3d Exercise 3
34
ua va wa θxa θya θza Fxa = 160.00 0.00 0.00 0.00 0.00 0.00 Fya 0.00 0.96 0.96 0.00 -2400.00 0.00 Fza 0.00 0.96 0.96 0.00 -2400.00 0.00 Mxa 0.00 0.00 0.00 1538.46 0.00 0.00 Mya 0.00 -2400.00 -2400.00 0.00 ### 0.00 Mza 0.00 0.00 0.00 0.00 0.00 0.00 Fxb -160.00 0.00 0.00 0.00 0.00 0.00 Fyb 0.00 -0.96 -0.96 0.00 2400.00 0.00 Fzb 0.00 -0.96 -0.96 0.00 2400.00 0.00 Mxb 0.00 0.00 0.00 -1538.46 0.00 0.00 Myb 0.00 -2400.00 -2400.00 0.00 ### 0.00 Mzb 0.00 0.00 0.00 0.00 0.00 0.00 Fxc 0.00 0.00 0.00 0.00 0.00 0.00 Fyc 0.00 0.00 0.00 0.00 0.00 0.00 Fzc 0.00 0.00 0.00 0.00 0.00 0.00 Mxc 0.00 0.00 0.00 0.00 0.00 0.00 Myc 0.00 0.00 0.00 0.00 0.00 0.00 Mzc 0.00 0.00 0.00 0.00 0.00 0.00
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Transcript of 3d Exercise 3
ua va wa θxa θya θza ub
Fxa
=
160.00 0.00 0.00 0.00 0.00 0.00 -160.00Fya 0.00 0.96 0.96 0.00 -2400.00 0.00 0.00Fza 0.00 0.96 0.96 0.00 -2400.00 0.00 0.00Mxa 0.00 0.00 0.00 1538.46 0.00 0.00 0.00Mya 0.00 -2400.00 -2400.00 0.00 8000000.00 0.00 0.00Mza 0.00 0.00 0.00 0.00 0.00 0.00 0.00Fxb -160.00 0.00 0.00 0.00 0.00 0.00 160.00Fyb 0.00 -0.96 -0.96 0.00 2400.00 0.00 0.00Fzb 0.00 -0.96 -0.96 0.00 2400.00 0.00 0.00Mxb 0.00 0.00 0.00 -1538.46 0.00 0.00 0.00Myb 0.00 -2400.00 -2400.00 0.00 4000000.00 0.00 0.00Mzb 0.00 0.00 0.00 0.00 0.00 0.00 0.00Fxc 0.00 0.00 0.00 0.00 0.00 0.00 0.00Fyc 0.00 0.00 0.00 0.00 0.00 0.00 0.00Fzc 0.00 0.00 0.00 0.00 0.00 0.00 0.00Mxc 0.00 0.00 0.00 0.00 0.00 0.00 0.00Myc 0.00 0.00 0.00 0.00 0.00 0.00 0.00Mzc 0.00 0.00 0.00 0.00 0.00 0.00 0.00
vb wb θxb θyb θzb uc vc wc θxc0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00-0.96 -0.96 0.00 -2400.00 0.00 0.00 0.00 0.00 0.00-0.96 -0.96 0.00 -2400.00 0.00 0.00 0.00 0.00 0.000.00 0.00 -1538.46 0.00 0.00 0.00 0.00 0.00 0.00
2400.00 2400.00 0.00 4000000.00 0.00 0.00 0.00 0.00 0.000.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.000.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
150.96 0.96 0.00 2400.00 0.00 0.00 -150.00 0.00 0.000.96 1.90 3750.00 2400.00 0.00 0.00 0.00 -0.94 3750.000.00 3750.00 20001538.46 0.00 0.00 0.00 0.00 -3750.00 10000000.00
2400.00 2400.00 0.00 8002884.62 0.00 0.00 0.00 0.00 0.000.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.000.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
-150.00 0.00 0.00 0.00 0.00 0.00 150.00 0.00 0.000.00 -0.94 -3750.00 0.00 0.00 0.00 0.00 0.94 -3750.000.00 3750.00 10000000.00 0.00 0.00 0.00 0.00 -3750.00 20000000.000.00 0.00 0.00 -2884.62 0.00 0.00 0.00 0.00 0.000.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
θyc θzc
0.00 0.00 ua
0.00 0.00 va
0.00 0.00 wa
0.00 0.00 θxa
0.00 0.00 θya
0.00 0.00 θza
0.00 0.00 ub
0.00 0.00 vb
0.00 0.00 wb
0.00 0.00 θxb
-2884.62 0.00 θyb
0.00 0.00 θzb
0.00 0.00 uc
0.00 0.00 vc
0.00 0.00 wc
0.00 0.00 θxc
2884.62 0.00 θyc
0.00 0.00 θzc
