# CS计算机代考程序代写 flex matlab python algorithm assembly scheme case study Advanced Structural Analysis and Earthquake Engineering ENG5274

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

1 / 131

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 Lecture 2b Lecture 2c Lecture 2d Lecture 2e Lecture 2f Lecture 2g

Course in pictures

FEA – strong form

FEA – weak form

FEA – continuity

FEA – discrete equations

FEA – assembly and the global problem FEA – example problem

FEA – Python code Structural elements

Time-dependent problems

Dr Andrew McBride

Course information

2021 – Second Semester

6 / 131

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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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

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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

>0

⌦

<0
tension
n = +1
x
n = t
[N/m2]
n = 1
compression
t>0 t<0
⌦
x =t[u t u = ;
Dr Andrew McBride The finite element method: a refresher 2021 - Second Semester 20 / 131
Weak form in 1D
Strong form for linear elasticity
d AEdu+b=0 dx dx
du σn= Edx n=t
0

■ ForstockybeamDsr(ALtn

cp =

Shear or S (secondary) waves

P waves travel faster than S waves!

Dr Andrew McBride Time-dependent problems: Dynamics 2021 – Second Semester 108 / 131

Where are we?

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 Time-dependent problems: Dynamics 2021 – Second Semester 109 / 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

Consistent mass matrix

If the mass matrix is integrated using a sufficient quadrature rule and computed from an element contribution:

Me= ρNTeNedx Ωe

then it is consistent. For a linear element in 1D, p = 2 in the mass matrix and ngp ≡ 2. Sometimes it’s good to break the rules: mass lumping

a diagonal mass matrix is attractive as it can lead to very efficient algorithms when used in conjunction with explicit time-integration schemes

Dr Andrew McBride Time-dependent problems: Dynamics 2021 – Second Semester 119 / 131

Mass lumping by nodal quadrature Recall: consistent mass matrices

e ρle2 1 e ρle4 2 −1 Mcon=6 1 2 and Mcon=302 16 2

−1 2 4

To construct a lumped element mass matrix, the general idea is to choose a quadrature scheme where the quadrature points are at the nodes of an element.

For example, using the trapezoidal rule with linear shape functions:

Melumped = ρle 1 0 201

Using Simpson’s rule with quadratic shape functions:

e ρle1 0 0 Mlumped = 6 0 4 0

001

More generally we can use a Lobatto quadrature rule.

In both cases were are concentrating the nodal contributions into a single diagonal term!

Dr Andrew McBride Time-dependent problems: Dynamics 2021 – Second Semester 120 / 131

Estimating the critical time-step size

The stability of conditionally-stable time-integration schemes is dependent on the time step size ∆t. Such schemes become unstable if ∆t > ∆tcrit.

We can estimate ∆tcrit using the maximum eigenvalue λneq from the global eigenvalue problem

this is however expensive! It can be shown that:

λneq ≤ max(λemax) =: λhmax e

where λemax is the maximum eigenvalue of element e, and λhmax is the maximum eigenvalue of the whole system.

Element eigenvalue problem

[Ke −λeMe]Ψe =0 To solve: convert to a standard eigenvalue problem:

[Me]−1Ke −λeIΨe = 0 where Me−1M = I

Ae

=⇒ det [Ae − λeI] = 0 (characteristic polynomial)

Dr Andrew McBride Time-dependent problems: Dynamics 2021 – Second Semester 121 / 131

Example: Eigenvalue problem for linear element and consistent mass matrix

e AeEe 1 −1 e ρle 2 1 (ρAe)le 2 1 K=le−11 andM=612=612.

6E1−1 21e

ρle2 −11−λ12Ψ=0

̃22 d:=6c /le

convert to a standard eigenvalue problem

̃12 −11 −1 1 =⇒ d3 −1 2 −1 1 −λ 0

̃1 −1 1 =⇒ d −1 1 −λ 0

0 e

1 Ψ =0

0 e

1 Ψ =0

− d ̃ ̃2 ̃

d ̃ − λ

=⇒ det −d ̃ d ̃−λ =0 =⇒ (d−λ)(d−λ)−d2 =0

−2dλ+λ = λ(λ−2d) = 0 2 e ̃ 12c2 c2

=⇒ ωmax =λmax =2d= le2 =12 le

̃ ̃ ̃

Note: the critical frequency depends on the ratio of the wave speed to the element length c/le. Its inverse relates to how long it takes a stress wave to pass through an element.

Dr Andrew McBride Time-dependent problems: Dynamics 2021 – Second Semester 122 / 131

Algorithms for the problem of elastodynamics

24d035 v0 a0

t0

t1

t2

24dn35 ? 24dn+135 vn vn+1 an an+1

tn t tn+1 T

[t0 = 0, …,tn,tn+1,…,T] and ∆t := tn+1 − tn

Initial conditions

d(0) = d0 and v(0) = v0 We can compute a0 from

Ma0 =F(t=0)−Cv0 −Kd0

Discrete problem

Given the state [dn, vn, an] and Fn+1 solve:

Man+1 +Cvn+1 +Kdn+1 =Fn+1

for [dn+1, vn+1, an+1]

Dr Andrew McBride Time-dependent problems: Dynamics 2021 – Second Semester 123 / 131

One-step algorithm (a-form) with a Newmark time-integration scheme Discrete problem

Giventhestate[dn,vn,an]andFn+1 solve: Man+1 +Cvn+1 +Kdn+1 =Fn+1 (⋆)

Define predictors (β and γ determine the stability and accuracy): ∆t2

d = dn +∆tvn + 2 (1−2β)an and v = vn +(1−γ)∆tan . Correctors for Newmark scheme

dn+1 = d + β∆t2an+1 and vn+1 = v + γ∆tan+1 . Finite-difference formulae for evolution of approximate solution:

∆t2 2 dn+1 =dn +∆tvn + 2 (1−2β)an +β∆t an+1

vn+1 = vn + (1 − γ)∆tan + γ∆tan+1

Substitute into (⋆) – solve for an+1 and update state from correctors

M+γ∆tC+β∆t2Kan+1 =Fn+1−Cv−Kd.

