lecture 5 ch - eth zürich · remember that is ... the timoshenko beam element ... dx title:...
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Institute of Structural Engineering Page 1
Method of Finite Elements I
Chapter 5
Structural Elements:The truss & beam elements
Institute of Structural Engineering Page 2
Method of Finite Elements I30-Apr-10
Today’s Lecture Contents• The truss element
• The Euler/Bernoulli beam element
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Method of Finite Elements I30-Apr-10
FE Classification
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Method of Finite Elements I30-Apr-10
Uniaxial Elementi. The longitudinal direction is sufficiently larger than the
other two ii. The bar resists an applied force by stresses developed
only along its longitudinal direction Prismatic Element
i. The cross-section of the element does not change along the element’s length
Assumptions
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Method of Finite Elements I30-Apr-10
Variational Formulation
Prismatic Element:
Uniaxial Element: Only the longitudinal stress and strain components are taken into account. The traction loads degenerate to distributed line loads
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Method of Finite Elements I30-Apr-10
Finite Element Idealization
Truss Element Shape Functions
(in global coordinates)
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Method of Finite Elements I30-Apr-10
Strain-Displacement relation
define The truss element stress displacement matrix
Constant Variation of strains along the element’s length
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Method of Finite Elements I30-Apr-10
Or we may use the isoparametric formulation:
Once again: The truss element stress displacement matrix
Constant Variation of strains along the element’s length
Isoparametric Shape Functions
1
1 2 2
uu N N
u
11 12
N
21 12
N
1 1 1 1N x N
B Jx L
-11
-1 1
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Method of Finite Elements I30-Apr-10
From the Continuous to the Discrete form – from Strong to Weak form
It is convenient to re-write the Principle of Virtual Work in matrix form
where:
and:
Elastic material:
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Method of Finite Elements I30-Apr-10
From the Continuous to the Discrete form
Therefore the relation turns to
Rearranging terms:
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Method of Finite Elements I30-Apr-10
From the Continuous to the Discrete form
Therefore the relation turns to
Rearranging terms:
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Method of Finite Elements I30-Apr-10
From the Continuous to the Discrete form
Finally the discrete form of the truss element equilibrium equation reduces to
or more conveniently
where:
Performing the integration, the following expression is derived
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Method of Finite Elements I30-Apr-10
Example 1: Truss Element under concentrated tensile force
Strong Form Solution Finite Element Solution
Boundary Conditions
Solution
Boundary Conditions
Solution
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Method of Finite Elements I30-Apr-10
Example 2: Truss Element under uniform distributed tensile force
Strong Form Solution
Boundary Conditions
Solution
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Method of Finite Elements I30-Apr-10
Example 2: Truss Element under uniform distributed tensile force
Finite Element Solution
But now we need to account for the distributed loading
Boundary Conditions
Solution
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Method of Finite Elements I30-Apr-10
Example 2: Truss Element under uniform distributed tensile force
Strong Form Solution Finite Element Solution
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Method of Finite Elements I30-Apr-10
Example 2: Truss Element under uniform distributed tensile force
Strong Form Solution Finite Element Solution
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Method of Finite Elements I30-Apr-10
Example 2: Truss Element under uniform distributed tensile force
0
0.5
1
1.5
2
2.5
0 0.5 1 1.5 2Length (m)
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Method of Finite Elements I30-Apr-10
Example 2: Truss Element under uniform distributed tensile force
0
0.5
1
1.5
2
0 0.5 1 1.5 2Length (m)
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Method of Finite Elements I30-Apr-10
Uniaxial Elementi. The longitudinal direction is sufficiently larger than the
other two Prismatic Element
i. The cross-section of the element does not change along the element’s length
Euler/ Bernoulli assumptioni. Upon deformation, plane sections remain plane AND
perpendicular to the beam axis
Assumptions
The 2−node Euler/Bernoulli beam element
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Method of Finite Elements I30-Apr-10
The 2−node Euler/Bernoulli beam element
Uniaxial Elementi. The longitudinal direction is sufficiently larger than the
other two Prismatic Element
i. The cross-section of the element does not change along the element’s length
Euler/ Bernoulli assumptioni. Upon deformation, plane sections remain plane AND
perpendicular to the beam axis
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Method of Finite Elements I30-Apr-10
Finite Element Idealization
We are looking for a 2-node finite element formulation
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Method of Finite Elements I30-Apr-10
Kinematic Field
Two deformation components are considered in the 2-dimensional case1. The axial displacement2. The vertical displacement
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Method of Finite Elements I30-Apr-10
Kinematic FieldPlane sections remain plane AND perpendicular to the beam axis
