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Method of Finite Elements I
Chapter 5
Structural Prismatic Elements:The truss & beam elements
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Method of Finite Elements I
Chapter Goals
Learn how to formulate the Finite Element Equations for 1D elements, and specifically The bar element (review)
The Euler/Bernoulli beam element
What is the Weak form?
What order of elements do we use?
What is the isoparametric formulation?
How is the stiffness matrix formulated?
How are the external loads approximated?
![Page 3: Method of Finite Elements I - Homepage | ETH Zürich · 2019-03-31 · Method of Finite Elements I Chapter Goals ... What order of elements do we use? What is the isoparametric formulation?](https://reader030.vdocuments.mx/reader030/viewer/2022040920/5e986628af7051348c7bd6bc/html5/thumbnails/3.jpg)
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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
The 2-node truss element
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 Elementi. The cross-section of the element does not change along the element’s length
Assumptions
![Page 6: Method of Finite Elements I - Homepage | ETH Zürich · 2019-03-31 · Method of Finite Elements I Chapter Goals ... What order of elements do we use? What is the isoparametric formulation?](https://reader030.vdocuments.mx/reader030/viewer/2022040920/5e986628af7051348c7bd6bc/html5/thumbnails/6.jpg)
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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)
( ) [ ] 11 2
2
uu x N N
u
=
![Page 8: Method of Finite Elements I - Homepage | ETH Zürich · 2019-03-31 · Method of Finite Elements I Chapter Goals ... What order of elements do we use? What is the isoparametric formulation?](https://reader030.vdocuments.mx/reader030/viewer/2022040920/5e986628af7051348c7bd6bc/html5/thumbnails/8.jpg)
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Method of Finite Elements I30-Apr-10
Strain-Displacement relation
defineThe truss element
stress displacement matrix
Constant Variation of strains along the element’s length
![Page 9: Method of Finite Elements I - Homepage | ETH Zürich · 2019-03-31 · Method of Finite Elements I Chapter Goals ... What order of elements do we use? What is the isoparametric formulation?](https://reader030.vdocuments.mx/reader030/viewer/2022040920/5e986628af7051348c7bd6bc/html5/thumbnails/9.jpg)
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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
( ) ( ) ( ) 11 2
2
uu N N
uξ ξ ξ
=
( ) ( )11 12
N ξ ξ= −
( ) ( )21 12
N ξ ξ= +
( ) ( ) ( ) [ ]1 1 1 1N x N N
B Jx x L
ξ ξξξ ξ
−∂ ∂ ∂∂= = = = −
∂ ∂ ∂ ∂
-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:
and for concentrated loads on the element end nodes 1, 2, or:
( ) ( )0
LTF N x q x dx = ∫
for distributed loads q(x)
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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:
This has to hold xuδ∀ which implies that:
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Method of Finite Elements I30-Apr-10
From the Continuous to the Discrete form
the discrete form of the truss element equilibrium equation reduces to
or more conveniently
where:
Performing the integration, the following expression is derived
![Page 14: Method of Finite Elements I - Homepage | ETH Zürich · 2019-03-31 · Method of Finite Elements I Chapter Goals ... What order of elements do we use? What is the isoparametric formulation?](https://reader030.vdocuments.mx/reader030/viewer/2022040920/5e986628af7051348c7bd6bc/html5/thumbnails/14.jpg)
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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
11
22
1 11 1
x
x
uf EAuLf
− = −
1
2
01 11 1100
xf EAuL
− = −
( )
( ) ( ) ( )2
1 1 2 2
100 , Lu u L thenEA
u x N x u N x u
= =
= +
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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
1
20
1 112
Lx x
xx
xf q LL q dx
xfL
− − = =
∫
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
Displacement at node 2@ x=L
11
22
1 11 1
x
x
uf EAuLf
− = −
21
22
1 1 11 1 12 2
x xuq L q LEA uuL EA
− − = ⇒ = −
![Page 17: Method of Finite Elements I - Homepage | ETH Zürich · 2019-03-31 · Method of Finite Elements I Chapter Goals ... What order of elements do we use? What is the isoparametric formulation?](https://reader030.vdocuments.mx/reader030/viewer/2022040920/5e986628af7051348c7bd6bc/html5/thumbnails/17.jpg)
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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
Displacement at node 2@ x=L
2
2 2xq L
uEA
=
( ) ( ) ( )1 1 2 2 2xq L
u x N x u N x u xEA
= + =
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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
2
2 2xq L
uEA
=
( ) ( ) ( )1 1 2 2 2xq L
u x N x u N x u xEA
= + =
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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)
( ) ( ) ( )1 1 2 2 2xq L
u x N x u N x u xEA
= + =
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Institute of Structural Engineering Page 20
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)
( ) ( ) ( )1 21 2 2
xdN x dN x q Lu x u u
dx dx EA= + =
