Rak-54.3200
Numerical Methods in Structural Engineering
Spring 2016, periods III–IV, 5 credits, P level (MSc, DSc)
Department of Civil and Structural Engineering
School of Engineering
Aalto University
Jarkko Niiranen
Assistant Professor, Academy Research Fellow
First lecture: 14–16, Tuesday, January 5, 2016
Rak-54.3200
Numerical Methods in Structural Engineering
Topic
Numerical methods ─ mainly finite element methods (FEM) ─
for fundamental problems in structural mechanics, structural
engineering and building physics: theory, applications and
software tools (Matlab, Comsol)
Lecturer
Assistant
Jarkko Niiranen, assistant professor
Sergei Khakalo and Viacheslav Balobanov, doctoral students
Lectures
Exercises
Tuesdays 14─16 in R2 (from January 5 to March 5)
Thursdays 12─14 in R5 (advice hours for home exercises)
Mondays 14─16 in R266 (advice hours and presentations)
Web site
https://mycourses.aalto.fi/course/view.php?id=8491
Material
Lectures slides (2016, as pdfs in MyCourses) augmented by a
text book by Hughes Thomas J. R.: The Finite Element Method,
Linear Static and Dynamic Finite Element Analysis (1987/2000),
and lecture notes by Reijo Kouhia and Markku Tuomala:
Rakennetekniikan numeeriset menetelmät (2009, in Finnish).
Rak-54.3200 / 2016 / JN
2
Rak-54.3200
Numerical Methods in Structural Engineering
Attendance and grading
I. Attendance for lectures or exercises is not compulsory.
II. The final grade is built as a combination of examination (0–24 points), home
exercises (0–6 points) and computer+software exercises (0–6 points).
III. Grade 1 can be achieved by 15 points which is about 40% of the total maximum
(36), and about 60% of the examination maximum (24).
IV. Examinations dates in 2016 are April 7 (13–17) and May 24 (13–17).
Work load
Desired distribution of the total 133 hours (5 credits) is divided as follows:
Contact teaching 20 h (15 %)
Lectures: 10 x 1.5 h = 15 h
Computer class presentations: 10 x 0.5 h = 5 h
Instructed or independent studying 113 h (85 %)
Home exercises: 10 x 5 h = 50 h (incl. 10 x 1.5 advise hours)
Computer+Software exercises: 10 x 4.5 h = 45 h (incl. 10 x 1 advise hours)
Preparation for examination: 14 h
Examination: 4 h
Rak-54.3200 / 2016 / JN
3
Rak-54.3200
Numerical Methods in Structural Engineering
Contents
1. Modelling principles and boundary value problems in engineering sciences
2. Energy methods and basic 1D finite element methods
- bars/rods, beams, heat diffusion, seepage, electrostatics
3. Basic 2D and 3D finite element methods
- heat diffusion, seepage
4. Numerical implementation techniques of finite element methods
5. Abstract formulation and accuracy of finite element methods
6. Finite element methods for Euler−Bernoulli beams
7. Finite element methods for Timoshenko beams
8. Finite element methods for Kirchhoff−Love plates
9. Finite element methods for Reissner−Mindlin plates
10. Finite element methods for 2D and 3D elasticity
11. Extra lecture: other finite element applications in structural engineering
(time-dependent problems, nonlinearities, isogeometric methods)
Rak-54.3200 / 2016 / JN
4
Rak-54.3200
Numerical Methods in Structural Engineering
Contents
1. Modelling principles and boundary value problems in engineering sciences
2. Energy methods and basic 1D finite element methods
- bars/rods, beams, heat diffusion, seepage, electrostatics
3. Basic 2D and 3D finite element methods
- heat diffusion, seepage
4. Numerical implementation techniques of finite element methods
5. Abstract formulation and accuracy of finite element methods
6. Finite element methods for Euler−Bernoulli beams
Research activities
7. Finite element methods for Timoshenko beams
are going on at our
8. Finite element methods for Kirchhoff−Love plates
department in many
9. Finite element methods for Reissner−Mindlin plates
topics of the course!
10. Finite element methods for 2D and 3D elasticity
11. Extra lecture: other finite element applications in structural engineering
(time-dependent problems, nonlinearities, isogeometric methods)
Rak-54.3200 / 2016 / JN
5
Connections to other courses −
examples
Numerical hand calculation oriented methods (previous basic courses)
Basic (strong form) methods
 exsponential solutions for ordinary differential equations
 Navier and Levy solutions for partial differential equations of plates
Energy (weak form / variational) methods
 unit virtual force method – principle of (complementary) virtual work (/ forces)
 method of minimum potential energy – principle of minimum potential energy
Numerical computer oriented methods (current and other basic courses)
Energy (weak form / variational) methods
 finite element method, FEM – principle of virtual work (/ displacements)
Other (strong form) methods
 finite difference method, FDM
 finite volume method, FVM
 discrete element method, DEM
Rak-54.3200 / 2014 / JN
6
Commercial finite element software −
examples
Commercial analysis software usually provide a simulation environment facilitating
all the steps in the modelling process: (1) defining the geometry, material data,
loadings and boundary conditions; (2) choosing elements, meshing and solving
the problem; (3) visualizing and postprocessing the results.
Some common general purpose or multiphysics FEM software:

