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2D Truss Analysis with Stiffness Matrix

The document provides the details to model and analyze a simple truss structure using the stiffness matrix method. It includes: 1) The geometry and properties of the truss members. 2) The nodal coordinates and restraints. 3) Member end forces and nodal displacements from the analysis. 4) Verification of static equilibrium by checking the sum of vertical reactions.

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KABINET JALADARA NABDA
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49 views11 pages

2D Truss Analysis with Stiffness Matrix

The document provides the details to model and analyze a simple truss structure using the stiffness matrix method. It includes: 1) The geometry and properties of the truss members. 2) The nodal coordinates and restraints. 3) Member end forces and nodal displacements from the analysis. 4) Verification of static equilibrium by checking the sum of vertical reactions.

Uploaded by

KABINET JALADARA NABDA
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as DOCX, PDF, TXT or read online on Scribd
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M.

RHEZA FAHLEVI 022113144

SOAL 2 . RANGKA BATANG SEDERHANA

1. GAMBAR SOAL

5t 5t

3t

4 4 4 4

2. PERMODELAN

4 5
6
5 Data Batang (Semua)
7
3
E = 2E8 t/m2
1 3
A = 0.005 m2
1 2 2

I = 1 m4

3. PERSIAPAN DATA
 Data Propertis
1. E = 2e8 t/m2 A = 0.005 m2 I = 1 m4
 Data Koordinat

No. Titik X Y
1 0 0
2 8 0
3 16 0
4 4 4
5 12 4
M. RHEZA FAHLEVI 022113144

 Data Batang

No. Batang No. Titik Kiri No. Titik No.


(Bawah) Kanan (Atas) Proportis
1 1 2 1
2 2 3 1
3 1 4 1
4 4 5 1
5 3 5 1
6 2 4 1
7 2 5 1
 Data Tumpuan

No. Titik Restraint Restraint Restraint


RX RY RZ
1 1 1 0
3 0 1 0
 Data Beban Titik

No. Titik Beban Restraint


FX FY
4 0 -5
5 3 -5

4. INPUT DATA FILE

%****RANGKA BATANG 2D---TEKNIK SIPIL UNISSULA SEMARANG----


%****nama file : rangf----oleh :M. Rheza Fahlevi 26/9/2013----
%****struktur : rangka batang fleming -- kip.in
clear all
clc
%****General Data---
type='t2d'
nfile='rangf80'
%****Property---i E A -------
prop = [ 1 2e8 0.005 ];
%****Coordinates----------------
%****--- Joint X Y -------
coord=[ 1 0 0
2 8 0
3 16 0
4 4 4
5 12 4];
%****Element data---------------
%****--- Element J1 J2 prop---
element=[1 1 2 1
2 2 3 1
3 1 4 1
4 4 5 1
5 3 5 1
6 2 4 1
7 2 5 1];
%****Nodal Restraint-----------
%****---- Joint JR1 JR2 ---
Support=[ 1 1 1
M. RHEZA FAHLEVI 022113144

3 0 1 ];
%****Joint Load----------------
%****---Joint FX FY -------
JL=[ 4 0 -5
5 3 -5];
%****Element Load Data--------
%****---Elem. Fx1 Fy1 Fx2 Fy2 ---
AML=[ 1 0 0 0 0 ];
%****Call function for analysis t2d-----------------------
[dof,index,coord,element]=TRUSS2D_sdata(prop,element,coord,type);
[S,Sm,SmS,Cx,Cy,RT,L,A,Joint,Xj,Xk,Yj,Yk]=...
TRUSS2D_stiff(prop,element,coord,index,nfile);
[IR,IF,Support]=TRUSS2D_ldata(Support,dof);
[DF,AR,AM]=TRUSS2D_analysis_result...
(element,dof,index,IF,IR,S,Sm,JL,AML,RT,Support,type,nfile);
[joint_disp,support_reaction,beam_endforces]=TRUSS2D_print_result...
(JL,AML,dof,Support,element,IF,IR,DF,AR,AM,type,nfile);
%****end data------------------------------------------------
TRUSS2D_xplot
%

