Elasto-plastik zemine oturan plakların elasto-plastik analizi

Abstract

In this study, elasto-plastic analysis of rotationally symmetric reinforced concrete plates under concentrated loads at the middle resting on a Generalized Winkler Foundation is presented. For different plate yielding stages, loads, foundation yielding region, separation region have been investigated. The plate problem, resting on Winkler Elastic Foundation was first examined by Hertz in 1884. in Winkler Foundation it is assumed that the foundation is composed of independent springs. in Generalized Winkler Foundation, yielding starts at a certain subgrade pressure (Q0) and the yielding area expands under increasing loads. in engineering applications, there are some important problems which can be handled succesfully by means of Winkler and Generalized Winkler Hypothesis, for example, transportation on içe, platforms floating on a heavy liquid, mat foundations resting on relatively soft soil, airfield concrete pavements. in the first chapter the problem has been introduced, relevant literatüre and hypothesis have been given and the purpose of the work has also been stated. in the second chapter the problem has been formulated: Characteristic length of a plate resting on an elastic subgrade is L=(D/k)1/4 in which D is flexural rigidity of plate, and k is the foundation modulus. The flexural rigidity of an isotropically reinforced cocrete plate is defined by D=-j?^Clc+(n-l)l], n = Es/Ec where Es is Young1 s modulus of steel, Ec is that of concrete, v is Poisson' s ratio for concrete, and Ic is the moment of inertia of cracked concrete section, Is is that of reinforcement taken about the neutral axis. in a symmetrically loaded-isotropic plate, the unit radial and circumferential bending moments, M,., MQ are generalized stresses; subscripts r and 6 denote polar coordinates along the radial and tangential directions respectively. The shearing force, S, which is treated only as a reaction, is assumed to have no influence on plastic behaviour of the plate material. ix -In long plates (> 2L), separation from foundation is usually observed. In these types of plates, failure mechanism forms when a circumferential yield line occurs in the yield region, -At small moment values, while there is no yielding in the foundation, the plate may reach failure mechanism, -For pb=0.1 circumferential yield line radius is around 0.65L. For infinite plates, this value is approximately 0.8L, -From the first hinge, to failure mechanism, for typical plate and foundation parameters, load increments are between %100 and %300, -In the foundation elastic region, limit load increases linearly with increasing moment while separation radius, radial yield line radius and circumferential yield line radius do not change with increasing moment, -Need of upper reinforcement is obvious for plates under concentrated loads, -Although around concentrated load, radial curvatures become pozitive and at a distance from load it is negative, circumferential curvatures are always pozitive anywhere on the plate, -For plate and foundation materials, in engineering applications, yielding in the foundation does not happen, -If there is yielding in the foundation, depending on increment in load, foundation yielding region, radial yield line region and circumferential yield line radius become larger, -If there is no yielding in the foundation, depending on increment in load, concentration of the reactions occurs at the central region of the plate, -Radial and circumferential moments are equal to each other at the plate center. Radial bending moment is zero at the plate edge. Circumferential bending moment is positive anywhere on the plate, -If reinforcement ratios of the faces are different, solution may be obtained using different flexural rigidities for different regions, -With some small changes, it is possible to use equations obtained in this study for uniformly loaded plates with different boundary conditions. xvi -Hooke's law mr = (kre + vkS), me = (kS + vkre) or k? = ^r(mr - vme), k| = ^(me - vmr) For different plate parameters ultimate loads are different. There are two different failure patterns. The first of these patterns shows that in short or short-medium plates, radial yield lines reach the plate edge. The second on the other hand shows that in long and long-medium plates, before radial yield lines reach the plate edge, a circumferential yield line occurs in the radial yield line region and the failure mechanism forms. Until reaching failure mechanism, if it is intended to obtain loads at different yielding stages, for plate yielding, foundation yielding and plate separation region from the foundation there are six different regions. Differential equations for upper side of the yield square (m^ = m,) for these regions are obtained: -Plate and foundation yielding:.,..v, 2 w"' _ (P-') W "*" P ~(l-v*) w= -fıp(lnp- l) + |p2 + f3p + f4 + ^7 j! mr = £L^-f2(l-v2) + Vm0-(p-l4 __ fzd-v2) (1-v>no /_ np s = p p (p-1)2 -Elastic foundation and plate yielding İM 1 III w O w lv+f w + -^r = -rr P 1_V2 1-v2 w = D0S0+D,S,+D2S2+D3[S,lnp+SS]+p Dimensionless quantities related to the plate: m, = D0S0m +D, S, m +D2S2Dm+D3Sy+vm0 s = D^^+D, S,'+D2 S^+D, S3s-(l-v)m0/p xi D0-D3 are integration constants, S0-S2, SS, S0m, S,m, S2""", S", S0°, S,8, Sj", S3S are functions of p and can be expressed using series expansion: f(n,i) = (4i+n) (4i+n-l)2(4i+n-2); n = 0,1,2 «, xk(l - v2)"R Sn=P»+ I (-1J \ <^ p* + " " A