1 9-jan-16 last course lecture plan and policies what is fem? brief history of the fem example of...
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1Apr 21, 2023
Last course Lecture plan and policies What is FEM? Brief history of the FEM Example of applications Discretization Example of FEM softwares Why study the theory of finite elements? Computational steps Introduction to Matlab Homework 1
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Today Stiffness equations for the linear
spring element and the bar element Application of the direct stiffness
method for structural systems made up of springs and bars in 1D space
Interpretation and properties of the stiffness matrix
Discretize the structure (problem domain) Divide the structure or continuum into finite elements
Once the structure has been discretized, the computational steps faithfully follow the steps in the direct stiffness method.
The direct stiffness method (DSM): The global stiffness matrix of the discrete structure are obtained
by superimposing (assembling) the stiffness matrices of the element in a direct manner.
Computational steps of the FEM- the direct stiffness method
Idealization Process
Source: C. Felippa (2012)
DSM: Breakdown Steps
Source: C. Felippa (2012)
DSM: Assembly & Solution Steps
Source: C. Felippa (2012)
72023.04.21
Any question before we discuss the simplest element, i.e. spring element?
Linear spring
Linear spring: Obeys Hooke’s law:
F : applied axial loadδ : elongation or contraction of the spring (deformation)
ks: spring constant (stiffness)
Resists forces only in the direction of the spring
F
F = ksδ
Linear spring as a finite element (1)
Points 1 and 2 are the finite element nodes. x: element (local) coordinate system f1, f2: element nodal forces, shown in the positive sense d1, d2: element nodal displacement, shown in the
positive sense Now let’s find the relationship between the nodal forces
and the nodal displacement when the spring is in equilibrium state.
f1, d1 f2, d2
x1 2
ks
Linear spring as a finite element (2)
First, let d1>0 and d2=0 (compression)
Nodal forces consistent with static equilibrium and Hooke’s law are
1 s 1f k d 2 s 1f k d
f1 f21 2
d1
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Linear spring as a finite element (3)
Next, let d1=0 and d2>0 (tension)
1 s 2f k d 2 s 2f k d
f1 f21 2
d2
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If both d1>0 and d2>0
In matrix format
1 s 1 2( )f k d d
2 s 1 2( )f k d d
1 1s
2 2
1 1
1 1
d fk
d f
orkd f
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This is equation is called the (discrete) element equilibrium equation or the element stiffness equation, refer to the element (local) coordinate system
kd f
s
1 1
1 1k
k
Spring element stiffness matrix
Linear spring as a finite element (4)
Example of a spring assemblage (1)
1, 2, 3: structural (global) node numbers (as opposed to element or local nodes)
X: global coordinate system F2 and F3 are the given applied forces, while F1 is
the unknown reaction force. D2 and D3 are the unknown nodal displacement,
while D1 is the given nodal displacement (in this case D1=0).
1 2 31 2
X
F1, D1
k1 k2F2, D2 F3, D3
Example of a spring assemblage (2)
The number of degree of freedom (DOF)= 3. The number of active DOF= 2.
We will establish the stiffness equation for the spring assemblage by using the element stiffness equations 1 and 2.
In this problem the local axes coincide with the global axis.
