29329 - Strength of Materials

Course contents

NOTIONS OF VECTOR CALCULUS  1) Vector; versor; null vector. 2) Equivalent vectors. 3) Bound vector. 4) Cursor. 5) Addition method. 6) Subtraction method. 7) Angular momentum. 8) Couple. 9) Torque-free resultant vector. 10) Equivalent sets of vectors. 11) Set of vectors in equilibrium. 12) Equations of equilibrium. 13) Equilibrant vector of a set of vectors. 14) Force. 15) Net force and resultant force-torque. 16) Equivalent sets of forces. 17) Set of forces in equilibrium. 18) 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) Polar second moment of area and perpendicular axis theorem. 5) Distributive property of first moment of area, planar second moment of area, product second moment of area. 6) Calculation of elastic section muduli and second moments of area of elementary shapes. 7) Parallel axis Theorem (Huygens-Steiner Theorem) and tensor generalization. 8) Mohr’s circle of second moments of area: construction, central principal second moments of area, pole method, central principal axes of inertia. 9) Radii of gyration; momental ellipse, central ellipse of inertia. 10) Conjugate diameter with respect to the attitude of a given line: definition and geometric construction. 11) Geometric construction of antipole and antipolar line of a given pole; 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) Infinitesimal strain tensor; Voigt notation representation of the infinitesimal strain tensor. 4) Principal strains; strain invariants; principal axes of strain. 5) Mutual orthogonality of the principal axes of strain; infinitesimal strain tensor and strain invariants in the principal reference frame. 

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) Cauchy’s stress theorem; symmetry of the matrix of the orthogonal normal stresses and orthogonal shear stresses; Cauchy stress tensor; Voigt notation representation of the Cauchy stress tensor. 4) Definitions of principal axes and principal planes of stress.

ELASTIC SYSTEMS  1) Configuration space; undeformed initial state; stress-strain state in an elastic body; deformation work. 2) Homogeneous materials; non-homogeneous materials; isotropic material; anisotropic material. 3) Hooke’s laws for the three-dimensional body; relationship between principal axes of stress and strain. 4) Physical meaning of Young’s modulus, E, Poisson’s ratio, n, and shear modulus, G. 5) 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). 

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; assumption of de Saint-Venant and its consequence (semi-inverse approach). 3) Centric axial loading (pure compression and pure extension): Cauchy stress tensor; displacement field; deformation of the cross-section; 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. 4) Pure bending along a principal axis of inertia: Cauchy stress tensor; Navier formula; neutral surface; neutral axis; diagram of the normal stresses on the cross-section. 5) Pure bending along an axis that is not principal of inertia: solution by means of the superposition principle; binomial Navier formula, neutral axis, special cases. 6) Pure bending along an axis that is not principal of inertia: relationship between neutral axis and direction of the bending moment: geometric method; diagram of the normal stresses on the cross-section. 7) Eccentric axial loading: solution by means of the superposition principle; trinomial Navier formula; equation of the neutral axis; 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; diagram of the normal stresses on the cross-section. 8) Torsion: thin-walled beams of closed and open cross-sections. 9) Shearing and bending in solid cross-sections: centre of torsion; approximated solution of Jourawski; shearing force along a symmetry axis . 10) Shearing and bending in thin-walled cross-sections.

YIELD CRITERIA FOR BEAMS 1) Verification of allowable stresses for centric axial loading. 2) Verification of allowable stresses for pure bending. 3) Verification of allowable stresses for eccentric axial loading. 4) Verification of allowable stresses for shearing in thin-walled and solid cross-sections. 5) Verification of allowable stresses for torsional loading in thin-walled cross-sections.

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

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