“Structural Analysis” course by Prof. Carlo Poggi - Academic year 2019-20 .
Tutor : Ing. Angelo Calabrese
Lectures - Theoretical items
Fundamentals of Structural Mechanics
The deformation and stress tensors – Tensor components and principal values
Representation of 2D stresses.
Elastic Constitutive Theory.
Elastic moduli and Limits of elasticity. Strength criteria.
The theory of beams
Kinematic hypothesis and models
Constitutive relations in terms of stress resultants
The Bernoulli-Euler Beam model
The effects of combined stresses (axial loads, bending moment, shear stresses and torsional moments)
The theory of plates – Introduction
Kinematic hypothesis and models
Equation of Equilibrium
Constitutive equations for resultants – The boundary conditions
The Kirchhoff-love plate equations
Introduction to the Finite Element Method for Structural Analysis
Analytical and numerical methods. What is a Finite Element
Bar structures . Matrix analysis of bar structures. Direct assembly of the global stiffness matrix.
Equilibrium equations using the principle of virtual work and the total potential energy principle.
1D finite elements for axially loaded trusses with constant cross sections.
Beam elements. Shape functions. Stiffness matrix.
Plate elements. Shape functions and stiffness matrix.
Static Stability of beams and frames
Introduction to geometric nonlinearities. Virtual work and Energy functionals
Bifurcation of geometrically perfect systems.
Nonlinear beam theory. Inplane buckling of axially compressed columns. The effects of imperfections.
The use of Finite Element models for buckling analysis of beams and frames
The geometric stiffness matrix.
Symmetric and asymmetric buckling modes of frames.
Layout of the final technical report
The students are expected to deliver a report with the calculations performed using FE models as suggested during the classes. The following items should be presented and discussed.
The proposed steel frame must be studied using finite element models.
The steel frame may be considered as part of a commercial building (for example supermarket, office, public building as schools or libraries) and the loads may be determined on this basis.
1. Geometry of the frame and materials
• Only the geometry of the frame is provided.
• The geometries of the slabs, roof, external and internal walls are not provided and must be defined by the students.
• The bracings can be assumed of rectangular section (flat plates) with dimensions that must be defined. See the picture of the proposed frame on the slides provided to design the bracing.
• The proposed geometry can be commented at the end of the analysis and improvements or optimizations can be suggested.
• Materials : report the properties of the materials used for the frame, roof, slabs, internal and external walls.
• Describe the building and report other information that can be useful for the structural design (stairs, elevator, bracing, roof).
2. FE models
Geometries of the FE models adopted :
• one 3D full model of the steel frame
• two 2D models of one frame in orthogonal directions (2Da longer side, 2Db)
• Bracings : must be modelled as trusses.
• Create two other models modifying 2Da or 2Db increasing the level of the first floor and therefore the length of the columns. This is reasonable in the hypothesis that the destination of the building is different. For example if the original level was 4m, consider 6m and 8m.
3. Static analysis using FE
Evaluation of loads : dead and live loads, wind and snow. Seismic actions are not considered at this stage.
• Static analysis of the frame with rigid joints considering different combinations of the live loads. Displacements and internal actions must be analyzed and compared.
Strength analysis of the most significant sections. Calculations and comments.
• Possible changes of the frame geometry and optimization of the sections (as a possible reduction) may be proposed on the basis of the obtained results.
• Compare the results of a 2D model with and without bracing when an horizontal load is applied.
• Comparison of the results obtained with 2D and 3D models and evaluate the convenience of using 2D instead of 3D models.
4. Buckling analysis of columns
FE models of a single column subject to axial compression. The geometry of the column must be similar to the ground columns.
• Evaluation of bifurcation loads.
• Parametric analysis of the effects of geometric imperfections (similar to the first and second buckling modes). Comment the results.
• At least two different boundary conditions must be assumed on the basis of the frame geometry.
• Analysis of a columns subject to compression and bending actions.
5. Geometrically nonlinear analysis of frames
• Effects of the geometric imperfections. Analyze the basic frame with and without geometric imperfections and comment the behavior.
• Change of geometry : modify one of the two 2D geometries. Two models can be obtained changing the column lengths of the columns at ground level. Factors of 1.5 or 2.0 can be adopted. (ex. 2Da6 and 2Da8)
The initial geometric imperfections and horizontal loads should be varied accordingly.
The results in terms of generalized stresses and displacements should be commented.