Exercise 041515

MEMBER ab LOCAL - ELEMENT STIFFNESS MATRIX [K']Notation for Node 1 a

Notation for Node 2 b ua va θza

Fxa

=

120 0 0

AREA AND GEOMETRIC PROPERTIES Fya 0 0.48 2400

Length (mm) 10000 L Mza 0 2400 16000000

Area (mm^2) 6000 A Fxb -120 0 0

Iy (mm^4) 0 Iy Fyb 0 -0.48 -2400

Iz (mm^4) 200000000 Iz Mzb 0 2400 8000000

J (mm^4) 300000000 J

E (GPa) 200 E TRANSFORMATION MATRIX [Γ]G (GPa) 76.9230769230769 G

v 0.3 v Fxa Fya Mza

Fx'a

=

0.000000 1.000000 0.000000

Local : x', y', z' Global: x, y, z Fy'a -1.000000 0.000000 0.000000

x to x' 90 Mz'a 0.000000 0.000000 1.000000

y to x' 0 Fx'b 0.000000 0.000000 0.000000

x to y' 180 Fy'b 0.000000 0.000000 0.000000

y to y' 90 Mz'b 0.000000 0.000000 0.000000

ua va θza

Fxa

=

7.3509E-15 -0.48 -2400

Fya 120 2.9404E-17 1.4702E-13

Mza 0 2400 16000000

Fxb -7.3509E-15 0.48 2400

Fyb -120 -2.9404E-17 -1.4702E-13

Mzb 0 2400 8000000

[K] - FOR GLOBAL

ua va θza

Fxa 0 0 -2400

Fya 0 120 0

Mza -2400 0 16000000

Fxb 0 0 2400

Fyb 0 -120 0

Mzb -2400 0 8000000

αx

βx

αy

βy

[Γ]T * [K']

[K] = [Γ]T * [K'] * [Γ]

LOCAL - ELEMENT STIFFNESS MATRIX [K']

ub vb θzb

-120 0 0 ua

0 -0.48 2400 va

0 -2400 8000000 θza

120 0 0 ub

0 0.48 -2400 vb

0 -2400 16000000 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

0.000000 1.000000 0.000000 Fxb

-1.000000 0.000000 0.000000 Fyb

0.000000 0.000000 1.000000 Mzb

ub vb θzb

-7.3509E-15 0.48 -2400 ua

-120 -2.9404E-17 1.4702E-13 va

0 -2400 8000000 θza

7.3509E-15 -0.48 2400 ub

120 2.9404E-17 -1.4702E-13 vb

0 -2400 16000000 θzb

[K] - FOR GLOBAL

ub vb θzb

0 0 -2400 ua

0 -120 0 va

2400 0 8000000 θza

0 0 2400 ub

0 120 0 vb

2400 0 16000000 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

=

75 0 0

AREA AND GEOMETRIC PROPERTIES Fyb 0 0.234375 937.5

Length (mm) 8000 L Mzb 0 937.5 5000000

Area (mm^2) 3000 A Fxc -75 0 0

Iy (mm^4) 0 Iy Fyc 0 -0.234375 -937.5

Iz (mm^4) 50000000 Iz Mzc 0 937.5 2500000

J (mm^4) 70000000 J


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' 90 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

=

75 1.4357E-17 5.7429E-14

Fyb 4.5943E-15 0.234375 937.5

Mzb 0 937.5 5000000

Fxc -75 -1.4357E-17 -5.7429E-14

Fyc -4.5943E-15 -0.234375 -937.5

Mzc 0 937.5 2500000

[K] - FOR GLOBAL

ub vb θzb

Fxb 75 4.6087E-15 5.7429E-14

Fyb 4.6087E-15 0.234375 937.5

Mzb 5.7429E-14 937.5 5000000

Fxc -75 -4.6087E-15 -5.7429E-14

Fyc -4.6087E-15 -0.234375 -937.5

Mzc 5.7429E-14 937.5 2500000

αx

βx

αy

βy

[Γ]T * [K']

[K] = [Γ]T * [K'] * [Γ]


uc vc θzc

-75 0 0 ub

0 -0.234375 937.5 vb

0 -937.5 2500000 θzb

75 0 0 uc

0 0.234375 -937.5 vc

0 -937.5 5000000 θzc


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

-75 -1.4357E-17 5.7429E-14 ub

-4.5943E-15 -0.234375 937.5 vb

0 -937.5 2500000 θzb

75 1.4357E-17 -5.7429E-14 uc

4.5943E-15 0.234375 -937.5 vc

0 -937.5 5000000 θzc

[K] - FOR GLOBAL

uc vc θzc

-75 -4.6087E-15 5.7429E-14 ub

-4.6087E-15 -0.234375 937.5 vb

-5.7429E-14 -937.5 2500000 θzb

75 4.6087E-15 -5.7429E-14 uc

4.6087E-15 0.234375 -937.5 vc

-5.7429E-14 -937.5 5000000 θzc

[Γ]T * [K']

