SME 3033 FINITE ELEMENT METHOD
Two-Dimensional Problems Using CST Elements (Initial (Initial notes notes are designed designed by Dr. Nazri Kamsah) Kamsah)
SME 3033 FINITE ELEMENT METHOD
8-1 Introduction A thin plate of thickness thickness t , with a hole in the middle, is subjected to a uniform traction load, T as shown. shown. This 3-D plate can be analyzed as a two-dimensional problem.
y T x
2-D problems generally fall into two stress and plane strain. categories: plane stress stress problem A plane stress
a) Plane Stress
stress problem, where the normal and The thin plate can be analyzed as a s a plane stress shear stresses perpendicular to the x-y plane are assumed to be zero, i.e. z
0; xz 0; yz 0
The nonzero stress components are x
0; y 0; xy 0
SME 3033 FINITE ELEMENT METHOD
8-1 Introduction A thin plate of thickness thickness t , with a hole in the middle, is subjected to a uniform traction load, T as shown. shown. This 3-D plate can be analyzed as a two-dimensional problem.
y T x
2-D problems generally fall into two stress and plane strain. categories: plane stress stress problem A plane stress
a) Plane Stress
stress problem, where the normal and The thin plate can be analyzed as a s a plane stress shear stresses perpendicular to the x-y plane are assumed to be zero, i.e. z
0; xz 0; yz 0
The nonzero stress components are x
0; y 0; xy 0
SME 3033 FINITE ELEMENT METHOD
y
b) Plane Strain A dam subjected subjected to uniform pressure and a pipe under a uniform internal pressure can be analyzed in twodimension as plain strain problems. The strain components perpendicular to the x-y plane are assumed to be zero, i.e. z 0; xz 0; yz 0
x A dam subjected to a uniform un iform pressure
z
y
Thus, the nonzero strain components are x , y , and xy . x
x
0; y 0; xy 0 Pipe under a uniform internal pressure
z
SME 3033 FINITE ELEMENT METHOD
8-2 General Loading Condition A two-dimensional body can be subjected subjected to three types of forces: a) Concent Concentrat rated ed forces forces,, P x & P y at a point, i; b) Body forces, es, f b,x & f b,y acting at its centroid ;
c) Tract ractio ionn forc force, e, T (i.e. force per unit length), acting along a perimeter .
SME 3033 FINITE ELEMENT METHOD
The 2-dimensional body experiences a deformation due to the applied loads.
At any point in the body, there are two components of displacement, i.e. u = displacement in x-direction; v = displacement in y-direction.
SME 3033 FINITE ELEMENT METHOD
Stress-Strain Relation Recall, at any point in the body, there are three components of strains, i.e. u
x x v y y xy u v y x
The corresponding stress components at that point are
x y xy
SME 3033 FINITE ELEMENT METHOD
The stresses and strains are related through,
D where [D] is called the material matrix, given by
1 v E D v 1 2 1 v 0 0
0
0 1v 2
for plane stress problems and v 1 v E D v 1 v 1 v 1 2v 0 0
for plane strain problems.
0
0 1 v 2
SME 3033 FINITE ELEMENT METHOD
8-3 Finite Element Modeling The two-dimensional body is transformed into finite element model by subdividing it using triangular elements. Note: 1. Unfilled region exists for curved boundaries, affecting accuracy of the solution. The accuracy can be improved by using smaller elements. 2. There are two displacement components at a node. Thus, at a node j, the displacements are: Q2 j 1 in x-direction Q2 j in y -direction
SME 3033 FINITE ELEMENT METHOD
Finite element model of a bracket.
SME 3033 FINITE ELEMENT METHOD
8-4 Constant-Strain Triangle (CST) Consider a single triangular element as shown. The local node numbers are assigned in the counterclockwise order.
The local nodal displacement vector for a single element is given by,
q q1 ,
q2 , ...,
T
q6
Within the element, displacement at any point of coordinate ( x, y), is represented by two components, i.e. u in the xdirection and v in the y-direction. Note: We need to express u and v in terms of the nodal displacement components, i.e. q1 , q2 , …, q6 .