1 2 31 2
X
F1, D1
k1 k2F2, D2 F3, D3
Example of a spring assemblage (3)
Equilibrium condition for element 1:
Equilibrium condition for element 2:
12f
x1 2
k111f
11d
12d
1 11 1 1 1
1 11 1 2 2
k k d f
k k d f
22f
x1 2
k221f
21d
22d
2 22 2 1 1
2 22 2 2 2
k k d f
k k d f
Example of a spring assemblage (4)
Continuity or displacement compatibility condition:
1 2 31 2
X
F1, D1
k1 k2F2, D2 F3, D3
12f
x1 2
k111f
11d
12d
1 22 1 2d d D
11 1d D
22 3d D
22f
x1 2
k221f
21d
22d
The equilibrium equations expressed in terms of the global nodal displacements:
Example of a spring assemblage (5)
1 11 1 1 1
1 11 1 2 2
k k d f
k k d f
22 2 2 1
22 2 3 2
k k D f
k k D f
11 1 1 1
11 1 2 2
k k D f
k k D f
2 22 2 1 1
2 22 2 2 2
k k d f
k k d f
Expanding the matrix equations to the size of the DOF:
11 1 1 1
11 1 2 2
3
0
0
0 0 0 0
k k D f
k k D f
D
12
2 2 2 12
2 2 3 2
0 0 0 0
0
0
D
k k D f
k k D f
22 2 2 1
22 2 3 2
k k D f
k k D f
11 1 1 1
11 1 2 2
k k D f
k k D f
1 2 31 2
D1 D2 D3
1 2 31 2
D1 D2 D3
The addition of the two equations yields
11 1 1 1
1 21 1 2 2 2 2 1
22 2 3 2
0
0
k k D f
k k k k D f f
k k D f
Example of a spring assemblage (7)
The equilibrium conditions for nodes 1, 2, and 3:
12f
x1 2
k111f
11d
12d
22f
x1 2
k221f
21d
22d
Example of a spring assemblage (8)
11f1F 1
1 22 1f f
2F222f 3F3
11 1f F 1 2
2 1 2f f F 22 3f F
Example of a spring assemblage (9)
Substituting the last equations, we obtain:
or
1 1 1 1
1 1 2 2 2 2
2 2 3 3
0
0
k k D F
k k k k D F
k k D F
KD F
Structural or global nodal force matrix
Structural or global nodal displacement matrix
Structural or global stiffness matrix
To facilitate the assembling of the global stiffness equation, let us define a matrix in which each column contains the numerical labels of the global nodal displacement associated with each element.
Let call this matrix “code matrix”, denote Mcode
The size of Mcode is Ndofe x Nelem, where Ndofe: number of degrees of freedom per element Nelem: total number of elements in the structure
Assembling process (1)
Assembling process (2)
1 2 31 2
X
F1, D1
k1 k2F2, D2 F3, D3
1 2
d1 d2
Assembling process (3)
The rows and columns of each stiffness matrix are labeled according to the DOFs associated with them; the labels are copied from Mcode.
1 1
1 1
k k
k k
1 2
1
2
2 2
2 2
k k
k k
2 32
3
1 1
1 1 2 2
2 2
0
0
k k
k k k k
k k
1 2 3
1
2
3
Boundary Conditions (1)
Given F1, F2 and F3, can you now solve this equation to obtain D1, D2, and D3?
The support or boundary conditions must be specified, otherwise K will be singular.
Types of boundary conditions: Homogeneous b.c.’s: zero values of nodal displacement are
specified. Non-homogeneous b.c.’s: nonzero values are specified.
1 1 1 1
1 1 2 2 2 2
2 2 3 3
0
0
k k D F
k k k k D F
k k D F
Example of homogeneous b.c.’s: D1=0 The global stiffness equation can be written as
The unknown values: F1, D2, D3
1 1 1
1 1 2 2 2 2
2 2 3 3
0 0
0
k k F
k k k k D F
k k D F
Boundary Conditions (2)
Boundary Conditions (3)
Considering the 2nd and the 3rd equations, we can write the matrix equation as
This equation is called reduced stiffness equation. For all homogeneous b.c.’s, the reduced stiffness equation can be
obtained by simply deleting the rows and columns corresponding to the zero-displacement DOF’s from the original stiffness equation.
