Monday 10:45-12:30 hours Room F, Tuesday 13:45-15:30 hours Room D
4 Credits (ECTS)
Energy Principles (EP)
Derivation of the principles of virtual work and virtual complementary work; principles of minimum potential energy and minimum complementary energy; both laws of Castigliano; reciprocal theorem of Maxwell-Betti.
| 5 September | Systematics in truss analysis. Derivation of the kinematic equations, constitutive equations and equilibrium equations, Solution by the force method and displacement method, Symmetry of the stiffness matrix according to Maxwell, Selecting redundants, Computer implementation, Example of a bicycle wheel (2 dofs, 34 redundants), Example of a small truss (9 dofs, 1 redundant). | DM, Chap. 1 |
| 11 September | Explanation of the word "linear elastic", Systematics and redundancy of continuous systems, Glued connection (Exam 26 Jan. 2005), Kinematic equations, constitutive equations and equilibrium equations, Solution by the force method and displacement method, Derivation of the differential equations, Boundary equations, Coupled shear walls, Frame model versus continuous model, Derivation of the kinematic equations, Solution method. Formula for the largest shear force. | DM, Chap. 2 |
| 12 September | Plates loaded in plane, Frame work of quantities, Kinematic equations, Hooke's law, Equilibrium equations, Plane stress and plane strain, Derivation of Hooke's law for plane strain, Displacement method (Navier's equations) and force method (biharmonic equation), Airy stress function, Compatibility equation, Solutions for a cantilever and deep beam, Stress in a plate corner loaded by shear stress. Shear lag in a semi infinite strip (open problem). | DM, Chap. 3 |
| 18 September | Principle of De Saint Venant, Axisymmetric plates loaded in plane, Derivation of the equations (kinematic, constitutive and equilibrium), Force method and displacement method, Differential equations, Stresses around a hole (Exam Jan. 12, 1998 Problem 3), More stress states around holes, Thick-wall tubes, Curved beams, Point load on a plate edge, Superposition, Brazilian splitting test | DM, Chap. 3 |
| 19 September | Axisymmetric plates loaded perpendicular to their plane, Evenly distributed load, Moment distributions for pinned and fixed edges, Point load on a fixed plate, Validity of the plate theory around a point load, Timoshenko's solution for 3D stresses under a point load, Finite element mesh around point loads | DM, Chap. 4 |
| 25 September | Torsion of prismatic bars, Overview of 3D frame analysis, The 6 stiffnesses of a frame element, Torsion stiffness and torsion stresses, De Saint Venant theory, Displacement method (warping function) and force method (phi bubble), Closed form solutions, Finite element method, Results for rectangular cross-sections, Finite element example of a rectangular beam using ESA Prima Win. | DM, Chap. 5 and 6 |
| 26 September | Torsion stresses in typical cross-sections, Software for cross-section analysis, Interpretation of the phi-bubble, Qualitative analysis of cross-sections, Soap film analogy of Prandtl, Analogy for sections with holes, Derivation of both formulas of Bredt, Calculation of a two-cell box girder | DM, Chap. 6 |
| 2 October | Vlasov torsion theory, Shear centre, Distributed torsion moment, Warping restraints, Bi-moment, Warping constant, Derivation of Vlasov's differential equation for solid sections and thin-wall sections, Example of a box-girder bridge, Stresses due to the bi-moment, Example of a cold-formed section cantilever, Adding bi-moment to frame program software, Trick to account for restrained warping in a frame program. | Hand out |
| 3 October | Energy and work, Definitions of strain energy and complementary energy, Tensor notation using Einstein’s convention, Vector notation, Derivation of shear stiffness and largest shear stress of a rectangular cross-section, Formulas for shear in other cross-sections, Derivation of the torsion stiffness and largest stress of square cross-sections | EP, Chap. 1 |
| 9 October | Brazilian splitting test simulated by a particle model, Potential energy definition, Minimum principle, Energy in a mass-spring system, Dynamic behaviour and damping, Potential energy in a double pendulum, stability and dynamic behaviour, Exam example (4 Nov. 2005, Problem 2) | EP, Chap. 2 |
| 10 October | Exam example continued, Alternative displacement functions, Recipe for applying potential energy, Restrictions of the minimum principle, Exam example (26 Oct. 2001, Problem 3), Total complementary energy, Definition, Comparison to potential energy, Minimum principle, Restrictions, Ring loaded by two opposite forces (EP p.55), calculation of the moment distribution and the deformation | EP, Chap. 3 |
| 16 October | Derivation of Casigliano's first prinicple. Symmetry of the stiffness matrix (Maxwell), Point load on an initially straight cable (EP p. 52), Application to a stainless steel tank roof, Distributed load on an ininitially straight cable, Derivation of Castigliano's second theorem, Exam problem (12 Jan. 1998, Problem 4), Reciprocity of load and displacement (Betti), Application to influence lines | EP, Chap. 4 |
| 17 October | Simply supported rectangular plate; exact and energy solution, Method of Ritz, Nature of energy approximations; too stiff or too flexible, Proof for potential energy, Selecting strut-and-tie models (Exam 4 Nov. 2005, Problem 3) | EP, Section 6.4, Chap. 8 |