Aims

The aims of the course are to:

• Provide students with an understanding of elastic methods of structural analysis.
• Ensure students recognise that stability and failure of structures by buckling is a key part of understanding structural behaviour.

Objectives

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

• Understand and compute the biaxial bending stress distribution in asymmetric sections.
• Calculate the section properties of complex sections with different techniques.
• Determine the shear stress distribution and shear centre in asymmetric sections.
• Understand and determine the torsional stresses in thin-walled open cross-sections.
• Analyse statically determinate and indeterminate space frames.
• Understand the application of virtual work principles.
• Explain the reciprocal theorem and the importance of influence lines.
• Understand and apply the displacement method.
• Recognise the shortcomings of the structural analysis learnt in Part I and appreciate the need to include stability in a complete theory of structures.
• Draw stable and unstable paths on a load/displacement diagram for bifurcation and snap through.
• Understand how elastic stability may be determined from the total potential energy and described by the eigenvalues of the stiffness matrix.
• Determine elastic critical loads for simple structures by eigenvalue analysis, whilst appreciating the importance of imperfection sensitivity.
• Apply approximation methods based on energy to determine the stability of simple systems.
• Understand second-order beam theory, using s and c functions.
• Understand how the tangent modulus and double modulus theories of inelastic buckling led to the column paradox, and how this was resolved.
• Understand the importance of lateral-torsional buckling of beams.

Content

There are two themes in this module: elastic analysis & stability and buckling of structures. Each section leads on from and extends a corresponding section of the first or second year courses in Structures.

In the first, the course aims are to extend the elastic analysis of beams to cover asymmetric sections in bending, to revise the determination of shear stresses, to consider the torsion of thin-walled open sections and to introduce the concept of shear centre. After that, the course will introduce the analysis of beams via differential equations and the reciprocal theorem,  which will lead to the study of influence lines. The course will also cover the displacement method of structural analysis and some new applications of virtual work and curved beams.

In the second, the course aims are to understand the fundamental principles of structural stability, to become familiar with common types of bifurcation and buckling phenomena and to formulate methods capable of dealing with geometrically non-linear structural behaviour. Once the general concept of stiffness degradation and the various post-buckling possibilities are understood, the course addresses the specific problem of column and beam design, taking account of initial imperfections, coexistent end-moments, residual stresses and material inelasticity.

Elastic Theory (8L) (Prof F Cirak)

• Asymmetric beams; principal axes
• Bending and shear stress distribution in asymmetric sections
• Torsion and warping of thin-walled open sections
• Analysis of space frames
• Virtual work
• Reciprocal theorem and influence lines
• Displacement method

Stability and Buckling (8L) (Prof A McRobie)

• Fundamentals of buckling and stability: total potential energy approach and direct equilibrium approach
• Classification of instabilities into snap-through tpe and bifurcation type
• Eigenvalues and eigenvectors of stiffness matrix
• Buckling of elastic structures; approximate estimates of buckling load; Rayleigh quotient
• Lateral buckling of columns: Euler strut, imperfections, Southwell plot, beam-columns, stability coefficients, buckling of frames
• Elasto-plastic buckling: tangent-modulus, double-modulus, Shanley's analysis
• Design of columns
• Lateral-torsional buckling of beams

Aims

The aims of the course are to:

• Provide students with an understanding of elastic methods of structural analysis.
• Ensure students recognise that stability and failure of structures by buckling is a key part of understanding structural behaviour.

Objectives

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

• Understand and calculate the biaxial bending stress distribution in asymmetric sections.
• Calculate the section properties of complex sections with different techniques.
• Understand and determine the torsional stresses in thin-walled open cross-sections.
• Analyse statically determinate and indeterminate space frames.
• Solve beam bending problems using the Macaulay’s method.
• Understand the application of virtual work principles.
• Explain the reciprocal theorem and the importance of influence lines.
• Understand and apply the displacement method.
• Recognise the shortcomings of the structural analysis learnt in Part I and appreciate the need to include stability in a complete theory of structures.
• Draw stable and unstable paths on a load/displacement diagram for bifurcation and snap through.
• Understand how elastic stability may be determined from the total potential energy and described by the eigenvalues of the stiffness matrix.
• Determine elastic critical loads for simple structures by eigenvalue analysis, whilst appreciating the importance of imperfection sensitivity.
• Apply approximation methods based on energy to determine the stability of simple systems.
• Understand second-order beam theory, using s and c functions.
• Understand how the tangent modulus and double modulus theories of inelastic buckling led to the column paradox, and how this was resolved.
• Understand the importance of lateral-torsional buckling of beams.

