A finite element solution of the beam equation via matlab s. Ublishing mart materials and structures smart mater. Similarily, displacement vector can be divided into two components. Finally, several numerical examples are shown to illustrate the efficiency and simplicity of the current approach. While in this study we only consider a simplysupported members it may be possible to extend this method of solution to other boundary conditions. For simply supported plates the displacement and bending moment are zero on the corresponding boundaries, i.
Equation chapter 1 section 1plosca podprta v robnih tockah arxiv. We consider the kirchho love model for the supported plate, that is, the fourth order di erential equation 2u fwith appropriate boundary conditions. The stress resultants m,, m, and t, in figure 2 are defined with respect to coordinate directions n and s, normal to and along the boundary curve. Rectangular plate, free on one edge, simplysupported on other edges, uniform loading. The proposed method offers a more accurate solution for the bending moment at the critical section and therefore serves as a better analytical solution for design purposes. Simply supported plate with a central hole of radius b, uniformly load by pressure q o over region c boundary conditions of clamped, simply supported, while those of free, guided, clamped, simply supported and elastically supported edges of various peripheral edge ranges shall be treated at a later point. Simply supported plate an overview sciencedirect topics. Rectangular plate, simplysupported on all edges, point loading. Hinged and supported plates with corners serguei a. A segment of a plate can be fixed or encastred left, simply supported center, or mixed supported right, or have a free edge.
Zero stiffness corresponds to a simply supported plate, whereas very high stiffness comes close to the case of a clamped plate. The sandwich plate is assumed to consist of isotropic facings separated by and bonded to an. The deflection of a beam must often be limited in order to provide integrity and stability of a structure or machine, or. Clamped boundary simply supported boundary free boundary r a w 0 w 0. Bending solution for simply supported annular plates using. Implementation of clamped and simply supported boundary. Relevant formulas valid in this natural system can be found in the appendix. Bending analysis of simply supported and clamped circular plate. Check the boundary conditions in the weak saintvenant sense on the shorter left and right hand sides for both normal and shear stress. In this chapter, equations that express moments and forces in terms of deflection are developed first. The plate is simply supported on its edges and connected with torsional springs to its neighbourhood, fig. The supports, or boundary conditions, play an important. The sliding boundary conditions will convert the eigenvalue problem into the equilibrium problem and therefore are not considered in the buckling analysis of plates. Response of a rectangular plate to base excitation revision e.
A unied scheme is proposed for the treatment of simply supported, clamped and transversely supported with nonuniform elastic rotational restraint boundary conditions. To prevent any attached brittle materials from cracking 2 beam deflection by integration. Once the stresses are known, failure theories can be. Rectangular plate, simplysupported on all edges, uniform loading. Difference between a clamped and a simply supported plate.
I also have looked at ac ugurals book, but the plate with my boundary conditions two edges simply supported and the other two free is also not there. Bending of plates, or plate bending, refers to the deflection of a plate perpendicular to the plane of the plate under the action of external forces and moments. Plate made of orthotropic materials under simply supported and clamped boundary conditions in both symmetric and antisymmetric modes of vibration, international journal of mechanical engineering and technology 88, 2017, pp. The robustness and reliability of the present approach are tested by a number of numerical experiments. The boundary conditions for a clamped plate generally indicate that the edge deflection and edge slopes are both equal to zero similar boundary conditions are used for beams with fixed ends. The applied pressure is indicated by a central arrow. Horn 6 derived the exact controllability of the eulerbernoulli plate with a simply supported boundary condition only via bending moments on the space of optimal regularity. In this model, the kirchhoff assumption is relaxed to allow for. Nonlinear vibration of beams and rectangular plates by joe g. But how can i simulate the simply supported boundary conditions. If the flanges are also prone to buckling, then the corners will rotate as shown in fig. The vibration of circular plates was also analyzed with clamped boundary conditions 26.
