structure analysis

1. Lecture #4
Analytical and Numerical Methods
of Structural Analysis

2. FLOWCHART OF STRUCTURAL ANALYSIS
2
Real object
Structural
layout
Design model
Only load-carrying structure
is kept
Assumptions and simplifications
are applied, loads are calculated
according to the problem
Results of
analysis
Implementation
on real object
This step is beyond the scope
of structural analysis
Structural analysis

3. FLOWCHART OF STRUCTURAL ANALYSIS
3
Real object
Structural
layout
Design model
Results of
analysis
Implementation
on real object

4. FLOWCHART OF STRUCTURAL ANALYSIS
4
Real object
Structural
layout
Design model
Results of
analysis
Implementation
on real object

5. FLOWCHART OF STRUCTURAL ANALYSIS
5
Real object
Structural
layout
Design model
Results of
analysis
Implementation
on real object
Depending on the kind of
problem which is solved, the
design model could be either as
detailed as structural layout, or
as generalized as below:

6. METHODS OF STRUCTURAL ANALYSIS
Analytical methods Numerical methods
Best for designing
calculations, suit for
checking calculations with
certain limitations
Best for checking
calculations, practically
effete for designing
calculations
Solutions exist for partial
cases (specific objects)
Versatile and flexible
Need much work to be
developed, but only
simple software for
application
Need expensive and
complex software and
hardware
6

7. ANALYTICAL METHODS OF STRUCTURAL ANALYSIS
7
Widely used nowadays:
• methods of Mechanics of Materials;
• methods for statically indeterminate structures:
• method of forces;
• method of displacements;
• beam theory for thin-walled structures;
• method of reduction coefficients.
Rarely used nowadays (mostly due to the progress
of numerical methods):
• methods based on Calculus of Variations;
• methods of Theory of Elasticity.

8. EXAMPLE TO COMPARE ANALYTICAL AND
NUMERICAL METHODS
8
Aim:
Design the steel
bracket to minimize
the self-weight
displacement of the
load.
Given data:
D2=80 mm;
L2=500 mm;
L1=200 mm;
mload=0.5 kg.
Objectives:
D1, δ1, δ2.
load
L1, D1, δ1
L2, D2, δ2
base

9. BASIC EQUATIONS OF SOLID MECHANICS
9
Equilibrium
equations
Constitutive
equations
Compatibility
equations
This is not only the sum of
forces or moments, but applies
for elementary volume as well
Physical law, expresses
the relation between
stress and strain
Solid body should
remain continuous
while being deformed

10. BASIC EQUATIONS OF SOLID MECHANICS
10
Equilibrium
equations
Constitutive
equations
Compatibility
equations
This is not only the sum of forces
or moments, but applies for
elementary volume as well.
0
0
0
xyx xz
xy y yz
yzxz z
X ,
x y z
Y ,
x y z
Z ,
x y z
where X ,Y ,Z external forces.
τσ τ
τ σ τ
ττ σ
∂∂ ∂
+ + + =
∂ ∂ ∂
∂ ∂ ∂
+ + + =
∂ ∂ ∂
∂∂ ∂
+ + + =
∂ ∂ ∂
−

11. BASIC EQUATIONS OF SOLID MECHANICS
11
Equilibrium
equations
Constitutive
equations
Compatibility
equations
Physical law, expresses the
relation between stress and strain.
( )( )
( )( )
( )( )
1
1
1
x x y z
y y x z
z z x y
xy yzxz
xy xz yz
,
E
,
E
,
E
, ,
G G G
ε σ ν σ σ
ε σ ν σ σ
ε σ ν σ σ
τ ττ
γ γ γ
= − +
= − +
= − +
= = =

12. BASIC EQUATIONS OF SOLID MECHANICS
12
Equilibrium
equations
Constitutive
equations
Compatibility
equations
Solid body should remain
continuous while being deformed.
2 22 2 22
2 2 2 2
2 22
2 2
2
2
2
2
2
2
y xyx x xzz
y yzz
yz xyxz x
yz xy yxz
yz xyxz z
, ,
y x x y z x x z
,
z y y z
,
x x y z y z
,
y x y z x z
.
z x y z x y
ε γε ε γε
ε γε
γ γγ ε
γ γ εγ
γ γγ ε
∂ ∂∂ ∂ ∂∂
+ = + =
∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂
∂ ∂∂
+ =
∂ ∂ ∂ ∂
∂ ∂ ∂ ∂∂
− + + = 
∂ ∂ ∂ ∂ ∂ ∂ 
∂ ∂ ∂ ∂∂
− + = 
∂ ∂ ∂ ∂ ∂ ∂ 
∂ ∂ ∂ ∂∂
+ − = 
∂ ∂ ∂ ∂ ∂ ∂ 