GAMBAR DEFORMASI
M. RHEZA FAHLEVI 022113144

5. OUTPUT PROGRAM DAN CEK

STIFFNESS MATRIX METHOD : PROGRAM TRUSS2D


File Name : rangf80_1.txt
Number of Members = 7
Number of Joints = 5
Modulus of Elasticity and Cross-sectional area :
No.Mat. E A
1 2.000e+008 5.000e-003

Coordinates of Joints :
Joint X Y
1 0.000 0.000
2 8.000 0.000
3 16.000 0.000
4 4.000 4.000
5 12.000 4.000

Element Information :
Elem. Ni Nj NM Cx Cy L
1 1 2 1 1.000 0.000 8.000
2 2 3 1 1.000 0.000 8.000
3 1 4 1 0.707 0.707 5.657
4 4 5 1 1.000 0.000 8.000
5 3 5 1 -0.707 0.707 5.657
6 2 4 1 -0.707 0.707 5.657
7 2 5 1 0.707 0.707 5.657

<<< end of file >>>

NAMA FILE_2.txt

STIFFNESS MATRIX METHOD : PROGRAM TRUSS2D


File Name : rangf80_2.txt

Joint Loads :
Joint Fx Fy
4 0.000 -5.000
5 3.000 -5.000

Member Loads :
Member Fx1 Fy1 Fx2 Fy2
1 0.000 0.000 0.000 0.000

Support Restraints :
Joint Rx Ry
1 1 1
3 0 1

Joint Displacements :
M. RHEZA FAHLEVI 022113144

Joint Dx Dy
1 0.0000e+000 0.0000e+000
2 5.8000e-005 -1.3657e-004
3 1.0400e-004 0.0000e+000
4 7.7485e-005 -1.2557e-004
5 4.9485e-005 -1.1957e-004

Support Reactions :
Joint Rx Ry
1 -3.0000e+000 4.2500e+000
3 0.0000e+000 5.7500e+000

Member End Forces (axial/normal):


Member axial force : + tension - compression
1 7.250
2 5.750
3 -6.010
4 -3.500
5 -8.132
6 -1.061
7 1.061

<<< end of file >>>

NAME FILE_4.txt

Member End Forces :


[AM]mx4 = [AML]mx4 + [AML]mx4[Sm]4x4[RT]4x4[Dj]4x4
Member F1x F1y F2x F2y
1 -7.250 0 7.250 0
2 -5.750 0 5.750 0
3 6.010 0 -6.010 0
4 3.500 0 -3.500 0
5 8.132 0 -8.132 0
6 1.061 0 -1.061 0
7 -1.061 0 1.061 0

Reaction of Supports :
{AR}IFx1 = -{ARC}IRx1 + [SRF]IRxIF {DF}IFx1
1 -3.000e+000
2 4.250e+000
6 5.750e+000

<<< end of file >>>

6. CEK KESETIMBANGAN

∑KY = 0
5 + 5 = 4,25 + 5,75
10 = 10
( OKE )
M. RHEZA FAHLEVI 022113144

7. FREE BODY DIAGRAM

8. PENYUSUNAN MATRIK
 MATRIK KEKAKUAN ELEMEN SUMBU BATANG (SM)

Stiffness Matrix Element (Local) [Sm]4x4 :


EA/L 0 -EA/L 0
0 0 0 0
-EA/L 0 EA/L 0
0 0 0 0

Stiffness Matrix Element Local [Sm]= 1 # 7


125000.0000 0.0000 -125000.0000 0.0000
0.0000 0.0000 0.0000 0.0000
-125000.0000 0.0000 125000.0000 0.0000
0.0000 0.0000 0.0000 0.0000

Stiffness Matrix Element Local [Sm]= 2 # 7


125000.0000 0.0000 -125000.0000 0.0000
0.0000 0.0000 0.0000 0.0000
-125000.0000 0.0000 125000.0000 0.0000
0.0000 0.0000 0.0000 0.0000

Stiffness Matrix Element Local [Sm]= 3 # 7


176776.6953 0.0000 -176776.6953 0.0000
0.0000 0.0000 0.0000 0.0000
-176776.6953 0.0000 176776.6953 0.0000
0.0000 0.0000 0.0000 0.0000

Stiffness Matrix Element Local [Sm]= 4 # 7


125000.0000 0.0000 -125000.0000 0.0000
0.0000 0.0000 0.0000 0.0000
-125000.0000 0.0000 125000.0000 0.0000
0.0000 0.0000 0.0000 0.0000
M. RHEZA FAHLEVI 022113144

