by s. ziaei rad · unknowns are and the reaction force f1 solving the equations, we obtain the...
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The Direct Method
ByS. Ziaei Rad
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Model Based Simulation
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Types of Finite Elements1-D (Line) Element
2-D (Plane) Element
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Types of Finite Elements3-D (Solid) Element
(3-D fields - temperature, displacement, stress, flow velocity)
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Spring ElementOne Spring Element
Spring force-displacement relationship:
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Spring ElementConsider the equilibrium of forces for the spring. At node i,we have
and at node j,
In matrix form,
or,
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Spring Element
where
k = (element) stiffness matrix
u = (element nodal) displacement vector
f = (element nodal) force vector
Note that k is symmetric. Is k singular or nonsingular? That is, can we solve the equation? If not, why?
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Spring System
For element 1,
For element 2,
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Assemble the stiffness matrix for the whole system
where is the (internal) force acting on local node i of Element m (i = 1, 2).
Consider the equilibrium of forces at node 1,
at node 2,
and node 3,
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Spring System (Assembly)That is,
In matrix form,
or
K is the stiffness matrix (structure matrix) for the spring system.
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An alternative way of assembling the whole stiffness matrix
“Enlarging”the stiffness matrices for elements 1 and 2, we have
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An alternative way of assembling the whole stiffness matrixAdding the two matrix equations (superposition), we have
This is the same equation we derived by using the force equilibrium concept.
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Boundary and load conditionsAssuming,
we have
which reduces to
and
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SolutionsUnknowns are and the reaction force F1
Solving the equations, we obtain the displacements
and the reaction force
Checking the Results·Deformed shape of the structure·Balance of the external forces·Order of magnitudes of the numbers
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Notes About the Spring Elements
Suitable for stiffness analysisNot suitable for stress analysis of the spring itselfCan have spring elements with stiffness in the lateral direction, spring elements for torsion, etc.
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Example 1.1
Given:For the spring system shown above,
Find:(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
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Example 1.1 : Solution(a) The element stiffness matrices are
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Example 1.1 : SolutionApplying the superposition concept, we obtain the global stiffnessmatrix for the spring system as
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Example 1.1 : Solutionor
which is symmetric and banded.Equilibrium (FE) equation for the whole system is
(*)
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Example 1.1 : Solution(b) Applying the BC
or deleting the 1st and 4th rows and columns, we have
Solving, we obtain
(c) From the 1st and 4th equations in (*), we get the reaction forces
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Example 1.1 : Solution(d) The FE equation for spring (element) 2 is
Here i = 2, j = 3 for element 2. Thus we can calculate the springforce as
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Example 1.2Problem: For the spring system with arbitrarily numbered nodesand elements, as shown above, find the global stiffness matrix.
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Example 1.2 : SolutionFirst we construct the following
which specifies the global node numbers corresponding to thelocal node numbers for each element.
Then we can write the element stiffness matrices as follows
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Example 1.2 : Solution
Finally, applying the superposition method, we obtain the Global stiffness matrix as follows
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Example 1.2 : Solution
The matrix is symmetric, banded, but singular. Why?