Advanced Structural Analysis and Earthquake Engineering ENG5274
Dr Andrew McBride
University of Glasgow Room 733 Rankine Building andrew.mcbride@glasgow.ac.uk
2021 – Second Semester
Dr Andrew McBride
2021 – Second Semester
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Course information
Dr Andrew McBride Course information 2021 – Second Semester 2 / 131
Course information
The course is composed of two main parts:
Part I is on structural elements; Part II on time-dependent problems
The aims of the course are to develop:
Structural elements:
an understanding of the mathematical models for structural elements including beams, plates and shells
an appreciation of boundary conditions and locking-related phenomena
an understanding of how to solve these models using the finite element method
Time-dependent problems: dynamics and earthquake engineering
a knowledge of mathematical models for waves
an understanding of elastodynamics (stress wave propagation)
an understanding and appreciation of the solution of time-dependent problems using
the finite difference and finite element methods
Intended learning outcomes
By the end of the course you should be able to:
apply the finite element method to systematically derive the solution procedure for a range of structural elements with an emphasis on beams
develop a basic finite element code to compute the deflection of a beam and the dynamic response of a rod
appreciate key aspects of earthquake engineering
Dr Andrew McBride Course information 2021 – Second Semester 3 / 131
Contents
The course is broken into two main sections (3–4)
1 Course information
2 The course in pictures
3 The finite element method: a refresher
4 Structural elements
Introduction and overview
Beam theory: notation and geometrically exact theory Euler–Bernoulli beam theory
Timoshenko beams
Summary of plate and shell theory
5 Time-dependent problems: Dynamics Overview
Introduction to wave propagation in elastic media Finite element method for wave propagation
Dr Andrew McBride Course information 2021 – Second Semester 4 / 131
Course management
We will use Moodle to manage the course
All announcements will be on Moodle
All submissions will be electronic and via Moodle
The majority of the lectures are pre-recorded and the links available on Moodle Weekly problem sets (not for submission) will be uploaded to Moodle
Some of the scheduled lecture slots will be used to answer questions and for worked examples
The primary mechanism to ask questions is via the Moodle forum
Due to the size of the class, please ask questions via the forum and not email The complete set of course notes is available on Moodle
These will be updated regularly
Dr Andrew McBride Course information 2021 – Second Semester 5 / 131
Lectures and laboratory sessions
Preliminary material
Lecture 1a Lecture 2a
Course in pictures
FEM for elasticity in 1D – Python code
Structural elements
Lecture 3a Lecture 3b
Overview
Notation and full theory
Time-dependent problems
Dr Andrew McBride
Course information
2021 – Second Semester
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Course requirements and assessment
Refer to the course specification document on Moodle for more information
Minimum requirements for award of credits
Must attend the degree examination
Should attend the computational laboratory classes
Must submit the two reports (additional to requirements in course specification)
The programming language Python will be used to illustrate key aspects of the theory and for the reports. A basic understanding of Python and of the finite element method is assumed.
Assessment
Examination 70%
Two reports (assignments) 30% of overall mark (equally weighted)
Dr Andrew McBride Course information 2021 – Second Semester 7 / 131
Reading material and resources
Comprehensive lecture notes will be provided on the Moodle site. These will be updated regularly.
