第一章 有限元基础1

writing the equilibrium equation for each node.
To illustrate, via a simple example shown in Figure 1.2.
Figure 1.2 System of two springs with node numbers, element numbers, nodal displacements, and nodal forces.
Thus, element-to-element displacement continuity is enforced at nodal connections.
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Substituting Equations 1.9 into Equations 1.8, we obtain
k1 k 1
k2 k 2
k1 U 1 f1(1) U (1) k1 2 f 2
k 2 U 2 f 2( 2 ) k 2 U 3 f 3( 2 )
(1.10a)
(1.10b)
Equation 1.10 is the equilibrium equations for each spring element We expand both matrix equations to 3 x 3 as follows
k k
k u1 f1 u f k 2 2
[1.4]
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Element 1
( k1 k1 u11) f1(1) k (1) (1) 1 k1 u2 f 2
nodal forces
Figure 1.1 (a) Linear spring element with nodes, nodal displacements, and nodal forces. (b) Load-deflection curve.
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the elongation or contraction of the spring is directly proportional to the applied axial load. The constant of proportionality between deformation and load is referred to as the spring constant , or spring stiffness, denoted as k, and has units of force per unit length.
the stiffness matrix indicates the resistance of the element
to deformation when subjected to loading. the stiffness matrix contains the geometric and material behavior information
(1.8a)
Element 2
k2 k 2
( k 2 u1 2 ) f 2( 2 ) ( k 2 u22 ) f 3( 2 )
(1.8b)
To begin “assembling” the equilibrium equations the displacement compatibility conditions, which relate element displacements to system displacements, are written as
we apply the first theorem of Castigliano and the principle of minimum potential energy.
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1.2 LINEAR SPRING AS A FINITE ELEMENT
nodal displacements
nodes
(1.4)
i.e.,
ke u f
(1.5)
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the element stiffness matrix is
k k e k
k k
(1.6)
the objective is to solve for the unknown nodal displacements, so we may have
these are denoted as f1 and f 2
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nodal displacements are denoted by
u1 and u 2
,Leabharlann Baiduthe deflection is
u2 u1
the resultant axial force in the spring is
stiffness matrix is identically zero. the element stiffness matrix is singular. The physical significance of the singular nature of the element stiffness matrix is that no displacement constraint whatever has been imposed on motion of the spring element: that is, the spring is not connected to any physical object that would prevent or limit motion of either node. With no constraint, it is not possible to solve for the nodal displacements individually.
(1.1)
f k k (u2 u1 )
For equilibrium,
(1.2)
f1 f 2 0
f1 k (u2 u1 )
(1.3a) (1.3b)
f 2 k (u2 u1 )
in matrix form
k k
k u1 f1 u f k 2 2
u1 1 f k e 1 u 2 f2
(1.7)
where
k e 1
is the inverse of the element stiffness matrix.
The element stiffness matrix for the linear spring element is a 2 x 2 matrix. the matrix is symmetric. A symmetric matrix has off-diagonal terms such that ,
k ij k ji
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■ Can we solve equation (1.7) and obtain the displacements No!
u1
and
u2
?
The inverse matrix k e
1
does not exist, since the determinant of the element
The global nodal displacements are identified as U 1 , U 2 , and U 3 The applied nodal forces are F1 , F2 , and F3 .
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The springs have different spring constants k1 and k 2 .
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step 2. global analysis:
1.2.1 System Assembly in Global Coordinates
For a connected system of spring elements, derivation of the global stiffness matrix is based on equilibrium conditions.
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Finite element analysis is based on the mathematic/physical principles.
■ For simple elements,
we utilize the principle of static equilibrium,
■ For more complicated structural systems,
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step 1. element analysis:
Two types of forces: 1. concentrated forces, 2. distributed forces. Nodal displacements
Nodal forces
The forces are applied to the element only at the nodes . (distributed forces are accommodated for other element types later),
Figure 1.3 Free-body diagrams of elements and nodes for the two-element system of Figure 1.2.
The equilibrium conditions for each spring, using Equation 1.4, as
( u11) U 1
( u21) U 2
( u1 2 ) U 2
( u22 ) U 3
(1.9)
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The compatibility conditions state the physical fact that the springs are connected at node 2, remain connected at node 2 after deformation.
A linear relationship
ykx
Constant, spring stiffness
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The general solution procedure:
For example:
Two big steps: 1. element analysis
2. global analysis
As an elastic spring supports axial loading only , we select an element coordinate system (also known as a local coordinate system) as an x axis oriented along the length of the spring, The global coordinate system is that system in which the behavior of a complete structure is to be described.
Chapter 1 Spring and Bar Elements
1
2
A piece of spring, as a spring element
A bar element:
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The bar elements used for trusses.
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1.1 INTRODUCTION
The finite element method (FEM) is a computational technique. Finite elements: (1) Spring element --- a linearly elastic spring (2) Bar element --- an elastic tension-compression member