Comsol http://www.comsol.com/
http://www.comsol.com/video/thermal-stress-analysis-turbine-stator-blade
http://www.comsol.com/release/4.4

Adina http://www.adina.com/

Abaqus http://www.simulia.com/products/abaqus_fea.html

Ansys http://www.ansys.com/
Some structural engineering FEM software:

Scia http://www.scia-online.com/

Lusas http://www.lusas.com/
A fairly long list of FEM software in Wikipedia:
http://en.wikipedia.org/wiki/List_of_finite_element_software_packages
Rak-54.3200 / 2016 / JN
7
1 Modelling principles and boundary value
problems in engineering sciences
1 Modelling principles and boundary value
problems in engineering sciences
Contents
1. Modelling and computation in engineering design and analysis
2. Boundary and initial value problems in engineering sciences
Learning outcome
A. Understanding of the main implications of the approximate nature of
computational methods in engineering design and analysis
B. Ability to formulate and solve some basic 1D model problems
References
Lecture notes: chapter 1
Text book: chapters 1.1−2
Rak-54.3200 / 2014 / JN
9
1.0 Questioning the computational analysis
How well do the computational techniques − of
civil and structural engineering − simulate the real life?
Rak-54.3200 / 2014 / JN
10
1.1 Modeling and computation
in engineering design and analysis
step 0
Physical engineering
problem with
design criteria
Customer needs!
Dimensions!
Laws and regulations!
Time slot!
Technology available!
Price range!
...
solution uP = ?
How long?
How thick?
Which material?
How many?
Which joints?
How to construct?
...
How to get answers?
Rak-54.3200 / 2014 / JN
11
1.1 Modeling and computation
in engineering design and analysis
step 0
Physical engineering
problem with
design criteria
Customer needs!
Dimensions!
Laws and regulations!
Time slot!
Technology available!
Price range!
...
solution uP = ?
How long?
How thick?
Which material?
How many?
Which joints?
How to construct?
...
How to get answers?
Formulate the problem
Rak-54.3200 / 2014 / JN
12
1.1 Modeling and computation
in engineering design and analysis
step 0
Physical engineering
problem with
design criteria
Customer needs!
Dimensions!
Laws and regulations!
Time slot!
Technology available!
Price range!
...
solution uP = ?
How long?
How thick?
Which material?
How many?
Which joints?
How to construct?
...
How to get answers?
Formulate the problem
− and solve it!
Rak-54.3200 / 2014 / JN
13
1.1 Modeling and computation
in engineering design and analysis
step 1
4D nonlinear
”all inclusive” theory
Rak-54.3200 / 2014 / JN
Physical engineering
problem with
design criteria
solution uP = ?
General physicomathematical model
solution u4D = ?
+ Idealization error
uP  u4D
14
1.1 Modeling and computation
in engineering design and analysis
step 1
4D nonlinear
”all inclusive” theory
Physical engineering
problem with
design criteria
solution uP = ?
General physicomathematical model
solution u4D = ?
+ Idealization error
uP  u4D
NONLINEAR
ANISOTROPIC
TIME-DEPENDENT
MULTI-PHYSICAL
Rak-54.3200 / 2014 / JN
15
1.1 Modeling and computation
in engineering design and analysis
step 1
Physical engineering
problem with
design criteria
solution uP = ?
General physicomathematical model
solution u4D = ?
+ Idealization error
uP  u4D
4D nonlinear
”all inclusive” theory
z
B
F
à la university schools
LINEAR
L
a
x
Rak-54.3200 / 2014 / JN
C
O
a
D
c
y
ISOTROPIC
TIME-INDEPENDENT
STRUCTURAL
16
1.1 Modeling and computation
in engineering design and analysis
4D nonlinear theory
step 2
Physical engineering
problem with
design criteria
solution uP = ?
General physicomathematical model
solution u4D = ?
Simplified physicomathematical model
+ Idealization error
solution u3D = ?
+ Modeling error
u4D  u3D
3D linear elasticity theory
Equilibrium
Constitutive models
Kinematics
   σ  b & BCs
σ  Eε

