AIMS AND LEARNING OUTCOMES

The main objective of the course is to provide methods of understanding the behavior of structures made up of rods, beams, frames, trusses through quantitative and qualitative descriptions according to a methodological process already introduced in previous courses, but that is extended here to hyperstatic structures including new concepts concerning the dynamic and seismic behavior of constructions from the contemporary Architecture. The analysis methodologies are extended to the elastic-plastic behavior and to the limit strength up to the structural collapse according to an approach useful to the introduction to the method of limit states. The aforementioned objectives will be pursued through the identification and adoption of quantitative structural models that are the basis of structural design. The concept of model is intended as a rigorous conceptual tool of a mathematical nature capable of correlating causes with effects, forces applied with displacements and stresses. The Student will reach the awareness that the representative structural model of the building and its structure must be obtained through simplifying hypotheses in the mechanical description of the empirical behavior of structural elements and structural systems.

At the end of the course Students should be able to:

• Analyze the equilibrium of restrained rigid bodies, beams, frames, trusses, and their classification as statically determinate/indeterminate or statically impossible systems;
• Determine the reaction forces in statically determinate beams, frames and trusses together with the representation of the free body diagram.  Determine the axial and shearing forces and bending moment at any section and construct the diagrams of internal forces for statically determinate beams, frames with internal partial releases and to explain the relationship between them. Determine the axial forces in statically determinate trusses;
• Determine the deflection and rotation at a section of elastic beams and frames due to applied loads, thermal loads and support settlement; explain curvature and how it changes in an elastic beam when bending changes; understand the application of the equation of virtual work; explain the deformed configuration of beams and frames on account of the curvature within it;
• Solve statically indeterminate structures through both the force and the displacement method to obtain reactions, internal forces, displacements and qualitative deformation of the structure. Evaluate the internal forces and reactions in single and two stories frames under vertical or horizontal forces.
• Formulate the equation of horizontal motion of a single story frame and description of free vibration and introduction of the concept of fundamental period of vibration of a structure. Evaluation of sinusoidal forced vibration induced by base ground sinusoidal motion. Response spectrum analysis of simple one dof structures.
• Analyze the elastic -plastic response of simple structures; flexural elastic-plastic response of simple beams; the concept of plastic hinge and the definition of plastic limit load; fundamentals of Limit Analysis of elastic-plastic structures; Upper and lower bound theorem and the Safe theorem as a base of the limit state design. The plastic hinge at section under axial force and bending moment; limit state domain in case of n-M interaction.
• Introduction to Computer-aided structural design; introduction to the SAP program through examples of different structural typologies. Explanation of the potential of the method and discussion on some limits depending on potential errors by users.
• All the problems / exercises developed in the classroom are discussed with specific attention to the functional purposes of the construction, to the typological and morphological choices that derive from it. This is in order to enrich the student's ability to understand the relationships between form, functionality and resistance of the construction.

TEACHING METHODS

The course is given through lectures explaining the basic principles and theory of the discipline. Exercises are focused on practical topics presented in lectures.

This course is delivered in Italian. The final oral exam may be given in English. The students who would like to be examined in English must inform the teacher.

SYLLABUS/CONTENT

Introduction and Aims of the Course

Statically determinate beams, frames and trusses

The structural model of straight beam; generalized applied forces; supports and reactions; equilibrium equation of the in-plane loaded beam; static discussion on the efficiency of the supports; evaluation of the reaction forces;

Internal forces in the beam: axial, shearing forces and bending moment; the diagrams of the internal forces; local equation of equilibrium; plane frames and internal releases; static equation of the release; multiple connected structures; plane trusses: the method of joints and the method of sections (Ritter);

The virtual work theorem and it application to evaluate displacements and rotations in structures ;  thermal effects on the deformation of beams; qualitative description of the deformed configuration of beams and frames from the diagram of the curvature; evaluation of displacements in plane trusses;

How to simplify the structural analysis taking into account the structural symmetry and the principle of superposition for linear systems.

Statically indeterminate structures

Evaluation of the grade of redundancy of frames and trusses; the force method to solve in-plane loaded statically indeterminate structures; the equations of Muller-Breslau; the effects of thermal variation and support settlement on the statics of redundant beams, frames and trusses; symmetric structures. The displacement method to solve statically indeterminate frames, Application to single and two-story frames under vertical and horizontal forces. The shear-type structural model.

Introduction to the dynamic analysis and seismic response of simple structures

Formulation of the equation of motion of the shear-type one story frame; Equations of motion and initial conditions; free vibrations and period of vibration of the structure; forced vibrations induced by horizontal sinusoidal oscillations of the foundation; Seismic action; response spectrum; evaluation of the maximum stresses and displacements through the response spectrum technique.

Introduction to the limit analysis of elasto-plastic structures

Elasto-plastic response of simple structures as the applied forces increase; Moment-curvature response of a perfectly elastic-plastic rectangular section; Incremental elasto-plastic analysis; Concept of plastic hinge for beams; Limit state and ultimate collapse load; direct determination of the collapse load; limit analysis of simple framed structures; Limit Analysis Theorems of elasto-plastic structures with Examples. The static theorem and the limit state method. Evaluation of the N-M interaction limit states.

Introduction to aided structural design

Presentation of the SAP 2023 aided structural design environment for the structural analysis of framed structures. Classroom development of structural models in SAP environment and determination of the structural response to environmental and anthropogenic actions.

RECOMMENDED READING/BIBLIOGRAPHY

L. Gambarotta, L. Nunziante, A. Tralli. Scienza delle Costruzioni, 3a Edizione, McGraw-Hill, 2011.