### Course Leader

### Lecturer

### Lecturer

### Timing and Structure

Weeks 1-8 Michaelmas term and weeks 3-8 Lent term. 24 lectures

## Aims

The aims of the course are to:

- Inform students of the key role of structures in different branches of engineering
- Illustrate the way in which structural engineers use the principles of structural mechanics to understand the behaviour of structures and so to design structures in order to meet specified requirements
- Examine in detail certain simple structural forms, including triangulated frameworks, beams and cables; to understand how such structures carry applied loads, how they deform under load, and how slender members may buckle

## Objectives

As specific objectives, by the end of the course students should be able to:

- Describe, qualitatively, the way in which different kinds of structure (frameworks, beams, cables, pressure vessels, etc.) support the loads that are applied to them.
- Analyse the limiting equilibrium conditions of bodies in frictional contact.
- Determine the axial force in any member of a statically determinate pin-jointed framework, making use of structural symmetry and of the principle of superposition when appropriate
- Explain and determine the shape of an inextensional cable subject to concentrated and distributed loads, as well as the tension distribution and support reactions.
- Test the stability of a simple, statically determinate arch structure
- Determine the displacement of any point of a pin-jointed framework subject to prescribed bar extensions by using a displacement diagram
- Understand and apply the equation of virtual work for pin-jointed frameworks and know how to choose appropriate equilibrium and compatible sets
- Construct bending-moment and shearing-force diagrams for simple beam structures, and to explain the relationship between them.
- Explain curvature, and how it changes in an elastic beam when the bending moment changes.
- Explain and compute the geometry of deflection of an initially straight beam on account of curvature within it.
- Explain and compute the detailed distribution of bending stress in the cross-section of an elastic beam having a symmetrical cross-section, and sustaining a bending moment.
- Explain and compute the distribution of shearing stress in the cross-section of an elastic beam having a symmetrical cross-section, and sustaining a shearing force.
- Determine the buckling load of a column, and be able to approach the design of columns accounting for the effects of yielding of the material and geometric imperfections.

## Content

#### Introduction and Aims of the Course (1 Lecture)

#### 1. External forces (3L)

*Equilibrium of point forces, moments and couples*

- Forces as vectors
- Moments as vectors
- Couples [3, Sect 1/1-1/5, 1/7-2/5]
- Resultants [3, Sect 2/6]
- Equilibrium [3, Sect 2/6, 3/3]
- Accuracy in structural mechanics

*Distributed loads and friction forces*

- Forms of distributed load [3, Sect 1/6, 5/1- 5/3, 5/9]
- Contact forces (without friction)
- Contact forces (with friction) [3, Sect 6/1-6/3, 6/8]
- Distributed friction

*Supports and free-body diagrams*

- Pin-joints
- Roller supports
- Built-in or ‘encastré’ supports
- Catalogue of support options
- Free-body diagrams [3, Sect 3/1- 3/2, 3/4]

#### 2. Internal forces (3L)

*Pin-jointed trusses*

- Method of joints [3, Sect 4/3]
- Method of sections [3, Sect 4/4]
- Some simplifications in analysing planar pin-jointed trusses
- Superposition
- Symmetry

*Shear forces and bending moments*

- Beams with transverse loading
- Free-body diagrams with shear forces and bending moments
- Arches [4, Sect 5.1, 5.6], [7, Ch. 5]

*Sress*

- Two-dimensional plane stress in thin-walled shells
- Thin-walled shells with uniform stress

#### 3. Deflection (5L)

*Cables and compatibility*

- Cables subjected to concentrated loads
- Cables subjected to distributed loads

*Deflection of members in pin-jointed frames*

- Statically determinate frames
- Strains, Hooke’s Law and bar extensions [5, Sect 5.2, 5.3, 5.4]
- Internal states of stress [5, Sect 5.5]

*Displacement Diagrams*

- Procedure for drawing displacement diagrams [5, Sect 2.3]
- Displacement diagram used for analysing real structures
- Interpreting displacement diagrams

*Virtual work*

- Real work
- Derivation of virtual work for pin-jointed frames
- Using virtual work to find extensions or nodal displacements
- Using virtual work to find forces or bar tensions

*Structural design*

- Iterative design
- Structural optimisation

#### 4. Equilibrium of Beams (2L)

- Introduction, hypotheses, sign conventions
*(5) Sect. 3.1, 3.2* - Distortion produced by internal forces
- Calculation of M, S, and T by analysis of free bodies
*(5) Sect. 3.2-3.4* - Differential relationships between q, S, and M
*(5) Sect. 3.5* - Construction of bending moment diagrams
- Statical indeterminacy
- Case study

#### 5. Deflection of Straight Elastic Beams (2L)

- Curvature and change of curvature, integration of curvature to find deflection
*(5) Sect. 8.1,8.2* - Deflection of elastic beams by integration
*(5) Sect. 8.3* - Deflection of elastic beams by superposition of deflection coefficients
*(5) Sect. 8.4*

#### 6. Stresses in Elastic Beams (5L)

- Introduction, basic geometric concepts
*(5) Sect. 7.2* - Bending of beams with rectangular cross-section
*(5) Sect. 7.5* - Bending of beams with non-rectangular cross-section, centroid and second-moment of area
- Use of section tables
- Combined bending moment and axial force
- Bending stresses in composite beams, transformed section, bending of reinforced concrete beams
- Shear stresses in beams
*(5) Sect. 7.6*

#### 7. Buckling of Columns (3L)

- Introduction, examples, hypotheses
- Euler column, fixed-end conditions, effective length
*(5) Sect. 9.4 (6) Sect 5.1* - Critical stress
- Imperfections
*(6) Sect 5.2* - Design of columns

* *

#### REFERENCES

*(1) GORDON, J.E. STRUCTURES OR WHY THINGS DON'T FALL DOWN
(2) HEYMAN, J. THE SCIENCE OF STRUCTURAL ENGINEERING
(3) MERIAM,J.L. & KRAIGE,L.G. ENGINEERING MECHANICS.VOL.1:STATICS
(4) FRENCH, M. INVENTION AND EVOLUTION
(5) CRANDALL,S.H.DAHL,N.C. & LARDNER,T.J INTRODUCTION TO THE MECHANICS OF SOLIDS,with SI Units
(6) HEYMAN,J.BASIC STRUCTURAL THEORY
(7) HEYMAN, J. STRUCTURAL ANALYSIS: A HISTORICAL APPROACH*

### Booklists

Please refer to the Booklist for Part IA Courses for references to this module, this can be found on the associated Moodle course.

### Examination Guidelines

Please refer to Form & conduct of the examinations.

### UK-SPEC

The UK Standard for Professional Engineering Competence (UK-SPEC) describes the requirements that have to be met in order to become a Chartered Engineer, and gives examples of ways of doing this.

UK-SPEC is published by the Engineering Council on behalf of the UK engineering profession. The standard has been developed, and is regularly updated, by panels representing professional engineering institutions, employers and engineering educators. Of particular relevance here is the 'Accreditation of Higher Education Programmes' (AHEP) document which sets out the standard for degree accreditation.

The Output Standards Matrices indicate where each of the Output Criteria as specified in the AHEP 3rd edition document is addressed within the Engineering and Manufacturing Engineering Triposes.

Last modified: 26/08/2020 09:16