Programme

Introduction to Algebra and Vector and Tensor Calculus

Kinematics of Rigid Bodies

The Rigid Body Model
Rigid Displacements
General Formula for Infinitesimal Rigid Displacement
Scalar Representation
Planar Rigid Displacements
Systems of Rigid Bodies
Kinematic Characterization of Constraints
Definitions
Kinematic Characterization of External Constraints
Kinematic Characterization of Internal Constraints
Constraint Failures
The Kinematic Problem
Problem Statement
Analytical Kinematic Classification
Direct Kinematic Classification
Gradient Method for Solving the Kinematic Problem
Definitions. Kinematic Chains.
Statics of Rigid Bodies

Static Characterization of Constraints
Static Characterization of External Constraints
Static Characterization of Internal Constraints
The Static Problem
Cardinal Equations of Statics
Problem Statement
Static Classification
Static-Kinematic Duality
Beam Kinematics

Deformation Process
Displacements and Rotations
Displacement
Sectional Rotation
Hypothesis of Small Displacements
Boundary Conditions on Displacements and Rotations
External Constraints: Kinematic Characterization
Strain Measurements
Axial Deformation
Angular Shear
Bending Curvature
Implicit Congruence Equations
Timoshenko Model
Euler-Bernoulli Model
Kinematic Problem
Discontinuities in the Kinematic Problem
Static of Beams

Problem Statement
External Forces
Internal Actions
Indeterminate Equations of Equilibrium
Static Problem
Laws and Diagrams of Load Characteristics
Discontinuities in the Static Problem
General Rules for Tracing Load Characteristic Diagrams

Linear Elastic Relation for the One-Dimensional Beam
Axial Behavior
Flexural Behavior
Shear Behavior
Thermal Distortions
Uniform Temperature Variation
Butterfly Temperature Variation
Linear Temperature Variation
Constitutive Equations for the One-Dimensional Beam
Elastic Problem for the Beam

Problem Statement: Given and Unknown
Analytical Formulation:
Resolvent Equations
Solution
Euler-Bernoulli Model
Timoshenko Model
Solution Strategies
Systems of Beams
Method of Displacements: The Elastic Line

Elastic Line
Axial Problem
Flexural Problem: Euler-Bernoulli Model
Observations
Elastic Line in Systems of Beams
Kinematic and Static Performance of Internal Constraints
Identity of Virtual Works. Duality.

General Definitions
Work
Congruent System
Balanced System
External Virtual Work
Internal Virtual Work
Virtual Work Theorem
Calculation of Displacements and Rotations in Isostatic Structures
Method of Forces

Once Hyperstatic Systems
Multiple Times Hyperstatic Systems
Müller-Breslau Equations
Lattice Structures and Continuous Beams

Lattice Structures
Node Method
Ritter's Section Method
Continuous Beams: Three-Moment Equation
The Continuous Medium: Deformation Analysis

Deformation Process
Deformation Analysis in the Surroundings: Strain Tensor
Mechanical Interpretation of ε Components
Meaning of Diagonal Components εx, εy, εz
Meaning of Off-Diagonal Components γxy, γxz, γyz
Decomposition of the Deformation Process
Cubic Dilatation
Cauchy's Strain Formula - Principal Directions of Strain
Triaxial Strain State
Cylindrical Strain State
Principal Reference - Mohr's Circles
Congruence Equations
The Continuous Medium: Stress Analysis

Cauchy's Stress
Cauchy's Lemma
Decomposition of Cauchy's Stress Vector
Cauchy's Stress Formula
Indeterminate Equilibrium Equations
Stresses and Principal Directions
Principal Reference
Stress States
Isostatic Lines
Mean Stress, Deviatoric Stress, and Octahedral Stress
Mohr's Circles
Plane Stress State or Biaxial Stress
Purely Tangential Stress State
Uniaxial Stress State
Linear Elastic Relation

Isotropic Materials: Generalized Hooke's Law
Elastic Equilibrium Problem: Direct Formulation and Energetic Aspects

Saint-Venant's Problem

Problem Statement
Saint-Venant's Postulate
Simple and Composite Loads
Solution
Semi-Inverse Method
Stress State
Indeterminate Equilibrium Equations
Congruence and Constitutive Relation Equations
Static Equivalence
Centric Axial Force. Pure Bending

Centric Axial Force

Uniform Pure Bending
Pure Bending My
Deviated Bending. Torsion, Bending-Torsion

Uniform Deviated Bending

Torsion

Torsion in Circular Sections

Compact Circular Section

Hollow Circular Section

Torsion in Compact Sections of Any Shape

Hydrodynamic Analogy for Tangential Stresses
Thin Rectangular Section
Open Sections Composed of Thin Rectangles
Thin-Walled Sections: Bredt's Theory
Composite Thin Sections
Bending and Shear

Distribution of Normal Stresses
Distribution of Shear Stresses: Jourawsky's Approximate Treatment
Intuitive Considerations
Problem Equations
Jourawsky's Formula
Applicability of Jourawsky's Formula
Open Thin Sections
Thin Rectangular Section
Double-T Thin Section
U and H Thin Sections
Closed Thin Sections
Symmetric Box Section
Shear Straight Along y
Deviated Shear
Symmetric Compact Sections
Combined Shear and Torsion Loading
Shear Center
Shear and Torsional Tangential Stresses
Strength Criteria

Strength Criteria for Ductile Materials
Strength Criteria for Brittle Materials
Structural Instability Phenomenon

Stability Analysis in Rigid Beams with Elastic Constraints
Euler's Column
Stability Curves, Slenderness
Structural Verification

Verification of Beams in Operating Conditions
Extension of Saint-Venant's Theory
Strength Criteria for the Solid of Saint-Venant
Operational Procedure for Structural Verification
Geometry of Areas

Area and Centroid
Moments of Inertia
Transport Formulas (without Rotation Formulas)
Principal Moments of Inertia
Central Ellipse of Inertia
Notable Cases
Recurring Static Schemes

Cantilever
Simply Supported Beam
Fixed-Supported Beam
Two-End Fixed Beam
Continuous Beam
Frame"

Core Documentation

Hibbeler: Mechanics of Materials

Attendance

In the classroom

Type of evaluation

The written test consists of solving a series of problems aimed at verifying the ability to apply the concepts learned during the course. The oral examination is aimed at assessing the understanding of the theoretical foundations underlying problem-solving methods in the context of this discipline.