ENGINEERING TRIPOS PART IIA - 2012/2013
Module 3D4 Structural Analysis and Stability
Mr F A McRobie
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 F. Cirak)
- Asymmetric beams; principal axes;
- Shear centre; torsion and warping of open-sections;
- Virtual work;
- Differential equation of beam (Macaulay);
- Reciprocal theorem and influence lines;
- Beams on elastic foundations;
Stability and Buckling (8L) (Mr F.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.
On completion of the course students should be able to:
stress distributions in asymmetric open sections, taking account of
bending, shear, torsion and warping effects.
the relevant section properties for complex sections.
grillages under out-of-plane loading.
beam bending problems using Macaulay’s method, and to obtain influence
the reciprocal theorem and the importance of influence lines.
the shortcomings of the structural analysis learnt thus far and appreciate
the need to include stability as a fourth concept in any complete theory
stable and unstable paths on a load/displacement diagram for various
bifurcation and snap through models.
how elastic stability may be determined from the total potential energy
and may be described by the eigenvalues of the total stiffness matrix.
elastic critical loads for simple structures by eigenvalue analysis,
whilst appreciating the importance of imperfection sensitivity and the
limitations of such analysis.
approximation methods based on energy to
determine the stability of simple systems.
second-order beam theory, using s and c functions.
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
the importance of lateral-torsional buckling of beams.
how of the above ideas combine in the context of column design.
Please see the Booklist for Part IIA Courses for references for this module.
Last updated: June 2012