In addition, the following reference books are recommended:
J Fish and T Belytschko, “A First Course in Finite Elements” – this book is available online from the University of Glasgow library
TJR Hughes, “The Finite Element Method: Linear Static and Dynamic Finite Element Analysis”
Software:
We will be using Python version 3.8 or similar
Recommend that you download Anaconda Individual Edition We will also make use of Google Colaboratory
Dr Andrew McBride Course information 2021 – Second Semester 8 / 131
The course in pictures
Dr Andrew McBride The course in pictures 2021 – Second Semester 9 / 131
Course in pictures: structural elements
Beam structures
Shell structures
Approximate geometry, loading and boundary conditions to simplify analysis
3D continuum
2D approximation
1D approximation
Plate structures
Shell structures
Beam structures
Dr Andrew McBride The course in pictures 2021 – Second Semester 10 / 131
Plate structures
direct matrix solving methods are usually used with Newton-Rhapson for non-linearities. This section gives an introductory description of FEM and the relevant equations as used in this project. The underlying mechanics and procedure of formulating the FEM are explained and derived in more detail
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Fig. 4.2. FEM simulation superimposed on an deformation experiment (DHBW Stuttgart 2012) Numerical modelling (FEA)
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Nonlinear continuum mechanics for finite element analysis[13]. For a description of the approaches used by ANSYS MECHANICAL please refer to the Mechanical APDL documentation [14][15]
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Dr Andrew McBride The course in pictures
2021 – Second Semester
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Course in pictures: Part I – structural elements
2D approximation
1D approximation
Plate structures Shell structures
Kirchhoff-Love plate theory Reissner-Mindlin plate theory
Euler-Bernoulli* beam theory (slender beams)
nonlinearities & buckling
Beam structures
Timoshenko beam theory (account for shear – stocky beams)
locking (numerical issue)
3D continuum
*Implement FE model in Matlab (project 1)
Dr Andrew McBride The course in pictures 2021 – Second Semester 12 / 131
Course in pictures: Part II – dynamics
liquids
gases
solids
Dr Andrew McBride The course in pictures 2021 – Second Semester 13 / 131
image correlation (DIC) to visualize the transverse wave propagation. In this study, we applied three-point- bending (TPB) technique to Kolsky compression bar to facilitate dynamic transverse loading on a glass
Course i
3-point bendinTghe dinynaamicKTPoBlsexkpyerimceontsmweprerecosndsuciotednwibthathre Kolsky compression bar at Sandia National Laboratories, California. Figure 1 shows the schematic of the testing section. A random pattern was painted on the composite surface for the purpose of applying DIC technique. The high-rate deformation of the composite beam was photographed with a Cordin 550 high speed digital camera. The stain gage signals on the pressure bars provide a velocity/displacement boundary transversally loaded on the composite beam. In this case, the transmitted signal is nearly negligible in comparison to the incident pulse. The displacement boundary on the span side can be considered stationary. The displacement history at the wedge end are calculated with
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Fig. 1
X =C03t(εI(τ)−εR(τ))dτ ≈2C03tεI(τ)dτ (1) 00
and is shown in Fig. 2. The impact speed is measured as 9.0 m/s.
When the stress wave in the incident bar travels to the composite beam, a longitudinal stress wave is generated and then propagates through the thickness direction. When the longitudinal stress wave arrives at the free back side of the composite beam, it reflects back and doubles the particle velocity. Due to the fast longitudinal stress wave speed and relatively small thickness in the composite beam, it takes only a few microseconds to achieve a nearly equilibrated state in displacement. In other words, the displacement gradient through the thickness direction can be neglected after a few microseconds. However, along the beam direction, there exists a significant gradient in displacement due to the propagation of transverse wave. The DIC results shown in Fig. 3 confirm the transverse wave propagation.
Schematictof the testing section. Fig. 2 Displacement history at the impact wedge end. 0
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Fig. 3 Propagation of transverse wave (time interval: 5 microseconds)
Dr Andrew McBride
The course in pictures 2021 – Second Semester 14 / 131
Course in pictures: Part II – dynamics
Dr Andrew McBride The course in pictures 2021 – Second Semester 15 / 131
Course in pictures: Part II – dynamics
3D wave equation
1D wave equation
P-waves S-waves
Computational model for waves in 1D
Finite element method in space Finite difference method in time
Aspects of earthquake engineering
Eigenvalue analysis Stability and accuracy Time integration schemes Numerical implementation*
Dr Andrew McBride The course in pictures 2021 – Second Semester 16 / 131
The finite element method: a refresher
Dr Andrew McBride The finite element method: a refresher 2021 – Second Semester 17 / 131
Key steps in the finite element method
1 Strong form (governing equations and boundary conditions)
2 Weak form
3 Approximation of test function and trial solution
4 Discrete FE system of equations
5 Solution and postprocessing
Dr Andrew McBride The finite element method: a refresher 2021 – Second Semester 18 / 131
Recap: FE formulation an axially-loaded elastic bar
b : external body force per unit length [N/m] p : internal force [N]
equilibrium (balance of forces)
p(x)+b(x+ x/2) x+p(x+ x) = 0 p(x+ x) p(x) +b(x+ x/2) = 0
x
=) dp + b(x) = 0 dx
kinematic (strain – displacement relation)
p(x)
b(x+ x/2)
x
p(x + x)
u
A(x) tx
“(x) = =)
u(x + x) u(x) x
t
⌦
b(x)
du(x) “(x)= dx
[ ] (x) = p(x) [N/m2]
A(x) constitutive relation
[N/m2]
kinetics (stress)
l
governing equation and boundary conditions
d ⇣ du⌘
dx AEdx +b=0
du
n = Edxn = t
u = u
⌦:=0