1. Introduction to Finite Element Analysis for Skeletal Structures
Dr. Nick A Alexander © 2006
Department of Civil Engineering
University of Bristol
22 November, 2018

2. General Structural Problem
Dr Nick. A. Alexander, Department of Civil Engineering, University of Bristol
General Structural Problem
For the following structure
we want to compute
Bending moments, shear forces, axial forces
Bending, shear and axial stresses and strains
Deflections

3. Finite Element Analysis (FEA), The basic idea
Dr Nick. A. Alexander, Department of Civil Engineering, University of Bristol
Finite Element Analysis (FEA), The basic idea
Complex structures systems are often too complicated to simply derive relationships between applied loads, deflections and internal stresses.
Hence large structures are divided up into many individual finite elements; that have a much simpler structural form,
e.g. a beam or column
The relationship between load, displacement, stresses and strains in a finite element can be determined
Thus, it is computationally possible for a complex structure to be modelled by assembling (aggregation) many individual finite elements. The aggregation process must satisfy equilibrium and continuity.

4. Modelling Idealisation
Dr Nick. A. Alexander, Department of Civil Engineering, University of Bristol
Modelling Idealisation
Nodes
connection points
(not necessarily
hinges)
Loads
Elements
Restraints

5. For a finite element, we need to derive the relationship between
Dr Nick. A. Alexander, Department of Civil Engineering, University of Bristol
For a finite element, we need to derive the relationship between
External Loads
Deflections/deformations
Internal stresses and strains

6. Deriving force-displacement relationship for general finite element.
Dr Nick. A. Alexander, Department of Civil Engineering, University of Bristol
Deriving force-displacement relationship for general finite element.
Five basic steps
Conjecture a displacement function
Use nodal boundary conditions
Derive strain–displacement relationship
Derive stress–displacement relationship
Use principle of Virtual Work

7. Bar Element example u1 u2 Deformed shape f2 f1 x Node (a hinge)
Dr Nick. A. Alexander, Department of Civil Engineering, University of Bristol
Bar Element example u1 u2
Deformed shape f1 f2
Element
Node
(a hinge) x

8. Bar Element example Conjecture a displacement function u(x) x
Dr Nick. A. Alexander, Department of Civil Engineering, University of Bristol
Bar Element example
Conjecture a displacement function
u(x) x

9. Dr Nick. A. Alexander, Department of Civil Engineering, University of Bristol
Bar Element example
Express u(x) in terms of nodal displacements by using boundary conditions.
Deformed shape
u(0) = u1
u(L) = u2

10. Bar Element example Sub (2) into (1)
Dr Nick. A. Alexander, Department of Civil Engineering, University of Bristol
Bar Element example
Sub (2) into (1)
Displacement polynomial that satisfies boundary conditions

11. Dr Nick. A. Alexander, Department of Civil Engineering, University of Bristol
Bar Element example
Derive strain-displacement relationship by using mechanics theory
Axial Strain

12. Bar Element example Derive stress-displacement relationship by using
Dr Nick. A. Alexander, Department of Civil Engineering, University of Bristol
Bar Element example
Derive stress-displacement relationship by using
elasticity theory
Axial Stress
Elastic Modulus

13. Bar Element example Use principle of Virtual Work
Dr Nick. A. Alexander, Department of Civil Engineering, University of Bristol
Bar Element example
Use principle of Virtual Work
Work = Stress x Strain x Volume
Bar cross-sectional area A
Internal work
External work

14. Bar Element example Equate internal and external work Stiffness matrix
Dr Nick. A. Alexander, Department of Civil Engineering, University of Bristol
Bar Element example
Equate internal and external work
Stiffness matrix

15. Bar Element example Resultant stiffness matrix
Dr Nick. A. Alexander, Department of Civil Engineering, University of Bristol
Bar Element example
Resultant stiffness matrix

16. Assembling issue (1); Element coords.
Dr Nick. A. Alexander, Department of Civil Engineering, University of Bristol
Assembling issue (1); Element coords.
Element axes are not all the same.
So there is a need for a coordinate transformation u v
local axes u v
local axes X Y Z
Global axes

17. Assembling issue (1); Element coords.
Dr Nick. A. Alexander, Department of Civil Engineering, University of Bristol
Assembling issue (1); Element coords.
for forces FX FZ fu fv q
Fx1
Fz1
Fx2
Fz2
fu1
fv1
fv2
fu2 q
(8)

