Study Notes on Axial Members and Beams

Overview of Axial Members and Beams

• Focus on analysis under axial loading, beams, and frames

Truss vs Beam

• Truss: Only axial load and motion; no moments

• Beam: Carries moments and rotation; can have shear motion

Key Concepts

• Post-Processing Strain: Calculate strain from displacement in 1-D problems

• ANSYS Analysis Types: Static Structural, Modal, Harmonic, Transient Dynamic, Spectrum, Buckling

Displacement Equations

• Nodal displacements expressed in global and local coordinates

• Rotation and displacement relationships derived using transformation matrices

Minimum Total Potential Energy

• Energy formulation used in finite element models; relates external work to strain energy

• For stable systems, the equilibrium position minimizes total potential energy

Members Under Axial Loading

• Elements discretized into nodes; effects of axial loading assessed through strain energy

Shape Functions in FEA

• Interpolates solution between discrete mesh node values; typically low order polynomials

• Shape functions exhibit unique properties: equals 1 at its own node, 0 at neighboring nodes

Beam Analysis

• Beams are subjected to transverse loading causing bending; analyzed through neutral axis deflection

• Governing equations relate deflection to internal bending moment and shear

Finite Element Formulation of Beams

• Beam elements consist of nodes with vertical displacement and rotation angles represented by polynomials

• Stiffness matrix derived using specific formulas; essential components from properties E, L

Example Analysis

• Deflection calculations require applying boundary conditions and evaluating shape functions at specific points

• Example of a cantilevered beam illustrates application of loading and resulting displacements