Content

There are two themes in this module: elastic analysis & stability and buckling of structures. Each section leads on from and extends a corresponding section of the first or second year courses in Structures.

In the first, the course aims are to extend the elastic analysis of beams to cover asymmetric sections in bending, to revise the determination of shear stresses, to consider the torsion of thin-walled open sections and to introduce the concept of shear centre. After that, the course will introduce the analysis of beams via differential equations and the reciprocal theorem,  which will lead to the study of influence lines. The course will also cover the displacement method of structural analysis and some new applications of virtual work and curved beams.

In the second, the course aims are to understand the fundamental principles of structural stability, to become familiar with common types of bifurcation and buckling phenomena and to formulate methods capable of dealing with geometrically non-linear structural behaviour. Once the general concept of stiffness degradation and the various post-buckling possibilities are understood, the course addresses the specific problem of column and beam design, taking account of initial imperfections, coexistent end-moments, residual stresses and material inelasticity.

Elastic Theory (8L) (Dr F Cirak)

• Asymmetric beams; principal axes
• Torsion and warping of thin-walled open sections
• Analysis of space frames
• Solution of the beam differential equation (Macaulay)
• Virtual work
• Reciprocal theorem and influence lines
• Displacement method

Stability and Buckling (8L) (Dr A McRobie)

• Fundamentals of buckling and stability: total potential energy approach and direct equilibrium approach
• Classification of instabilities into snap-through tpe and bifurcation type
• Eigenvalues and eigenvectors of stiffness matrix
• Buckling of elastic structures; approximate estimates of buckling load; Rayleigh quotient
• Lateral buckling of columns: Euler strut, imperfections, Southwell plot, beam-columns, stability coefficients, buckling of frames
• Elasto-plastic buckling: tangent-modulus, double-modulus, Shanley's analysis
• Design of columns
• Lateral-torsional buckling of beams

Aims

The aims of the course are to:

• Provide students with an understanding of elastic methods of structural analysis.
• Ensure that students recognise that an understanding of stability and the failure of structures by buckling, is a key part of understanding structural behaviour.

Objectives

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

• Determine stress distributions in asymmetric open sections, taking account of bending, shear, torsion and warping effects.
• Calculate the relevant section properties for complex sections.
• Apply Virtual Work.
• Analyse grillages under out-of-plane loading.
• Solve beam bending problems using Macaulay’s method, and to obtain influence lines thereby.
• Explain the reciprocal theorem and the importance of influence lines.
• Recognise the shortcomings of the structural analysis learnt thus far and appreciate the need to include stability as a fourth concept in any complete theory of structures.
• Draw stable and unstable paths on a load/displacement diagram for various bifurcation and snap through models.
• Understand how elastic stability may be determined from the total potential energy and may be described by the eigenvalues of the total stiffness matrix.
• Understand how elastic stability may be determined from the total potential energy and may be described by the eigenvalues of the total stiffness matrix.
• Determine elastic critical loads for simple structures by eigenvalue analysis, whilst appreciating the importance of imperfection sensitivity and the limitations of such analysis.
• Apply approximation methods based on energy to determine the stability of simple systems.
• Understand second-order beam theory, using s and c functions.
• Understand how the tangent modulus and double modulus theories of inelastic buckling led to the column paradox, and how this was resolved by Shanley's analysis, thereby presenting further difficulties for a general theory of structures.
• Understand the importance of lateral-torsional buckling of beams.
• Understand how of the above ideas combine in the context of column design.

Content

There are two themes in this module: elastic analysis & stability and buckling of structures. Each section leads on from and extends a corresponding section of the first or second year courses in Structures.

