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The idealization of flat plates under flexure into equivalent grid frameworks of beams or shafts, or both, has developed since the original method was proposed. In this note, an improved version of Yettram and Hussain's model is established for a square configuration, so that possible linear variation of bending moment and the associated shearing force can be taken into account. As a consequence, a more realistic modeling can be made for thick plates where shear deformation is significant. Problems are presented which show a good measure of agreement with those from the finite element method of Pryor, et al., over a wide range of thicknesses.

Increase of tanker size may be accompanied with the problem that hull deformation will affect alignment of shafting and turbined gear in various conditions.We investigated deformation of hull structure by FEM calculation and measurement of 270, 000 & 410, 000 DWT turbine tankers. The results of FEM calculation agree rather well with the measurement of actual ship (270, 000 DWT tanker).In this paper, we would report some results as reference for structural design. The summary is as follows.(1) Shear rigidity of hull girder affects the response of double bottom deflection. Therefore, 3-dimensional calculation is useful for checking the bouble bottom design.(2) In way of narrower breadth of double bottom, vertical deflection is much affected by bending moment carried from pressure on side shell than by vertical pressure from bottom shell.(3) Variation of loading weight in engine room, and L. O. temperature seem to give some effects on hull deformation and alignment of shafting.(4) Reverse deflection in (2) due to side pressure works good effection on shafting as a cancellation factor of upward deflection of double bottom due to bottom pressure.

There are many deep girders in a ship structure as the main strength members. Under increasing loads, local failures, such as buckling, yielding etc., occur in these members. These phenomena decrease the stiffness of the members and these changes of relative stiffness in the structure induce redistribution of internal forces and the structure finally shows its ultimate strength.A ship structure is composed of so many such members and its behavior to the ultimate strength is highly nonlinear. There are powerful tools to analyze nonlinear behavior, such as the finite element method, finite strip method, etc. However, the analysis by these methods requires enormous computing time for calculation and in the present circumstances it is impossible to apply these methods directly.To overcome these difficulties, an analytical unit of large size and idealization of the highly nonlinear behaviors are developed in this paper. A “Girder Element” is proposed, of which stiffness matrices are derived for indivisual kinematic condition, such as after buckling, at the ultimate strength or plastic strength under compression, bending, and/or shear. Based on these idealized behaviors, a method of ultimate strength analysis is proposed. The result of analysis by this method represents well the detail behavior of example structures, of which simple one well coincides experimental behavior conducted in this study. The computing time is remarkably very short.These nature of the new method promises the validity and possibility of the method to the ultimate strength analysis of structure of large size.

A new technique for designing a phase lead compensator on the S-plane has been presented in this work. This technique eliminates the need for trial and error cycles which are a feature of the present technique." It can be carried out in part through a graphical construction or in an entirely analytical way. A digital computer program which implements the new technique can be written using the algorithm given by Eqs. (39-44).

Experimentally deformed single layers of Indiana limestone provide stress-strain curves and buckled fold shapes as guidelines for incremental analyses based on the finite element method. It is assumed that if computer simulations can approximate these stress-strain curves and fold shapes, then the stress, moment, and energy histories of the simulation are indicative of the behavior of the single layer of limestone during deformation. A simulation with a Young's modulus of 0.075 Mbar, a Poisson's ratio ν of 0.25, an initial sine wave perturbation of 0.2 cm, and hinged end conditions is the best fit to experimental data. A stress and moment analysis suggests that the bottom part of the layer carries the load, provides the driving bending moment, and governs the buckling instability during deformation. Maximum principal stresses change from compressional to extensional as buckling begins near the top center of the layer. In the experimental situation the top part of the layer undergoes microfracture during this extension. The simulation suggests that this behavior is due to a ‘roll over’ of stresses, a decrease of the superposed axial and bending stress as the bending stress becomes a large negative value, and a decrease in distortional strain energy prior to fracture. Distortional strain energy provides the mechanism of deformation, whereas dilatational strain energy is controlled by shortening of the limestone layer. Geometric change of shape is interpreted to be the major factor influencing buckling instability, volume and material property changes occupying a subordinate role.

