Architecture

The course offers a first introduction to the structural analysis and design of constructions of architectural interest. Classes attempt to show how structural analysis can be thought as a sequence of three conceptual operations: 1) modeling (representation of reality by means of models) 2) computation (prevision of the structural response through the solution of equations that model the interaction between the construction and the environment) 3) interpretation (elaboration of the information gathered from calculations for the structural checks of existing structures or the structural design of new constructions) Introduction to Structural Analysis The modeling phase (geometry, internal actions, environment, materials). The computational phase (statics, kinematics and constitutive laws). The interpretation phase (structural check and design both for strength and deformability). Main concepts of kinematics Kinematics of a single material point. Rectilinear motion: velocity and acceleration. Rotation: angular velocity and acceleration. Brief introduction to the kinematics of discrete structures and of tridimensional continua. Displacement field, deformation gradient. Kinematics of one-dimensional structural models. Kinematic hypotheses for the determination of the behavior of cross-sections. Introduction to the Euler-Bernouilli and Timoshenko beam models. Kinematics of rigid motions. Representation of finite and infinitesimal rigid displacement fields. Degrees of freedom, translation, rotation, center of instantaneous rotation. Restraints and external kinematic compatibility Modeling of structural restraints. External kinematic problem. Restraint conditions and equations of kinematic compatibility. Analytic and graphic methods for the determination of infinitesimal rigid displacements of kinematically determined structures. Matrix representation of kinematic equation. Kinematic classification of structures. Strains and internal kinematic compatibility of beams Strain measures in beam models. Thermal strains. Equations of internal kinematic compatibility for beams. Beams without shear strains. Internal kinematic problem for beams. Computation of displacements induced by thermal strains in kinematically determined beams. Main concepts of dynamics and statics Dynamics of a single material point. Newton’s laws. Causes of translation and rotation. Brief introduction to the dynamics of discrete structures and of three-dimensional continua. Third law of dynamics. Internal forces. Euler’s laws. Stresses. Forces and moments. Systems of force. Equilibrium in deformable structures. Infinitesimal displacements. Moment of a single force about a point. Resultant force and moments of a system of of forces. Formulas for the transport of resultant moments. Equivalence and reduction of systems of force. Systems of spatially distributed forces. Constraint reactions and external equilibrium Internal and external reactions. External static problem. Number of solutions in terms of the statical determination. Free-body diagrams. Different methods for the computation of reactions in statically determined structures. Matrix representation of equilibrium equations. Duality with the kinematic problem. Reactions of statically indetermined structures. Stresses and internal equilibrium of beams Definition of stresses in beams. Distributed and concentrated loads. Internal equilibrium. Sign conventions on stresses. Internal static problem. Stresses diagrams. Various methods for the computation of stresses in statically determined beams. Materials behavior Experimental tests for the determination of the mechanical response of materials. Typical behaviors: elasticity, plasticity, viscosity. Stiffness, elastic limit, strengh, ductility. Brief introduction to the most common materials for architectural constructions: steel, concrete, masonry, wood. Constitutive laws for beams. Brief introduction to de-Saint-Venant problem. Axial, bending and shear constitutive law. Duality between statics and kinematics Duality between external problems. Energy, work, power. Virtual work theorem for beams. Use of the theorem for the computation of constraint reaction and of single displacement components in statically determined beams. The structural problem for planar beams The “method of displacements” for linearly elastic beams: general solution, boundary conditions, axial and transversal problems. Influence of thermal strains and of shear. Brief overview of the finite elements method. The “method of forces” for statically indetermined beams. Structural checks and design The sources of uncertainity in structural analysis. Methods of structural analysis. Characteristic values of strenghts and loads. Axial and bending strength of beams. Check for strength and deformability. Design for strength and deformability. Statically determined beams Structural behavior of typical examples of statically determined structures. Comparison of the stresses and of displacements. The influence of constraint settlements and of thermal strains. Statically indetermined beams Statical redundance. Applications of the “methods of forces”. Selection of the “principal system”. Computation of the statically indetermined unknowns. Application of the virtual work theorem. Effect of constraint settlements and of thermal strains. Influence of stiffness ratios on the distribution of stresses. Typical examples of statically indetermined structures