Jul 16, 2011 the boundary conditions for a clamped plate generally indicate that the edge deflection and edge slopes are both equal to zero similar boundary conditions are used for beams with fixed ends. There are many additional complications in the case of more general boundary conditions, and so the analysis of 1 is not easily extended to the soft simply supported and free plates, for example. Rectangular plate, clamped on all edges, triangular loading. Simply supported plates by the boundary integral equation method. In this paper, a new methodology for implementing the clamped and simply supported boundary conditions is presented for the free vibration analysis of beams and plates using the generalized differential quadrature gdq method. The stresses in the plate can be calculated from these deflections. Vibration modes of a single plate with general boundary. For a corner with soft simply supported edges the boundary layer removes the singular behaviour of kirchhoff type from the stress resultants. Graphs for the beam equation using homogenous boundary conditions the boundary conditions were then changed to obtain results for a simplysupported beam. Plates have free, simply supported or fixed boundary conditions. A simply supported plate has edge deflection 0, but the. Yes that is the type of solution i was looking for. Circular plates with clamped and simply supported conditions a rectangular plate subject to concentrated loads at its corners a simply supported rectangular plate subject to a general pressure distribution a rectangular plate clamped on two edges and simply supported on the other two solutions to nonlinear plate problemscoupled bending and. The right end of the beam is supported by a fixed end support therefore the slope of the deflection curve is 0 and the deflection is 0 ei dv dx.
Simply supported plate with a central hole of radius b, uniformly load by pressure q o over region c load collector. Since the normal stress xx is not zero at the ends, but only its resultant. The amount of deflection can be determined by solving the differential equations of an appropriate plate theory. The static or dynamic loads carried by plates are predominantly perpendicular to the plate surface. I want to do some experiments on the dynamic behavior of the beam, plate and so on. The solution of the plate equation by separating it into two potential equations is attributed to marcus 8. Came across examples of rectangular plate analysed by considering just one quadrant of it. A simply supported plate has edge deflection 0, but the slope of the plate can vary depending on the loading. Specifications of boundary conditions for reissnermindlin.
Hinged and supported plates with corners universitat zu koln. Uniformly loaded rectangular thin plates with symmetrical. The boundary conditions are zero deflections and moment equilibrium on the edges fig. Along the sides of the plate the deflection and normal inplane stress components vanish at 0 and. Y 1 vr 23 in the following we prefer r to y since r has a physical interpretation. Rectangular plate, clamped on all edges, uniform loading. Assessment of a simply supported plate with uniformly distributed load a building has a floor opening that has been covered by a durbar plate with a yield stress of 275mpa. The figure shows the boundary conditions, soft simply supports on the external edges and symmetry conditions no normal rotation on the symmetry boundaries. In this study, naviers solution for the analysis of simply supported rectangular plates is extended to consider rigid internal supports. The other two sides can have any type of support or boundary con ditions. Next, the basic differential equation for the bending of rectangular plates is established together with corresponding boundary conditions. A finite element solution of the beam equation via matlab. To model the platesupport interaction, the patched areas representing the contact.
The problem is solved using homogenous and nonhomogenous boundary conditions with various numbers of elements. For clamped boundary, i can use some bolts to fix the beam or plate on a base. For the two types of boundary conditions viz all the sides of the plate are simply supported and opposite sides are clamped, the nondimensionalized frequencies for different exponent and wave numbers have been determined along with the damping effect. The static deformation of the beam can be modelled with the following fourth order differential equation. Introduction the influence of initial membrane stress on the free and forced nonlinear vibration. Assessment of a simply supported plate with uniformly. It is not usually possible to provide ideal boundary conditions, i. For the variable coe cient case, yao 22 used the riemannian geometry approach to give checkable conditions for. Analytical solution for free vibrations of simply supported transversally inextensible homogeneous rectangular plate milan batista university of ljubljana, faculty of maritime studies and transport, slovenia, eu milan.