ε  u
3D
LINEAR
ISOTROPIC
TIME-INDEPENDENT
Rak-54.3200 / 2014 / JN
17
1.1 Modeling and computation
in engineering design and analysis
3D linear theory
step 3
Physical engineering
problem with
design criteria
solution uP = ?
General physicomathematical model
solution u4D = ?
Simplified physicomathematical model
N times simplified
physico-mathematical
model
1D axially loaded elastic rod
E ( x), A( x), b( x)
0
Rak-54.3200 / 2014 / JN
NL
+ Idealization error
solution u3D = ?
+ Modeling error
solution u = ...
+ N x Modeling error
 N '  b, N ( x)  A( x) ( x)
  E
x, u ( x )   u '
L
u3D"" u
1D, LINEAR, ISOTROPIC, TIMEINDEPENDENT
… Hand calculations work!
18
1.1 Modeling and computation
in engineering design and analysis
step 1
4D nonlinear
”all inclusive” theory
Physical engineering
problem with
design criteria
solution uP = ?
General physicomathematical model
solution u4D = ?
+ Idealization error
uP  u4D
NONLINEAR
ANISOTROPIC
TIME-DEPENDENT
MULTI-PHYSICAL
Rak-54.3200 / 2014 / JN
19
1.1 Modeling and computation
in engineering design and analysis
4D nonlinear
”all inclusive” theory
step 2
Physical engineering
problem with
design criteria
solution uP = ?
General physicomathematical model
solution u4D = ?
Numerical method
+ Idealization error
solution uh = ...
+ Discretization error
u4 D  uh
uh ( x, t )  numerical_ method(4D theory; x, t )
Rak-54.3200 / 2014 / JN
20
1.1 Modeling and computation
in engineering design and analysis
4D nonlinear
”all inclusive” theory
step 2
Physical engineering
problem with
design criteria
solution uP = ?
General physicomathematical model
solution u4D = ?
Numerical method
+ Idealization error
solution uh = ...
+ Discretization error
Reliable & Efficient
u4D  uh
 Applicable
 Stable
 Accurate
 Cheap
uh ( x, t )  numerical_ method(4D theory; x, t )
Rak-54.3200 / 2014 / JN
21
1.1 Modeling and computation
in engineering design and analysis
4D nonlinear
”all inclusive” theory
step 2
Physical engineering
problem with
design criteria
solution uP = ?
General physicomathematical model
solution u4D = ?
Numerical method
+ Idealization error
solution uh = ...
+ Discretization error
Neither a black box nor
u4D  uh
 Inapplicable
 Unstable
 Inaccurate
 Expensive
uh ( x, t )  numerical_ method(4D theory; x, t )
Rak-54.3200 / 2014 / JN
22
1.1 Modeling and computation
in engineering design and analysis
3D linear
”B&B” theory
step 3
Physical engineering
problem with
design criteria
solution uP = ?
General physicomathematical model
solution u4D = ?
Simplified physicomathematical model
Numerical method
+ Idealization error
solution u3D = ?
+ Modeling error
solution uh = ...
+ Discretization error
u3D  uh
uh ( x)  numerical_ method(3D theory; x)
Rak-54.3200 / 2014 / JN
23
1.1 Modeling and computation
in engineering design and analysis
Physical engineering
problem with
design criteria
solution uP = ?