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Method of Finite Elements I30-Apr-10
Kinematic FieldPlane sections remain plane AND perpendicular to the beam axis
Point A displacement(that’s because the section remains plane)
(that’s because the plane remains perpendicular to the neutral axis)
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Method of Finite Elements I30-Apr-10
Kinematic FieldPlane sections remain plane AND perpendicular to the beam axis
Therefore, the Euler/ Bernoulli assumptions lead to the following kinematic relation
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Method of Finite Elements I30-Apr-10
Basic Beam Theory
, ,M Ey yI E R
: normal stress: bending moment
R : curvature
where
M
Reminder: from the geometry of the beam, it holds that:
w
: dwslopedx
2
2
1: d wcurvatureR dx
(that’s because the plane remains perpendicular to the neutral axis)Therefore:
2
2
d w Mdx EI
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Method of Finite Elements I30-Apr-10
Assume an infinitesimal element:
Force Equilibrium:
0
0
x
V x p x x V x x
V x x V xp x
x
Moment Equilibrium:
0
02
2x
xM x V x x M x x p x x
M x x M x xV x p xx
dM V x
dx
dV p xdx
Basic Beam Theory
2 4
2 4, ( ) ( )d M d wFinally p x EI p xdx dx
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Method of Finite Elements I30-Apr-10
Interpolation Scheme
Beam homogeneous differential equation
4
4 ( ) 0d wEI p xdx
: on
on
uDirichlet w wdwdx
2
2
3
: on
on
M
S
d wNeumann EI Mdxd wEI Sdx
Boundary Conditions
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Method of Finite Elements I30-Apr-10
Interpolation Scheme
Beam homogeneous differential equation
That’s a fourth order differential equation, therefore a reasonable assumption for the interpolation field would be at least a third order polynomial expression:
or in matrix form:
Therefore the rotation would be a second order polynomial expression:
(I)
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Method of Finite Elements I30-Apr-10
Interpolation Scheme
The relation must hold for the arbitrary displacements at the nodal points:
Therefore, by solving w.r.t the polynomial coefficients :
So now we can derive the 2-dimensional Euler/Bernoulli finite element interpolation scheme (i.e. a relation between the continuous displacement field and the beam nodal values):
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Method of Finite Elements I30-Apr-10
Interpolation Scheme
Therefore by substituting in (I):
Or more conveniently:
where:
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Method of Finite Elements I30-Apr-10
0.2 0.4 0.6 0.8 1.0
0.2
0.4
0.6
0.8
1.0N5
0.2 0.4 0.6 0.8 1.0
-0.14
-0.12
-0.10
-0.08
-0.06
-0.04
-0.02
N6
Interpolation Scheme in global coordinates
After some algebraic manipulation the following expressions are derived for the shape functions
0.2 0.4 0.6 0.8 1.0
0.2
0.4
0.6
0.8
1.0N2
0.2 0.4 0.6 0.8 1.0
0.02
0.04
0.06
0.08
0.10
0.12
0.14
N3
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Method of Finite Elements I30-Apr-10
Interpolation Scheme in isoparametric coordinates
21 2
21 3
22 5
22 6
1 1 221 1 121 1 221 1 12
Hermite Polynomials
w N
N
w N
N
2 1 3 1 5 2 6 2
2 3 5 6 1 1 2 2
with 2 2 2
, 2 2
e e
andl l dx Lw N w N N w N J
dl lN N N N N d w w
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Method of Finite Elements I30-Apr-10
Interpolation Scheme
What about the axial displacement component ??????
Since the axial and bending displacement fields are uncoupled
Shape Function Matrix [N]Shape Function Matrix [N]Shape Function Matrix [N]
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Method of Finite Elements I30-Apr-10
Remember that is the beam’s curvature
Strain-Displacement compatibility relations
2-dimensional beam element
The normal strain :
The shear strain :
The normal strain : Euler/ Bernoulli TheoryEuler/ Bernoulli Theory
The Euler/ Bernoulli assumptions predict zero variation of both the shear strain and the vertical component of the normal strain
The Euler/ Bernoulli assumptions predict zero variation of both the shear strain and the vertical component of the normal strain
Euler/ Bernoulli Theory
The Euler/ Bernoulli assumptions predict zero variation of both the shear strain and the vertical component of the normal strain
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Method of Finite Elements I30-Apr-10
Strain-Displacement relations in matrix form
We can re-write the strain displacement relation in the following form
and we can easily substitute the matrix interpolation form
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Method of Finite Elements I30-Apr-10
Strain-Displacement relations in matrix form
So the strain-displacement matrix assumes the following form
where: and:
and by substituting the expressions of the shape functions, the strain-displacement matrix Bassumes the following form
or in isoparametric coordinates:
2
2
1 6 61 3 1 1 3 1 with & e e edw dw d wB J B dL L L d dx d
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Method of Finite Elements I30-Apr-10
The evaluation of the Beam Element Stiffness Matrix
The beam element stiffness matrix is readily derived as:
Performing the integration with respect to x:
where: is the cross-sectional moment of inertia and is
e TK B EI Bd
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Method of Finite Elements I30-Apr-10
The evaluation of the Beam Element force vector
The beam element force vector is readily derived as:
M Se
e
T Te eTe e
ff
d N d Nf N p x dx M S
dx dx
Assuming constant distributed load p:
1
612
6
Te e e
LpLf N x pdx p N Jd
L
L
p
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Method of Finite Elements I30-Apr-10
The Timoshenko Beam Element
The transverse shear strain now also comes into play:dwdx