[K] = [Γ]T * [K'] * [Γ]

MEMBER cd LOCAL - ELEMENT STIFFNESS MATRIX [K']Notation for Node 1 c

Notation for Node 2 d uc vc θzc

Fxc

=

120 0 0

AREA AND GEOMETRIC PROPERTIES Fyc 0 0.48 2400

Length (mm) 10000 L Mzc 0 2400 16000000

Area (mm^2) 6000 A Fxd -120 0 0

Iy (mm^4) 0 Iy Fyd 0 -0.48 -2400

Iz (mm^4) 200000000 Iz Mzd 0 2400 8000000

J (mm^4) 300000000 J


v 0.3 v Fxc Fyc Mzc

Fx'c

=

0.000000 -1.000000 0.000000

Local : x', y', z' Global: x, y, z Fy'c 1.000000 0.000000 0.000000

x to x' 270 Mz'c 0.000000 0.000000 1.000000

y to x' 180 Fx'd 0.000000 0.000000 0.000000

x to y' 0 Fy'd 0.000000 0.000000 0.000000

y to y' 270 Mz'd 0.000000 0.000000 0.000000

uc vc θzc

Fxc

=

-2.2053E-14 0.48 2400

Fyc -120 -8.8211E-17 -4.4105E-13

Mzc 0 2400 16000000

Fxd 2.2053E-14 -0.48 -2400

Fyd 120 8.8211E-17 4.4105E-13

Mzd 0 2400 8000000

[K] - FOR GLOBAL

uc vc θzc

Fxc 0.48 2.1964E-14 2400

Fyc 2.1964E-14 120 -4.4105E-13

Mzc 2400 -4.4105E-13 16000000

Fxd -0.48 -2.1964E-14 -2400

Fyd -2.1964E-14 -120 4.4105E-13

Mzd 2400 -4.4105E-13 8000000

αx

βx

αy

βy

[Γ]T * [K']

[K] = [Γ]T * [K'] * [Γ]


ud vd θzd

-120 0 0 uc

0 -0.48 2400 vc

0 -2400 8000000 θzc

120 0 0 ud

0 0.48 -2400 vd

0 -2400 16000000 θzd


Fxd Fyd Mzd

0.000000 0.000000 0.000000 Fxc

0.000000 0.000000 0.000000 Fyc

0.000000 0.000000 0.000000 Mzc

0.000000 -1.000000 0.000000 Fxd

1.000000 0.000000 0.000000 Fyd

0.000000 0.000000 1.000000 Mzd

ud vd θzd

2.2053E-14 -0.48 2400 uc

120 8.8211E-17 -4.4105E-13 vc

0 -2400 8000000 θzc

-2.2053E-14 0.48 -2400 ud

-120 -8.8211E-17 4.4105E-13 vd

0 -2400 16000000 θzd

[K] - FOR GLOBAL

ud vd θzd

-0.48 -2.1964E-14 2400 uc

-2.1964E-14 -120 -4.4105E-13 vc

-2400 4.4105E-13 8000000 θzc

0.48 2.1964E-14 -2400 ud

2.1964E-14 120 4.4105E-13 vd

-2400 4.4105E-13 16000000 θzd

[Γ]T * [K']

[K] = [Γ]T * [K'] * [Γ]