Member abDisplaced Node
25 610000 30000
200 -6100000000 30000
Member bcDisplaced Node
-25 -5.33333333315000 -40000
200 5.3333333333300000000 -40000
Δ, mm Fy1, kNL, mm Mz1, kN mm∙
E, GPa Fy2, kNI, mm4 Mz2, kN mm∙
Δ, mm Fy1, kNL, mm Mz1, kN mm∙
E, GPa Fy2, kNI, mm4 Mz2, kN mm∙
Global Matrix for Fixed-End ForcesFxa 0Fya 6Fza 0Mxa 0Mya 0Mza 30000Fxb 0Fyb -11.33333Fzb 0Mxb 0Myb 0Mzb -10000Fxc 0Fyc 5.333333Fzc 0Mxc 0Myc 0Mzc -40000Fxd 0Fyd 0Fzd 0Mxd 0Myd 0Mzd 0
MEMBER 1 LOCAL - ELEMENT STIFFNESS MATRIX [K']Notation for Node 1 a
Notation for Node 2 b ua va wa
Fxa
=
160 0 0
AREA AND GEOMETRIC PROPERTIES Fya 0 0.96 0
Length (mm) 5000 L Fza 0 0 0
Area (mm^2) 4000 A Mxa 0 0 0
Iy (mm^4) 0 Iy Mya 0 0 0
Iz (mm^4) 50000000 Iz Mza 0 2400 0
J (mm^4) 100000 J Fxb -160 0 0
E (GPa) 200 E Fyb 0 -0.96 0
G (GPa) 76.9230769230769 G Fzb 0 0 0
v 0.3 v Mxb 0 0 0
Myb 0 0 0
Local : x', y', z' Global: x, y, z Mzb 0 2400 0
x to x' 0
y to x' 270 TRANSFORMATION MATRIX [Γ]z to x' 270
x to y' 90 Fxa Fya Fza
y to y' 90 Fx'a
=
1 -1.8377E-16 -1.8377E-16
z to y' 0 Fy'a 6.1257E-17 6.1257E-17 1
x to z' 270 Fz'a -1.8377E-16 -1 6.1257E-17
y to z' 180 Mx'a 0 0 0
z to z' 90 My'a 0 0 0
Mz'a 0 0 0
Fx'b 0 0 0
Fy'b 0 0 0
Fz'b 0 0 0
Mx'b 0 0 0
My'b 0 0 0
Mz'b 0 0 0
ua va wa
Fxa
=
160 5.8807E-17 0
Fya -2.9404E-14 5.8807E-17 0
Fza -2.9404E-14 0.96 0
Mxa 0 -4.4105E-13 0
Mya 0 -2400 0
Mza 0 1.4702E-13 0
Fxb -160 -5.8807E-17 0
Fyb 2.9404E-14 -5.8807E-17 0
Fzb 2.9404E-14 -0.96 0
Mxb 0 -4.4105E-13 0
αx
βx
δx
αy
βy
δy
αz
βz
δz
[Γ]T * [K']
Myb
=
0 -2400 0
Mzb 0 1.4702E-13 0
[K] - FOR GLOBAL
ua va wa
Fxa 160 -2.9404E-14 -2.9345E-14
Fya -2.9404E-14 5.4072E-30 5.8807E-17
Fza -2.9345E-14 5.8807E-17 0.96
Mxa -2.7018E-29 -2.7018E-29 -4.4105E-13
Mya -1.4702E-13 -1.4702E-13 -2400
Mza 9.0059E-30 9.0059E-30 1.4702E-13
Fxb -160 2.9404E-14 2.9345E-14
Fyb 2.9404E-14 -5.4072E-30 -5.8807E-17
Fzb 2.9345E-14 -5.8807E-17 -0.96
Mxb -2.7018E-29 -2.7018E-29 -4.4105E-13
Myb -1.4702E-13 -1.4702E-13 -2400
Mzb 9.0059E-30 9.0059E-30 1.4702E-13
[K] = [Γ]T * [K'] * [Γ]
LOCAL - ELEMENT STIFFNESS MATRIX [K']
θxa θya θza ub vb wb θxb θyb θzb
0 0 0 -160 0 0 0 0 0
0 0 2400 0 -0.96 0 0 0 2400
0 0 0 0 0 0 0 0 0
1538.46154 0 0 0 0 0 -1538.46154 0 0
0 0 0 0 0 0 0 0 0
0 0 8000000 0 -2400 0 0 0 4000000
0 0 0 160 0 0 0 0 0
0 0 -2400 0 0.96 0 0 0 -2400
0 0 0 0 0 0 0 0 0
-1538.46154 0 0 0 0 0 1538.46154 0 0
0 0 0 0 0 0 0 0 0
0 0 4000000 0 -2400 0 0 0 8000000
TRANSFORMATION MATRIX [Γ]
Mxa Mya Mza Fxb Fyb Fzb Mxb Myb Mzb
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
1 -1.8377E-16 -1.8377E-16 0 0 0 0 0 0
6.1257E-17 6.1257E-17 1 0 0 0 0 0 0
-1.8377E-16 -1 6.1257E-17 0 0 0 0 0 0
0 0 0 1 -1.8377E-16 -1.8377E-16 0 0 0
0 0 0 6.1257E-17 6.1257E-17 1 0 0 0
0 0 0 -1.8377E-16 -1 6.1257E-17 0 0 0
0 0 0 0 0 0 1 -1.8377E-16 -1.8377E-16
0 0 0 0 0 0 6.1257E-17 6.1257E-17 1
0 0 0 0 0 0 -1.8377E-16 -1 6.1257E-17