Knowledge of a few basic topics in mathematics: - main properties of powers of real numbers - algebraic operations with fractions - definition of function, graph of real.valued functions - geometrical interpretation of the concept of derivative of a real valued function - geometrical interpretation of the concept of definite integral of a real valued function - computation of derivatives and primitives of polynomial functions - measure of angles in radians - basic algebraic operations on vector in Euclidean spaces: sum, difference, scalar multiplication, norm - scalar product and cross product of vectors Knowledge of a few basic topics in physics: - units of measurement of the main SI physical quantities - multiples and sub-multiples of units and equivalences

D. Bernardini 'Introduzione alla Meccanica delle Strutture', Città Studi Edizioni, 2012

https://sites.google.com/uniroma1.it/db-did/meccanica-delle-strutture

https://sites.google.com/uniroma1.it/db-did/meccanica-delle-strutture

Prerequisites: thorough knowledge of the topics covered in basic courses in Mathematics. Theory of Vectors. Elements of classical mechanics. Kinematics of rigid systems: generalized displacement; kinematic definition of constraints (internal and external); algebraic study of kinematics of rigid systems. Statics of rigid systems: static definition of constraints (internal and external); cardinal equations of statics; static-kinematic duality; determination of constraint reactions; classification of systems of beams; line pressure; indefinite equations of equilibrium for plane beams; internal beam reactions and diagrams of characteristics of internal reactions; statically determinate systems of beams and trusses. More details can be found in: www.pcasini.it/disg/statica

Thorough knowledge of the topics covered in basic courses in Mathematics.

Paolo Casini, Marcello Vasta. Scienza delle Costruzioni (quarta edizione), DeAgostini Scuola-CittàStudi ed., ISBN 9788825174274 (2019) Additional information and teaching materials can be found at: www.pcasini.it/disg/statica

Class attendance is not mandatory but strongly recommended.

The learning process is verified during the course through exercises and written tests.

Frontal lessons and in-class exercises. Learning assessments with exercises on classroom, corrected in person. Videos regarding lessons and exercises available on the website www.pcasini.it/disg/statica under the 'video' section.

Linear Algebra Ambient space. Vector spaces of finite dimension. Operations among vector. Scalar product. Vector product. representation in an orthonormal basis. Mixed product. Systems of forces and couples Notion of force. Moment of a force about a point. Transport formula. Systems of forces, resultant, moment resultant. Elementary operations on forces and couples. Central axis. Kinematics and statics of rigid bodies Definition of rigid body. Kinematics of rigid bodies. Absolute and relative position. Absolute and relative velocity. Representation formula of rigid velocities. Infinitesimal displacement fields. External constraints. Kinematic and static characterization of rigid bodies. Statics of rigid bodies: cardinal equations. Power associated to systems of forces and couples. Principle of powers. Kinematic problem and kinematic classification: systems kinematically determined, indetermined, impossible, degenerate. Static problem and static classification: systems statically determined, indetermined, impossible, degenerate. Duality static-kinematic. Systems of rigid bodies Internal constraints. Methods of auxiliary equations for conditions of partial equilibrium: computing reactions of isostatic systems. Kinematic chains. Instantaneous center of rotation. Relative center of rotation. First and second theorem about centers of rotation. Diagrams of velocities/infinitesimal displacements. Method of powers for computing reactions in isostatic systems. Plane straight beams Beams: geometry, kinematics, applied loads, internal actions. Internal stress. Differential equilibrium equations. Systems of beams. Trusses. Elastic beams Axial deformations, of mechanical and thermal origin. Bending. Differential equation of elastica. Principle of virtual workings. Solution of hyperstatic structures. Principles of structural design.

1) P. Podio-Guidugli, Lezioni di Statica, Aracne, 2014. 2) P. Podio-Guidugli, Lezioni di Scienza delle Costruzioni, Aracne, 2012. 3) A. Luongo, A. Paolone, Meccanica delle strutture. Sistemi rigidi ad elasticità concentrata, Masson, 1997.

Lessons at the blackboard

In presence

Written and oral exams

Lessons at the blackboard