SME 3033 FINITE ELEMENT METHOD
8-5 Displacement Functions Displacement components u and v at any point ( x, y) within the element are related to the nodal displacement components through
u N 1q1 N 2 q3 N 3 q5
(i)
v N 1q2 N 2 q4 N 3 q6 where N 1, N 2 and N 3 are the linear shape functions, given by N1 ;
N2 ;
N 3 1
(ii)
in which and are the natural coordinates for the triangular element. Substituting Eq.(ii) into Eq.(i) and simplifying, we obtain alternative expressions for the displacement functions, i.e.
u q1 q5 q3 q5 q5 v q2 q6 q4 q6 q6
(iii)
SME 3033 FINITE ELEMENT METHOD
Eq.(i) can be written in a matrix form as,
u N q where
N1 0 N 0 N1
q q1 ,
N2
0
N 3
0
0
N2
0
N 3
q2 , ...,
T
q6
For the triangular element, the coordinates ( x, y) of any point within the element can be expressed in terms of the nodal coordinates, using the same shape functions N 1 , N 2 and N 3. We have, x N1 x1 N2 x2 N3 x3 y N1 y1 N2 y2 N3 y3
This is called an isoparametric representation.
SME 3033 FINITE ELEMENT METHOD
Substituting for N i using eq. (ii), we get
x x1 x3 x2 x3 x3 y y1 y3 y2 y3 y3 Using the notation, xij = xi – x j and yij = yi – y j, the above equations can then be written as
x x13 x23 x3 y y13 y23 y3 Note: The above equations relate the x- and y-coordinates to the - and coordinates (the natural coordinates). We observe that,
x13 x1 x3 y23 y2 y3
SME 3033 FINITE ELEMENT METHOD
8-6 The Shape Functions
The shape functions for the triangular element are illustrated in the figures. Recall, we have N1 ; N 2 ; N 3 1
Also,
N 1 + N 2 + N 3 = 1
SME 3033 FINITE ELEMENT METHOD
Area Coordinate Representation The shape functions can be physically represented by area coordinates,
N 1 N 2 N 3
where A is the area of the triangular element, i.e. A = A1 + A2 + A3
A1 A A2 A A3 A
; ;
SME 3033 FINITE ELEMENT METHOD
Exercise 8-1 Consider a triangular element shown below. Evaluate the shape functions N 1 , N 2, and N 3 at an interior point P .
y x
The triangular element for solution.
SME 3033 FINITE ELEMENT METHOD
Solution x N1 x1 N 2 x2 N 3 x3 1.5N1 7N 2 4N 3 3.85 y N1 y1 N 2 y2 N 3 y3 2N1 3.5N 2 7N 3 4.8
Using the notation, xij = xi – x j and yij = yi – y j, the above become x x1 x3 x2 x3 x3 2.5 3 4 3.85 y y1 y3 y2 y3 y3 5 3.5 7 4.8
Simplifying the equations yields, 2.5 3 0.15 5 3.5 2.2
Solving the equations simultaneously, we obtain = 0.3 and h = 0.2. Thus, the shape functions for the triangular element are, N 1 0.3
N 2 0.2
N 3 0.5
SME 3033 FINITE ELEMENT METHOD
8-7 Area of the Triangular Element The area, A of any arbitrarily oriented straight-sided triangular elements can be determined using a formula A
1 2
det J
where [J ] is a square matrix called the Jacobian, given by
x13 J x23
y13
y23
The determinant of the Jacobian [ J ] is det J x13 y23 x23 y13
Note: “l l” represents the “magnitude of”. Most computer software use counterclockwise order of local node numbering, and use det [ J ] for computing the area of the triangular element.
SME 3033 FINITE ELEMENT METHOD
8-8 Strain-Displacement Matrix The strains within the triangular element are related to the components of the nodal displacement by a relation
Bq where [B] is a (3 x 6) rectangular matrix called the strain-displacement matrix, given by
y23 1 0 B det J x32
0
y31
0
y12
x32
0
x13
0
y23
x13
y31
x21
0
y12
x21
Note: For the given magnitude of { q}, the strains within the element depend only on [B] matrix, which in turns depends on the nodal coordinates, which are constant. Hence the strains are the same everywhere within the element, thus the name constant-strain triangle (CST).
SME 3033 FINITE ELEMENT METHOD
Exercise 8-2 Consider a triangular element in Exercise 8-1. a) Write the Jacobian matrix; b) Find the determinant of the Jacobian matrix; c ) Compute the area of the triangular element; d) Establish the strain-displacement matrix for the element.
y x
The triangular element for solution.
