At the end of this course it is required that student easily handle some typical tools of analytical methods for the static behavior prediction at least of beamlike structures. Fundamental goals of the discipline are: •a comprehensive knowledge, understanding and distinguishing of mechanical properties and strength of ductile and brittle materials; linear and nonlinear elastic behaviour conditions; concepts of stress, strain, displacement and rotation, described in both principal and non principal reference frames; static failure criteria and safety factor; geometrical properties of plane figures, interpreted as cross section of beams; theory of beam and relations between stress and load for each static behavior foreseen by De St Venant. In addition, student will get acquainted with the performing of fatigue verification of structures subjected to time-varying uniaxial loading conditions. •providing some skills as the basic tools to: 1) simplify to a level of elementary scheme the layout of a beamlike mechanical component and perform a complete static analysis; 2) evaluate the degree of indeterminacy of the system; 3) calculate reaction forces of statically determinate structures; 4) calculate the internal stress resultant diagrams, stresses, strains, displacements and rotations of each cross section of one-dimensional elements; 5) identify the critical points of the structure and compute the equivalent stress to be compared to the strength of material or even to buckling threshold; 6) verify structures subject to uniaxial fatigue.

Some typical mathematical tools (study of functions, computation of derivatives and integrals, matrix algebra, solution of eigenvalue/eigenvectors problems) and physics (basic concepts of kinematics and statics) and some basics of materials sciences (materials classes and properties).

Topics dealt within this course are herein listed. 1. Statics : Basic concepts of static behavior of structures (force, moment, rigid and deformable bodies), loading conditions, constraints, static and kinematic determinacy, equilibrium conditions and equations. Computations of reactions, internal forces, diagrams. Beams, bars, trusses. Outlines of Virtual Work Principle and application to undetermined structures. 2. Stress : Stress vector, tensor, components. Principal stresses and direction, related computation. Mohr circles. Equivalent stress definition and computation. 3. Strain : Rigid body motion and strain definition in elastic body. Strain components, principal strain and direction. Stress-strain relations, Hooke’s law. Elastic properties of materials. Elastic energy storage. 4. Strength of materials : Tensile test, material behaviour and properties. Elastic coefficients. Yielding phenomenon, brittle and ductile materials. Safety factors in statics. 5. Beam theory : De Saint Venant principle, beam definition, loading conditions, axial, flexural, shear, torsional behaviors. Approximated solutions for torsion of rectangular cross sections, multiple rectangles and thin walled structures. Computation of stresses, strains, displacements and rotations. Shear centre. Coupled behavior. Buckling and elastic instability. 6. Uniaxial mechanical fatigue at high number of cycles (HCF) : fundamental parameters, SN diagram, effect of the mean stress (Haigh, Goodman-Smith, Ros and Master diagrams). From material to component: surface finishing effect, load type effect, dimension effect and notch effect. Component life and fatigue safety factor. Fatigue with varying amplitude stress (Palmgren-Miner cumulative damage rule).

This course is organized in two parts. Lectures will give a straight presentation of relevant topics to be studied to perform a complete structural static analysis of some mechanical structures. Practice hours will be offered to solve examples, numerical exercises and practical cases and even an exam simulation.

Textbooks: Some notes directly taken from the classes will be shared with students through the website. Theoretical aspects presented during the lectures can be found on the following textbooks: 1.D.Gross, W.Hauger, J.Schroder, W.A.Wall, N. Rajapakse - "Engineering Mechanics 1: Statics", Springer. 2.V. Da Silva - "Mechanics and strength of materials", Springer. 3.J.D. Renton, "Applied elasticity : matrix and tensor analysis of elastic continua", Chichester, Horwood, New York: Wiley, 1987 4.J.A. Bannantine, J.J. Comer, J.L. Handrock, “Fundamentals of metal fatigue analysis”, Prentice Hall, Englewood Cliffs, 1990. or evenly, but the textbook is just partially dedicated to the above topics: 1.J. Beer, S.Johnston - "Solid mechanics", McGraw-Hill.