13. BASIC EQUATIONS OF SOLID MECHANICS
13
Equilibrium
equations
Constitutive
equations
Compatibility
equations
Boundary
conditions:
• geometrical
features;
• supports and
other means of
fixation;
• applied
displacements
and forces.
Design
model
From analytical point of view, every design model can
be expressed as a set of three basic equations with
corresponding boundary conditions.

14. WAYS TO SOLVE A SOLID MECHANICS PROBLEM
14
Displacements
are set as
unknowns
Strains are
derived
Stresses are
derived
Compatibility
equations
Constitutive
equations
Equilibrium
equations are
solved
Equilibrium
equations
Stresses are
set as
unknowns
Strains are
derived
Constitutive
equations
Equilibrium
equations
Compatibility
equations are
solved
Compatibility
equations

15. EXAMPLE TO COMPARE ANALYTICAL AND
NUMERICAL METHODS
15
Analytical methods
used:
• Methods of
Mechanics of
Materials;
• Mohr’s integral to
calculate
displacements.
Software used:
MathCAD V14.0
load
L1, D1, δ1
L2, D2, δ2
base

16. EXAMPLE TO COMPARE ANALYTICAL AND NUMERICAL METHODS
16

17. EXAMPLE TO COMPARE ANALYTICAL AND NUMERICAL METHODS
17

18. EXAMPLE TO COMPARE ANALYTICAL AND NUMERICAL METHODS
18

19. EXAMPLE TO COMPARE ANALYTICAL AND NUMERICAL METHODS
19

20. EXAMPLE TO COMPARE ANALYTICAL AND NUMERICAL METHODS
20

21. EXAMPLE TO COMPARE ANALYTICAL AND NUMERICAL METHODS
21

22. EXAMPLE TO COMPARE ANALYTICAL AND NUMERICAL METHODS
22

23. EXAMPLE TO COMPARE ANALYTICAL AND NUMERICAL METHODS
23

24. EXAMPLE TO COMPARE ANALYTICAL AND NUMERICAL METHODS
24

25. EXAMPLE TO COMPARE ANALYTICAL AND NUMERICAL METHODS
25

26. METHODS OF STRUCTURAL ANALYSIS
Analytical methods Numerical methods
Best for designing
calculations, suit for
checking calculations with
certain limitations
Best for checking
calculations, practically
effete for designing
calculations
Solutions exist for partial
cases (specific objects)
Versatile and flexible
Need much work to be
developed, but only
simple software for
application
Need expensive and
complex software and
hardware
26

27. NUMERICAL METHODS OF STRUCTURAL ANALYSIS
27
Most widely used nowadays is
Finite Element Method (FEM).
The structural analysis using FEM is called finite
element analysis (FEA).
Generally, FEM is just a method to solve
differential equations.
The physical background of FEM is the same as in
analytical methods.

28. FINITE ELEMENT METHOD
28
The smooth field of stresses is calculated using the combination
of basic functions, which are constructed on a mesh.

29. FINITE ELEMENT METHOD
29
Software work on principle GIGO: “garbage in – garbage out”…

30. FINITE ELEMENT METHOD
30
Software work on principle GIGO: “garbage in – garbage out”…

31. FINITE ELEMENT METHOD
31
Software work on principle GIGO: “garbage in – garbage out”…

32. FINITE ELEMENT METHOD
32
Even if the result is realistic,
you should always check it.
It’s not a big problem
to create a “nice picture” using FEA.
The problem is to get the reliable and verified result.
That’s a difference between professional and
beginner.
FEA is a way to find a stress state,
but not the way to design or to check the strength!

33. EXAMPLE TO COMPARE ANALYTICAL AND NUMERICAL METHODS
33load
base
Object FEM model

34. EXAMPLE TO COMPARE ANALYTICAL AND NUMERICAL METHODS
34
Assignment
of parameters

35. EXAMPLE TO COMPARE ANALYTICAL AND NUMERICAL METHODS
35
Deformed
shape
35 finite elements
in model

36. EXAMPLE TO COMPARE ANALYTICAL AND NUMERICAL METHODS
36
Optimization
took 7
minutes on
average
office PC.
990 points
were studied.