SME 3033 FINITE ELEMENT METHOD
7-9 Potential Energy Approach The total potential energy of a 2-D body, discretized using triangular elements, is given by
e
1
D tdA 2 T
e
T
u f tdA u T tdL e L T
e
ui P i T
i
The first term represents the sum of internal strain energy of all elements, U e. For a single element, the internal strain energy is Ue
1
D t dA 2 T
e
SME 3033 FINITE ELEMENT METHOD
8-10 Element Stiffness Matrix We will derive the stiffness matrix of a triangular element using the potential energy approach. Recall, the internal strain energy of an element, U e is given by Ue
1
D t dA T
2
(i)
e
The strains { } are related to nodal displacements { q} by,
Bq
(ii)
Substituting Eq.(ii) into Eq.(i), we get Ue
1 2
q B D B qt dA T
T
(iii)
e
Taking all constants in Eq.(iii) out of the integral we obtain, Ue
1 2
q B D B t e dA q T
T
(iv)
SME 3033 FINITE ELEMENT METHOD
Note that,
dA A , i.e. the area of the triangular element. e
e
Substituting this into eq.(iv) and further simplifying, we get, U e
1 2
qT t e Ae BT D B q
(v)
The internal strain energy of the element can now be written as Ue
1 2
T
e
q k q
(vi)
From eq.(vi) we identify the stiffness matrix [ k ]e of the triangular (CST) element as, e
T
k te Ae B
D B
Note: Since there are 6 DOFs for a given element, [ k ]e will be a (6 x 6) rectangular symmetric matrix.
SME 3033 FINITE ELEMENT METHOD
Exercise 8-3 Determine the stiffness matrix for the straight-sided triangular element of thickness t = 1 mm, as shown. Use E = 70 GPa, n = 0.3 and assume a plane stress condition. Solution Element stiffness matrix is given by e T k te Ae B D B
(i)
where, t e 1 mm
Ae
1 2 1 2
det J
1 2
23.75
2 Ae 11.875 mm
x13 y23 x23 y13 (Dimension is in mm)
SME 3033 FINITE ELEMENT METHOD
The strain-displacement matrix, [B] is given by
y23 1 0 B det J x32
0
y31
0
y12
x32
0
x13
0
y23
x13
y31
x21
0
y12
x21
0 7 2 0 2 3.5 0 3.5 7 1 0 47 0 1.5 4 0 7 1.5 23.75 4 7 3.5 7 1.5 4 7 2 7 1.5 2 3.5 0 5 0 0 1.5 3.5 1 B 0 3 0 2.5 0 5.5 23.75 3 3.5 2.5 5 5.5 1.5
SME 3033 FINITE ELEMENT METHOD
The transpose of [ B] matrix is, 0 3 3.5 0 3 3.5 0 2.5 1 5 T B 5 2.5 23.75 0 1.5 0 5.5 5.5 1.5 0
For a plane stress condition, the material’s matrix [D] is given by
1 n E n 1 D 2 1 n 0 0
3 70 10 0 2 1 0.3 1 1 n 2 0
1 0.3 0.3 1 0 0
0 1 1 0.3 2 0
SME 3033 FINITE ELEMENT METHOD
Substituting all the terms into eq.(i) we have, 0 3 3.5 0 3 3.5 1 0.3 0 3 0 2.5 70 10 1 5 e k 1 11.875 0 . 3 1 0 5 1 0.32 2.5 23.75 0 0 0 0.35 1.5 0 5.5 5.5 1.5 0
1.5 0 5 0 0 3.5 1 0 3 0 2.5 0 5.5 23.75 3 3.5 2.5 5.5 1.5 5
SME 3033 FINITE ELEMENT METHOD
Multiplying and simplifying, we obtain q1
q2
q3
q4
q5
q6
2.494 1.105 2.409 0.425 0.085 0.68 2.152 0.233 0.223 0.873 2.374 4.403 1.316 1.994 1.549 e 4 k 10 2.429 1.741 2.652 2.079 0.868 5.026 symmetry
Note: Connectivity with the local DOFs is shown.