MEMBER ac LOCAL - ELEMENT STIFFNESS MATRIX [K']Notation for Node 1 a

Notation for Node 2 c ua va θza

Fxa

=

46.8521286 0 0

AREA AND GEOMETRIC PROPERTIES Fya 0 0.05713674 365.853659

Length (mm) 12806.24847 L Mza 0 365.853659 3123475.24

Area (mm^2) 3000 A Fxc -46.8521286 0 0

Iy (mm^4) 0 Iy Fyc 0 -0.05713674 -365.853659

Iz (mm^4) 50000000 Iz Mzc 0 365.853659 1561737.62

J (mm^4) 70000000 J


v 0.3 v Fxa Fya Mza

Fx'a

=

0.624695 0.780869 0.000000

Local : x', y', z' Global: x, y, z Fy'a -0.780869 0.624695 0.000000

x to x' 51.34019175 Mz'a 0.000000 0.000000 1.000000

y to x' 321.34019175 Fx'c 0.000000 0.000000 0.000000

x to y' 141.34019175 Fy'c 0.000000 0.000000 0.000000

y to y' 51.34019175 Mz'c 0.000000 0.000000 0.000000

ua va θza

Fxa

=

29.2682927 -0.0446163 -285.683711

Fya 36.5853659 0.03569304 228.546969

Mza 0 365.853659 3123475.24

Fxc -29.2682927 0.0446163 285.683711

Fyc -36.5853659 -0.03569304 -228.546969

Mzc 0 365.853659 1561737.62

[K] - FOR GLOBAL

ua va θza

Fxa 18.318597 22.8268253 -285.683711

Fya 22.8268253 28.5906684 228.546969

Mza -285.683711 228.546969 3123475.24

Fxc -18.318597 -22.8268253 285.683711

Fyc -22.8268253 -28.5906684 -228.546969

Mzc -285.683711 228.546969 1561737.62

αx

βx

αy

βy

[Γ]T * [K']

[K] = [Γ]T * [K'] * [Γ]


uc vc θzc

-46.8521286 0 0 ua

0 -0.05713674 365.853659 va

0 -365.853659 1561737.62 θza

46.8521286 0 0 uc

0 0.05713674 -365.853659 vc

0 -365.853659 3123475.24 θzc


Fxc Fyc Mzc

0.000000 0.000000 0.000000 Fxa

0.000000 0.000000 0.000000 Fya

0.000000 0.000000 0.000000 Mza

0.624695 0.780869 0.000000 Fxc

-0.780869 0.624695 0.000000 Fyc

0.000000 0.000000 1.000000 Mzc

uc vc θzc

-29.2682927 0.0446163 -285.683711 ua

-36.5853659 -0.03569304 228.546969 va

0 -365.853659 1561737.62 θza

29.2682927 -0.0446163 285.683711 uc

36.5853659 0.03569304 -228.546969 vc

0 -365.853659 3123475.24 θzc

[K] - FOR GLOBAL

uc vc θzc

-18.318597 -22.8268253 -285.683711 ua

-22.8268253 -28.5906684 228.546969 va

285.683711 -228.546969 1561737.62 θza

18.318597 22.8268253 285.683711 uc

22.8268253 28.5906684 -228.546969 vc

285.683711 -228.546969 3123475.24 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

=

19 23 -2686 0 0

Node 4 d Fya 23 149 229 0 -120

Node 5 Mza -2686 229 19123475 2400 0

Node 6 Fxb 0 0 2400 75 0

Node 7 Fyb 0 -120 0 0 120

Node 8 Mzb -2400 0 8000000 2400 937

Node 9 Fxc -18 -23 286 -75 0

Node 10 Fyc -23 -29 -229 0 0

Mzc -286 229 1561738 0 938

Fxd

Fyd

Mzd

Fx

Fy

Mz

(APPLY BOUNDARY AND COMPATIBILITY CONDITIONS)

θza ub vb

Mza

=

0

=

19123475 2400 0

Fxb 0 2400 75 0

Fyb 0 0 0 120

Mzb 0 8000000 2400 937

Fxc 0 286 -75 0

Fyc 0 -229 0 0

Mzc 0 1561738 0 938

DISPLACEMENTS

θza ub vb

θza

=

0.00000 -0.00001 0.00000

[ Kff ]

{ Δ } = [ Kff ] 1 ̶� * { F }

ub

=

-0.00001 0.07980 0.00008

vb 0.00000 0.00008 0.00832

θzb 0.00000 -0.00001 0.00000

uc -0.00001 0.06654 0.00007

vc 0.00000 -0.01031 0.00000

θzc 0.00000 -0.00001 0.00000


-0.0010 31.5989 -1.1729 -0.0154 31.2761

Fxa

=

-2685.7 -0.5 0.0 -2400.0 -18.3

Fya 228.5 0.0 -120.0 0.0 -22.8

Fxd 0.0 0.0 0.0 0.0 -0.5

Fyd 0.0 0.0 0.0 0.0 0.0

Mzd 0.0 0.0 0.0 0.0 2400.0

[ Ksf ]