Stiffness Matrix Element Local [Sm]= 5 # 7


176776.6953 0.0000 -176776.6953 0.0000
0.0000 0.0000 0.0000 0.0000
-176776.6953 0.0000 176776.6953 0.0000
0.0000 0.0000 0.0000 0.0000

Stiffness Matrix Element Local [Sm]= 6 # 7


176776.6953 0.0000 -176776.6953 0.0000
0.0000 0.0000 0.0000 0.0000
-176776.6953 0.0000 176776.6953 0.0000
0.0000 0.0000 0.0000 0.0000

Stiffness Matrix Element Local [Sm]= 7 # 7


176776.6953 0.0000 -176776.6953 0.0000
0.0000 0.0000 0.0000 0.0000
-176776.6953 0.0000 176776.6953 0.0000
0.0000 0.0000 0.0000 0.0000
 MATRIK ROTASI (RT)

Rotation Matrix Element [RT]4x4 :


Cx Cy 0 0
-Cy Cx 0 0
0 0 Cx Cy
0 0 -Cy Cx
---> Cx=Lx/L Cy=Ly/L

Rotation Matrix Element [RT]= 1 # 7


1.0000 0.0000 0.0000 0.0000
0.0000 1.0000 0.0000 0.0000
0.0000 0.0000 1.0000 0.0000
0.0000 0.0000 0.0000 1.0000

Rotation Matrix Element [RT]= 2 # 7


1.0000 0.0000 0.0000 0.0000
0.0000 1.0000 0.0000 0.0000
0.0000 0.0000 1.0000 0.0000
0.0000 0.0000 0.0000 1.0000

Rotation Matrix Element [RT]= 3 # 7


0.7071 0.7071 0.0000 0.0000
-0.7071 0.7071 0.0000 0.0000
0.0000 0.0000 0.7071 0.7071
0.0000 0.0000 -0.7071 0.7071

Rotation Matrix Element [RT]= 4 # 7


1.0000 0.0000 0.0000 0.0000
0.0000 1.0000 0.0000 0.0000
0.0000 0.0000 1.0000 0.0000
0.0000 0.0000 0.0000 1.0000

Rotation Matrix Element [RT]= 5 # 7


-0.7071 0.7071 0.0000 0.0000
-0.7071 -0.7071 0.0000 0.0000
M. RHEZA FAHLEVI 022113144

0.0000 0.0000 -0.7071 0.7071


0.0000 0.0000 -0.7071 -0.7071

Rotation Matrix Element [RT]= 6 # 7


-0.7071 0.7071 0.0000 0.0000
-0.7071 -0.7071 0.0000 0.0000
0.0000 0.0000 -0.7071 0.7071
0.0000 0.0000 -0.7071 -0.7071

Rotation Matrix Element [RT]= 7 # 7


0.7071 0.7071 0.0000 0.0000
-0.7071 0.7071 0.0000 0.0000
0.0000 0.0000 0.7071 0.7071
0.0000 0.0000 -0.7071 0.7071

 MATRIK KEKAKUAN ELEMEN SUMBU STRUCTURE (SMS)

Stiffness Matrix Element (Global):


[SmS]4x4= [RT]^T 4x4.[Sm]4x4.[RT]4x4

Stiffness Matrix Element Global [SmS]= 1 # 7


1 2 3 4
125000.0000 0.0000 -125000.0000 0.0000
0.0000 0.0000 0.0000 0.0000
-125000.0000 0.0000 125000.0000 0.0000
0.0000 0.0000 0.0000 0.0000

Stiffness Matrix Element Global [SmS]= 2 # 7


3 4 5 6
125000.0000 0.0000 -125000.0000 0.0000
0.0000 0.0000 0.0000 0.0000
-125000.0000 0.0000 125000.0000 0.0000
0.0000 0.0000 0.0000 0.0000

Stiffness Matrix Element Global [SmS]= 3 # 7


1 2 7 8
88388.3476 88388.3476 -88388.3476 -88388.3476
88388.3476 88388.3476 -88388.3476 -88388.3476
-88388.3476 -88388.3476 88388.3476 88388.3476
-88388.3476 -88388.3476 88388.3476 88388.3476