The imposed rotation method for the inelastic analysis of frames rests on the interpretation of the actual bending moment distribution as the superposition of linear elastic moment responses to loads and to unknown plastic rotations regarded as imposed strains. The method given herein is a generalized formulation, covering second-order geometric effects and moment-axial force interaction. Finite element models of frames and piecewise linear moment-rotation laws for critical sections are assumed. Recent algorithms for solving quadratic programming and linear complementarity problems are shown to be efficiently applicable to the analysis of multistory frames. The safety factor with respect to local failure because of “brittle” flexural behavior turns out to be attainable by a suitably modified linear programming procedure.

An automatic incremental iterative finite element procedure for the analysis of post-tensioned box girders subject to combined bending and torsion in the uncracked and post- cracking range is presented. The analysis includes the non- linear behaviour of the materials, the change in the structural topology due to the initiation and propagation of cracks, and makes provision for the gradual but progressive breakdown of bond. The three-dimensional character of the girders is synthesized and structural actions such as shear lag, load diffusion and distortion are automatically taken into account. The effect of diaphragms of given shear and warping stiffness placed anywhere in the structure can be investigated. The results of the analysis of two post-tensioned single cell box girders, each subjected to a different mid-span bending moment/torsion moment ratio, are given and compared with the experimental results obrained from the corresponding concrete model tests. The prediction of strain distribution, first cracking load, crack pattern, mode of failure and collapse load is satisfactory. /Author/

Openings are often introduced into shear walls in multistoried buildings for either architectural or environmental reasons. These openings reduce the efficiency and change the deformation characteristics of the shear walls, depending on the distribution of the loads and the geometry of the walls with openings. Variations of the material properties also affect the stress distribution. The use of the finite-element method in analyzing these problems is found to be convenient and more accurate than conventional approximate methods. A model shear wall problem is selected and analyzed by the finite-element method and the results are presented. It was observed that local distortions occurred at the junctions of the lintel beams and the shear walls due to the bending moments, shear forces, and axial loads carried by the lintel beams. Because of these distortions, the bending moments on the lintel beams were found to be much less than the bending moments obtained from other approximate methods of analysis.

The concepts of the matrix displacement approach to finite element structural analysis are extended to analyze the large deflection behavior of plates with initial deflections. The initial deflections are considered to be of the order of magnitude of the plate thickness. The large deflection phenomena are accounted for by considering the influence of the element membrane forces due to initial deflection and large bending on element flexure stiffness matrices. The effects of element membrane forces due to initial deflection and large bending are characterized in the nonlinear stiffness formulations by the zero, first, and second order incremental stiffness matrices. Representations of the nonlinear effects are formulated for a conforming rectangular plate finite element. The formulations are appropriate for both the iterative and incremental approaches. Large deflection behaviors are predicted for rectangular plates with different length-width ratios and various boundary conditions. Comparisons with some available alternative analytic solutions are provided. Examples are performed by a piecewise linear incremental approach. A reduction scheme which removes certain degrees of freedom from the problem statement is incorporated into the numerical procedure.

The paper describes a theoretical and experimental study of the behaviour of corner supported slabs. The finite element method of analysis was used to determine the deflections and bending moments in three slabs: 1. (1) a square slab supported at the corners, 2. (2) a 45° skew slab supported at the corners, and 3. (3) a 30° continuous skew slab supported at the corners and at the centres of the long sides. The effects of three loading patterns were investigated 1. (a) a uniformly distributed load over the whole slab, 2. (b) a concentrated load at the centre of the slab, and 3. (c) concentrated loads at the centres of two opposite sides. The results of the analysis showed good agreement with measurements taken on model perspex plates with appropriate loading and boundary conditions.

