00890 - Mechanics of solids and structures

Learning outcomes

The course's aim is to teach the basic concepts of solid mechanics and the techniques for structural analysis and verification.

Course contents

AIM, HYPOTHESES AND MODELS IN STRUCTURAL MECHANICS
1)Structural mechanics models. Elastic equilibrium problem for axially loaded beam: hypotheses, equations, boundary conditions, scheme of physical theories. 2)Elastic equilibrium problem for axially loaded beam: total potential energy. Example of finite element method application. 3)Elastic equilibrium problem for bending beam: hypotheses, equations, boundary conditions. 4)Elastic equilibrium problem for bending beam: scheme of physical theories. Mixed problem and problem in terms of displacements. Total potential energy functional and meaning of its minimum. 5)Integration of the differential equations for simply supported beam with uniformly distributed load. 6)Integration of the differential equations for cantilevered beam with a force applied at the free end. 7)Funicular polygon, polygon of successive resultants. 8)Influence lines for isostatic systems: constraint reaction forces and internal forces resultants (exercise 6.11.1).

CALCULUS OF VARIATIONS
1)Definition of functional and varied function. Neighborhood of a function. First and second variation of a function. Commutative property of the variational operator. 2)First and second variation of a functional. Stationary points of a functional. Euler-Lagrange equations for axially loaded beam. 3)Differential and variational formulations of the elastic equilibrium problem for bending beam. Natural and essential boundary conditions. 4)Differential and variational formulations of elastic equilibrium problem for axially loaded beam. Natural and essential boundary conditions.

STRAIN ANALYSIS
1)Displacement field and its properties. Local deformation problem. Displacement gradient and deformation gradient. Affine transformation and its geometric representation. 2)Rigid body motion components and deformation components. Congruence operator. Coordinate transformation matrix and strain tensor. 3)Physical meaning of the strain tensor components. 4)Spherical neighborhood of unitary radius: properties of dilatations and shear strains. Expressions of dilatations and shear strains. 5)Principal directions and principal strains. Strain state in the principal reference system. 6)Physical meaning of the first strain invariant. Hydrostatic tensor, deviatoric tensor and their properties. 7)Plane and uniaxial strain states. Internal and external congruence. Compatibility equations and compatibility operator. 8)Representation of finite strain tensor in terms of displacement components. Deduction of infinitesimal and finite deformations. Application to bending cantilevered beam for the calculation of axial displacement caused by bending.

STRESS ANALYSIS
1)Surface and volume forces. Equilibrium equations for the rigid body. Internal stress. Cartesian and special stress components and their relationship. 2)Stress on planes parallel to the coordinate planes. Local properties of stress state: Cauchy's equations and symmetry of tangential stresses. 3)Stress tensor. Theorem of reciprocity of mutual components. 4)Principal directions and principal stresses. Stress state in the principal reference system. Octahedral shearing stress. 5)Mohr's circles: triaxial stress states (outline and results), spherical neighborhood of unitary radius. 6)Mohr's circles for plane stress. 7)Statically admissible stress states. Field equilibrium equations and boundary equilibrium equations (different notations). Equilibrium operator. 8)Classification of stress states. Plane stress state. Hydrostatic and deviatoric stress states. 9)Application of Mohr's circles for determining fissure directions.

GENERAL RELATIONSHIPS
1)Virtual work theorem for the three dimensional continuum. Internal and external virtual work. The rigid body case. 2)Alternative formulations for equilibrium and congruence (exercise 4.4). 3)Principles of virtual forces and virtual displacements. 4)Application of the principle of unitary force to beams. Calculation of displacements and rotations for the simply supported beam with uniformly distributed load or with couple. 5)Virtual work theorem for axially loaded beam.

ELASTIC BODY
1)Real transformations. Internal and external deformation work. Example in the one dimensional case. 2)Elastic potential energy and complementary potential energy. 3)Linear elastic body: constitutive equations, stiffness and compliance matrices for the material. Elastic potential function. 4)Expressions for elastic potential energy and complementary potential energy. Hessian matrix. Elastic constants. 5)Isotropy hypothesis, Hooke's generalized laws and inverse laws. 6)Principal stress and strain directions. Alternative notation for Hooke's laws. Elastic constants for the isotropic medium. 7)Alternative form for Hooke's inverse laws. 8)Elastic equilibrium problem in index notation and operator notation (par. 5.8.1. and 5.8.5). 9)Field equilibrium equations in terms of displacements (Navier's equations). Deduction of the dynamic equilibrium equations: vibrating string and membrane, longitudinal forced vibration, three dimensional and distortional wave equations. 10)Superposition principle. Uniqueness of the solution. 11)Clapeyron, Betti and Maxwell's theorems. Calculation of displacements and rotations with Clapeyron's theorem. 12)Equations of elastic equilibrium problem in operator notation in the primal scheme of physical theories.

VARIATIONAL PRINCIPLES
1)Principles of stationary and minimum of the total potential energy. Discreet set of forces. 2)Deduction of Navier's equations (Euler-Lagrange equations) from the first variation of total potential energy (outline). Displacement method for trusses. 3)First Castigliano theorem. Engesser theorem. Second Castigliano theorem.4) Principles of stationary and minimum complementary potential energy. Principle of minimum of elastic energy. 5)Force method for trusses. Scheme of physical theories for the force and displacement methods for trusses. 7)Integral formulation of equilibrium and congruence for the bending beam. Deduction of the total potential energy. 8)Rayleigh-Ritz method. Example 6.2. 9)Methods for evaluating the stiffness matrix. Application to bending beams.