θxa θya θza ub vb wb θxb θyb θzb
0 0 1.4702E-13 -160 -5.8807E-17 0 0 0 1.4702E-13
0 0 1.4702E-13 2.9404E-14 -5.8807E-17 0 0 0 1.4702E-13
0 0 2400 2.9404E-14 -0.96 0 0 0 2400
1538.46154 0 -1.4702E-09 0 4.4105E-13 0 -1538.46154 0 -7.3509E-10
-2.8273E-13 0 -8000000 0 2400 0 2.8273E-13 0 -4000000
-2.8273E-13 0 4.9006E-10 0 -1.4702E-13 0 2.8273E-13 0 2.4503E-10
0 0 -1.4702E-13 160 5.8807E-17 0 0 0 -1.4702E-13
0 0 -1.4702E-13 -2.9404E-14 5.8807E-17 0 0 0 -1.4702E-13
0 0 -2400 -2.9404E-14 0.96 0 0 0 -2400
-1538.46154 0 -7.3509E-10 0 4.4105E-13 0 1538.46154 0 -1.4702E-09
[Γ]T * [K']
2.8273E-13 0 -4000000 0 2400 0 -2.8273E-13 0 -8000000
2.8273E-13 0 2.4503E-10 0 -1.4702E-13 0 -2.8273E-13 0 4.9006E-10
[K] - FOR GLOBAL
θxa θya θza ub vb wb θxb θyb θzb
-2.7018E-29 -1.4702E-13 9.0059E-30 -160 2.9404E-14 2.9345E-14 -2.7018E-29 -1.4702E-13 9.0059E-30
-2.7018E-29 -1.4702E-13 9.0059E-30 2.9404E-14 -5.4072E-30 -5.8807E-17 -2.7018E-29 -1.4702E-13 9.0059E-30
-4.4105E-13 -2400 1.4702E-13 2.9345E-14 -5.8807E-17 -0.96 -4.4105E-13 -2400 1.4702E-13
1538.46154 1.4699E-09 -2.8273E-13 2.7018E-29 2.7018E-29 4.4105E-13 -1538.46154 7.3537E-10 2.8273E-13
1.4699E-09 8000000 -4.9006E-10 1.4702E-13 1.4702E-13 2400 7.3537E-10 4000000 -2.4503E-10
-2.8273E-13 -4.9006E-10 3.0072E-26 -9.0059E-30 -9.0059E-30 -1.4702E-13 2.8273E-13 -2.4503E-10 1.4958E-26
2.7018E-29 1.4702E-13 -9.0059E-30 160 -2.9404E-14 -2.9345E-14 2.7018E-29 1.4702E-13 -9.0059E-30
2.7018E-29 1.4702E-13 -9.0059E-30 -2.9404E-14 5.4072E-30 5.8807E-17 2.7018E-29 1.4702E-13 -9.0059E-30
4.4105E-13 2400 -1.4702E-13 -2.9345E-14 5.8807E-17 0.96 4.4105E-13 2400 -1.4702E-13
-1538.46154 7.3537E-10 2.8273E-13 2.7018E-29 2.7018E-29 4.4105E-13 1538.46154 1.4699E-09 -2.8273E-13
7.3537E-10 4000000 -2.4503E-10 1.4702E-13 1.4702E-13 2400 1.4699E-09 8000000 -4.9006E-10
2.8273E-13 -2.4503E-10 1.4958E-26 -9.0059E-30 -9.0059E-30 -1.4702E-13 -2.8273E-13 -4.9006E-10 3.0072E-26
[K] = [Γ]T * [K'] * [Γ]
LOCAL - ELEMENT STIFFNESS MATRIX [K']
ua
va
wa
θxa
θya
θza
ub
vb
wb
θxb
θyb
θzb
TRANSFORMATION MATRIX [Γ]
Fxa
Fya
Fza
Mxa
Mya
Mza
Fxb
Fyb
Fzb
Mxb
Myb
Mzb
ua
va
wa
θxa
θya
θza
ub
vb
wb
θxb
θyb
θzb
[K] - FOR GLOBAL
ua
va
wa
θxa
θya
θza
ub
vb
wb
θxb
θyb
θzb
MEMBER 2 LOCAL - ELEMENT STIFFNESS MATRIX [K']Notation for Node 1 b
Notation for Node 2 c ub vb wb
Fxb
=
150 0 0
AREA AND GEOMETRIC PROPERTIES Fyb 0 0.9375 0
Length (mm) 8000 L Fzb 0 0 0
Area (mm^2) 6000 A Mxb 0 0 0
Iy (mm^4) 0 Iy Myb 0 0 0
Iz (mm^4) 200000000 Iz Mzb 0 3750 0
J (mm^4) 300000 J Fxc -150 0 0
E (GPa) 200 E Fyc 0 -0.9375 0
G (GPa) 76.9230769230769 G Fzc 0 0 0
v 0.3 v Mxc 0 0 0
Myc 0 0 0
Local : x', y', z' Global: x, y, z Mzc 0 3750 0
x to x' 90
y to x' 0 TRANSFORMATION MATRIX [Γ]z to x' 270
x to y' 270 Fxb Fyb Fzb
y to y' 90 Fx'b
=
6.1257E-17 1 -1.8377E-16
z to y' 0 Fy'b -1.8377E-16 6.1257E-17 1
x to z' 0 Fz'b 1 -1.8377E-16 6.1257E-17