Bending analysis of simply supported and clamped circular. The university of texas at arlington, 2011 supervising professor. The determination of natural frequencies of rectangular. The owner has been instructed by his insurers that for safety the load carrying capacity of the plate needs to be assessed. For the variable coe cient case, yao 22 used the riemannian geometry approach to. Seriesbased solution for analysis of simply supported. Specifications of boundary conditions for reissnermindlin plate. For the extension of the method to plates with a curvilinear boundary under any type of boundary conditions. What are boundary conditions for simply supported plate on. Extremely accurate dsc results were obtained for beam bending, vibration and buckling33,34. Seiichi nomura this thesis work deals with the study of plates with various geometries and different boundary conditions.
Response of a rectangular plate to base excitation revision e by tom irvine email. The vertical deflection of a simply supported and clamped beam is considered under a uniform load using the finite element method. For free boundary condition, i can use a string to tie the beam or plates up and hang them on some place. Analytical solution for free vibrations of simply supported. Deflection and moments for plate two simply supported. Pdf uniformly loaded rectangular thin plates with symmetrical. How to simulate the simply supported boundary condition in.
Pdf analysis of simply supported rectangular kirchhoff. This report is one of a series anc23, item a7 prepared and distributed by the forest products laboratory under u. The considered plate has dimensions l x, l y and thickness h. Plate analysis with different geometries and arbitary boundary conditions ashwin balasubramanian, m. The derivation of plate governing equations of free vibration modes is introduced. The graphical results were obtained for the same parameters as described for a clamped beam. The maximum deflection of the plate for various geometries and boundary conditions is determined.
In the case of supported plates, the boundary conditions permit the uncoupling of the field equation into two harmonic equations that originate, due to the nature. Full analytical solutions for k as a function of ab and m may be found in timoshenko and gere 1961, allen and bulson 1980 and others. Mechanics of materials civl 3322 mech 3322 deflection of beams the elastic curve. Nonlinear vibration of beams and rectangular plates. Therefore the real structures are mostly constrained by elastic supports. Show that the stress function 3 2 2 5 2 2 2 3 3 2 3 20 4 15 2 5 20 y l x y h x y h y h x h p satisfies the boundary conditions for the simply supported beam subjected to a. Simply supported pzxy simply supported differential equation boundary conditions on x ony 0 w0,y wx, 0 pzx, y ox oy oy on x ony a b wx, b only even derivatives in the boundary value problem. Kirchhoff plate theory and the reissner plate theory. Therefore, the solutions of the differential equation of equilibrium will depend on certain integration constants that are determined by imposing two boundary conditions on each edge. The unloaded edges of rectangular plates can be either simply supported ss, clamped c or free.
The thickness is much smaller than other dimensions. An analytical solution for a multilayered magnetoelectroelastic circular plate under simply supported lateral boundary conditions rwang 1,qhan,2 and e pan1 1 computer modelling and simulation group, college of engineering, university of akron, akron, oh 443253905,usa 2 school of mechanical engineering and automation, northeastern university. The loaded edges could be either simply supported or clamped. The typical boundary conditions for the axisymmetric bending of annular plates are listed in table 1. Bending of simply supported rectangular plates springerlink. A simply supported beam is obtained by balancing the beam on a pivot at each of its ends. The governing differential equation is that predescribed by the bernoulli beam. Reissner plate theory w reissner developed a first order shear deformable plate model to analyze the effects of increasing thickness. The boundary conditions of the problem are the following srinivas, rao et al. Deflection and moments for plate two simply supported edges. Plate is simply supported on two opposite sides as shown below. Pdf boundary condition identification of a plate on elastic support. These expressions are then used to solve various plate configurations and loading conditions.
In this paper, we analyze the boundary layer for all the boundary conditions mentioned above in a uni ed fashion. The proposed approach directly substitutes the boundary conditions into the governing equations. Both simply supported and clamped boundary conditions subjected to uniformly distributed load and center concentrated point load have been considered in the present study. The solution of these equations is suggested and expressed. Nazarov, athanasios stylianou and guido sweers abstract. Plate calculators, rectangular and circular plates with various boundary and loading conditions. Bending of simply supported rectangular plates bending of simply. Since we consider a simplysupported beam and plate, the solution is either a sine or double sine in.
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