GLOBAL STIFFNESS MATRIX

θzb uc vc θzc ud vd θzd u v θz

-2400 -18 -23 -286

0 -23 -29 229

8000000 286 -229 1561738

2400 -75 0 0

937 0 0 938

21000000 0 -938 2500000

0 94 23 2686 0 0 2400

-938 23 149 -1166 0 -120 0

2500000 2686 -1166 24123475 -2400 0 8000000

0 0 -2400 0 0 -2400

0 -120 0 0 120 0

2400 0 8000000 -2400 0 16000000


θzb uc vc θzc FIXED-END FORCES

8000000 286 -229 1561738

+

0.0

2400 -75 0 0 0.0

937 0 0 938 120.0

21000000 0 -938 2500000 160000.0

0 94 23 2686 -537.6

-938 23 149 -1166 120.0

2500000 2686 -1166 24123475 -928000.0

DISPLACEMENTS

θzb uc vc θzc DISPLACEMENTS

0.00000 -0.00001 0.00000 0.00000 0

=

-0.0010 rad

[ Kff ]

{ Δ } = [ Kff ] 1 ̶� * { F }

-0.00001 0.06654 -0.01031 -0.00001 0

=

31.5989 mm0.00000 0.00007 0.00000 0.00000 -120 -1.1729 mm0.00000 0.00000 0.00000 0.00000 -160000 -0.0154 rad0.00000 0.06659 -0.01031 -0.00001 537.6 31.2761 mm0.00000 -0.01031 0.00832 0.00000 -120 -5.4182 mm0.00000 -0.00001 0.00000 0.00000 928000 0.0364 rad


-5.4182 0.0364

-22.8 -285.7 -0.0010

=

-435.15582

-28.6 228.5 31.5989 -410.18731

0.0 -2400.0 -1.1729 -102.44418

-120.0 0.0 -0.0154 650.187313

0.0 8000000.0 31.2761 366501.495

-5.4182

0.0364

[ Ksf ]

GLOBAL STIFFNESS MATRIX

FIXED-END FORCES

ua

+

va

θza

ub

vb

θzb

uc

vc

θzc

ud

vd

θzd

u

v

θz

θza ub vb θzb uc vc θzc θza ub

19123475 2400 0 8000000 286 -229 1561738 24000000 24002400 75 0 2400 -75 0 0 2400 120

0 0 120 937 0 0 938 0 08000000 2400 937 21000000 0 -938 2500000 8000000 2400

286 -75 0 0 94 23 2686 1874 -120-229 0 0 -938 23 149 -1166 -1499 0

1561738 0 938 2500000 2686 -1166 24123475 4000000 0

0 0 0 0 0 0 0θza 0.00000 -0.00001 0.00000 0.00000 -0.00001 0.00000 0.00000ub -0.00001 0.07980 0.00008 -0.00001 0.06654 -0.01031 -0.00001vb 0.00000 0.00008 0.00832 0.00000 0.00007 0.00000 0.00000θzb 0.00000 -0.00001 0.00000 0.00000 0.00000 0.00000 0.00000uc -0.00001 0.06654 0.00007 0.00000 0.06659 -0.01031 -0.00001vc 0.00000 -0.01031 0.00000 0.00000 -0.01031 0.00832 0.00000θzc 0.00000 -0.00001 0.00000 0.00000 -0.00001 0.00000 0.00000

Mza -0.001048Fxb 31.59888Fyb -1.17292Mzb -0.015357Fxc 31.27613Fyc -5.418228Mzc 0.03643

vb θzb uc vc θzc

2400 24000000 1874 -1499 40000000 2400 -120 0 0

121 2400 0 -1 24002400 24000000 0 -2400 4000000

0 0 168 58 4274-1 -2400 58 195 -3899

2400 4000000 4274 -3899 32000000

0 00 0

120 -120160000 -160000-537.6 537.6

120 -120-928000 928000

θza ub θza ub vb θzb uc vc θzc

19123475 2400 -2686 0 0 -2400 -18 -23 -2862400 75 229 0 -120 0 -23 -29 229

0 0 -24000 -120 0

2400 0 8000000

FIXED-END FORCES

FIXED-END FORCES

Member bcDISPLACED NODE

Member cdDISPLACED NODE

Global Matrix for Fixed-End ForcesFxa 0Fya 6Mza 30000Fxb 0Fyb -32000Mzb 235.2Fxc 0Fyc -768000Mzc 608000Fxd 0Fyd 0Mzd 0Fxe 0Fye 0Mze 0

Member bcDISPLACED NODE

q 30 120

L 8 160000200 120

50000000 -160000

Member cdDISPLACED NODE

P -600 -62.4

L 10 -192000

a 2 115.2

b 8 -768000

608000

Fy1, kNMz1, kN∙mm

E, GPa Fy2, kN

I, mm4 Mz2, kN∙mm

Fy1, kNMz1, kN∙mmFy2, kNMz2, kN∙mm

Exercise 041515

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Transcript of Exercise 041515