y to z' 270 Mx'b 0 0 0
z to z' 90 My'b 0 0 0
Mz'b 0 0 0
Fx'c 0 0 0
Fy'c 0 0 0
Fz'c 0 0 0
Mx'c 0 0 0
My'c 0 0 0
Mz'c 0 0 0
ub vb wb
Fxb
=
9.1886E-15 -1.7229E-16 0
Fyb 150 5.7429E-17 0
Fzb -2.7566E-14 0.9375 0
Mxb 0 3750 0
Myb 0 -6.8915E-13 0
Mzb 0 2.2972E-13 0
Fxc -9.1886E-15 1.7229E-16 0
Fyc -150 -5.7429E-17 0
Fzc 2.7566E-14 -0.9375 0
Mxc 0 3750 0
αx
βx
δx
αy
βy
δy
αz
βz
δz
[Γ]T * [K']
Myc
=
0 -6.8915E-13 0
Mzc 0 2.2972E-13 0
[K] - FOR GLOBAL
ub vb wb
Fxb 5.9453E-31 9.1886E-15 -1.7229E-16
Fyb 9.1886E-15 150 -2.7508E-14
Fzb -1.7229E-16 -2.7508E-14 0.9375
Mxb -6.8915E-13 2.2972E-13 3750
Myb 1.2665E-28 -4.2215E-29 -6.8915E-13
Mzb -4.2215E-29 1.4072E-29 2.2972E-13
Fxc -5.9453E-31 -9.1886E-15 1.7229E-16
Fyc -9.1886E-15 -150 2.7508E-14
Fzc 1.7229E-16 2.7508E-14 -0.9375
Mxc -6.8915E-13 2.2972E-13 3750
Myc 1.2665E-28 -4.2215E-29 -6.8915E-13
Mzc -4.2215E-29 1.4072E-29 2.2972E-13
[K] = [Γ]T * [K'] * [Γ]
LOCAL - ELEMENT STIFFNESS MATRIX [K']
θxb θyb θzb uc vc wc θxc θyc θzc
0 0 0 -150 0 0 0 0 0
0 0 3750 0 -0.9375 0 0 0 3750
0 0 0 0 0 0 0 0 0
2884.61538 0 0 0 0 0 -2884.61538 0 0
0 0 0 0 0 0 0 0 0
0 0 20000000 0 -3750 0 0 0 10000000
0 0 0 150 0 0 0 0 0
0 0 -3750 0 0.9375 0 0 0 -3750
0 0 0 0 0 0 0 0 0
-2884.61538 0 0 0 0 0 2884.61538 0 0
0 0 0 0 0 0 0 0 0
0 0 10000000 0 -3750 0 0 0 20000000
TRANSFORMATION MATRIX [Γ]
Mxb Myb Mzb Fxc Fyc Fzc Mxc Myc Mzc
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
6.1257E-17 1 -1.8377E-16 0 0 0 0 0 0
-1.8377E-16 6.1257E-17 1 0 0 0 0 0 0
1 -1.8377E-16 6.1257E-17 0 0 0 0 0 0
0 0 0 6.1257E-17 1 -1.8377E-16 0 0 0
0 0 0 -1.8377E-16 6.1257E-17 1 0 0 0
0 0 0 1 -1.8377E-16 6.1257E-17 0 0 0
0 0 0 0 0 0 6.1257E-17 1 -1.8377E-16
0 0 0 0 0 0 -1.8377E-16 6.1257E-17 1
0 0 0 0 0 0 1 -1.8377E-16 6.1257E-17
θxb θyb θzb uc vc wc θxc θyc θzc
0 0 -6.8915E-13 -9.1886E-15 1.7229E-16 0 0 0 -6.8915E-13
0 0 2.2972E-13 -150 -5.7429E-17 0 0 0 2.2972E-13
0 0 3750 2.7566E-14 -0.9375 0 0 0 3750
1.767E-13 0 20000000 0 -3750 0 -1.767E-13 0 10000000
2884.61538 0 -3.6754E-09 0 6.8915E-13 0 -2884.61538 0 -1.8377E-09
-5.3011E-13 0 1.2251E-09 0 -2.2972E-13 0 5.3011E-13 0 6.1257E-10
0 0 6.8915E-13 9.1886E-15 -1.7229E-16 0 0 0 6.8915E-13
0 0 -2.2972E-13 150 5.7429E-17 0 0 0 -2.2972E-13
0 0 -3750 -2.7566E-14 0.9375 0 0 0 -3750
-1.767E-13 0 10000000 0 -3750 0 1.767E-13 0 20000000
[Γ]T * [K']
-2884.61538 0 -1.8377E-09 0 6.8915E-13 0 2884.61538 0 -3.6754E-09
5.3011E-13 0 6.1257E-10 0 -2.2972E-13 0 -5.3011E-13 0 1.2251E-09
[K] - FOR GLOBAL
θxb θyb θzb uc vc wc θxc θyc θzc
-6.8915E-13 1.2665E-28 -4.2215E-29 -5.9453E-31 -9.1886E-15 1.7229E-16 -6.8915E-13 1.2665E-28 -4.2215E-29
2.2972E-13 -4.2215E-29 1.4072E-29 -9.1886E-15 -150 2.7508E-14 2.2972E-13 -4.2215E-29 1.4072E-29
3750 -6.8915E-13 2.2972E-13 1.7229E-16 2.7508E-14 -0.9375 3750 -6.8915E-13 2.2972E-13