In the first course the aims are to extend the elastic analysis of beam elements as given in Part I to cover asymmetric sections in bending, to revise the determination of shearing stresses in beams, to consider the torsion of open section beams, including effects due to restraint of warping) and to introduce the concept of shear centre.  The course will cover the analysis of beams via differential equations, and their efficient solution using Macaulay’s method.  This will be applied to beams on elastic foundations, as well as to normal beams.  The reciprocal theorem will be introduced which will lead to the study of influence lines.  The course will also cover some new applications of virtual work, grillages and other beams curved in plan, and the cable catenary. 

In the second course the aims are to understand the fundamental principles of structural stability, to become familiar with common types of bifurcation and buckling phenomena and to formulate methods capable of dealing with geometrically non-linear structural behaviour. Once the general concept of stiffness degradation and the various post-buckling possibilities are understood, the course addresses the specific problem of column and beam design, taking account of initial imperfections, coexistent end-moments, residual stresses and material inelasticity.

Elastic Theory (8L) (Dr S Stanier)

• Asymmetric beams; principal axes;
• Shear centre; torsion and warping of open-sections;
• Virtual work;
• Grillages;
• Differential equation of beam (Macaulay);
• Reciprocal theorem and influence lines;
• Beams on elastic foundations;
• Catenary.

Stability and Buckling (8L) (Dr A McRobie)

• Fundamentals of buckling and stability: total potential energy approach and direct equilibrium approach;
• Classification of instabilities into snap-through tpe and bifurcation type;
• Eigenvalues and eigenvectors of stiffness matrix;
• Buckling of elastic structures; approximate estimates of buckling load; Rayleigh quotient;
• Lateral buckling of columns: Euler strut, imperfections, Southwell plot, beam-columns, stability coefficients, buckling of frames;
• Elasto-plastic buckling: tangent-modulus, double-modulus, Shanley's analysis;
• Design of columns;
• Lateral-torsional buckling of beams.

Objectives

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

• Analyse stress and strain in three dimensional conditions and define pore pressure parameters
• Understand the applications of elasto-plastic models with isotropic volumetric hardening to the behaviour of soils
• Use the Cam Clay model to predict changes of stress and volume in simple shear and triaxial tests
• Predict the onset of yield, failure and ultimate critical states of soil elements subject to any stress path
• Recognise the origins of the undrained strength of clay, and estimate excess pore pressures induced by shearing
• Recognise the origins of the super-critical strength of dense sand and overconsolidated clay in terms of interlocking and dilatancy
• Assess the influence of effective stress history on lateral earth pressure
• Assess the stability of slopes
• Recognise the potential sources of brittle failure in dilatant sands and overconsolidated clay
• Diagnose the delayed failure of overconsolidated clay slopes, and suggest counter-measures
• Diagnose quick clay flowslides, and suggest counter-measures
• Compute active and passive earth pressure with different theories
• Understand the effects of water on the stability of earth retaining structures
• Recognise the main types of retaining structures and construction methods
• Design basic retining structures

Content

Whereas module 3D1 was concerned chiefly with the limiting equilibrium of plastic soil bodies and soil consolidation, 3D2 aims to address modelling of the mechanical behaviour of soils and geotechnical structures. Soils are an order of magnitude more compliant than steel or concrete, so designers have to limit the mobilisation of soil strength to keep ground strains small enough to guarantee the serviceability of adjacent structures. Furthermore, some soils are inherently brittle, and their strength can deteriorate if they are permitted to strain excessively; this can lead to unexpected catastrophic failures. In geotechnical engineering, therefore, strains are often more important than stresses.

The Cam Clay model of soil behaviour is introduced to link concepts of consolidation and shearing, to envision drained and undrained soil behaviour within a single conceptual framework, to distinguish between yielding and failure, and to contrast stress paths that lead to brittle softening with those that lead to stable hardening. These comparisons and contrasts are central to the correct characterisation of soils for geotechnical decision-making. They are the subject of the first example paper.

The module continues with the assessment of the stability of natural slopes and cuts, the characterisation of in-situ stress states as a function of the previous stress history of the site, and considers the stress paths which they will follow as a result of construction. Particular materials, stress paths, and changes in environmental conditions can lead to catastrophic failures. The key to avoiding such failures is either to improve the ductility and continuity of materials and structures, or to take the utmost care in controlling soil strains in service. This material is the subject of the second example paper.