General physicomathematical model
solution u4D = ?
Simplified physicomathematical model
Changes
to the methods:
verification
step 4
Rak-54.3200 / 2014 / JN
Numerical method
+ Idealization error
solution u3D = ?
+ Modeling error
solution uh = ...
+ Discretization error
u3D  uh
Observations and
conclusions
+ Human errors
24
1.1 Modeling and computation
in engineering design and analysis
Changes
to the models:
validation
Changes
to the methods:
verification
step 4
Rak-54.3200 / 2014 / JN
Physical engineering
problem with
design criteria
solution uP = ?
General physicomathematical model
solution u4D = ?
Simplified physicomathematical model
Numerical method
+ Idealization error
solution u3D = ?
+ Modeling error
solution uh = ...
+ Discretization error
u3D  uh
Observations and
conclusions
+ Human errors
25
1.1 Modeling and computation
in engineering design and analysis
Changes
to the problem
and design
Changes
to the models:
validation
Changes
to the methods:
verification
step 4
Rak-54.3200 / 2014 / JN
Physical engineering
problem with
design criteria
solution uP = ?
General physicomathematical model
solution u4D = ?
Simplified physicomathematical model
Numerical method
+ Idealization error
solution u3D = ?
+ Modeling error
solution uh = ...
+ Discretization error
u3D  uh
Observations and
conclusions
+ Human errors
26
1.1 Modeling and computation
in engineering design and analysis
Changes
to the problem
and design
Changes
to the models:
validation
Changes
to the methods:
verification
Physical engineering
problem with
design criteria
solution uP = ?
General physicomathematical model
solution u4D = ?
Simplified physicomathematical model
Numerical method
Rak-54.3200 / 2014 / JN
solution u3D = ?
+ Modeling error
solution uh = ...
+ Discretization error
u3D  uh
Observations and
conclusions
step 5
+ Idealization error
+ Human errors
Acceptance
27
1.1 Modeling and computation
in engineering design and analysis
Break exercise 1
Formulate an error estimate for the total error present in a typical
design and analysis process in terms of the error terms described above.
uP "" uh  ...
Rak-54.3200 / 2014 / JN
28
1.2 Boundary and initial value problems in
engineering sciences
step 1
Physical engineering
problem with
design criteria
solution uP = ?
General physicalmathematical model
solution u4D = ?
+ Idealization error
uP  u4D
z
B
F
”General” physico-mathematical model. A vertical
profile (pipe or tube) mast OB is supported by a balland-socket joint O and cables BC and BD. A force F
is acting on the the mast at point B.
L
a
x
O
a
C
Rak-54.3200 / 2016 / JN
D
c
y
Problem. Determine the displacements and stresses
in the mast for given length L, cross-sectional area A,
density ρ, and Young’s modulus E, distances a and c,
as well as force F.
29
1.2 Boundary and initial value problems in
engineering sciences
 details in 
 exercises 