20000000 -3.6753E-09 1.2251E-09 6.8915E-13 -2.2972E-13 -3750 10000000 -1.8379E-09 6.1257E-10
-3.6753E-09 2884.61538 -5.3011E-13 -1.2665E-28 4.2215E-29 6.8915E-13 -1.8379E-09 -2884.61538 5.3011E-13
1.2251E-09 -5.3011E-13 7.5147E-26 4.2215E-29 -1.4072E-29 -2.2972E-13 6.1257E-10 5.3011E-13 3.7427E-26
6.8915E-13 -1.2665E-28 4.2215E-29 5.9453E-31 9.1886E-15 -1.7229E-16 6.8915E-13 -1.2665E-28 4.2215E-29
-2.2972E-13 4.2215E-29 -1.4072E-29 9.1886E-15 150 -2.7508E-14 -2.2972E-13 4.2215E-29 -1.4072E-29
-3750 6.8915E-13 -2.2972E-13 -1.7229E-16 -2.7508E-14 0.9375 -3750 6.8915E-13 -2.2972E-13
10000000 -1.8379E-09 6.1257E-10 6.8915E-13 -2.2972E-13 -3750 20000000 -3.6753E-09 1.2251E-09
-1.8379E-09 -2884.61538 5.3011E-13 -1.2665E-28 4.2215E-29 6.8915E-13 -3.6753E-09 2884.61538 -5.3011E-13
6.1257E-10 5.3011E-13 3.7427E-26 4.2215E-29 -1.4072E-29 -2.2972E-13 1.2251E-09 -5.3011E-13 7.5147E-26
[K] = [Γ]T * [K'] * [Γ]
LOCAL - ELEMENT STIFFNESS MATRIX [K']
ub
vb
wb
θxb
θyb
θzb
uc
vc
wc
θxc
θyc
θzc
TRANSFORMATION MATRIX [Γ]
Fxb
Fyb
Fzb
Mxb
Myb
Mzb
Fxc
Fyc
Fzc
Mxc
Myc
Mzc
ub
vb
wb
θxb
θyb
θzb
uc
vc
wc
θxc
θyc
θzc
[K] - FOR GLOBAL
ub
vb
wb
θxb
θyb
θzb
uc
vc
wc
θxc
θyc
θzc
MEMBER 2 LOCAL - ELEMENT STIFFNESS MATRIX [K']Notation for Node 1 b
Notation for Node 2 c ub vb wb
Fxb
=
150 0 0
AREA AND GEOMETRIC PROPERTIES Fyb 0 0.9375 0
Length (mm) 8000 L Fzb 0 0 0
Area (mm^2) 6000 A Mxb 0 0 0
Iy (mm^4) 0 Iy Myb 0 0 0
Iz (mm^4) 200000000 Iz Mzb 0 3750 0
J (mm^4) 300000 J Fxc -150 0 0
E (GPa) 200 E Fyc 0 -0.9375 0
G (GPa) 76.9230769230769 G Fzc 0 0 0
v 0.3 v Mxc 0 0 0
Myc 0 0 0
Local : x', y', z' Global: x, y, z Mzc 0 3750 0
x to x' 90
y to x' 0 TRANSFORMATION MATRIX [Γ]z to x' 270
x to y' 270 Fxb Fyb Fzb
y to y' 90 Fx'b
=
6.1257E-17 1 -1.8377E-16
z to y' 0 Fy'b -1.8377E-16 6.1257E-17 1
x to z' 0 Fz'b 1 -1.8377E-16 6.1257E-17
y to z' 270 Mx'b 0 0 0
z to z' 90 My'b 0 0 0
Mz'b 0 0 0
Fx'c 0 0 0
Fy'c 0 0 0
Fz'c 0 0 0
Mx'c 0 0 0
My'c 0 0 0
Mz'c 0 0 0
ub vb wb
Fxb
=
9.1886E-15 -1.7229E-16 0
Fyb 150 5.7429E-17 0
Fzb -2.7566E-14 0.9375 0
Mxb 0 3750 0
Myb 0 -6.8915E-13 0
Mzb 0 2.2972E-13 0
Fxc -9.1886E-15 1.7229E-16 0
Fyc -150 -5.7429E-17 0
Fzc 2.7566E-14 -0.9375 0
Mxc 0 3750 0
αx
βx
δx
αy
βy
δy
αz
βz
δz
[Γ]T * [K']
Myc
=
0 -6.8915E-13 0
Mzc 0 2.2972E-13 0
[K] - FOR GLOBAL
ub vb wb
Fxb 5.9453E-31 9.1886E-15 -1.7229E-16
Fyb 9.1886E-15 150 -2.7508E-14
Fzb -1.7229E-16 -2.7508E-14 0.9375
Mxb -6.8915E-13 2.2972E-13 3750
Myb 1.2665E-28 -4.2215E-29 -6.8915E-13
Mzb -4.2215E-29 1.4072E-29 2.2972E-13
Fxc -5.9453E-31 -9.1886E-15 1.7229E-16
Fyc -9.1886E-15 -150 2.7508E-14
Fzc 1.7229E-16 2.7508E-14 -0.9375
Mxc -6.8915E-13 2.2972E-13 3750
Myc 1.2665E-28 -4.2215E-29 -6.8915E-13
Mzc -4.2215E-29 1.4072E-29 2.2972E-13
[K] = [Γ]T * [K'] * [Γ]
LOCAL - ELEMENT STIFFNESS MATRIX [K']
θxb θyb θzb uc vc wc θxc θyc θzc
0 0 0 -150 0 0 0 0 0
0 0 3750 0 -0.9375 0 0 0 3750
0 0 0 0 0 0 0 0 0