SME 3033 FINITE ELEMENT METHOD
8-11 Element Force Vector We will derive the force vector for a single element, which is contributed by a) body force, f and b) traction force, T . We need to convert both f and T into the equivalent nodal forces. Note: The concentrated forces can be included directly into the global load vector, appropriate DOF direction.
a) Body Force Suppose body force components, f x and f y, act at the centroid of a triangular element. The potential energy due to these forces is given by,
e
T
u
f t dA te e uf x vf y dA
(i)
SME 3033 FINITE ELEMENT METHOD
u N 1q1 N 2 q3 N 3 q5
Recall,
v N 1q2 N 2 q4 N 3 q6 1 Ni dA Ae e 3
Also,
Substituting the above into eq.(i), we get
u
T
e
T
e
f t dA q f
where { f }e is the element body force vector, given by e
f
te Ae
f x , 3
fy,
fx ,
fy,
fx ,
f y
Note: Physical representation of force vector { f }e is shown.
T
SME 3033 FINITE ELEMENT METHOD
b) Traction Force Suppose a linearly varying traction components act along edge 1-2 of a triangular element.
The potential energy due to the traction force is,
e
T
u
(ii)
T tdL l uT x vTy tdL 12
Using the relations, u N1q1 N 2q3 v N1q2 N 2q4 T x N1Tx1 N 2Tx 2
with,
T y N1Ty1 N 2Ty 2 1 Also, l N dl l12 , 12 3
2 1
l12
1 N dl l12 , l12 3
2 2
2
x2 x1 y2 y1 1 N1N 2dl l 12 l 12 6
2
SME 3033 FINITE ELEMENT METHOD
Substitution into eq.(ii) yields,
u
T
e
T tdL q1,
q2 ,
T
e
q3 , q4 T
where {T }e is the equivalent nodal force vector due to traction force, given by
T e
t e l 12 6
2T
x1 T x 2
2T
y1 T y 2
Note:
The physical representation of the nodal force vector {T }e is shown.
T x1 2T x2 T y1 2T y 2 T
(iii)
SME 3033 FINITE ELEMENT METHOD
Special Case: If the traction forces are uniform, then
T x1 Tx 2 Tx ;
Ty1 Ty 2 Ty
Thus, the nodal force vector in eq.(iii) becomes e
T
tel 12
T
T x , Ty , Tx , Ty 2
(iv)
SME 3033 FINITE ELEMENT METHOD
8-12 Concentrated Force The concentrated force term can be easily considered by having a node at the point of application of the force. If concentrated load components P x and P y are applied at a point i, then T
ui Pi Q2i 1Px Q2i Py Thus, P x and P y, i.e. the x and y components of { P }i get added to the ( 2i - 1)th component and (2i)th components of the global force vector, { F }.
Note: The contribution of the body, traction and concentrated forces to the global force vector, { F } is represented by,
F
e
e
e
f T P
SME 3033 FINITE ELEMENT METHOD
Exercise 8-4 Consider a portion of finite element model of a plate as shown. A uniform traction force of 2 kN/m2 acts along the edges 7-8 and 8-9 of the model. Determine the equivalent nodal forces at nodes 7, 8, and 9.
SME 3033 FINITE ELEMENT METHOD
Solution We will consider the two edges, 7-8 and 8-9 separately, and then merge the final results. cos sin
20 15 15 25
4
T x T cos 2
5 3
T y T sin 2
5
4
5 3 5
1.6 kN/m 1.6 N/mm
1.2 kN/m 1.2 N/mm
For edge 7-8 (edge 1-2 local) l 12
100 852 20 402 25 mm
t e 10 mm
Equivalent nodal forces due to uniform traction force T = 2 kN/m2 is,
T 1
1025 2
1.6
1.2 1.6 1.2 200 150 T
200 150
T
N
SME 3033 FINITE ELEMENT METHOD
For edge 8-9 (edge 1-2 local) l 12
85 702 40 602 25 mm
t e 10 mm
Equivalent nodal forces due to uniform traction force T = 2 kN/m2 is, 2
T 2
10 25 2
T
1.6 1.2 1.6 1.2
T 200 150
T
200 150 N
These loads add to global forces F 13, F 14 , …,F 18 as shown.
