CVL757: Finite Element Methods: IIT Delhi
CVL757: Finite Element Methods: IIT Delhi
CVL757: Finite Element Methods: IIT Delhi
Anoop Krishnan
IIT Delhi
N. M. Anoop Krishnan
Block IV, Room No. 314
Department of Civil Engineering
Indian Institute of Technology Delhi
Email: krishnan@iitd.ac.in
IIT Delhi CVL757 – Finite elements 1
Lecture 3 N. M. Anoop Krishnan
FEA Methodology
Computational procedure
• Computational procedure of FE is
– Generate matrices describing element behavior (element
matrix)
– Assemble the element matrices to obtain a structure matrix
or a global matrix
– Provide the respective nodes with loads arising from the
forces acting on the body
– Provide the respective nodes with boundary conditions
arising from the support conditions
– Solve the following algebraic equation to obtain the nodal
values of the field quantity
𝐾𝑈 = 𝐹
Elements
• Typical elements used in FE include
– one-dimensional: bar element, beam element,
– two-dimensional: linear triangle (or constant-strain triangle,
CST), quadratic triangle, bilinear rectangle, quadratic
rectangle,
– three-dimensional: rectangular solid elements, tetrahedron
Bar element
• Represented by a straight line of length L (with a finite area A
and elastic modulus E) with nodes on each ends
• E.g.: axially loaded bar can be represented by a bar element with
nodal loads F1 and F2 and nodal (axial) displacements u1 and u 2
Bar element
𝐴𝐸 𝐴𝐸
𝑢) − 𝑢+ = 𝐹) and 𝑢+ − 𝑢) = 𝐹+
𝐿 𝐿
𝑘 −𝑘 𝑢) 𝐹)
𝑢 = or 𝐤 𝐮 = −{𝐫}
−𝑘 𝑘 + 𝐹+
where k = AE/L
Bar element
• 𝐤 is the element stiffness matrix. Size depends on the degrees
of freedom. For a two-noded bar element with only axial
displacements 𝐤 is a 2 x 2 matrix.
• −{𝐫} represents the loads applied by an element to the
structure. Here, –ve sign is because F1 and F1 are applied to the
elements.
• For a stiffness matrix: A column of 𝐤 is the vector of loads that
must be applied to an element at its nodes to maintain a
deformation state in which the corresponding nodal d.o.f has
unit value while all other nodal d.o.fs are zero
• Example: let u1 = 0 and u2 = 1
𝑘 −𝑘 𝑢) 𝑘 −𝑘 0 −1 𝐹)
𝑢 = =𝑘 =
−𝑘 𝑘 + −𝑘 𝑘 1 1 𝐹+
Element to structure
Assembly
• An alternative way to obtain [K] is as follows.
• Number each of the nodes, while assuming they are not yet
connected.
• Expand each of the matrix to the structure size consistent
with the numbering as
Boundary conditions
• Support conditions are generally called as the boundary
conditions
• This will be reflected in the structure matrix by restricting
certain displacements
• For example, if the left end of the structure is fixed, then u1 = 0
• Thus, the structure matrix can be reduced as
𝑘) + 𝑘+ −𝑘+ 𝑢+ 𝐹+
𝑢 =
−𝑘+ 𝑘+ = 𝐹=
• In computer software, the [K] matrix is assembled using the
addition process. Process involves generation of a null matrix,
followed by insertion of coefficients for each element at the
respective locations in [K]. The efficiency of a finite element
code is determined largely by this process.
Beam element
• A beam element has two nodes with two degrees of freedom
per node: translation and rotation
Beam element
• Element stiffness matrix can be constructed using the rule
mentioned earlier, that is, a column of the stiffness matrix
corresponds to the loads required to produce unit displacement
in the corresponding node
• Assume the element as a
cantilever beam fixed at 2 and
loaded at 1 with force k11 and
moment k21 such that v1 = 1 and θ?) = 0
𝑘)) 𝐿= 𝑘+) 𝐿+
𝑣) = 1: − =1
3𝐸𝐼D 2𝐸𝐼D
𝑘)) 𝐿+ 𝑘+) 𝐿
𝜃D) = 0: − + =0
2𝐸𝐼D 𝐸𝐼D
• Solve for 𝑘)) and 𝑘+) . Then use equilibrium equations to obtain
𝑘=) and 𝑘G) . Continue the procedure for other d.o.f
IIT Delhi CVL757 – Finite elements
Lecture 3 N. M. Anoop Krishnan
Beam element
• The final element stiffness matrix is obtained as
Summary
Ø Computational methodology of FE
Ø Bar element