Physical engineering
problem with
design criteria
solution uP = ?
General physicomathematical model
solution u4D = ?
Simplified physicomathematical model
step 2
3D linear elasticity
z
B
F
FB = Fz + TCz + TDz
b = Ag
x
O
a
C
Rak-54.3200 / 2016 / JN
solution u3D = ?
+ Modeling error
u4D  u3D
Under ”reasonable” assumptions:
L
a
+ Idealization error
D
c
y
equilibrium eq.    σ  b & BCs
constitutive eq.
FA = Oz
kinematics
σ  Eε

ε  u
30
1.2 Boundary and initial value problems in
engineering sciences
 σ  b
σ  Eε

ε  u
3D linear theory
 details in 
 exercises 


step 2N
Physical engineering
problem with
design criteria
solution uP = ?
General physicomathematical model
solution u4D = ?
Simplified physicomathematical model
N times simplified
physico-mathematical
model
1D axially loaded elastic rod
E ( x), A( x), b( x)
0
Rak-54.3200 / 2014 / JN
NL
+ Idealization error
solution u3D = ?
+ Modeling error
solution u = ...
+ N x Modeling error
 N '  b, N ( x)  A( x) ( x)
  E
x, u ( x )
  u'
L
u3D"" u
Assumptions: symmetry,
pointwise joints and actions etc.
31
1.2 Boundary and initial value problems in
engineering sciences
 σ  b
σ  Eε

ε  u
3D linear theory
step 2N
  AEu'  ' ( x )  b( x )
u(0)  u0
N ( L)  N L
Rak-54.3200 / 2014 / JN
Physical engineering
problem with
design criteria
solution uP = ?
General physicomathematical model
solution u4D = ?
+ Idealization error
solution u3D = ?
Simplified physicomathematical model
+ Modeling error
N times simplified
physico-mathematical
model
 details in 
 exercises 


solution u = ...
+ N x Modeling error
x
 u ( x )  u0  N L 
0
x
 1 L

ds
  
b dr  ds

AE 0  AE s

32
1.2 Boundary and initial value problems in
engineering sciences
 σ  b
σ  Eε
Physical engineering
problem with
design criteria
solution uP = ?
General physicomathematical model
solution u4D = ?

ε  u
3D linear theory
Simplified physicomathematical model
N times simplified
physico-mathematical
model
step 2N
Numerical method
step 3
L
 uˆ ' AEu ' dx  uˆ ( L) N   uˆ b dx,
h
0
Rak-54.3200 / 2014 / JN
h
L
solution u3D = ?
+ Modeling error
solution u = ...
+ N x Modeling error
solution uh = ...
+ Discretization error
L
h
+ Idealization error
h
uh (0)  u0
u  uh
0
33
1.2 Boundary and initial value problems in
engineering sciences
Axially loaded linearly elastic bar problem (in a displacement form):
(1)  EAu ' ' ( x)  b( x), 0  x  L
(2a)
differential equation
u (0)  u0
essential BC
(2b) ( EAu ' )( L)  N L
natural BC
Other 1D problems of the same form:
Problem
I variable
Data
Load
II variable
Axially loaded bar
displacement
EA
extensional force
force
Torsionally loaded bar
angle
GJ
torsional moment torque
Seepage in soils
head (pressure)
k
infiltration
velocity
Heat diffusion
temperature
k
heat generation
heat
Electrostatics
electric potential
ε
charge density
electric flux
Rak-54.3200 / 2016 / JN
34
1.2 Boundary and initial value problems in
engineering sciences
Axially loaded linearly elastic bar problem (in a displacement form):
(1)  EAu ' ' ( x)  b( x), 0  x  L
u (0)  u0
(2a)
differential equation
essential BC
(2b) ( EAu ' )( L)  N L
natural BC
1D generalization. Axially, torsionally and transversally loaded beam (uncoupled):
extension : (1)  EAu ' ' ( x)  b( x), 0  x  L
torsion :
(2)
u (0)  u0 , ( EAu ' )( L)  N L
(4)
 (0)   0 , (GJ ' )( L)  TL
(3)  GJ ' ' ( x)  r ( x), 0  x  L
bending : (5)  EIw' '' ' ( x)  q ( x), 0  x  L This one is of a different form!
(6)
w(0)  w0 , w' (0)   0 , ( EIw' ' )( L)  M L , ( EIw' ' )' ( L)  QL
Rak-54.3200 / 2014 / JN
35
1.2 Boundary and initial value problems in
engineering sciences
1D modification and 2D generalization. Heat diffusion:
(1)  EAu ' ' ( x)  b( x), 0  x  L
(2a)
u (0)  u0
(2b) ( EAu ' )( L)  N L