The final part of the course addresses the fundamentals of earth pressures and earth retaining structures. This will start from a review of the main tools available for the calculation of earth pressure, including upper bound, lower bound and limit equilibrium methods, followed by consideration of the main retaining structures types and construction methods (gravity, embedded, composite walls and other support systems).  Finally the course will address the basic design of simple retaining structures.

Topic 1: Basics - Soil Stress-strain, 3D Stresses & strains and their invariants, pore pressure parameters

Modelling in geortechnical engineering (Lecture 1) 

Modelling forms an implicit part of all engineering design but many engineers are not aware either of the fact that they are making assumptions as part of the modelling or of the nature and consequences of those assumptions. The lecture is an introduction to the course iproviding an overview of the evolution of modelling and the shift of modelling paradigms in science and engineering and in soil mechanics.

Stress/strain invariants (Lecture 2)

3D stresses and strains, Lode's coordinates, strain and strain invariants, work conjugates, pore pressure parameters, stress paths.

Topic 2: Strain hardening elastoplasticity

1D elasto-plasticity (Lecture 3)

Additive decomposition of strain, elasticity, admissible stress, yield criterion, elastic range, flow rule. Kuhn Tucker condition, consistency condition, plastic multiplier. Isotropic and kinematic hardening. Elasto-plastic stiffness.

Linear elasticity and Mohr Coulomb strength criterion (Lecture 4)

Isotropic linear elasticity. Mohr Coulomb Yield criterion with associative flow rule. predicted behaviour for drained and undrained triaxial compression and triaxial extension. Limitations and possible ways to overcome them.

Plane strain stress paths (Lecture 5)

Stress paths in the ground arising from a variety of construction processes, and relating to a range of representative locations. Use of vertical and horizontal equilibrium equations to estimate total stress paths due to simplified cases of vertical loading or horizontal unloading. Correlation with effective stress paths dictated either by undrained or drained soil conditions. Predicting the approach of soil states to limiting strength envelopes.

Topic 3: Cam-Clay

Shearing of soils: work and dissipation, yield surface and normality (Lecture 6)

Taylor’s work equation. Yield surface in effective stress space. Normality principle guarantees maximum dissipation, providing a plastic flow rule. Derivation of the Cam Clay yield surface. Compressibility and volumetric hardening.

Critical states, normal compression, and yield  (Lecture 7)

Stress dilatancy and critical state. Radial compression lines, critical state line. 3D state surface of shear stress, effective normal stress and specific volume. Drained and undrained shearing of soil at a given density, from points of normal consolidation and overconsolidation

Undrained shear strength,  predicting behaviour of geotechnical structures using using Cam-clay model (Lecture 8)

Undrained shear strength. Predicting behaviour of smooth retaining wall and embankment  on soflt clay. Staged loading. Development of stress-strain relationship of Cam clay model. Application of numerical programs for modern geotechnical analysis.

Topic 4: Slope stability - avoiding catstrophic failure

Slope stability analysis (Lectures 9 and 10)

Occurrence of slope failure in the UK and worldwide. Examples. Modes of movement: falls, topples, slides, and flows. Analysis methods to assess the stability of slopes in sands and clays. Infinite slope, effect of groundwater flow. Finite slope undrained. General Limit equilibrium methods. 

Avoiding catastrophic failure on the dry side (Lecture 11)

Selection of mechanical parameters for the design of engineered slopes. Factors promoting failure on the dry side: brittle failure for dilatant sand and overconsolidated clay. Need to design for critical state friction and worst pore water pressures. 

Delayed failure in clay slopes and catastrophic failure on the wet side (Lecture 12)

Delayed failure in clay slopes due to progressive softening on cycles of wetting/drying.  Factors promoting brittle failure on the dry side: quick clay flowslides, volumetric collapse on saturation for partly saturated slopes. 

Topic 5. Geotechnical Investigation

Geotechnical Site Investigation (Lecture 13)

Requirements of geotechnical site investigation. Objectives, extent, frequency and layout of investigations for the geotechnical characterisation of a site. 