Considerable attention has been devoted in the literature on numerical methods towards securing energy convergence of solutions for, say, linearly elastic plate bending problems. Although energy convergence is necessary it by no means follows that the derived bending moments and shearing forces converge uniformly at a given point and it is this kind of feature which the engineer is really seeking. This question is examined in the context of a problem which is of particular interest to the civil engineering field and concerns the bending of a square plate under uniformly distributed load; the plate has simply supported edges and contains a central square hole with free edges. The solution to this multiply connected and mixed boundary value problem is obtained through a recently developed modification to the Rayleigh–Ritz method which has very general application and renders the solution mathematically valid up to the internal corner points where the bending moments are singular. Use is made of triangular equilibrium finite elements in conjunction with continuous eigenfunctions. Although it is already known that the order (i.e. the eigenvalue) of the singularity at the internal corners is available by inspection, it is an interesting feature of the present solution that a good approximation to the amplitude is also obtained by an inspection of the finite element results.

For the problem of elastic plate bending, a finite element procedure based on the minimum of complementary energy principle is developed. Primary variable of the problem is the distribution of the bending moments m x , m y and m xy . These vary parabolically inside each triangular element and have continuous values throughout the plate. In order to satisfy the equilibrium conditions, Lagrangean multipliers are introduced as secondary variables. It is then shown that these multipliers are generalized displacement parameters. The final solution will therefore provide direct information about both the stress distribution and the displaced shape of the plate. Equilibrium conditions are satisfied exactly. A strain energy bound is found. For several cases numerical results are given and compared with known solutions.

The plate bending problem in the stationary creep state was investigated by means of the finite element method. The Stationary creep problem was transformed into a non linear elastic problem in accordance with the HOFFS' analogy and a REISSNER type variational principle was formulated. Based on the above-mentioned principle, the finite element method was applied to the plate bending problem and a computer program was made to solve the bending problem of arbitrary shape plate subjected to arbitrary load. The bending moment and stationary creep strain rate were calculated for circular and rectangular plates under uniform loading. The results obtained here for circular plates show a good agreement with the results calculated by using the RUNGE-KUTTA method.

The Finite Element Method of analysis was used to determine the buckling loads of the web plate under patch loading together with simple supporting shear forces. Combinations of patch loading with shear loading or with in-plane bending moment are also studied. The results are presented in the form of interaction curves which show that the applied edge load necessary to buckle the panel is reduced by the presence of either an additional in-plane moment or shear. Tests are carried out on light gage cold formed troughs. The test results show that there is a considerable amount of post buckling strength for panels having high slenderness ratios. A linear relationship exists between the ratio of the ultimate load to the buckling load and the relative length of the loaded patch and the slenderness ratio of the panel. Failure of all test panels is associated with the formation of a localized yield curve under the load.

The lateral-torsional buckling strength of beam columns in both the elastic and the inelastic ranges is compared for (1) the influence of the residual stress patterns which are idealized for as-rolled and as-welded flange sections, and (2) the effect of the deterioriation in the St. Venant stiffness as a result of yielding. In this note, the cross-sectional properties in the inelastic range are numerically calculated by dividing the cross section into finite elements for given bending moment under an axial thrust. Additionally, the lateral-torsional buckling strength is calculated by the finite difference method applying the cross-sectional properties numerically obtained for the moment along the length of a beam-column. The numerical results are shown in several examples.

Bending analysis of elastoplastic circular plates for small deflection and axisymmetric loadings and support conditions, based on Kirchhoff hypothesis

THE APPLICATION OF A GENERAL FINITE ELEMENT PROGRAM INCORPORATING TRIANGULAR PLATES, BEAM AND SUPPORT COLUMN 'ELEMENTS' IS DESCRIBED. THE OBJECTIVE IS THE ASSESSMENT OF ACCURACY ATTAINABLE IN EVALUATING BENDING MOMENTS AND SUPPORT REACTIONS FOR COMPLEX SITUATIONS ENCOUNTERED IN MODERN BRIDGES. PARTICULAR ATTENTION IS GIVEN TO THE PROBLEM OF COLUMN SUPPORTS. COMPARISON WITH EXACT SOLUTIONS AND MODEL INVESTIGATION SHOWS EXCELLENT AGREEMENT AND INDICATES THAT A GENERAL, INEXPENSIVE, METHOD OF SOLUTION IS NOW AVAILABLE. THE PAPER ALSO PRESENTS AN EXPLICIT FORM OF THE STIFFNESS MATRIX FOR TRIANGULAR PLATE ELEMENTS USED TO MAKE THIS WIDELY AVAILABLE. /AUTHOR/