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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 Elementi. 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
![Page 23: Method of Finite Elements I - Homepage | ETH Zürich · 2019-03-31 · Method of Finite Elements I Chapter Goals ... What order of elements do we use? What is the isoparametric formulation?](https://reader030.vdocuments.mx/reader030/viewer/2022040920/5e986628af7051348c7bd6bc/html5/thumbnails/23.jpg)
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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
Finite Element Idealization
©Carlos Felippa
DOFs relating to pure bending
1w2w
2w
1w
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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
dw wdx
θ ′= =
![Page 26: Method of Finite Elements I - Homepage | ETH Zürich · 2019-03-31 · Method of Finite Elements I Chapter Goals ... What order of elements do we use? What is the isoparametric formulation?](https://reader030.vdocuments.mx/reader030/viewer/2022040920/5e986628af7051348c7bd6bc/html5/thumbnails/26.jpg)
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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
![Page 29: Method of Finite Elements I - Homepage | ETH Zürich · 2019-03-31 · Method of Finite Elements I Chapter Goals ... What order of elements do we use? What is the isoparametric formulation?](https://reader030.vdocuments.mx/reader030/viewer/2022040920/5e986628af7051348c7bd6bc/html5/thumbnails/29.jpg)
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Method of Finite Elements I30-Apr-10
Basic Beam Theory
, ,M Ey yI E R
σσ ε σ= = = −
: normal stress : bending moment
R : curvature
where
Mσ
Reminder: from the geometry of the beam, it holds that:
w
: dwslopedx
θ =
θ2
2
1: d wcurvatureR dx
κ = =
(that’s because the plane remains perpendicular to the neutral axis)Therefore:
2
2
d w Mdx EI
= −
p q
bax
Δx
y
![Page 30: Method of Finite Elements I - Homepage | ETH Zürich · 2019-03-31 · Method of Finite Elements I Chapter Goals ... What order of elements do we use? What is the isoparametric formulation?](https://reader030.vdocuments.mx/reader030/viewer/2022040920/5e986628af7051348c7bd6bc/html5/thumbnails/30.jpg)
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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 θθ θ
= Γ
= = Γ
2
2
3
: on
on
M
S
d wNeumann EI Mdx
d wEI Sdx
− = Γ
− = Γ
Boundary Conditions
![Page 32: Method of Finite Elements I - Homepage | ETH Zürich · 2019-03-31 · Method of Finite Elements I Chapter Goals ... What order of elements do we use? What is the isoparametric formulation?](https://reader030.vdocuments.mx/reader030/viewer/2022040920/5e986628af7051348c7bd6bc/html5/thumbnails/32.jpg)
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Method of Finite Elements I30-Apr-10
Interpolation Scheme – the Galerkin Method
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)
![Page 33: Method of Finite Elements I - Homepage | ETH Zürich · 2019-03-31 · Method of Finite Elements I Chapter Goals ... What order of elements do we use? What is the isoparametric formulation?](https://reader030.vdocuments.mx/reader030/viewer/2022040920/5e986628af7051348c7bd6bc/html5/thumbnails/33.jpg)
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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 :
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):
![Page 34: Method of Finite Elements I - Homepage | ETH Zürich · 2019-03-31 · Method of Finite Elements I Chapter Goals ... What order of elements do we use? What is the isoparametric formulation?](https://reader030.vdocuments.mx/reader030/viewer/2022040920/5e986628af7051348c7bd6bc/html5/thumbnails/34.jpg)
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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 1 241 1 141 1 241 1 14
Hermite Polynomials
w N
N
w N
N
ξ ξ ξ
θ ξ ξ ξ
ξ ξ ξ
θ ξ ξ ξ
→ = − +
→ = − +
→ = + −
→ = + −
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) [ ]
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
ξ ξ ξ θ ξ ξ θξ
ξ ξ ξ θ θ
= + + + = =
= =
![Page 37: Method of Finite Elements I - Homepage | ETH Zürich · 2019-03-31 · Method of Finite Elements I Chapter Goals ... What order of elements do we use? What is the isoparametric formulation?](https://reader030.vdocuments.mx/reader030/viewer/2022040920/5e986628af7051348c7bd6bc/html5/thumbnails/37.jpg)
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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]
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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 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
ξ ξξ ξξ ξ
= − − + = =
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Institute of Structural Engineering Page 42
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= Ω∫
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Institute of Structural Engineering Page 43
Method of Finite Elements I30-Apr-10
The evaluation of the Beam Element force vector
The beam element force vector is readily derived as:
( ) ( ) ( ) T
S
Me
e
TeTe e e
ff
d Nf N p x dx M N S
dx
Ω
Γ
ΓΩ
Γ
= + +
∫
Assuming constant distributed load p:
( ) ( )
1
612
6
Te e e
LpLf N x pdx p N Jd
Lξ
ξ ξΩΩ
= = = −
∫ ∫L
p