2 21 2 2
3 32 2
D Fk k k
D Fk k
Boundary Conditions (4)
Solving the reduced stiffness equation, we can obtain
Substituting D2 and D3 into the original stiffness equation, we obtain
2 32
1
F FD
k
2 3 3
31 2
F F FD
k k
1 2 3( )F F F
Example of nonhomogeneous b.c.’s: D1=δ The global stiffness equation can be written as
The unknown values: F1, D2, D3
1 1 1
1 1 2 2 2 2
2 2 3 3
0
0
k k F
k k k k D F
k k D F
Boundary Conditions (5)
Boundary Conditions (6)
Considering the 2nd and the 3rd equations,
or
When dealing with non-homogeneous b.c.’s, we must transform the terms associated with the known displacements to the right-side force matrix.
2 21 2 21
3 32 20
D Fk k kk
D Fk k
2 21 2 2 1
3 32 2 0
D Fk k k k
D Fk k
Boundary Conditions (7)
In general, specified boundary conditions are treated by partitioning the global stiffness equation as follows:
Subscript a: associated with active (free, unconstrained) DOF’s Subscript p: associated with passive (locked, constrained) DOF’s Unknown vectors: Da, Fp
aa ap a a
pa pp p p
K K D F
K K D F
Boundary Conditions (8)
From this equation, we have
aa ap a a
pa pp p p
K K D F
K K D F
aa a a ap p K D F K D
p pa a pp p F K D K D
Example
Given a system of three springs supporting three equal weights W.
W=1 kN, k= 0.2 kN/mm Treating the springs as finite
elements, determine the vertical displacement of each weight and the support reaction.
Repeat the problem if the second weight is “forced” to move 5 mm upward.
3k
2k
W
W
k
W
Homework 2 (due date next class)
A system of three rigid carts on a horizontal plane interconnected by a system of linear elastic springs (adopted from Bathe, 1996)
The spring stiffness k1=1 kN/mm, k2=2 kN/mm, k3=3 kN/mm, k4=4 kN/mm, k5=5 kN/mm.
Homework 2 (cont’d)
Applying a force of 2 kN at cart 2 in the positive x direction, determine the nodal displacements, the reactions, and the internal spring forces. Use the direct stiffness method, implemented in Matlab.
Instead of applying the force, now at node 2 is given a fixed, known displacement – 2 mm. Repeat the calculations.
For exercise (not compulsary homework)
The spring stiffness k1=1 kN/mm, k2=2 kN/mm, k3=3 kN/mm, k4=4 kN/mm, k5=5 kN/mm.
Applying a force of 2 kN at node 2 in the positive x direction, determine the nodal displacements, the reactions, and the forces in each element. Use the direct stiffness method, implemented in Matlab.
Instead of applying the force, now at node 2 given a fixed, known displacement – 2 mm. Repeat the calculations.
Any questions before we proceed to 1D bar element?
Linear-elastic Bar-- Assumptions
The bar is geometrically straight. The material obeys Hooke’s law, i.e.
A: sectional area of the bar E: modulus of elasticity of the material σ: axial stress in the bar, σx = F/A
εx: axial strain of the bar
x xE FF
A, E
Linear-elastic Bar-- Assumptions
Forces are applied only at the ends of the bar. The bar supports axial loading only.
Bending, torsion, and shear are not transmitted to the element
FFx xE
Bar element (1)
Now let’s find the relationship between the nodal forces and the nodal displacement when the spring is in equilibrium state.