A comparison between pipe bends with assumed cross sections, namely elliptic and semi oval to include ovality along with wall thinning, was performed to determine the plastic collapse load under in-plane closing bending moment using finite element limit analysis based on an elastic-perfectly plastic material considering geometric nonlinearity. Twice-elastic-slope method was used to obtain collapse load from the moment-rotation curve drawn for each pipe bend model considered. The effect of ovality on collapse load is significant and higher for elliptic cross sections for almost all cases while the thinning effect is negligible for both the cross sections. The study concludes that the use of elliptic cross section is suitable for analyzing pipe bend with ovality. Therefore, a closed-form solution is proposed to determine collapse load of pipe bend based on the finite element results of elliptic cross sections.

An equivalent grid-framework procedure is presented for the analysis of skew plates in flexure. A rhomboid grid model with side and diagonal members is used to represent a similar element of uniform thickness in the plate. The real value of Poisson's ratio is taken into account. Results for deflexion and bending moment predicted by the equivalent skew grid are given for two examples and compared with those obtained by other, notably finite element, methods.

This article gives the mathematical solutions of deflection, bending moment, and longitudinal fibre stress of seven different configurations of single- and multilayer roofs in horizontally bedded rock. The work, based on the theory of elastic beams on elastic supports, shows the importance of abutment compression in contributing to the deflection of the roof layers. When the ratio of the thickness of the individual layers, H , to the span of the working room, l , exceeds l /2 the theory becomes inexact. By using the finite element method it was found that the extreme fibre tensile stress is equal to the uniformly distributed load on the roof layer for H/l greater than 1.5. For ratios between these limits the actual tensile stresses can be determined from stress concentration diagrams. The theories for single- and double-layer roofs have been verified by model experiments. Two types of experimental equipment set-ups were used: ( 1 ) involving a uniform load on the top surface of the model; and ( 2 ) involving runs under controlled conditions in a high-capacity centrifuge. The model experiments in a centrifuge together with finite element theories display the difference in failure mode between thin and thick roof layers. Vertical to subvertical fissures in regions with high bending moments, that is in the roof centre and above the abutment, precede the collapse of a thin roof layer. High shear stresses appear above the abutment in thick roof layers ( H > l/2 ) which give rise to arching over the opening. The theories and results from model experiments are now being put into practice at the lead mine of Laisvall, northern Sweden. Results from present precise leveling and differential roof sag (sag differences from layer to layer) measurements in the face excavations are in accordance with results from proposed theories.

Many floating marine structures of various kinds have been constructed for varied purposes ; however, researches and technical experiences for such structures, some of which are quite different from conventional ships, are not so exhaustive and comprehensive, as those for the ships, and many problems still remain unsolved.Taking an opportunity of construction of newly developed iron ore loading station with submersible, column stabilized, catamaran hulls, a systematic study was made on the motion and strength of the structure in waves. Theoretical analysis of the motion and strength of the structure was carried out first. Secondly, the theoretical results of its motion in waves were compared with that of model tank test. Thirdly, in order to check the theoretical calculation the stress measurements were made at the time of launching. Finally, during the ocean towage of the structure from Japan to India, comprehensive measurements of motion of and stress in the structure in waves were conducted simultaneously with the accurate measurements of waves by means of a clover-leaf buoy.Conclusions of the present study are summerized as follows : 1) Motions of floating marine structure with catamaran hulls in irregular waves can be estimated with sufficient accuracy for engineering purpose by the spectral computations of responses utilizing the linear superposition principle using the response functions which are obtained by the strip method usually used for ship motions in oblique waves and are corrected by the results of model tank test.2) Longitudinal stress of the structure can be estimated with fairly good accuracy, by the finite element method using the response function of longitudinal bending moment in waves which is also calculated by the strip method.