2884.61538 0 0 0 0 0 -2884.61538 0 0
0 0 0 0 0 0 0 0 0
0 0 20000000 0 -3750 0 0 0 10000000
0 0 0 150 0 0 0 0 0
0 0 -3750 0 0.9375 0 0 0 -3750
0 0 0 0 0 0 0 0 0
-2884.61538 0 0 0 0 0 2884.61538 0 0
0 0 0 0 0 0 0 0 0
0 0 10000000 0 -3750 0 0 0 20000000
TRANSFORMATION MATRIX [Γ]
Mxb Myb Mzb Fxc Fyc Fzc Mxc Myc Mzc
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
6.1257E-17 1 -1.8377E-16 0 0 0 0 0 0
-1.8377E-16 6.1257E-17 1 0 0 0 0 0 0
1 -1.8377E-16 6.1257E-17 0 0 0 0 0 0
0 0 0 6.1257E-17 1 -1.8377E-16 0 0 0
0 0 0 -1.8377E-16 6.1257E-17 1 0 0 0
0 0 0 1 -1.8377E-16 6.1257E-17 0 0 0
0 0 0 0 0 0 6.1257E-17 1 -1.8377E-16
0 0 0 0 0 0 -1.8377E-16 6.1257E-17 1
0 0 0 0 0 0 1 -1.8377E-16 6.1257E-17
θxb θyb θzb uc vc wc θxc θyc θzc
0 0 -6.8915E-13 -9.1886E-15 1.7229E-16 0 0 0 -6.8915E-13
0 0 2.2972E-13 -150 -5.7429E-17 0 0 0 2.2972E-13
0 0 3750 2.7566E-14 -0.9375 0 0 0 3750
1.767E-13 0 20000000 0 -3750 0 -1.767E-13 0 10000000
2884.61538 0 -3.6754E-09 0 6.8915E-13 0 -2884.61538 0 -1.8377E-09
-5.3011E-13 0 1.2251E-09 0 -2.2972E-13 0 5.3011E-13 0 6.1257E-10
0 0 6.8915E-13 9.1886E-15 -1.7229E-16 0 0 0 6.8915E-13
0 0 -2.2972E-13 150 5.7429E-17 0 0 0 -2.2972E-13
0 0 -3750 -2.7566E-14 0.9375 0 0 0 -3750
-1.767E-13 0 10000000 0 -3750 0 1.767E-13 0 20000000
[Γ]T * [K']
-2884.61538 0 -1.8377E-09 0 6.8915E-13 0 2884.61538 0 -3.6754E-09
5.3011E-13 0 6.1257E-10 0 -2.2972E-13 0 -5.3011E-13 0 1.2251E-09
[K] - FOR GLOBAL
θxb θyb θzb uc vc wc θxc θyc θzc
-6.8915E-13 1.2665E-28 -4.2215E-29 -5.9453E-31 -9.1886E-15 1.7229E-16 -6.8915E-13 1.2665E-28 -4.2215E-29
2.2972E-13 -4.2215E-29 1.4072E-29 -9.1886E-15 -150 2.7508E-14 2.2972E-13 -4.2215E-29 1.4072E-29
3750 -6.8915E-13 2.2972E-13 1.7229E-16 2.7508E-14 -0.9375 3750 -6.8915E-13 2.2972E-13
20000000 -3.6753E-09 1.2251E-09 6.8915E-13 -2.2972E-13 -3750 10000000 -1.8379E-09 6.1257E-10
-3.6753E-09 2884.61538 -5.3011E-13 -1.2665E-28 4.2215E-29 6.8915E-13 -1.8379E-09 -2884.61538 5.3011E-13
1.2251E-09 -5.3011E-13 7.5147E-26 4.2215E-29 -1.4072E-29 -2.2972E-13 6.1257E-10 5.3011E-13 3.7427E-26
6.8915E-13 -1.2665E-28 4.2215E-29 5.9453E-31 9.1886E-15 -1.7229E-16 6.8915E-13 -1.2665E-28 4.2215E-29
-2.2972E-13 4.2215E-29 -1.4072E-29 9.1886E-15 150 -2.7508E-14 -2.2972E-13 4.2215E-29 -1.4072E-29
-3750 6.8915E-13 -2.2972E-13 -1.7229E-16 -2.7508E-14 0.9375 -3750 6.8915E-13 -2.2972E-13
10000000 -1.8379E-09 6.1257E-10 6.8915E-13 -2.2972E-13 -3750 20000000 -3.6753E-09 1.2251E-09
-1.8379E-09 -2884.61538 5.3011E-13 -1.2665E-28 4.2215E-29 6.8915E-13 -3.6753E-09 2884.61538 -5.3011E-13
6.1257E-10 5.3011E-13 3.7427E-26 4.2215E-29 -1.4072E-29 -2.2972E-13 1.2251E-09 -5.3011E-13 7.5147E-26
[K] = [Γ]T * [K'] * [Γ]
LOCAL - ELEMENT STIFFNESS MATRIX [K']
ub
vb
wb
θxb
θyb
θzb
uc
vc
wc
θxc
θyc
θzc
TRANSFORMATION MATRIX [Γ]
Fxb
Fyb
Fzb
Mxb
Myb
Mzb
Fxc
Fyc
Fzc
Mxc
Myc
Mzc
ub
vb
wb
θxb
θyb
θzb
uc
vc
wc
θxc
θyc
θzc
