A trabecular bone consists of trabeculae whose mechanical properties differ significantly from the surrounding marrow, therefore an ultrasonic wave propagating within the bone structure is strongly scattered. The aim of this paper was to evaluate the contribution of the first, second and higher order scattering (multiple scattering) into the total scattering of ultrasound in a trabecular bone. The scattering due to the interconnections between thick trabeculae, usually neglected in trabecular bone models, has been also studied. The basic element in our model of the trabecular bone was an elastic cylinder with a various finite-length and diameter as well as orientation. The applied model was taking into account variation of both, elements size and their spatial configuration. The field scattered on the bone model was evaluated by solving numerically the integral form of the generalized Sturm-Liouville equation describing a scalar wave in inhomogeneous and lossy media. For the scattered fields calculated numerically the effective cross-sections were determined. The influence of absorption on the scattering coefficients was demonstrated. The results allowed to conclude that within the frequency range from 0.5 to 1.5 MHz contribution of the second order scattering to the effective backscattering cross-section is at least 500 times lower than that due to the first order scattering. It was noticed that for a frequency higher than 1.5 MHz fast growth of the backscattering (reflection) coefficients, calculated for the second order scattering, occurs.

The paper presents the method of probabilistic optimisation of load bearing capacity and reliability of statically indeterminate bar structures, and of coupling of members in kinematically admissible failure mechanisms (KAFM), which contain minimal critical sets of elements (MCSE). The latter are characterised by the fact that if only a single element is operational, the whole set is operational too. A method of increasing load bearing capacity and reliability of KAFM built from bars dimensioned in accordance with the code is presented. The paper also shows estimation of load bearing capacity and reliability of KAFM of the optimised structures containing elastic-plastic bars with quasi-brittle connections with nodes. The necessity of increasing connection of load bearing capacity and reliability in relation to bar reliability in order to prevent bars from being excluded from MCSE due to connection fracture is estimated.

The paper provides a solution to the problem of dimensioning decisive bars on the basis of the conditions of meeting the recommended reliability classes [9] of statically determinate structures composed of n members. A theorem was formulated:if a statically determinate structure composed of n decisive members is to attain the reliability greater than, or equal to, the recommended relia-bility p = 1 – q, it is necessary and suffi cient that the damage frequency sum qᵢ of decisive members is smaller than the admissible damage frequency q of thestructure: ∑qᵢ < q. On the basis of this theorem, s coeffi cients that recommend increase of the load bearing capacity of the decisive bars in a statically determinate structure constructed in order to meet the recommended class [9] of the structure reliability, are estimated and presented in a tabular form.