Exam: Compulsory oral exam; Computer-based written test using the PoliTo platform;

Learning outcomes: The written part focuses on exercise related to the calculation of reactions and internal actions of either a beam assembly or a truss; to the stress field in beams sections or on stress-strain relationships and Mohr circles; to the static verification at the maximum stressed points; to fatigue verification of structures subjected to time-varying uniaxial loading conditions. The oral part focuses on demonstrations and theoretical concepts that have been taught in the course. Exam type: Remote exam Type and number of tests: Written exam (Test 1, mandatory) and oral exam (Test 2, mandatory). Written exam: Test 1 is given through the EXAM platform by Politecnico with automated proctoring through the software ‘Respondus’. Students must follow these rules: Test 1 lasts for 2 hours. Students can consult notes or other didactic material. Test 1 consists of 3 exercises that can be solved independently (i.e., the solution of one exercise does not depend on the solutions of the previous problems). Exercise #1 focuses on the calculation of reactions and internal actions of either a beam assembly or a truss. Exercise #2 focuses either on the stress field in beams sections or on stress-strain relationships and Mohr circles. Static verification at the maximum stressed points might be asked. Exercise #3 focuses on fatigue verification of structures subjected to time-varying uniaxial loading conditions. Each exercise has a limited solution time. Students must answer to the questions related to each exercise within the provided solution time. Students are allowed to take Test 2 only if they get a minimum mark for Test 1 equal to 18/30. The maximum mark for Test 1 is 30/30. Oral exam: Test 2 is given through Virtual Classroom and the whole exam is recorded. The rules for the Test 2 are: Test 2 lasts for 45 minutes; Test 2 is a closed-book test; Students that attended Dexpilab must answer to the first 2 questions (15 points each); Students that did not attend Dexpilab must answer to all the 3 questions (10 points each). Final mark: The final mark is computed by taking into account the marks in Test 1 and in Test 2, according to the following rule: F=2/3 A+1/3 B, where F denotes the final mark, A is the mark in Test 1 and B is the mark in Test 2. The final mark F is rounded to the nearest integer value. The minimum mark for passing the exam is 18/30. In case of A=30 and B=30, the final mark F is equal to 30L.

Exam: Compulsory oral exam; Computer-based written test using the PoliTo platform;

Learning outcomes: The written part focuses on exercise related to the calculation of reactions and internal actions of either a beam assembly or a truss; to the stress field in beams sections or on stress-strain relationships and Mohr circles; to the static verification at the maximum stressed points; to fatigue verification of structures subjected to time-varying uniaxial loading conditions. The oral part focuses on demonstrations and theoretical concepts that have been taught in the course. Exam type: Remote exam Type and number of tests: Written exam (Test 1, mandatory) and oral exam (Test 2, mandatory). Written exam: Test 1 is given through the EXAM platform by Politecnico with automated proctoring through the software ‘Respondus’. Students must follow these rules: Test 1 lasts for 2 hours. Students can consult notes or other didactic material. Test 1 consists of 3 exercises that can be solved independently (i.e., the solution of one exercise does not depend on the solutions of the previous problems). Exercise #1 focuses on the calculation of reactions and internal actions of either a beam assembly or a truss. Exercise #2 focuses either on the stress field in beams sections or on stress-strain relationships and Mohr circles. Static verification at the maximum stressed points might be asked. Exercise #3 focuses on fatigue verification of structures subjected to time-varying uniaxial loading conditions. Each exercise has a limited solution time. Students must answer to the questions related to each exercise within the provided solution time. Students are allowed to take Test 2 only if they get a minimum mark for Test 1 equal to 18/30. The maximum mark for Test 1 is 30/30. Oral exam: Test 2 is given through Virtual Classroom and the whole exam is recorded. The rules for the Test 2 are: Test 2 lasts for 45 minutes; Test 2 is a closed-book test; Students that attended Dexpilab must answer to the first 2 questions (15 points each); Students that did not attend Dexpilab must answer to all the 3 questions (10 points each). Final mark: The final mark is computed by taking into account the marks in Test 1 and in Test 2, according to the following rule: F=2/3 A+1/3 B, where F denotes the final mark, A is the mark in Test 1 and B is the mark in Test 2. The final mark F is rounded to the nearest integer value. The minimum mark for passing the exam is 18/30. In case of A=30 and B=30, the final mark F is equal to 30L.