[K] - FOR GLOBAL
ub
vb
wb
θxb
θyb
θzb
uc
vc
wc
θxc
θyc
θzc
GLOBAL STIFFNESS MATRIXNODE NOTATIONS
Node 1 a
Node 2 b ua va wa θxa θya
Node 3 c Fxa
=
160 -2.94E-14 -2.934E-14 -2.702E-29 -1.47E-13
Node 4 Fya -2.94E-14 5.4072E-30 5.8807E-17 -2.702E-29 -1.47E-13
Node 5 Fza -2.934E-14 5.8807E-17 0.96 -4.411E-13 -2400
Node 6 Mxa -2.702E-29 -2.702E-29 -4.411E-13 1538.46154 1.4699E-09
Node 7 Mya -1.47E-13 -1.47E-13 -2400 1.4699E-09 8000000
Node 8 Mza 9.0059E-30 9.0059E-30 1.4702E-13 -2.827E-13 -4.901E-10
Node 9 Fxb -160 2.9404E-14 2.9345E-14 2.7018E-29 1.4702E-13
Node 10 Fyb 2.9404E-14 -5.407E-30 -5.881E-17 2.7018E-29 1.4702E-13
Fzb 2.9345E-14 -5.881E-17 -0.96 4.4105E-13 2400
Mxb -2.702E-29 -2.702E-29 -4.411E-13 -1538.4615 7.3537E-10
Myb -1.47E-13 -1.47E-13 -2400 7.3537E-10 4000000
Mzb 9.0059E-30 9.0059E-30 1.4702E-13 2.8273E-13 -2.45E-10
Fxc 0 0 0 0 0
Fyc 0 0 0 0 0
Fzc 0 0 0 0 0
Mxc 0 0 0 0 0
Myc 0 0 0 0 0
Mzc 0 0 0 0 0
Fx 0 0 0 0 0
Fy 0 0 0 0 0
Fz 0 0 0 0 0
Mx 0 0 0 0 0
My 0 0 0 0 0
Mz 0 0 0 0 0
(APPLY BOUNDARY AND COMPATIBILITY CONDITIONS)
ub θzb uc
0
=
0
=
310 0 -160
0 0 0 28000000 0
3.53553391 3.53553391 -160 0 160
-3.5355339 -3.5355339 0 -2400 0
0 0 0 4000000 0
[ Kff ]
DISPLACEMENTS
0 0 3.53553391
ub
=
0.00666667 0 0.00666667
θzb 0 0.00000005 0
uc 0.00666667 0 0.01291667
vc 0 0.00025 0
θzc 0 0.00000005 0
(APPLY BOUNDARY AND COMPATIBILITY CONDITIONS)
ub θzb uc
0
=
0
=
310 0 -160
0 0 0 28000000 0
3.53553391 3.53553391 -160 0 160
-3.5355339 -3.5355339 0 -2400 0
0 0 0 4000000 0
{ Δ } = [ Kff ] 1 ̶� * { F }
[ Kss ]
GLOBAL STIFFNESS MATRIX
θza ub vb wb θxb θyb θzb uc vc wc θxc
9.0059E-30 -160 2.9404E-14 2.9345E-14 -2.702E-29 -1.47E-13 9.0059E-30 0 0 0 0
9.0059E-30 2.9404E-14 -5.407E-30 -5.881E-17 -2.702E-29 -1.47E-13 9.0059E-30 0 0 0 0
1.4702E-13 2.9345E-14 -5.881E-17 -0.96 -4.411E-13 -2400 1.4702E-13 0 0 0 0
-2.827E-13 2.7018E-29 2.7018E-29 4.4105E-13 -1538.4615 7.3537E-10 2.8273E-13 0 0 0 0
-4.901E-10 1.4702E-13 1.4702E-13 2400 7.3537E-10 4000000 -2.45E-10 0 0 0 0
3.0072E-26 -9.006E-30 -9.006E-30 -1.47E-13 2.8273E-13 -2.45E-10 1.4958E-26 0 0 0 0
-9.006E-30 160 -2.021E-14 -2.952E-14 -6.891E-13 1.4702E-13 -5.122E-29 -5.945E-31 -9.189E-15 1.7229E-16 -6.891E-13
-9.006E-30 -2.021E-14 150 -2.745E-14 2.2972E-13 1.4702E-13 5.0658E-30 -9.189E-15 -150 2.7508E-14 2.2972E-13
-1.47E-13 -2.952E-14 -2.745E-14 1.8975 3750 2400 8.2698E-14 1.7229E-16 2.7508E-14 -0.9375 3750
2.8273E-13 -6.891E-13 2.2972E-13 3750 20001538 -2.205E-09 1.2249E-09 6.8915E-13 -2.297E-13 -3750 10000000
-2.45E-10 1.4702E-13 1.4702E-13 2400 -2.205E-09 8002884.62 -4.906E-10 -1.266E-28 4.2215E-29 6.8915E-13 -1.838E-09
1.4958E-26 -5.122E-29 5.0658E-30 8.2698E-14 1.2249E-09 -4.906E-10 1.0522E-25 4.2215E-29 -1.407E-29 -2.297E-13 6.1257E-10
0 -5.945E-31 -9.189E-15 1.7229E-16 6.8915E-13 -1.266E-28 4.2215E-29 5.9453E-31 9.1886E-15 -1.723E-16 6.8915E-13