37. Analytical method Finite element analysis
Displacement is 7.66 µm
(or 8.07 µm in FEA)
Displacement is 7.81 µm
(or 8.22 µm in FEA)
D1=67 µm, δ1=2 µm,
δ2= 11.23 mm
D1=58 µm, δ1=2 µm,
δ2= 13 mm
Need only simple
software for application
Need expensive and
complex software
37
EXAMPLE TO COMPARE ANALYTICAL AND NUMERICAL METHODS

38. METHODS OF STRUCTURAL ANALYSIS
Analytical methods Numerical methods
Best for designing
calculations, suit for
checking calculations with
certain limitations
Best for checking
calculations, practically
effete for designing
calculations
Solutions exist for partial
cases (specific objects)
Versatile and flexible
Need much work to be
developed, but only
simple software for
application
Need expensive and
complex software and
hardware
38

39. EXAMPLE ILLUSTRATING THE DESIGN OF FINITE ELEMENT MODEL
39
Shell model.
12K elements in
model, around 5
minutes to solve
on poor office PC

40. EXAMPLE ILLUSTRATING THE DESIGN OF FINITE ELEMENT MODEL
40
Solid model.
250K elements, around 20
minutes to solve on powerful PC
(4x4.2GHz processor,
16 GB memory).

41. EXAMPLE ILLUSTRATING THE DESIGN OF FINITE ELEMENT MODEL
41
Difference?
Certain there is a difference, but for some results it is negligible.
Thus, the model should be built according to problem,
as in analytical methods

42. FINITE ELEMENT SOFTWARE
42
Widely used nowadays:
• ANSYS;
• Patran/Nastran;
• ABAQUS;
• Solidworks Simulation (former COSMOS);
And many-many others, freeware and proprietary…
Any difference? Generally, no difference. FEA
packages are often customized to be more convenient
for some specific problems. But for most cases,
differences between packages and limitations have
marketing but not the technical reason.

43. 1940
Theoretical aspects were developed for the
first time
1950,
1960
Theoretical background was developed, the
rise of computer power allowed to make first
specific software
1970
Large general-purpose packages appeared
(ANSYS, NASTRAN)
1980 Graphical pre- and post-processing
1990
Rise of the role of automatic meshing, and
thus the model scale
2000 Multi-disciplinary and large-scale problems
43
FINITE ELEMENT METHOD HISTORICAL PROGRESS

44. PRINCIPAL COMPONENTS OF FEA SOTWARE
44
Pre-processor
Solver
Post-precessor
Construction of the model, setting
the properties, loads, constraints
etc.
Solution itself. Two main types of
solvers are iterative or sparse .
Looking on results, processing of
results, generating “nice figures”

45. 45
FINITE ELEMENT METHOD – OLD INTERFACE

46. 46
FINITE ELEMENT METHOD – MODERN INTERFACE

47. CONCLUSIONS
• “Analytical method” does not mean “by hand” or “on
paper”, modern software can give large benefits.
• Numerical methods are very exacting in terms of
software and hardware.
• FEA is not a magic key that opens every door. It’s
just a versatile and effective tool to calculate stresses.
• Deep insight is required for successful structural
analysis, no matter is it analytics of FEA.
• Generally, analytical methods are good for
designing, FEA – for checking. However, in many
cases both of methods can be used for both phases of
analysis. 47

48. CONCLUSIONS
What does it mean to be a professional in structural
analysis?
Knowledge of complex formulas? Usage of modern
software?
Neither of it is an answer. Answer is a deep
understanding of how the structure works, and
application of proper methods according to the given
problem.
Analytical and numerical methods loss their
disadvantages when they are combined together.
48

49. WHERE TO FIND MORE INFORMATION?
BOOKS AND USEFUL LINKS
49
Good book about structural analysis and Ansys is
E. Madenci, I.Guven. The Finite Element Method and Applications in
Engineering Using ANSYS, 2006
Very useful links are:
Official website resource library ansys.com
Resourse for Ansys users ansys.net
Perfect forum (not for beginners, though) xansys.org
Youtube (it’s not a joke, search “Ansys” for amazing training
videos) youtube.com
… Internet is boundless …

50. TOPIC OF THE NEXT LECTURE
50
Statically determinate trusses
All materials of our course are available
at department website k102.khai.edu
1. Go to the page “Библиотека”
2. Press “Structural Mechanics (lecturer Vakulenko S.V.)”