4. spring element e-mail: dr. ahmet zafer Şenalp e-mail: [email protected]@gmail.com...
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4. Spring Element
Dr. Ahmet Zafer Şenalpe-mail: [email protected]
Mechanical Engineering DepartmentGebze Technical University
ME 520Fundamentals of Finite Element Analysis
1-D Line Element
(Spring, truss, beam, pipe, ...,etc.)
2-D Plane Element
(Membrane, plate, shell, ...,etc.)
Types of Finite Elements
ME 520 Dr. Ahmet Zafer Şenalp 2Mechanical Engineering Department, GTU
4. Spring Element
3-D Solid Element
(3-D fields - temperature, displacement, stress, flow velocity, ...,etc.)
Types of Finite Elements
ME 520 Dr. Ahmet Zafer Şenalp 3Mechanical Engineering Department, GTU
4. Spring Element
One Spring Element:
Two nodes: i, jNodal displacements: ui, uj (in, m, mm)Nodal forces: fi, fj (lb, Newton)Spring constant (stiffness): k (lb/in, N/m, N/mm)
Spring force-displacement relationship:
Spring Element
ME 520 Dr. Ahmet Zafer Şenalp 4Mechanical Engineering Department, GTU
4. Spring Element
linearnonlinear
; is the force needed to produce a unit stretch.
We only consider linear problems in this introductory course.
Consider the equilibrium of forces for the spring. At node i,we have
and at node j,
In matrix form,
or,
wherek = (element) stiffness matrixu = (element nodal) displacement vectorf = (element nodal) force vectorNote that k is symmetric. Is k singular or nonsingular? That is,can we solve the equation? If not, why?
Spring Element (Spring Element)
ME 520 Dr. Ahmet Zafer Şenalp 5Mechanical Engineering Department, GTU
4. Spring Element
Spring System
ME 520 Dr. Ahmet Zafer Şenalp 6Mechanical Engineering Department, GTU
4. Spring Element
For element 1,
element 2,
: is the (internal) force acting on local node i of element m (i = 1, 2).
System Stiffness Matrix
ME 520 Dr. Ahmet Zafer Şenalp 7Mechanical Engineering Department, GTU
4. Spring Element
Method 1 – Force Balance:
Consider the equilibrium of forces at node 1,
consider the equilibrium of forces at node 2,
consider the equilibrium of forces at node 3
K : Stiffness matrix (structure matrix) for the spring system.
In martix form:
System Stiffness Matrix
ME 520 Dr. Ahmet Zafer Şenalp 8Mechanical Engineering Department, GTU
4. Spring Element
Method 2 – Enlarging the Element Stiffness Matrices :
u1 u2 u3
+
=
332313
322212
312111
uu all ofsummation uu all ofsummation uu all ofsummation
uu all ofsummation uu all ofsummation uu all ofsummation
uu all ofsummation uu all ofsummation uu all ofsummation
System Stiffness Matrix
ME 520 Dr. Ahmet Zafer Şenalp 9Mechanical Engineering Department, GTU
4. Spring Element
Method 3 – Assembling by Using Row and Column Addresses :
K=
u2 u3u1 u2
u1
u2
u1
u2
u3
u1 u2 u3
u2
u3
K=
Example 1
ME 520 Dr. Ahmet Zafer Şenalp 10Mechanical Engineering Department, GTU
4. Spring Element
Connectivity table:
Boundary conditions:a) Displacement boundary conditions:
b) Force boundary conditions: 0u ,0u ,0u 321
PF ,PF ,0F 321
E#Element Number
N1 Node 1
N2 Node 2
1 1 2
2 2 3
1 2
Example 1
ME 520 Dr. Ahmet Zafer Şenalp 11Mechanical Engineering Department, GTU
4. Spring Element
Reaction force:
Nodal displacement values:
Applying boundary conditions;
1 2
· Deformed shape of the structure· Balance of the external forces· Order of magnitudes of the numbers
Checking the Results
ME 520 Dr. Ahmet Zafer Şenalp 12Mechanical Engineering Department, GTU
4. Spring Element
Notes About the Spring Elements· Suitable for stiffness analysis· Not suitable for stress analysis of the spring itself· Can have spring elements with stiffness in the lateral direction, spring elements for torsion, etc.
Notes About the Spring Elements
ME 520 Dr. Ahmet Zafer Şenalp 13Mechanical Engineering Department, GTU
4. Spring Element
Example 2
ME 520 Dr. Ahmet Zafer Şenalp 14Mechanical Engineering Department, GTU
4. Spring Element
Given; k1=100N/mm, k2=200 N/mm, k3=100 N/mm, P=500 NFind; (a) the global stiffness matrix(b) displacements of nodes 2 and 3(c) the reaction forces at nodes 1 and 4(d) the force in the spring 2
Solution:
Connectivity table:
1 2 3
E# N1 N2
1 1 2
2 2 3
3 3 4
Example 2
ME 520 Dr. Ahmet Zafer Şenalp 15Mechanical Engineering Department, GTU
4. Spring Element
Boundary conditions: Displacement boundary conditions:
Force boundary conditions:
a) Element Stiffness Matrices (N/mm):
0u ,0u ,0u ,0u 4321
0F ,PF ,0F ,0F 4321
1 2 3
Example 2
ME 520 Dr. Ahmet Zafer Şenalp 16Mechanical Engineering Department, GTU
4. Spring Element
Construction of global stiffness matrix :
Equilibrium (FE) equation for the whole system is;
1 2 3
4
3
2
1
4
3
2
1
F
F
F
F
u
u
u
u
10010000
1003002000
0200300100
00100100symmetric and banded.
Example 2
ME 520 Dr. Ahmet Zafer Şenalp 17Mechanical Engineering Department, GTU
4. Spring Element
b) Applying boundary conditions; 0u ,0u ,0u ,0u 4321 0F ,PF ,0F ,0F 4321
1 2 3
4
1
3
2
F
P
0
F
0
u
u
0
10010000
1003002000
0200300100
00100100
4
1
3
2
F
P
0
F
0
u
u
0
10010000
1003002000
0200300100
00100100
P
0
u
u
300200
200300
3
2
Example 2
ME 520 Dr. Ahmet Zafer Şenalp 18Mechanical Engineering Department, GTU
4. Spring Element
c) From the 1st and 4th equations in FE equation for the whole system, we get the reaction forces :
d) FE equation for 2. Spring Element:
1 2 3
P
0
u
u
300200
200300
3
2
i=2,j=3
Force in the spring 2: F
Example 2
ME 520 Dr. Ahmet Zafer Şenalp 19Mechanical Engineering Department, GTU
4. Spring Element
c) From the 1st and 4th equations in FE equation for the whole system, we get the reaction forces :
d) FE equation for 2. Spring Element:
1 2 3
P
0
u
u
300200
200300
3
2
i=2,j=3
Force in the spring 2: F
Spring System Example 3
ME 520 Dr. Ahmet Zafer Şenalp 20Mechanical Engineering Department, GTU
4. Spring Element
Given;
Find; the global stiffness matrix
Solution :
Connectivity table:E# N1 N2
1 4 2
2 2 3
3 3 5
4 2 1
Spring System Example 3
ME 520 Dr. Ahmet Zafer Şenalp 21Mechanical Engineering Department, GTU
4. Spring Element
Element Stiffness Matrices :
Global stiffness matrix :
Symmetric and bandedSingular as boundary conditionsare not applied; det(K)=0