0 -9.189E-15 -150 2.7508E-14 -2.297E-13 4.2215E-29 -1.407E-29 9.1886E-15 150 -2.751E-14 -2.297E-13
0 1.7229E-16 2.7508E-14 -0.9375 -3750 6.8915E-13 -2.297E-13 -1.723E-16 -2.751E-14 0.9375 -3750
0 -6.891E-13 2.2972E-13 3750 10000000 -1.838E-09 6.1257E-10 6.8915E-13 -2.297E-13 -3750 20000000
0 1.2665E-28 -4.222E-29 -6.891E-13 -1.838E-09 -2884.6154 5.3011E-13 -1.266E-28 4.2215E-29 6.8915E-13 -3.675E-09
0 -4.222E-29 1.4072E-29 2.2972E-13 6.1257E-10 5.3011E-13 3.7427E-26 4.2215E-29 -1.407E-29 -2.297E-13 1.2251E-09
0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0
(APPLY BOUNDARY AND COMPATIBILITY CONDITIONS)
vc θzc
0 0 ub
-2400 4000000 θzb
0 0 uc
0.96 -2400 vc
-2400 8000000 θzc
[ Kff ]
DISPLACEMENTS
-3.5355339 0
0 0 0
=
0.02357023
0.00025 0.00000005 0 -0.0008839
0 0 3.53553391 0.04566731
5.41666667 0.0015 -3.5355339 -19.150809
0.0015 0.00000055 0 -0.0053033
(APPLY BOUNDARY AND COMPATIBILITY CONDITIONS)
vc θzc
0 0 ub
-2400 4000000 θzb
0 0 uc
0.96 -2400 vc
-2400 8000000 θzc
{ Δ } = [ Kff ] 1 ̶� * { F }
[ Kss ]
GLOBAL STIFFNESS MATRIX
θyc θzc u v w θx θy θz
0 0 0 0 0 0 0 0 ua
0 0 0 0 0 0 0 0 va
0 0 0 0 0 0 0 0 wa
0 0 0 0 0 0 0 0 θxa
0 0 0 0 0 0 0 0 θya
0 0 0 0 0 0 0 0 θza
1.2665E-28 -4.222E-29 0 0 0 0 0 0 ub
-4.222E-29 1.4072E-29 0 0 0 0 0 0 vb
-6.891E-13 2.2972E-13 0 0 0 0 0 0 wb
-1.838E-09 6.1257E-10 0 0 0 0 0 0 θxb
-2884.6154 5.3011E-13 0 0 0 0 0 0 θyb
5.3011E-13 3.7427E-26 0 0 0 0 0 0 θzb
-1.266E-28 4.2215E-29 0 0 0 0 0 0 uc
4.2215E-29 -1.407E-29 0 0 0 0 0 0 vc
6.8915E-13 -2.297E-13 0 0 0 0 0 0 wc
-3.675E-09 1.2251E-09 0 0 0 0 0 0 θxc
2884.61538 -5.301E-13 0 0 0 0 0 0 θyc
-5.301E-13 7.5147E-26 0 0 0 0 0 0 θzc
0 0 0 0 0 0 0 0 ud
0 0 0 0 0 0 0 0 vd
0 0 0 0 0 0 0 0 wd
0 0 0 0 0 0 0 0 θxd
0 0 0 0 0 0 0 0 θyd
0 0 0 0 0 0 0 0 θzd
wa θxa θya wb θxb θyb wc
Fza 0.96 0.00 -2400.00 -0.96 0.00 -2400.00 0.00Mxa 0.00 1538.46 0.00 0.00 -1538.46 0.00 0.00Mya -2400.00 0.00 8000000.00 2400.00 0.00 4000000.00 0.00Fzb -0.96 0.00 2400.00 1.90 3750.00 2400.00 -0.94Mxb 0.00 -1538.46 0.00 3750.00 20001538.46 0.00 -3750.00Myb -2400.00 0.00 4000000.00 2400.00 0.00 8002884.62 0.00Fzc 0.00 0.00 0.00 -0.94 -3750.00 0.00 0.94Mxc 0.00 0.00 0.00 3750.00 10000000.00 0.00 -3750.00Myc 0.00 0.00 0.00 0.00 0.00 -2884.62 0.00
θxc θyc
0.00 0.00 wa
0.00 0.00 θxa
0.00 0.00 θya
3750.00 0.00 wb
10000000.00 0.00 θxb
0.00 -2884.62 θyb
-3750.00 0.00 wc
20000000.00 0.00 θxc
0.00 2884.62 θyc
FIXED-END FORCES
FIXED-END FORCES
Member abDISPLACED NODE
Member bcDISPLACED NODE
Global Matrix for Fixed-End ForcesFxa 0Fya
Mza
Fxb 0Fyb #DIV/0!Mzb #DIV/0!Fxc 0Fyc #DIV/0!Mzc 0Fxd 0Fyd 0Mzd 0Fxe 0Fye 0Mze 0
Member abDISPLACED NODE
q 1 25005000 2400200 -2500
50000000 2400
Member bcDISPLACED NODE
#DIV/0!#DIV/0!#DIV/0!
#DIV/0!
Fy1, kNL, mm Mz1, kN∙mmE, GPa Fy2, kN
I, mm4 Mz2, kN∙mm
Δ, mm Fy1, kNL, mm Mz1, kN∙mmE, GPa Fy2, kN
I, mm4 Mz2, kN∙mm
