30978 - Mechanics of Solids and Structures

Course Unit Page

SDGs

This teaching activity contributes to the achievement of the Sustainable Development Goals of the UN 2030 Agenda.

Quality education Sustainable cities

Academic Year 2018/2019

Learning outcomes

Basics of solid and structure mechanics, with applications to elementary cases of recurrent structures.

Course contents

REQUIRED KNOWLEDGE 1) Mathematical Analysis T. 2) Geometry and Algebra T.

NOTIONS OF VECTOR CALCULUS 1) Vector. 2) Equivalent vectors. 3) Bound vector. 4) Cursor. 5) Dot product. 6) Cross product. 7) Addition method. 8) Subtraction method. 9) Angular momentum. 10) Couple. 11) Torque-free resultant vector. 12) Equivalent sets of vectors. 13) Set of vectors in equilibrium. 14) Equations of equilibrium. 15) Equilibrant vector of a set of vectors. 16) Invariance of some vector operations. 17) Force. 18) Net force and resultant force-torque. 19) Equivalent sets of forces. 20) Set of forces in equilibrium. 21) Conditions for a set of vectors in the plane to be equal to the zero vector. 22) Finding the resultant force and torque of a set of forces in a plane.

GEOMETRIC PROPERTIES OF AN AREA 1) First moment of area. 2) Locating the centroid of a shape by geometric decomposition and integral formula. 3) Planar second moment of area and product second moment of area. 4) Distributive property of first moment of area, planar second moment of area, product second moment of area. 5) Polar second moment of area and perpendicular axis theorem. 6) Calculation of elastic section muduli and second moments of area of elementary shapes (rectangle, circle, right triangle, semicircle). 7) Using the perpendicular axis theorem for calculating the planar second moment of area of a circle with respect to a diameter. 8) Parallel axis Theorem (Huygens-Steiner Theorem) and tensor generalization. 9) Coordinate transformation in two dimensions for rotation of the reference frame; planar and product second moments of area in the rotated reference frame; parametric equations in 2 a of the planar and product second moments of area; principal and central principal second moments of area; plotting of the parametric equations. 10) Mohr's circle of second moments of area: construction, central principal second moments of area, pole method, central principal axes of inertia. 11) Analytical method for finding central principal second moments of area and central principal axes of inertia: rotation angles a0 about which the planar second moments of area Ixand Iy are found to be maximum and minimum values and the product second moment of area Ixy is found to be zero; relationship between the two second derivatives of Ix and Iy. 12) Central principal second moments of area for Ix0>Iy0 and Ix0< Iy0; geometric meaning. 13) Parametric equations in different reference frames. 14) Radii of gyration; momental ellipse, central ellipse of inertia; polar line of a given polar point (pole) with respect to the central ellipse of inertia; reciprocation; polarity. 15) Conjugate points with respect to a polarity; conjugate lines with respect to a polarity; Principle of Duality for points of the Euclidean plane and points at infinity. 16) Conjugate diameter with respect to the attitude of a given line: definition and geometric construction. 17) Antipole; antipolar line; antipolarity; conjugate antipoles with respect to an antipolarity. 18) Geometric construction of antipole and antipolar line of a given pole; special cases of pole at the centroid and pole at infinity. 19) Central antipolar curve and central antipolar area: properties and constructions.

DESCRIPTION OF DEFORMATION 1) The displacement field and its properties. Engineering normal strain; true and engineering shear strain. 2) Linearization of the displacement field; affinity; decomposition of the displacement field in rigid-body displacement and deformation; meaning of the decomposition in the assumption of small displacement gradients. 3) Physical and geometric meaning of the normal strain. 4) Physical and geometric meaning of the shear strain. 5) Normal and shear strains for a given rotation of the reference frame; infinitesimal strain tensor; Voigt notation representation of the infinitesimal strain tensor; partial differential operator of compatibility; kinematically admissible system of strains-displacements. 6) Invariance of bilinear and quadratic forms for rotation of the reference frame. 7) Proof of the tensor transformation law under a change in the system of coordinates starting from the displacement components of the points on the surface of the spherical neighbourhood of unit radius. 8) Reciprocity of the displacement components. 9) Principal strains; strain invariants; principal axes of strain. 10) Mutual orthogonality of the principal axes of strain; infinitesimal strain tensor and strain invariants in the principal reference frame; deformation of a rectangular parallelepiped for which all faces are orthogonal to the principal axes of strain. 11) Volumetric strain; volumetric strain tensor; strain deviator tensor. 12) Biaxial state of strain (plane strain); uniaxial state of strain.

