E-raamat: Convex Models of Uncertainty in Applied Mechanics

1. Probabilistic Modelling: Pros and Cons. Preliminary considerations.
Probabilistic modelling in mechanics. Reliability of structures. Sensitivity
of failure probability. Some quotations on the limitations of probabilistic
methods.
2. Mathematics of Convexity. Convexity and Uncertainty. What is
convexity? Geometric convexity in the Euclidean plane. Algebraic convexity in
Euclidean space. Convexity in function spaces. Set-convexity and
function-convexity. The structure of convex sets. Extreme points and convex
hulls. Extrema of linear functions on convex sets. Hyperplane separation of
convex sets. Convex models.
3. Uncertain Excitations. Introductory examples.
The massless damped spring. Excitation sets. Maximum responses. Measurement
optimization. Vehicle vibration. Introduction. The vehicle model. Uniformly
bounded substrate profiles. Extremal responses on uniformly bounded
substrates. Duration of acceleration excursions on uniformly bounded
substrates. Substrate profiles with bounded slopes. Isochronous obstacles.
Solution of the Euler-Lagrange equations. Seismic excitation. Vibration
measurements. Introduction. Damped vibrations: full measurement. Example:
2-dimensional measurement. Damped vibrations: partial measurement. Transient
vibrational acceleration.
4. Geometric Imperfections. Dynamics of thin bars.
Introduction. Analytical formulation. Maximum deflection. Duration above a
threshold. Maximum integral displacements. Impact loading of thin shells.
Introduction. Basic equations. Extremal displacement. Numerical example.
Buckling of thin shells. Introduction. Bounded Fourier coefficients:
first-order analysis. Bounded Fourier coefficients: second-order analysis.
Uniform bounds on imperfections. Envelope bounds on imperfections. Estimates
of the knockdown factor. First and second-order analyses.
5. Concluding
Remarks. Bibliography. Index.