In-situ testing (Lecture 14)

Procedures and interpretation of Standard Penetration Test (SPT), Cone Penetration Test (CPT), Field Vane, and pressuremeter tests

Topic 6: Elasto-plastic radial solutions

Cavity Expansion (Lecture 15)

Cavity expansion in elastic perfectly plastic medium. Application to the interpretation of pressuremeter test. Estimation of soil properties from pressuremeter tests in clay: in-situ total horizontal stress, shear modulus, undrained shear strength.

Cavity Contraction (Lecture 16)

Cavity contraction in elastic perfectly plastic medium. Applications to bored tunnelling. Estimation of support pressure required for tunnel stability. Tunnel convergence and settlements above tunnels.

This syllabus contributes to the following areas of the UK-SPEC standard:

Toggle display of UK-SPEC areas.

 
Last modified: 04/06/2025 13:18

Objectives

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

• Analyse stress and strain in three dimensional conditions and define pore pressure parameters
• Understand the applications of elasto-plastic models with isotropic volumetric hardening to the behaviour of soils
• Use the Cam Clay model to predict changes of stress and volume in simple shear and triaxial tests
• Predict the onset of yield, failure and ultimate critical states of soil elements subject to any stress path
• Recognise the origins of the undrained strength of clay, and estimate excess pore pressures induced by shearing
• Recognise the origins of the super-critical strength of dense sand and overconsolidated clay in terms of interlocking and dilatancy
• Assess the influence of effective stress history on lateral earth pressure
• Assess the stability of slopes
• Recognise the potential sources of brittle failure in dilatant sands and overconsolidated clay
• Diagnose the delayed failure of overconsolidated clay slopes, and suggest counter-measures
• Diagnose quick clay flowslides, and suggest counter-measures
• Compute active and passive earth pressure with different theories
• Understand the effects of water on the stability of earth retaining structures
• Recognise the main types of retaining structures and construction methods
• Design basic retining structures

Content

Whereas module 3D1 was concerned chiefly with the limiting equilibrium of plastic soil bodies and soil consolidation, 3D2 aims to address modelling of the mechanical behaviour of soils and geotechnical structures. Soils are an order of magnitude more compliant than steel or concrete, so designers have to limit the mobilisation of soil strength to keep ground strains small enough to guarantee the serviceability of adjacent structures. Furthermore, some soils are inherently brittle, and their strength can deteriorate if they are permitted to strain excessively; this can lead to unexpected catastrophic failures. In geotechnical engineering, therefore, strains are often more important than stresses.

The Cam Clay model of soil behaviour is introduced to link concepts of consolidation and shearing, to envision drained and undrained soil behaviour within a single conceptual framework, to distinguish between yielding and failure, and to contrast stress paths that lead to brittle softening with those that lead to stable hardening. These comparisons and contrasts are central to the correct characterisation of soils for geotechnical decision-making. They are the subject of the first example paper.

The module continues with the assessment of the stability of natural slopes and cuts, the characterisation of in-situ stress states as a function of the previous stress history of the site, and considers the stress paths which they will follow as a result of construction. Particular materials, stress paths, and changes in environmental conditions can lead to catastrophic failures. The key to avoiding such failures is either to improve the ductility and continuity of materials and structures, or to take the utmost care in controlling soil strains in service. This material is the subject of the second example paper.

The final part of the course addresses the fundamentals of earth pressures and earth retaining structures. This will start from a review of the main tools available for the calculation of earth pressure, including upper bound, lower bound and limit equilibrium methods, followed by consideration of the main retaining structures types and construction methods (gravity, embedded, composite walls and other support systems).  Finally the course will address the basic design of simple retaining structures.

Topic 1: Basics - Soil Stress-strain, 3D Stresses & strains and their invariants, pore pressure parameters

Modelling in geortechnical engineering (Lecture 1) 

Modelling forms an implicit part of all engineering design but many engineers are not aware either of the fact that they are making assumptions as part of the modelling or of the nature and consequences of those assumptions. The lecture is an introduction to the course iproviding an overview of the evolution of modelling and the shift of modelling paradigms in science and engineering and in soil mechanics.