An analytical formulation of the generalized stress-strain relationships for biaxially loaded column segments is presented. The generalized stresses are an axial force, two bending moments about the principal axes, and a bimoment. The corresponding generalized strains are an axial strain at the centroid, two bending curvatures, and a warping curvature. The material characteristics are assumed to be trilinear, i.e., elastic, plastic, and linear strain hardening. Elastic unloading of yielded fibers and “Bauschinger effects” are included in the analysis. Based upon the equations formulated, a computer program was developed to provide numerical results. The element of the tangent stiffness matrix was evaluated numerically by dividing the wide flange section into finite element.

Comprehensive details are given for a fully conforming plate bending triangular finite element which has cubically varying normal displacements along the sides and which employs rational functions to supplement the ten term cubic description of the displaced surface. The mid-side connections are eliminated by means of extended interpolation and explicit ‘best’ linear approximations are derived for the final valuation of the element bending moments. The element is efficient and is found to be agreeably accurate in dealing with the classical comparison problems of square and circular plates. No difficulty arises in approximating curved boundaries provided that the associated prescribed directional derivatives are referred strictly to the true boundary and not to its rectilinear approximation.

Triangular plate bending element using Herrmann variational method, deriving matrices for finite elements

The analysis, repair, and further static testing of the previously failed graphite horizontal stabilizer are described. The design, analysis, and development testing of redesign concepts to correct the understrength condition at the main pivot fitting joint are discussed. Finite element analyses of the joint region were conducted to quantify the interlaminar shears and bending moments induced in the laminated skin panel by secondary bending effects. The calculated interlaminar shear and tensile stresses were of the order of 5 to 10 percent of allowable stresses determined in quality control specimens. The failed static test article was repaired with bolted and bonded steel splice plates and additional static tests were conducted to verify the strengths of the three elevator hinge brackets and supporting structures. These tests successfully demonstrated the design integrity of the hinge brackets since loads in excess of design ultimate load (150 percent design limit load) were sustained in each case. Joint specimens were tested to evaluate two redesign concepts (i.e., bonded external scarf and internal stepped-lap titanium doublers) to correct the understrength condition at the pivot fitting joint. The latter concept was selected for further development because it eliminated design reliance on laminate properties in the immediate vicinity of the major bolted joint. (Author Modified Abstract)

The results of twelve tests of hull-deckhouse interaction are presented and discussed. Deflections of hull and deckhouse were measured. Stress distributions along three transverse sections and along the deck to deckhouse joint were calculated from measured strains. Three bending moment distributions and two deck support arrangements were used in the test program. The experimental results are compared with results from finite element analysis and two other theoretical methods of analysis of hull-deckhouse interaction.

THE COMMENTS BY ALTERMAN ARE APPRECIATED, AS IS HIS EMPHASIS ON RELIABLE MEASUREMENTS ON PROTOTYPE SOIL-STRUCTURE SYSTEMS. THE FIELD COMPACTION PROCEDURE FOR THE SOIL BACKFILL AND THE CULVERT BEDDING CONDITIONS INFLUENCE GREATLY THE MAGNITUDE OF BENDING MOMENTS WHICH DEVELOP IN THE CULVERT WALL. SINCE IT IS IMPOSSIBLE TO MODEL THE FIELD INSTALLATION IN SMALL SIZE EXPERIMENTS, THE INFLUENCE OF BEDDING CONDITIONS WAS DISCUSSED. THE WRITER DOES NOT BELIEVE THAT THE TANGENTIAL WALL DISPLACEMENTS WERE SIGNIFICANTLY RESTRAINED BY THE WAY THE TEST SECTION WAS DESIGNED. THE EXPERIMENTAL DATA SHOWED THAT CONCEPTS FROM CONTINUUM MECHANICS LEAD TO REALISTIC ANALYSIS OF SOIL-STRUCTURE INTERACTION. HOWEVER THE FINITE ELEMENT METHOD OF ANALYSIS CAN NOW BE USED TO MAKE SUCH A STUDY MORE GENERAL AND TO CONSIDER FIELD SITUATIONS. REPORTS ISSUED: STRESSES AGAINST UNDERGROUND STRUCTURAL CYLINDERS, KAARE HOEG, ASCE PROC. PAPER 6022, JULY 1968.