 kT ' ' ( x)  f ( x), 0  x  L
T (0)  T0
(kT ' )( L)  qL
EA, b
NL
u  u0 
 T 


 qx 
q x q y

x


T  T ,   q 

, q   

qy 


x
y

 y 
T
T  T0


k, f
(1)    (k ( x, y )T ( x, y ))  f ( x, y ), ( x, y )  

(2a)
T ( x, y )  T0 ( x, y ), ( x, y )  T
q0 n q
(2b)
q( x, y )  n  q0 ( x, y ), ( x, y )  q
Rak-54.3200 / 2014 / JN
36
1.X Continuum mechanics in
civil engineering
Building blocks of boundary value problems in civil engineering
Deformation and motion is defined by the continuum mechanics concepts as
(1) Kinematics (displacements and strains)
2D/3D elasticity
(2) Kinetics (conservation of linear and angular momentum)
   σ  b & BCs
(3) Thermodynamics (I and II laws)
(4) Constitutive equations (stresses vs. strains)
σ  Eε

ε  u
The main mathematical tools are
1D elasticity
(i) Vector and tensor algebra and analysis
 N '  b, N  A , & BCs
(ii) Differential, integral and variational calculus
(iii) Partial differential equations
  E
  u'
Altogether, physical conservation principles, i.e., the laws of conservation of mass,
momenta and energy as well as constitutive responses of materials or other
observed relations, are covered by a combination of the theoretical tools above.
Rak-54.3200 / 2014 / JN
37
1.X Continuum mechanics in
civil engineering
Matter (or material) is composed of particles ─ from
electrons and atoms up to molecules ─ which can be,
under certain assumptions, modelled as a continuum,
however.
Idealizations of physics and chemistry are further
simplified – or homogenized – by the theory of
continuum mechanics.
Rak-54.3200 / 2014 / JN
38
1.X Continuum mechanics in
civil engineering
Continuum is a hypothetical tool with specific assumptions and features
─ overlooks particles up to the molecular size (homogenity)
─ scales of interest are large enough (practicality)
─ physical quantities of interest are continuously differentiable (mathematicality)
─ applicaple for all materials (generality)
Within continuum mechanics, a wide spectrum of physical phenomena can be
studied, however.
Many variations, modifications or extensions for the classical continuum theories
exist as well: discontinuum-continuum, pseudo-continuum or Cosserat continuum
etc. often applied to capture microstructural effects of granular materials, for
instance.
Rak-54.3200 / 2014 / JN
39
1.X Continuum mechanics in
civil engineering
Continuum mechanics studies not only the deformation of solids but the
deformation and flow of a continuum covering solids, liquids and gases.
Engineering sciences as structural engineering study particular tailorings of
continuum mechanics: bars, beams, plates and shells within elasticity,
plasticity, viscoelasticity or viscoplasticity, for instance.
Problems formulated in terms of continuum mechanics are transformed by
mathematical tools into the form of computational mechanics: continuum
mechanics and numerical methods with the corresponding computer
implementations – referred as numerical simulation tools.
Rak-54.3200 / 2016 / JN
40
QUESTIONS?
ANSWERS”
LECTURE BREAK!