Stress/strain invariants (Lecture 2)

3D stresses and strains, Lode's coordinates, strain and strain invariants, work conjugates, pore pressure parameters, stress paths.

Topic 2: Strain hardening elastoplasticity

1D elasto-plasticity (Lecture 3)

Additive decomposition of strain, elasticity, admissible stress, yield criterion, elastic range, flow rule. Kuhn Tucker condition, consistency condition, plastic multiplier. Isotropic and kinematic hardening. Elasto-plastic stiffness.

Linear elasticity and Mohr Coulomb strength criterion (Lecture 4)

Isotropic linear elasticity. Mohr Coulomb Yield criterion with associative flow rule. predicted behaviour for drained and undrained triaxial compression and triaxial extension. Limitations and possible ways to overcome them.

Plane strain stress paths (Lecture 5)

Stress paths in the ground arising from a variety of construction processes, and relating to a range of representative locations. Use of vertical and horizontal equilibrium equations to estimate total stress paths due to simplified cases of vertical loading or horizontal unloading. Correlation with effective stress paths dictated either by undrained or drained soil conditions. Predicting the approach of soil states to limiting strength envelopes.

Topic 3: Cam-Clay

Shearing of soils: work and dissipation, yield surface and normality (Lecture 6)

Taylor’s work equation. Yield surface in effective stress space. Normality principle guarantees maximum dissipation, providing a plastic flow rule. Derivation of the Cam Clay yield surface. Compressibility and volumetric hardening.

Critical states, normal compression, and yield  (Lecture 7)

Stress dilatancy and critical state. Radial compression lines, critical state line. 3D state surface of shear stress, effective normal stress and specific volume. Drained and undrained shearing of soil at a given density, from points of normal consolidation and overconsolidation

Undrained shear strength,  predicting behaviour of geotechnical structures using using Cam-clay model (Lecture 8)

Undrained shear strength. Predicting behaviour of smooth retaining wall and embankment  on soflt clay. Staged loading. Development of stress-strain relationship of Cam clay model. Application of numerical programs for modern geotechnical analysis.

Topic 4: Slope stability - avoiding catstrophic failure

Slope stability analysis (Lectures 9 and 10)

Occurrence of slope failure in the UK and worldwide. Examples. Modes of movement: falls, topples, slides, and flows. Analysis methods to assess the stability of slopes in sands and clays. Infinite slope, effect of groundwater flow. Finite slope undrained. General Limit equilibrium methods. 

Avoiding catastrophic failure on the dry side (Lecture 11)

Selection of mechanical parameters for the design of engineered slopes. Factors promoting failure on the dry side: brittle failure for dilatant sand and overconsolidated clay. Need to design for critical state friction and worst pore water pressures. 

Delayed failure in clay slopes and catastrophic failure on the wet side (Lecture 12)

Delayed failure in clay slopes due to progressive softening on cycles of wetting/drying.  Factors promoting brittle failure on the dry side: quick clay flowslides, volumetric collapse on saturation for partly saturated slopes. 

Topic 5. Geotechnical Investigation

Geotechnical Site Investigation (Lecture 13)

Requirements of geotechnical site investigation. Objectives, extent, frequency and layout of investigations for the geotechnical characterisation of a site. 

In-situ testing (Lecture 14)

Procedures and interpretation of Standard Penetration Test (SPT), Cone Penetration Test (CPT), Field Vane, and pressuremeter tests

Topic 6: Elasto-plastic radial solutions

Cavity Expansion (Lecture 15)

Cavity expansion in elastic perfectly plastic medium. Application to the interpretation of pressuremeter test. Estimation of soil properties from pressuremeter tests in clay: in-situ total horizontal stress, shear modulus, undrained shear strength.

Cavity Contraction (Lecture 16)

Cavity contraction in elastic perfectly plastic medium. Applications to bored tunnelling. Estimation of support pressure required for tunnel stability. Tunnel convergence and settlements above tunnels.

This syllabus contributes to the following areas of the UK-SPEC standard:

Toggle display of UK-SPEC areas.

 
Last modified: 01/11/2023 08:59