3 Equilibrium States

· Stable equilibrium

· Neutral equilibrium: limit of stability

· Unstable equilibrium: static & dynamic instability

Governing equations

Solution for instability criterion

Constitutive Equations

Incorporate boundary conditions: Essential & Natural

Kinematics

Identify the system’s degrees of freedom

Buckling

Bifurcation of displacement under compression & bending

Analytical

Combining all constitutive, kinematic & governing equations, solve analytically for buckling load, mode shapes, etc.

Influencing Factors

· Structure, member or beam-column dimensions

· End conditions

· Kinematics: degrees of freedom

· Material properties

Energy Methods

· Single d.o.f. / rigid systems: Theorem of Stationary P.E.

· Multiple d.o.f. / flexible systems: Ritz Method

Lapunov’s Stability Criterion

When an equilibrium condition that is being analyzed for stability continues to deform without increase in load or with decrease in load, the system is unstable dynamically.

D’Alambert’s Principle

System may be idealized in to lumped masses with inertia forces acting opposite in direction to the dynamic displacements.

Discrete systems

Distributed mass creates complexity of problem formulation & calculations. Lumped masses at discrete locations are idealized for simplicity but retaining critical components of system’s dynamics

Continuous systems

Mass is distributed realistically, increasing computation but improving accuracy

Equations of motion

Using free-body diagrams (FBDs), N equations are defined for N discrete masses - Linear system

Continuous systems require integration techniques - Governing equations with analytical or numerical Gaussian integration

Equation components

· Inertia forces

· Damping forces

· Stiffness

· Forcing function

Solution techniques for single d.o.f.

· General D.E solution

· w²=K/M

· x=C/C_cr=C/2mw

Solution techniques for multiple d.o.f.

· Matrix operation, Cramer’s rule

· Eigenvalue (w) & eigenvectors (f)

· D.E. solution for homogeneous

· Dirac function

· Duhamel integral for steady state

· Orthogonality

· Mode Transposition Method - X(t) = {Y }{q(t)}

Total stress

Dynamic stress + static stress

Structural Dynamics

· Google

· Directhit

· Metacrawler

Structural Stability

· Google

· Metacrawler