STRESS ANALYSIS 1) Forces in a continuum body; equilibrium of the free body in deformed and undeformed configurations. 2) Cauchy stress vector; Cartesian components of the Cauchy stress vector; special components of the Cauchy stress vector: normal and shear stresses. 3) Transformation matrix between special and Cartesian components of the Cauchy stress vector; local reference frames on the faces of the infinitesimal cube. 4) Cauchy's stress theorem; symmetry of the matrix of the orthogonal normal stresses and orthogonal shear stresses. 5) Reciprocity of the stress components. 6) Tensor transformation law under a change in the system of coordinates; Cauchy stress tensor; Voigt notation representation of the Cauchy stress tensor. 7) Invariance of bilinear and quadratic forms for rotation of the reference frame; spherical neighbourhood of unit radius. 8) Principal stresses; stress invariants; principal axes and principal planes of stress. 9) Mutual orthogonality of the principal axes of stress; Cauchy stress tensor and stress invariants in the principal reference frame. 10) Relationship between points on the surface of the spherical neighbourhood of unit radius and points of the Mohr plane; domain of Mohr. 11) Relationship between mean stress and first stress invariant; mean hydrostatic stress tensor and stress deviator tensor (definitions, stress invariants, Mohr's circles, principal stresses, principal axes of stress). 12) Biaxial state of stress (plane stress); uniaxial state of stress. 13) Plane stress: Mohr's circles; fundamental property of the pole on the Mohr circle; pole approach; geometric approach for finding the principal stresses and the principal axes and planes of stress. 14) Equilibrium equations and boundary conditions for the three-dimensional body (scalar and matrix notions); partial differential operator of equilibrium; statically admissible stress field.

MECHANICS OF MATERIALS 1) Axial testing in load control and strain control. 2) Tensile and compressive axial testing of brittle materials: stress vs. strain diagram, strain-softening, Cauchy stress tensor, Mohr's circles, sliding fracture planes for cubic specimens. 3) Compressive triaxial testing of brittle materials: effect of the confining pressure on the stress vs. strain diagrams, Cauchy stress tensor, Mohr's circles, stress path, envelope of the Mohr circles at failure, sliding fracture planes for cubic specimens. 4) Brittle-ductile transition. 5) Failure domain in the Mohr plane for brittle materials; Mohr/Coulomb yield surface and Mohr/Coulomb yield (failure) criterion; cohesion; angle of internal friction; granular materials. 6) Mohr's circles for pure shear. 7) Effect of the interstitial fluid on the Mohr circles of granular materials; effective stress. 8) Effect of the rate of deformation on the stress vs. strain diagrams; creep (primary, secondary, and tertiary). 9) Tensile and compressive axial testing of steel: stress vs. strain diagram; unloading behaviour; strain-hardening; simplified model of Prandtl; permissible (allowable) stress design; safety factor; rigid-plastic model. 10) Tresca yield surface and Tresca criterion.

FUNDAMENTAL EQUALITY IN SOLID MECHANICS 1) Virtual work theorem for a deformable body; virtual work principle for a deformable body; virtual work principle for a rigid body. 2) Work of elastic deformation; quasi-static loading process; complementary work of elastic deformation.

ELASTIC SYSTEMS 1) Configuration space; stress-strain state in an elastic body; strain energy density; strain energy; elastic potential energy; complementary energy density; complementary energy. 2) Elasticity formulation: Maclaurin series expansions of strain energy density and stress components; homogeneous materials; non-homogeneous materials; assumption of small displacement gradients; assumption of natural undeformed state to which all deformed states must return upon unloading; elastic constants. 3) Quadratic forms of elastic potential energy and complementary elastic potential energy; principle of minimum potential energy. 4) Isotropic material; anisotropic material; linear theory of elasticity for an isotropic material; Hooke's laws for the three-dimensional body; relationship between principal axes of stress and strain. 5) Physical meaning of Young's modulus, E, Poisson's ratio, n, and shear modulus, G. 6) Linear elastic boundary value problem; linear elastic boundary value problem for a homogenous material; linear elastic boundary value problem for an isotropic-homogenous material; superposition principle; principle of the uniqueness of solution (Kirchhoff).