FAILURE CRITERIA
1)Limit state and safety coefficient. Limit surface. Tests of ductile and brittle materials. Equally dangerous stress states. 2)Tresca criterion: crisis condition and limit surface. Plane stress. Probabilistic interpretation of the domain and conditional collapse probability function. 3)Huber-Hencky-Mises criterion: yield, resistance and safety conditions. Representations for the case of beams.

SAINT-VENANT PROBLEM
1)General layout. Saint-Venant hypotheses. Internal stress resultants. Equivalence between stress distributions and internal stress resultants.The four fundamental cases. Strain energy. 2)Axial force resultant: problem solution, stress and strain analysis. Strain energy. Alternative solution. Mohr's circle. 3)Bending of a beam about a principal axis: layout and experience. Solution. 4)Strain state. Stress state. Strain energy. 5)General case of pure bending: outline and decomposition in bending about the principal axes. Analytical and graphical definition of the neutral axis. 6)Representation of the stress state. Strain energy. 7)Eccentric axial force resultant: outline and trinomial formula. Analytical and graphical definition of the neutral axis. Shape of the stress diagram and safety assessment. 8)Core of the section. Strain energy. 9)Torsion for cylinder with arbitrary cross-section: assumed solution, congruence, constitutive and equilibrium equations. Neumann's and Dirichlet's problems. Statical equivalence. 10)Definition of the centre of torsion. Tension function and its properties. 11)Elliptical cross-section. 12)Cylinder with circular cross-section as a particular case of generical cross-section cylinder. 13)Torsion equations written in the primal scheme of physical theories. 14)Thin-walled closed cross sections: unitary torsion angle. 15)Bredt's formula obtained through statical equivalence. 16)Approximate formulation for shear internal force: medium shear stress. Stress component directed along the chord. Shear stress along a generical chord. 17)Shear factor and strain energy. 18)Approximate evaluation of the shear centre. Thin walled closed cross-section. 19)Space beam: technical theory, stress resultants and strain components. Congruence equations. Strain energy: evaluation through Clapeyron's theorem. 20)Strain energy: internal deduction. 21)Unitary force principle (Virtual work principle). Clapeyron's and Castigliano's theorems. 22)Layout of the calculation of the displacement of a statically determined structure (4 methods). 23)Field equilibrium equations, congruence and constitutive equations in extended and matrix notation. Equations for the space beam in the primal scheme of physical theories.

STRUCTURAL THEORY
1)Plane sets of forces. Statical and kinematical analysis of the rigid body. Restraints for plane systems. 2)Calculation of the restraints' reactions through static equations and kinematic chains method. 3)Static and kinematic analysis of plane structures. Closed systems. Auxiliary equations. 4)Stress resultants in plane and space problems. 5)Evaluation of the stress resultants through virtual work principle. 6)Thrust line. 7)Plane trusses. 8)Centroids and moments of inertia. 9)Symmetry and antisymmetry in structures. 10)Equation of bending beam. 11)Mohr's corollaries. 12)Kinematic method for straight axis beams. Composition of displacements and rotations. 13)Force and displacement methods for the resolution of non statically determined structures. Scheme of physical theories. 14)Constraint displacements. 15)Continuous beam. 16)Fundamental schemes for axial, shear and bending stiffnesses and compliances. 17)Kinematic general method: negligible effects. 18)Principle of virtual work: calculation of generalized displacements and rotations, resolution of non statically determined structures. 19)Clapeyron's, Betti's, Castigliano's and Menabrea's theorems on beams' strain energy. 20)Stability of elastic equilibrium: systems with concentrated elasticity. 21)Stability of elastic equilibrium: systems with diffused elasticity. Limits of validity for Euler's formula. Omega method. 22)Verification of resistance for axial force resultant, bending, shear force resultant and torsion. Composition of different internal force resultants.

STATICS OF CONCRETE
1)Strenght of material assumed as random variable: histogram and frequency polygon, probability density function and distribution function. 2)Methods for structural safety assessment. Mean, characteristic and design values for loads and resistance. 3)Verification and design problem of the rectangular cross-section for an axially loaded solid. 4)Bending: verification and project and mixed problem for rectangular cross-section. 5)Verification and design problem for T shaped section for a solid under bending moment. 6)Axial and bending combined loads: verification and mixed problem for the rectangular cross-section. 7)Introduction to verification and design with the limit states method.

Readings/Bibliography

• Slides and lecture notes
• VIOLA E., Scienza delle Costruzioni, voll. 1,3, Pitagora, Bologna.
• VIOLA E., Esercitazioni di Scienza delle Costruzioni, voll.1,2,  Pitagora, Bologna.

Teaching methods

The course syllabus is covered entirely during classes. Exercises will be solved in class in order to guide students in the resolution of specific structural mechanics problems, based on the knowledge acquired in class.

 

Assessment methods

Assessment is composed of a written and an oral exam. The written test consists on the resolution of simple exercises, similar to those presented during classes. The oral test is based on some questions aimed at assessing the student's knowledge of the topics covered during classes.

Teaching tools

In the classroom, transparencies and projected slides will be used.

Office hours

See the website of Erasmo Viola