Nonlinear rigid-plastic analysis of stiffened plates under blast loads

The large ductile deformation response of stiffened plates subjected to blast loads is investigated and simplified methods of analysis of such response are developed. Simplification is derived from modelling stiffened plates as singly symmetric beams or as grillages thereof. These beams are further assumed to behave in a rigid, perfectly plastic manner and to have piecewise linear bending moment-axial force capacity interaction relations, otherwise known as yield curves. A blast loaded, one-way stiffened plate is modelled as a singly symmetric beam comprised of one stiffener and its tributary plating, and subjected to a uniformly distributed line load. For a stiffened plate having edges fully restrained against rotations and translations, both transverse and in-plane, use of the piecewise linear yield curve divides the response of the beam model into two distinct phases: an initial small displacement phase wherein the beam responds as a plastic hinge mechanism, and a final large displacement phase wherein the beam responds as a plastic string. If the line load is restricted to be a blast-type pulse, such response is governed by linear differential equations and so may be solved in closed form. Examples of a one-way stiffened plate subjected to various blast-type pulses demonstrate good agreement between the present rigid-plastic formulation and elastic-plastic beam finite element and finite strip solutions. The response of a one-way stiffened plate is alternatively analysed by approximating it as a sequence of instantaneous mode responses. An instantaneous mode is analogous to a normal mode of linear vibration, but because of system nonlinearity exists for only the instant and deformed configuration considered. The instantaneous mode shapes are determined by an extremum principle which maximizes the rate of change of the stiffened plate's kinetic energy. This approximate rigid-plastic response is not solved in closed form but rather by a semi-analytical time-stepping algorithm. Instantaneous mode solutions compare very well with the closed-form results. The instantaneous mode analysis is extended to the case of two-way stiffened plates, which are modelled by grillages of singly symmetric beams. For two examples of blast loaded two-way stiffened plates, instantaneous mode solutions are compared to results from super finite element analyses. In one of these examples the comparison between analyses is extremely good; in the other, although the magnitudes of displacement response differ between the analyses, the predicted durations and mechanisms of response are in agreement. Incomplete fixity of a stiffened plate's edges is accounted for in the beam and grillage models by way of rigid-plastic links connecting the beams to their rigid supports. Like the beams, these links are assumed to have piecewise linear yield curves, but with reduced bending moment and axial force capacities. The instantaneous mode solution is modified accordingly, and its results again compare well with those of beam finite element analyses. Modifications to the closed-form and instantaneous mode solutions to account for strain rate sensitivity of the panel material are presented. In the closed-form solution, such modification takes the form of an effective dynamic yield stress to be used throughout the rigid-plastic analysis. In the time-stepping instantaneous mode solution, a dynamic yield stress is calculated at each time step and used within that time step only. With these modifications in place, the responses of rate-sensitive one-way stiffened plates predicted by the present analyses once again compare well with finite element and finite strip solutions.Applied Science, Faculty ofCivil Engineering, Department ofGraduat

Finite deflection dynamic response of axially restrained beams

The deformation response of symmetrically supported, axially restrained beams subjected to uniformly distributed pulse loads is studied herein, leading to the development of an analytical procedure to predict the character and magnitude of such response. The procedure is valid for beams of any singly symmetric or doubly symmetric cross-section, and is based upon the assumption that the beam material can be approximated as behaving in a rigid-perfectly plastic manner. The governing equations of motion are derived from variational statements consisting of the principle of virtual work and d'Alembert's principle, and include the effects of finite geometry changes. From the static analysis of axially restrained beams it is found that the yield curve of a beam section may be replaced by a linear approximation thereof to obtain a good estimate of the beam's load capacity. Incorporating the linear yield curve approximation in a dynamic analysis of an axially restrained beam results in the uncoupling of the response into two distinct phases — an initial small deflection phase in which the beam retains bending resistance and deforms as a mechanism formed by plastic hinges, and a subsequent large deflection phase in which the beam has no bending resistance and deforms as a plastic string. The results of such an analysis for a rectangular beam subjected to a rectangular load pulse compare well with the results of a previous solution which used the true quadratic yield curve. The linear yield curve approximation further results in linear differential equations of motion, and the response to load pulses of general load-time history may be solved in closed form. Blast-type pulses of varying shape are found to induce significantly different permanent deflections in a beam than a rectangular pulse. On the other hand, the effect of finite rise time of the pulse's load intensity is found to be small if the the rise time is less than about twenty to thirty percent of the pulse duration. A procedure developed by CK. Youngdahl is used to obtain rough estimates of the permanent deformation response by converting a pulse of triangular shape to an "effective" rectangular pulse. These estimates compare well with results obtained by the complete analysis of a triangular pulse developed herein. The use of Youngdahl's procedure combined with the analysis of a rectangular pulse developed herein can provide a quick, simple solution to the permanent deformation of a dynamically loaded beam which is amenable to hand computation.Applied Science, Faculty ofCivil Engineering, Department ofGraduat