SME 3033 FINITE ELEMENT METHOD
8-13 Strains and Stress Calculations a) Strains The strains in a triangular element are,
x du dx e dv y dy du dv xy dy dx e
Bq Note: We observed that { }e depends on the [B] matrix, which in turn depends only on nodal coordinates ( x ,i yi), which are constant. Therefore, for a given nodal displacements {q}, the strains { }e within the element are constant.
Hence the triangular element is called a constant-strain triangle.
SME 3033 FINITE ELEMENT METHOD
b) Stresses The stresses in a triangular element can be determined using the stress-strain relation,
x e e e y D D Bq xy Note: 1. Since the strains { }e are constant within the element, the stresses are also the same at any point in the element.
2. Stresses for plane stress problem differ from those for plane strain problem by the material’s matrix [D]. 3. For interpolation purposes, the calculated stresses may be used as the values at the centroid of the element.
4. Principal stresses and their directions are calculated using the Mohr circle.
SME 3033 FINITE ELEMENT METHOD
Example 8-1 Consider a thin plate having thickness t = 0.5 in. being modeled using two CST elements, as shown. Assuming plane stress condition, (a) determine the displacements of nodes 1 and 2, and (b) estimate the stresses in both elements.
SME 3033 FINITE ELEMENT METHOD
Solution Element connectivity Local Nodes Element No
1
2
3
1
1
2
4
2
3
4
2
For plane stress problem, the materials matrix is given by
1 n 0 E n 1 0 D 2 1 n 0 0 12 1 n 0.25 0 1 0 D 32 106 0.25 1 0 0 0.375
SME 3033 FINITE ELEMENT METHOD
Element 1 A1
Area of element,
1 2
det J
1 2
6 3 in 2
The strain-displacement matrix,
y23 1 0 B det J x32
0
y31
0
y12
x32
0
x13
0
y23
x13
y31
x21
2 0 3 0 2 0 1 x21 0 3 0 3 0 0 6 3 2 3 0 0 2 y12 0
Multiplying matrices [ D][ B] we get,
(1)
D B
0 1.067 0.4 0 0.4 1.067 0 107 0.267 1.6 0 1.6 0.267 0.6 0.4 0.6 0 0.4 0
SME 3033 FINITE ELEMENT METHOD
The stiffness matrix is given by, (1)
T
k t1 A1 B1 D B1 Substitute all parameters and multiplying the matrices, yields Q1
(1)
k
Q2
Q3
Q4
Q7
Q8
0.5 0.45 0.2 0.533 0.3 0.983 1.4 0.3 1.2 0.2 0.2 0.3 0.45 0 0 7 10 1.2 0.2 0 symmetric 0.533 0 0.2
Note: Connectivity with global DOFs are shown.
SME 3033 FINITE ELEMENT METHOD
Element 2 A2
Area of element,
1 2
det J
1 2
6 3 in 2
The strain-displacement matrix is
y23 1 0 B det J x32
0
y31
0
y12
x32
0
x13
0
y23
x13
y31
x21
2 0 0 0 2 0 1 0 3 0 3 0 0 x21 6 3 2 3 0 0 2 y12 0
Multiplying matrices [ D][ B] we get,
(2 )
D B
0 0.4 1.067 0 1.067 0.4 107 0.267 1.6 1.6 0.267 0 0 0.6 0 0.4 0.4 0.6 0
SME 3033 FINITE ELEMENT METHOD
The stiffness matrix is given by, (2 ) T k t2 A2 B2 D B2
Substituting all parameters and multiplying the matrices yield Q5
(2)
k
Q6
Q7
Q8
Q3
Q4
0.5 0.45 0.2 0.533 0.3 0.983 1.4 0.3 1.2 0.2 0.2 0.3 0.45 0 0 7 10 1.2 0.2 0 symmetric 0.533 0 0.2
Note: Connectivity with global DOFs are shown.
SME 3033 FINITE ELEMENT METHOD
Write the global system of linear equations, [ K ]{Q} = { F }, and then apply the boundary conditions: Q2 , Q5 , Q6 , Q7 , and Q8 = 0.
The reduced system of linear equations are,
0.983 0.45 0.2 Q1 0 7 10 0.45 0.983 0 Q3 0 0.2 0 1.4 Q4 1000 Solving the reduced SLEs simultaneously yields,
Q1 1.913 5 Q 0.875 in. 3 10 Q 7.436 4