A

R

Dr Andrew McBride

Time-dependent problems: Dynamics

2021 – Second Semester

124 / 131

The Newmark family of schemes

Method Average acceleration

Linear acceleration Fox Goodwin Central difference

Type β γ

Stability∗ Unconditional

Ωcrit = 2√3 Ωcrit = √6 Ωcrit = 2

Order accuracy 2

2 2 2

Implicit Implicit Implicit Explicit

11 42 11 62 11

12 2

01 2

∗Stability is based on analysis of

Damping increases the ∆tcrit. Ωcrit = (γ/2 − β)−1/2 (critical sampling frequency).

Stability

a single degree of freedom undamped problem.

Unconditional stability: 2β ≥ γ ≥ 1 2

Conditional stability: γ ≥ 1 , 22

ωh ∆t ≤ Ωcrit ↔ max

β < γ
∆t ≤ Ωcrit T 2π
Dr Andrew McBride Time-dependent problems: Dynamics
2021 - Second Semester
125 / 131
Accuracy
A time-integration scheme is n-th order accurate if the error e is proportional to the time-step size ∆t to the power of n
e(∆t) = C∆tn
All the schemes we considered were second order accurate: the errors decreases by a
where C is a constant.
factor of 4 if we halve the time step size.
Dr Andrew McBride Time-dependent problems: Dynamics 2021 - Second Semester 126 / 131
Explicit versus implicit schemes
Consider an undamped problem C = 0
Explicit scheme: e.g. Central difference (β = 0)
dn+1 = d
Man+1 =Fn+1 −Kd
The displacement dn+1 is computed using only information at tn If M is lumped (diagonal) then no matrix inversion required
leads to a very efficient scheme but requires a small time step
well suited for high strain rate problems (e.g. impact) that require a small time step
Dr Andrew McBride Time-dependent problems: Dynamics 2021 - Second Semester 127 / 131
Stability: some examples
Solved the element eigenvalue problem:
and have λhmax
[Ke −λeMe]Ψe =0 λneq ≤ max(λemax) =: λhmax
e
Assume we are using a central difference scheme: Ωcrit = 2 Two-node linear element with a lumped mass matrix:
ωh =2c thus ∆t≤ 2 =le max le ωmhax c
Two-node linear element with a consistent mass matrix:
h 2√3c 2 le ωmax = le thus ∆t ≤ ωh = √3c
max
Consistent mass matrices tend to yield smaller critical time steps than lumped mass matrices. Another advantage of mass lumping.
Exercise: Compute the critical time step for a three-node quadratic element.
Dr Andrew McBride Time-dependent problems: Dynamics 2021 - Second Semester 128 / 131
Algorithm for elastodynamics using Newmark scheme
Data: Material and load, Domain, Mesh, Time Assemble M and K;
foretone do
M ← Me; K ← Ke;
end
Result: A := M + β∆t2K
Data: Initial conditions d0 and v0 and loading F 0 Solveforinitialaccelerationa0: Ma0 =F0 −Kd0; Assemble R and solve Aan+1 = R for accelerations for all n; forn=1tont do
Data: an ←a,vn ←v,dn ←d d = dn + ∆tvn + ∆t2 (1 − 2β)an;
2
v = vn + (1 − γ)∆tan; R=Fn+1 −Kd;
Solve Aan+1 = R subject to boundary conditions;
dn+1 ← d + β∆t2an+1;
vn+1 ← v + γ∆tan+1; end
Result: Complete time history of problem at all nodal points
Dr Andrew McBride Time-dependent problems: Dynamics 2021 - Second Semester 129 / 131
Some comments on damping
The higher modes of the discrete equations are artefacts of the spatial discretisation (the mesh)
It is generally desirable and often necessary to remove the high-frequency modes The Newmark algorithm itself can introduce algorithmic damping:
γ > 1/2 introduces high-frequency dissipation

for fixed γ > 1/2 , select β to maximise high-frequency dissipation

an ideal value of β = (γ + 1/2)2/4

one drawback: only γ = 1/2 is second-order accurate!

Viscous damping (i.e. C ̸= 0):

Viscous damping damps an intermediate band of frequencies without having a

significant effect on the all-important high modes

Dr Andrew McBride Time-dependent problems: Dynamics 2021 – Second Semester 130 / 131

Stress waves and boundary conditions

A longitudinal stress wave can be compressive (- sign) or tensile (+ sign) A wave will reflect without inverting off a fixed (essential) boundary

A wave will reflect and invert of a free boundary

L

t=0

–

t=L

2c –

t = 3L 2c

t = 5L 2c

–

+

Dr Andrew McBride Time-dependent problems: Dynamics 2021 – Second Semester 131 / 131