1
Preliminaries
1.1
Preface
1.2
Acknowledgement
2
Syllabus
Teaching team
2.1
Schedule
2.2
Catalog Description
2.3
Prerequisites
2.4
Textbook/Suggested Resources
2.5
Reference materials:
2.6
Course topics
2.7
Course Objective
2.8
Learning Outcomes
2.8.1
Extended description
2.9
Method of instruction:
2.10
Grading Standard
2.10.1
Major graded items
2.11
Assignments and communication
2.11.1
Expectations on student conduct for assignments and assessments
2.12
Academic integrity
2.13
Attendance, illness, and absences
Additional preliminaries
2.14
Mathematical Rigor
3
Introduction and historical background
3.1
Sub-disciplines within aerospace engineering
3.2
Challenges associated with structural analysis
3.2.1
Typical task
3.2.2
Flight loads
3.2.3
Weight drives geometry
3.3
Principal structural elements in an aircraft
3.4
Historical background
3.4.1
Prelude
3.4.2
Galileo’s scaling problem
3.4.3
Aero structures history
3.4.4
Truss structures in early airframes
3.4.5
Cantilevered wings
3.4.6
Metallic aircraft
3.4.7
Monocoque Structures
3.4.8
Modern transport aircraft
3.5
Challenges in modern aerostructures
3.5.1
Sandwich structures and composite materials
3.5.2
Tailored stiffness–composite materials
3.5.3
Wing warping
3.6
Other safety innovations in modern aircraft
3.6.1
Material innovations
3.6.2
Temperature resistance
3.6.3
Environmental impact
3.6.4
3D Printing
3.7
International politics
4
Stress
4.1
A simplified description
4.2
Traction vector
4.3
Stress tensor and its components
4.3.1
Tensor representation
4.3.2
Traction vector on arbitrary plane (Cauchy’s formula)
4.4
Relationship between boundary conditions and tractions
4.5
Principal stress
4.5.1
Principal stress values
4.5.2
Solution to the eigenvalue problem
4.5.3
Characteristic equation and invariants
4.5.4
Principal directions
4.5.5
Notes about principal stresses and orientations
4.5.6
Example
4.5.7
Additional thoughts on principal stresses
4.6
Equations of motion: equilibrium and stress
4.6.1
2D conservation of linear momentum (translational equilibrium equations)
4.6.2
2D moment equilibrium equations (conservation of angular momentum)
4.6.3
Equilibrium in cylindrical coordinates
4.6.4
Equilibrium in spherical coordinates
5
Strain
5.1
Small deformation definition of strain
5.2
Normal strain for larger deformations
5.3
Definition of shear strain and its relation to linear strain
5.3.1
Shear strain for larger deformations
5.3.2
Summary of the engineering strain-deformation equations
5.3.3
Engineering strain vs tensor strain
5.4
Summary of the tensor strain-deformation equations
5.5
The principal strains
5.6
Displacement Compatibility
5.6.1
Compatibility relationships
6
Constitutive relationships
6.1
General concepts of material constitutive response
6.1.1
Intuitive development of a constitutive matrix
6.2
Linear elastic
6.2.1
Anisotropic behavior
6.2.2
Shear modulus
6.2.3
Other material descriptions
6.2.4
Number of equations to find the 18 (15 unique) unknowns
6.3
Plane stress
6.4
Plane strain
6.5
Summary of elasticity
7
2D Elasticity Example Problem
7.1
Example: 2D Elasticity Problem
7.1.1
Summary of the 2D Elasticity Problem
7.1.2
The displacements are thus:
8
Beam Bending
8.1
Observations of beam physical deflection
8.2
Definition of a beam
8.3
Kinematics of an Euler-Bernoulli beam
8.3.1
Assumed Euler-Bernoulli beam bending kinematics
8.3.2
Strain in a beam
8.3.3
Stress in a beam
8.4
Beam loads
8.4.1
Axial load in a beam
8.4.2
Definitions
8.4.3
Choice of coordinate system
8.5
8.5.1
Bending moments about
\(x\)
8.5.2
Bending moments about
\(y\)
8.6
Differential equations of static equilibrium
8.6.1
Axial static equilbrium
8.6.2
Bending static equilibrium
8.7
Summary of beam equations
8.8
Example: Uniform beam
8.8.1
With uniform loading
8.8.2
With uniform loading
8.8.3
Example of a uniform beam with off-centroid point load
9
Torsion of solid shafts
9.1
Torsion of uniform bars (shafts)
9.2
Torsion boundary condition
9.3
Integration of torque
9.3.1
Notes
9.3.2
Summary of key equations
9.3.3
Examples
10
Thin beam in shear
10.1
Thin beam in shear
10.2
Shear flow
10.2.1
Example: past homework problem
10.3
10.4
Shear center
10.4.1
Steps to take to find the shear center
10.5
Shear of single cell thin-walled cross sections
10.6
Torque and shear flow in a closed cell
10.6.1
Torque of a closed cell
10.6.2
Torque twist rate relationship
10.6.3
Example: torsional rigidity of a circle and a square
10.6.4
Same material area (cross section area), same thickness
10.6.5
Example:
10.6.6
Torsion of thin open sections
10.7
Closed section vs open section
10.7.1
Closed section vs open section
10.7.2
Recall these equations
10.7.3
Multicell thin walled torsion
10.7.4
Example:
10.8
Shear flow for stringer-web sections
10.8.1
Example
10.9
Shear flow in multicells with stringer-web beams
11
Energy principles–Background
11.1
Work
11.1.1
Definition of conservative force
11.1.2
Definition of a potential field
11.1.3
Thoughts on potential
11.1.4
External work
11.2
Strain energy and complementary strain energy
11.2.1
Strain energy
11.2.2
Linear elastic material
11.3
Complementary energy
11.3.1
Note: for Linear elastic materials
11.4
Castigliano’s Method
11.4.1
Castigliano’s theorums
11.4.2
Example: Castigliano’s method for a cantilever beam with a tip load
11.4.3
Castigliano’s method for displacement at the mid-point load due to a tip load
11.4.4
Castigliano for a statically indeterminate system
11.4.5
(Castigliano for more complex systems)
11.5
Virtual Work
11.5.1
Definitions
11.5.2
Rigid body example
11.6
Deformable body
11.6.1
Principle of Virtual Work for Deformable Bodies
11.7
Minimum potential energy
11.7.1
Principle of minimum total potential energy
11.7.2
Deformable body example
12
Approximate solutions using energy methods
12.1
Rayleigh-Ritz method
12.1.1
Boundary conditions
12.1.2
Rayleigh-Ritz example one: cantilever beam with distributed end load
12.1.3
Example two: simply supported beam with sinusoidal load
12.1.4
Galerkin Method
12.1.5
Example One – Beam with distributed load on a portion
12.1.6
Example Two – Choosing a Polynomial – Robust Approach
12.1.7
Example Two Revisited – Choosing a Polynomial – Shortcut Approach
12.1.8
Example Two Revisited – Choosing a Polynomial – Longcut Approach
12.1.9
Hints at choosing the Galerkin approximating function
12.1.10
Reciprocal theorem
12.2
The purpose of this lecture is to introduce the Abaqus finite element package. There are no slides.
13
References
Aerospace Structural Design
8
Beam Bending
The content of this section was delivered over several lectures:
Part 1
Part 2
Part 3