The student will be able to determine the constraint reactions and the diagrams of internal actions (normal stress, shear stress and bending moment), and to plot the deflection curve for any plane system of isostatic beams; to calculate the stresses in the beams on the basis of De Saint Venant theory; to apply the strength criteria for generic stress states; to investigate a slender bar subjected to a buckling load.

The student must know the kinematic, static and dynamic theory of the material point, the operations on vectors (sum, multiplication by a scalar, scalar product and vector product) and on matrices, the fundamentals of linear algebra and analytic/differential geometry. For functions of one variable, he/she must know the limits, derivatives, integrals, developments in Taylor series and solutions of differential equations with constant coefficients. For functions of several variables, he/she must know Taylor's rules of derivation, integrations and development in series.

STATICALLY DETERMINATE BEAM SYSTEMS: statics and kinematics, plane constraints, hypostatic systems, determination of constraint reactions with auxiliary equations and with the graphical method; internal beam reactions; indefinite equations of equilibrium for plane beams; trusses. ANALYSIS OF STRAIN AND STRESS: strain tensor; dilations and shearing strains; principal directions of strain; cubic dilation. Measures by extensometers. Stress tensor; principal directions of stress; plane stress condition; Mohr’s circle. Indefinite equations of equilibrium; boundary equations of equivalence; principle of virtual work for deformable bodies. ELASTIC CONSTITUTIVE LAW AND STRENGTH CRITERIA: experimental techniques to characterize the mechanical behavior of materials. Traction test. Linear elasticity; elastic potential; Young modulus and Poisson’s coefficient; problem of a linear elastic body: Clapeyron’s theorem; Betti’s reciprocal theorem; isotropy; Tresca’s and Von Mises’ strength criteria. GEOMETRY OF AREAS: centroid, static moment, moment of inertia, product of inertia, principal axes and moments of inertia. SAINT-VENANT PROBLEM: fundamental hypotheses; centered axial force; flexure; eccentric axial force and biaxial flexure; central cores of inertia; torsion; shearing force; beam strength analysis. CALCULUS OF ELASTIC DISPLACEMENTS AND SIMPLE STATICALLY UNDETERMINATE BEAM SYSTEMS: equation of the elastica; determination of elastic displacements; method of forces; equations of congruence written with the principle of virtual works; Simpson’s integration rule. ELASTIC INSTABILITY: compressed beams with different constrain conditions.

The course is based on lecture and pratical classess. Lectures are intended to present the theoretical basis of the topics; practical classess show the solutions of sample problems. One lecture is supposed to be held at the Laboratory Mastrlab of DISEG, where students can attend some experimental tests to assess material strength.

Notes, exercises, formularies will be downloadable from the website of the course. Suggested textbook: Carpinteri, A. (1997) Structural Mechanics: A Unified Approach, E. & F.N. Spon, London. -

Exam: Written test; Individual essay; Computer-based written test in class using POLITO platform;

The final examination assesses the student's theoretical understanding and practical problem-solving skills in Structural Mechanics. The exam is held in-person over two separate days/sessions (one dedicated to practical problem-solving and one to theoretical concepts). The final grade is expressed on a scale of 30. To pass the course, students must obtain a minimum grade of 16/30 in both parts and achieve an overall weighted average of at least 18/30. The final grade is calculated as: Final Mark = 0.6 × Written Practical Exam Mark + 0.4 × Theory Mark ------------------------- # Part 1: Written Practical Exam (Problems on Paper) Total duration: 3 hours and 30 minutes (allocated as 1h 10m per problem). Contents: The exam consists of three independent problems: • Problem 1 – Statically Determinate System of Beams (Isostatics). • Problem 2 – Statically Indeterminate System of Beams (Hyperstatics). • Problem 3 – Geometric Properties of Cross-Sections and Saint-Venant Strength Verification. Exam Stages and Midterm Exemption: The exam session is split into two distinct stages: • Stage 1 (2 hours and 20 minutes): Solution of Problem 2 (Hyperstatics) and Problem 3 (Cross-Section & Saint-Venant). • Stage 2 (1 hour and 10 minutes): Solution of Problem 1 (Isostatics). Midterm Test Waiver: During the semester, students have the option to take an intermediate test on Isostatics (2 problems in 2h 20m). Students who achieve a passing mark (at least 16/30) in this midterm test are exempt from Stage 2 during the final written exam; their midterm grade will be transferred directly to replace Problem 1. Permitted Materials: This exam is NOT open-book. Students are not permitted to bring external textbooks, personal exercise notebooks, or bound volumes. Scientific non-programmable calculators are permitted. Electronic communication devices are strictly forbidden. Official Reference Sheets: The teaching team provides a standardized 5-page template containing integral tables and geometric properties of standard cross-sections. Personal Handwritten Notes: Students are authorized to use the blank back sides of these 5 official pages to handwrite personal notes, synthesis formulas, and reference diagrams. No other loose papers or attachments are allowed. Minimum Pass Requirement: The combined score of the written practical part must be at least 16/30. Experimental Laboratory Bonus: During the semester, an in-person experimental laboratory activity is offered. Students who participate and successfully complete the laboratory assignment (a set of 10 questions) can earn up to +3 points (3/30) added directly as a bonus to the practical written exam mark. ------------------------- # Part 2: Theoretical Examination (Online Quiz + Open Question) The theoretical exam evaluates conceptual rigor, physical interpretations of structural behavior, and mathematical derivations. It is conducted in a single session divided into two consecutive parts: Stage 1: • Online Multiple-Choice Quiz (POLITO Platform - 30 minutes) • Consists of 15 multiple-choice questions (4 options per question). Scoring criteria: o Correct answer: +2.0 points (+100%) o Wrong answer: -0.5 points (-25% penalty) o Blank / Unanswered: 0.0 points Stage 2: • Open Conceptual Question (Paper-based - 15 minutes) Immediately following the online quiz, students are given one comprehensive open question to be answered by hand on paper. This question assesses in-depth theoretical understanding and derivations. It is evaluated up to +5.0 additional points, acting as a bonus added directly to the quiz score (allowing a theoretical maximum of 35/30, functioning as an aid to compensate for penalties in the quiz). Permitted Materials: The theoretical exam is strictly closed-book. No consultation of books, notebooks, formula sheets, or electronic devices is allowed. Minimum Pass Requirement: The combined score of the theoretical part must be at least 16/30. Exam results and final grade proposals will be published directly on the Teaching Portal (“Portale della Didattica”).

In addition to the message sent by the online system, students with disabilities or Specific Learning Disorders (SLD) are invited to directly inform the professor in charge of the course about the special arrangements for the exam that have been agreed with the Special Needs Unit. The professor has to be informed at least one week before the beginning of the examination session in order to provide students with the most suitable arrangements for each specific type of exam.