f1, d1 f2, d2
x1 2
EA
L
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Bar element (2)
First, let d1>0 and d2=0 (compression)
Nodal forces consistent with static equilibrium and Hooke’s law are
1 1
EAf d
L 2 1
EAf d
L
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Bar element (3)
Next, let d1=0 and d2>0 (tension)
1 2
EAf d
L 2 2
EAf d
L
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If both d1>0 and d2>0
In matrix format
1 1 2( )EA
f d dL
2 1 2( )EA
f d dL
1 1
2 2
1 1
1 1
d fEA
d fL
orkd f
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Example – A bar structure (1)
The area of a cross section at a distance x:
A=A(x)
0x Tx L
03A A0A A
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Example – A bar structure (2) Discretization
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Example – A bar structure (3) Generate element stiffness matrices
Apply the element stiffness matrix formula to the bar problem
1 0
T
1 15
1 1
EA
L
k 2 0
T
1 13
1 1
EA
L
k
Element 1 Element 2
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Example – A bar structure (4) Element 1
D1 D2 D3
1 0
T
1 1 05
1 1 0
0 0 0
EA
L
K
D1 D2 D3
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Example – A bar structure (4) Element 2
D1 D2 D3
D1 D2 D3
2 0
T
0 0 03
0 1 1
0 1 1
EA
L
K
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Example – A bar structure (5) All elements are present
1 2 0
T
5 5 0
5 8 3
0 3 3
EA
L
K K K
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Example – A bar structure (6) Applying the loads
Nodal force vector
0
H
F
F
Unknown reaction force
D1
D2
D3
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Example – A bar structure (7) Imposing the displacement boundary
conditions D1=0
In consequence, now there are only two unknown parameters D2 and D3
Thus the “active” DOF=2
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Solving the global equation,
The unknowns are D2, D3, and H
Discarding row 1 and column 1 from K, we obtain
02
T3
5 5 0 0
5 8 3 0
0 3 3
HEA
DL
D F
20
3T
8 3 0
3 3
DEA
D FL
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Solution
Support reaction
Example – A bar structure (9)
2 T
3 0
3/15
8 /15
D FL
D EA
02
T
5EA
D HL
T2
05
FLD
EA
Thus H F
Agree with the global equilibrium
F F
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Example – A bar structure (10) “Disassembly” of the structural nodal
displacement
1
T2
03
0
3/15
8 /15
DFL
DEA
D
D
111 1 T
12 02
0
3/15
Dd FL
D EAd
d
222 1 T
23 02
3/15
8 /15
Dd FLD EAd
d
Element 2
Element 1
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Example – A bar structure (11) Calculating stresses
Element 1
Element 2
1 11 1 2 1
T 0
2
0.5 5x x
d d FE E
L A
2 22 2 2 1
T 0
2
0.5 3x x
d d FE E
L A
1 T
0
0
3/15
FL
EA
d
2 T
0
3/15
8 /15
FL
EA
d
Any questions before we discuss the interpretation and properties of the stiffness matrix?
Element stiffness matrix
Recall the bar element
The stiffness equation for the element
The stiffness matrix
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1 1
2 2
1 1
1 1
d fEA
d fL
1 1
1 1
EA
L
k
From this eq., can you give a definition of stiffness matrix?
A stiffness matrix is a matrix relating the nodal displacement vector to nodal force vector
How did we derive this matrix?
By directly applying the basic physical laws, i.e. the Hooke’s and equilibrium laws
This direct approach is limited to simple elements
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kd f
Consider the stiffness equation
If d1=1 and d2=0, then
On the other hand, if d1=0 and d2=1
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1 1
2 2
1 1
1 1
d fEA
d fL
1
2
1
1
f EA
f L
The 1st column of k
1
2
1
1
f EA
f L
The 2nd column of k
Thus, a column of k is the vector of loads that must be applied to an element at its nodes to maintain a deformation state in which the corresponding nodal d.o.f. has unit value while all other nodal d.o.f are zero
This concept is general; it is applicable to all stiffness matrices, including the structural stiffness matrix
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Properties of stiffness matrices
Nonnegative main diagonal values
This is because it is physically unreasonable that a single load in a given direction would produce a displacement component in the opposite direction
Symmetry
k (or K) is symmetric if loads are linearly related to displacements
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0iik 0iiK
Properties of stiffness matrices (cont’d) Singularity
This is because the element or structure can under go “rigid body motion”
Sparsity (for the global or structural stiffness matrix) A matrix is called “sparse” if it contains many zeros A practical FE structure may have thousands of d.o.f, and more than 99% of
coefficients in its stiffness matrix may be zero http://web-ext.u-aizu.ac.jp/~niki/javaappl/jassem/jassem.html
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