Learning objectives
To give the basic concepts and analysis tools for the mechanical design of components and structures.
Prerequisites
Students are recommended to have attended the following courses: Mathematical analysis, Geometry, theoretical Mechanics
Course unit content
Geometry of areas. Simple (beams) and complex (frames) structural systems
Statically determinate framed structures
Internal beam reactions
Analysis of stresses
Analysis of strains
The theorem of virtual work
Theory of elasticity
The problem of a linear elastic body: solution uniqueness theorem (or Kirchhoff’s theorem)
Work of deformation
De Saint-Venant’s problem
Strength criteria
Computation of displacements for beams with rectilinear axes
Statically indeterminate framed structures
Instability of elastic equilibrium.
Full programme
Geometry of areas. Introduction. Static moment and centroid. Moments of inertia. Laws of transformation. Principal axes and moments of inertia. Mohr’s circle for moments of inertia.Simple (beams) and complex (frames) structural systems. Plane beams and frames. Problem of structural system equilibrium: kinematic definition of plane constraints; static definition of plane constraints (constraint reactions) and cardinal equations of statics. Framed structures: statically determinate (or isostatic); hypostatic; statically indeterminate (or hyperstatic). Principle of superposition.Statically determinate framed structures. Equilibrium equations. Three methods: cardinal equations of statics; auxiliary equations; the principle of virtual work.Internal beam reactions. Three methods: direct method; differential method (indefinite equations of equilibrium for plane beams); the principle of virtual work. Diagrams of characteristics of internal beam reactions.Particular problems. Closed-frame structures. Plane trusses. Symmetric frames.Analysis of stresses (for three-dimensional solids). Stress tensor, equations of Cauchy, law of reciprocity. Principal stress directions, Mohr’s circles. Plane stress condition and Mohr’s circle. Boundary conditions of equivalence and indefinite equations of equilibrium.Analysis of strains (for three-dimensional solids). Rigid displacements, strain tensor. Strain components: dilatations and shearing strains. Principal strain directions.The theorem of virtual work (for three-dimensional solids).Theory of elasticity (for deformable three-dimensional solids). Real work of deformation, elastic material, linear elasticity, homogeneity and isotropy, linear elastic constitutive equations. Real work of deformation: Clapeyron’s theorem; Betti’s theorem.The problem of a linear elastic body: solution uniqueness theorem (or Kirchhoff’s theorem).The problem of De Saint-Venant. Fundamental hypotheses, indefinite equations of equilibrium, elasticity equations and boundary conditions. Centred axial force, flexure (bending moment), biaxial flexure, eccentric axial force, torsion, bending and shearing force.Strength criteria. Criteria for brittle materials (by Rankine, Grashof) and for ductile materials (by Tresca, von Mises).Computation of displacements for beams with rectilinear axes. Differential equation of the elastic line; theorem of virtual work for deformable beams with rectilinear axes; thermal distortions and constraint settlements.Statically indeterminate framed structures. Theorem of virtual work: structures subjected to loads, thermal distortions and constraint settlements.Instability of elastic equilibrium. Second order geometrical effects. Euler’s critical load and free length of deflection; omega method.
Bibliography
Documentation provided by the teacher.
Carpinteri A.: "Scienza delle Costruzioni", Vol. 1 e 2, Ed. Pitagora, Bologna.
Corradi dell’Acqua, L.: "Meccanica delle strutture", Vol. 1,2 e 3, Mc Graw-Hill, 1995.
Viola E.: “Esercizi di Scienza delle Costruzioni”, Ed. Patron, Bologna.
Teaching methods
Theoretical and practical lessons.
Assessment methods and criteria
Written and oral examination.
Other information
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