THE DE SAINT-VENANT'S PROBLEM 1) The de Saint-Venant's solid: geometric model, model of the external loads, constitutive model; problem statement. 2) Boussinesq's postulate; de Saint-Venant's principle. 3) Consequence of Kirchhoff's principle; assumption of de Saint-Venant and its consequence (semi-inverse approach). 4) Equations of static equivalence; integral boundary conditions; the four fundamental classes of problems (relaxed de Saint-Venant's problem); vector of the total shear stress for the points of the lateral surface. 5) Centric axial loading (pure compression and pure extension): integral boundary conditions; semi-inverse solution; Cauchy stress tensor; verification of assumptions. 6) Centric axial loading: displacement field; setting to zero the components of rigid motion; deformation of the cross-section. 7) Centric axial loading: Mohr's circles; maximum shear stresses and orientation of their planes of action; relationship between principal axes of stress and strain; volumetric strain; cases for which the solution of the relaxed de Saint-Venant's problem is exact also at the loaded ends and their neighbourhoods; diagram of the normal stresses on the cross-section. 8) Pure bending along a principal axis of inertia: integral boundary conditions; semi-inverse solution; Cauchy stress tensor; verification of assumptions. 9) Pure bending along a principal axis of inertia: displacement field; deformation of the centroidal axis; deformation of the cross-section; physical meaning of a shear stress equal to zero on the cross-section; relative rotation between cross-sections. 10) Pure bending along a principal axis of inertia: normal strains on the cross-section; neutral surface; neutral axis; diagram of the normal stresses on the cross-section; relationship between neutral axis and direction of the bending moment. 11) Pure bending along an axis that is not principal of inertia: solution by means of the superposition principle; relationship between neutral axis and direction of the bending moment: analytical method, geometric method, special cases; diagram of the normal stresses on the cross-section. 12) Eccentric axial loading: centre of stress; solution by means of the superposition principle; trinomial formula in terms of radii of gyration; equation of the neutral axis; geometric relationship between neutral axis and centre of stress. 13) Eccentric axial loading: diagram of the normal stresses on the cross-section for a centre of stress situated upon or within the central antipolar curve; diagram of the normal stresses on the cross-section for a centre of stress situated outside of the central antipolar area; relationship between neutral axis and direction of pure bending; graphic constructions of the neutral axis. 14) Torsion: thin-walled beams of closed and open cross-sections (shear stresses distribution across the wall of the cross-section, factor of torsional rigidity, angle of twist); hydrodynamic analogy; stress concentrations in solid and thin-walled cross-section. 15) Torsion: Mohr's circles; tensile and compressive force lines. 16) Shearing and bending: centre of torsion; approximated solution of Jourawski. 17) Shearing and bending: shearing force along a symmetry axis.

YIELD CRITERIA FOR BEAMS 1) Von Mises stress; Tresca criterion. 2) Verification of allowable stresses for centric axial loading. 3) Verification of allowable stresses for pure bending. 4) Verification of allowable stresses for eccentric axial loading. 5) Verification of allowable stresses for shearing and bending. 6) Verification of allowable stresses for shearing and torsional loading.

Readings/Bibliography

  • A. DI TOMMASO, FONDAMENTI DI SCIENZA DELLE COSTRUZIONI, PARTE I, 1981; PARTE II, 1993, PATRON ED., BOLOGNA

  • E. VIOLA, ESERCITAZIONI DI SCIENZA DELLE COSTRUZIONI, 1/ STRUTTURE ISOSTATICHE E GEOMETRIA DELLE MASSE, 2/ STRUTTURE IPERSTATICHE E VERIFICHE DI RESISTENZA, PITAGORA ED., BOLOGNA, 1993

Teaching methods

Classroom lectures.

Assessment methods

Written and oral examinations.

Teaching tools

The teacher handles a course and e-learning website.

Office hours

See the website of Elena Ferretti