPORT REACTIONS
[ Kff ]
{ Δ } = [ Kff ] 1 ̶� * { F }
FIXED-END FORCES SPPRT REACTIONS
0
=
14.02 kips20.125 62.25 kips
1350.000 1928.10 kip-in0 -54.27 kips
15 8.00 kips-600 -57.65 kip-in
GLOBAL STIFFNESS MATRIX
ua
va
θza
ub
vb
θzb
uc
vc
θzc
u
v
θz
u
v
θz
ub vb θzb
-260 -508 -670-508 -1021 335670 -335 67008
-1426 0 00 -8 -9360 936 74917
FIXED-END FORCES
FIXED-END FORCES
Member abDISPLACED NODE
Member bcDISPLACED NODE
Global Matrix for Fixed-End ForcesFxa 0
Fya 20.125
Mza 1350.000
Fxb -40.24922359
Fyb 35.125
Mzb -750.000
Fxc 0
Fyc 15
Mzc -600
Member abDISPLACED NODE
40.24922359 20.124611795268.3281573 1350
20.124611795-1350
Member bcDISPLACED NODE
0.125 15.000240 600.00
15.000-600.00
P, kips Fy1, kipsL, in Mz1, kip-in
Fy2, kipsMz2, kip-in
q, kips/in Fy1, kipsL, in Mz1, kip-in
Fy2, kipsMz2, kip-in