18. Assembling issue (1); Element coords.
Dr Nick. A. Alexander, Department of Civil Engineering, University of Bristol
Assembling issue (1); Element coords.
Similarly for displacement X1 Z1 X2 Z2 u1 v1 v2 u2 q
(9)

19. Assembling issue (1); Element coords.
Dr Nick. A. Alexander, Department of Civil Engineering, University of Bristol
Assembling issue (1); Element coords.
Element force-displacement in global coordinate
Element nodal forces and displacements in local coordinates
Local coordinates
Element stiffness matrix
in local coordinates (extended)
Global coordinates
Element nodal forces and displacements
in global coordinates
Element stiffness matrix
in global coordinates

20. Assembling issue (1); Element coords.
Dr Nick. A. Alexander, Department of Civil Engineering, University of Bristol
Assembling issue (1); Element coords.
Element stiffness matrix in global coordinates
(10)

21. Assembling issue (2); Structure matrix
Dr Nick. A. Alexander, Department of Civil Engineering, University of Bristol
Assembling issue (2); Structure matrix
Element and nodal numbering
Node number 1 2 3
-45 90
P [kN]
Element
number 1 2 3
L m

22. Assembling issue (2); Structure matrix
Dr Nick. A. Alexander, Department of Civil Engineering, University of Bristol
Assembling issue (2); Structure matrix
Create Structure stiffness matrix
from element stiffness matrices
1,1
1,3
3,1
3,3

23. Assembling issue (2); Structure matrix
Dr Nick. A. Alexander, Department of Civil Engineering, University of Bristol
Assembling issue (2); Structure matrix
2,2
2,3
3,2
3,3

24. Assembling issue (2); Structure matrix
Dr Nick. A. Alexander, Department of Civil Engineering, University of Bristol
Assembling issue (2); Structure matrix
1,1
2,1
2,2
2,1

25. Assembling issue (2); Structure matrix
Dr Nick. A. Alexander, Department of Civil Engineering, University of Bristol
Assembling issue (2); Structure matrix a b i
ith Element
ith Element in Structure
Stiffness Matrix with
Node numbers a and b
ith Element’s
Stiffness Matrix with
Node numbers a and b

26. Assembling issue (2); Structure matrix
Dr Nick. A. Alexander, Department of Civil Engineering, University of Bristol
Assembling issue (2); Structure matrix
vector of nodal forces
vector of nodal displacements
Structure stiffness matrix

27. Assembling issue (3); Supports
Dr Nick. A. Alexander, Department of Civil Engineering, University of Bristol
Assembling issue (3); Supports
How to deal with the problem of supports (restraints) ?
These are nodes where the displacements are known, zero in the perfectly rigid support case.
unknown
support
Reactions R
known
support
Displacements Ds
known applied nodal loads P
unknown nodal
Displacements D

28. Assembling issue (3); Supports
Dr Nick. A. Alexander, Department of Civil Engineering, University of Bristol
Assembling issue (3); Supports
This system of equations needs to be partitioned to determine the unknown nodal displacements and support reactions.
Solve system of algebraic equations
key numerical process
Determine
nodal
displacements
Determine
support
reactions

29. Dr Nick. A. Alexander, Department of Civil Engineering, University of Bristol
Finishing off; calculate element internal stresses, strains and actions.
Calc. nodal displacements in local coordinates, eqn (9)
Calc. element strains, equation (4)
For
Element 1
For
Element 1
Negative sign indicates
Compression

30. Finishing off; calculate element stresses, strains and actions.
Dr Nick. A. Alexander, Department of Civil Engineering, University of Bristol
Finishing off; calculate element stresses, strains and actions.
Calc. element stresses, equation (5)
Calc. element axial forces
For
Element 1
Negative sign indicates
Compression
For
Element 1

31. Summary of FEA * * key numerical process Divide complex structure into
Dr Nick. A. Alexander, Department of Civil Engineering, University of Bristol
Summary of FEA
Divide complex structure into
many finite elements
connected at nodes
Compute
Element Stiffness
Matrices (global Coords.)
Assemble
Structure Stiffness Matrix
and applied load vector
Introduce Supports
(partitioning)
Compute element internal
stresses, strains and actions
Compute element
nodal displacements
in local coords *
* key numerical process
Solve partitioned system
determine reactions and
nodal displacements