Portale di Ateneo - En.Unibs.it

The Total Potential Energy
Brief introduction to the Calculus of Variations with focus on Solid and Structural Mechanics. Functional definition and essential boundary conditions. Functional stationarity, natural boundary conditions, and Euler-Lagrange Equations. The Total Potential Energy for the Cauchy continuum and the Euler-Bernoulli beam.

Structural Theories for plane beams
Kinematics assumptions and determination of the balance equations from the minimisation of the Total Potential Energy. The Timoshenko beam. The Newmark theory applied to composite concrete-timber beams.

Introduction to the Finite Element Method
Methods for the discretisation of the linear elastic problem and direct minimisation of the Total Potential Energy. The Rayleigh-Ritz Method. The Finite Element Method. Interpolation and shape functions. Transformation from the parent to the real finite element. Isoparametric finite elements. Numerical integration: Gauss-Legendre method. Convergence criteria.

Limit Analysis of plane frames
Elastic-perfectly plastic constitutive law. The case of beams subject to axial force combined with bending moment. Pure axial elastoplastic behaviour. Pure bending elestoplastic behaviour. M-N interaction curves. Assumption of plastic hinge. Step-by-step analysis of perfectly plastic plane frames. The plastic collapse. Determination of the limit load. Static and kinematic theorems of Limit Analysis.

Introduction to the Stability Analysis of plane frames
Method based on the Total Potential Energy minimisation. Analysis of discrete problems under arbitrarily large displacements and strains. Second order theory. Methods for the analysis of stability of discrete problems. Determination of the critical load for continuous beams in the elastic range. Hints to more complex problems: analysis of frames; elastic-plastic material; imperfect or transversally loaded beams.