Stiffness Matrix Element Global [SmS]= 4 # 7


7 8 9 10
125000.0000 0.0000 -125000.0000 0.0000
0.0000 0.0000 0.0000 0.0000
-125000.0000 0.0000 125000.0000 0.0000
0.0000 0.0000 0.0000 0.0000

Stiffness Matrix Element Global [SmS]= 5 # 7


5 6 9 10
88388.3476 -88388.3476 -88388.3476 88388.3476
M. RHEZA FAHLEVI 022113144

-88388.3476 88388.3476 88388.3476 -88388.3476


-88388.3476 88388.3476 88388.3476 -88388.3476
88388.3476 -88388.3476 -88388.3476 88388.3476

Stiffness Matrix Element Global [SmS]= 6 # 7


3 4 7 8
88388.3476 -88388.3476 -88388.3476 88388.3476
-88388.3476 88388.3476 88388.3476 -88388.3476
-88388.3476 88388.3476 88388.3476 -88388.3476
88388.3476 -88388.3476 -88388.3476 88388.3476

Stiffness Matrix Element Global [SmS]= 7 # 7


3 4 9 10
88388.3476 88388.3476 -88388.3476 -88388.3476
88388.3476 88388.3476 -88388.3476 -88388.3476
-88388.3476 -88388.3476 88388.3476 88388.3476
-88388.3476 -88388.3476 88388.3476 88388.3476

 MATRIK KEKAKUAN STRUKTURE TOTAL (SF)

Stiffness Matrix Structure Total [S]=Total [SmS]i 10 x 10


1 2 3 4
5 6 7 8 9
10
213388.3476 88388.3476 -125000.0000 0.0000
0.0000 0.0000 -88388.3476 -88388.3476 0.0000
0.0000
88388.3476 88388.3476 0.0000 0.0000
0.0000 0.0000 -88388.3476 -88388.3476 0.0000
0.0000
-125000.0000 0.0000 426776.6953 0.0000
-125000.0000 0.0000 -88388.3476 88388.3476
-88388.3476 -88388.3476
0.0000 0.0000 0.0000 176776.6953
0.0000 0.0000 88388.3476 -88388.3476 -88388.3476
-88388.3476
0.0000 0.0000 -125000.0000 0.0000
213388.3476 -88388.3476 0.0000 0.0000
-88388.3476 88388.3476
0.0000 0.0000 0.0000 0.0000
-88388.3476 88388.3476 0.0000 0.0000
88388.3476 -88388.3476
-88388.3476 -88388.3476 -88388.3476 88388.3476
0.0000 0.0000 301776.6953 0.0000 -125000.0000
0.0000
-88388.3476 -88388.3476 88388.3476 -88388.3476
0.0000 0.0000 0.0000 176776.6953 0.0000
0.0000
0.0000 0.0000 -88388.3476 -88388.3476
-88388.3476 88388.3476 -125000.0000 0.0000
301776.6953 0.0000
M. RHEZA FAHLEVI 022113144

0.0000 0.0000 -88388.3476 -88388.3476


88388.3476 -88388.3476 0.0000 0.0000
0.0000 176776.6953

<<< end of file >>>

 MATRIK TOTAL (AC)

Joint Loads Total {AC} 10x1


1 0.000
2 0.000
3 0.000
4 0.000
5 0.000
6 0.000
7 0.000
8 -5.000
9 3.000
10 -5.000

Joint Loads Free Joint {AFC} 7x1


3 0.000
4 0.000
5 0.000
7 0.000
8 -5.000
9 3.000
10 -5.000

Joint Displacements :
{DF}IFx1 = [SRFI]IFxIF {AFC}IFx1
3 5.8000e-005
4 -1.3657e-004
5 1.0400e-004
7 7.7485e-005
8 -1.2557e-004
9 4.9485e-005
10 -1.1957e-004

Degree of Freedom of Joints :


Joint Nx Ny
1 1 2
2 3 4
3 5 6
4 7 8
5 9 10

Index Degree of Freedom of Members :


Member N1x N1y N2x N2y
1 1 2 3 4
2 3 4 5 6
3 1 2 7 8
4 7 8 9 10
M. RHEZA FAHLEVI 022113144

5 5 6 9 10
6 3 4 7 8
7 3 4 9 10

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