A FINITE ELEMENT ANALYSIS OF A BRIDGE USING SAP2000
Introduction
The finite element research was pioneered in the University of California in Berkely. This was in 1957-1970 the pioneer research, and it was a direct extension of the classical methods of structural analysis that had previously been restricted to uni-dimensional elements. Most of the research was driven by the desire to solve the problems practically that involved Aerospace, Mechanical and Civil Engineering. During this time, the finite element period extended the solution of linear and non-linear problems that were associated with the creep, incremental construction or excavation, as well as the flow of water in porous mediums. This also applied in crack closing, dynamic response analysis as well as computer assisted learning in structural analysis. It also applied in the consolidation of soil.
In the past six years, the computer programs that were developed in Berkley were distributed freely globally. These were done by all the practicing engineers, and were used to solve the many problems that were emerging in the field of structural economics. The research was eventually transferred to the engineering profession. It was used professionally before the formal paper was published. Before 1952, structural analysis was confined to the elements that connected to only two points in space. The structural engineers utilized the lattice analogy, which prior to 1952 structural analysis was restricted to elements connected to only two points in space. The Structural engineers used the lattice analogy, which was developed by Hrennikoff and McHenry, to model membrane and plate bending parts of the structure. Needless to say, this analogy was not applicable to non-rectangular areas. Ray Clough experienced this problem first handled during the summers in 1952 and 1953. This was after the Boeing Summer Faculty Program. At this time, he was working with Jon Turner, who was the head of the Structural Dynamics Unit, and he was requested to calculate the bending and torsional flexibility influence coefficients on low aspect wings. They had obtained Static experimental for the swept-back box. They disagreed with the results that were produced from a structural analysis model that used a single dimension solely.
Turner then presented the Boeing pioneering work at the January 1954 meeting of the Institute of Aeronautical Sciences in New York. This paper was published until September 1956. More so, there was triangular constant strain membrane element, a rectangular which was based on the equilibrium patterns of stress which avoided shear locking. The direct stiffness methods were formed by the node equilibrium equations. The aim of the two dimensional elements that was developed at Boeing was used to model the properties of the dynamic stiffness accurately and also the displacements of the structure. In the year 1956 and 1957 Clough went on a sabbatical leave in Norway. During that time, Clough had the time to reflect back on his work and to study about the new developments in those fields.
Through the use of matrix transformation methods, it was shown evidently that structural analysis methods could be categorized as either a force or a displacement method. Clough came to a conclusion in Norway that the two dimensional elements connected to more than the two nodes and could be used in solving problems in continuum mechanics. The stress relationship within the turner triangular element is displaced between the adjacent elements and also on the force equilibrium on an integral basis within the structure and the finite number of node points. These three equations proved that the convergence of the exact elasticity solution and the mesh was refined. Another approach to the continuum mechanic problems was the discreet idealization; hence the terminology finite element was coined by Clough himself. Therefore, the analysis of both models from continuous structures and the frame structures were modeled as a case of elements interconnected at the joints or the nodes. In structural analysis, other researchers have proved that solving problems in mechanics of continuum by the use of discreet elements.
Actual Frame Finite Element Model: Finite Element Model Actual Dam
It should be out during nineteen sixties at Berkeley. The department of Defense was not only learning on how to strengthen buildings so as to ensure they resist nuclear missiles, as well as the underground structures but the cost of the entire activity and their ability also. This was first while Cold War was at its peak. In the second phase, a research on Earthquake Engineering was conducted. It also included the creation of the prime international shaking table. This was a program birthed by several professors including Penzien and Seed. The Federal Government in association with The Transportation Department of California sponsored a research authorizes by Professors Scordelis and Minismith at Berkeley. The research focused on the behavior of bridges as well as overpass structures. They did this alongside swift expansion of the freeway system as the third event. Professors Penzien, Pister, Popop, Sackman, Taylor and Wilson actively conducted a research on manned space as the fourth. This was a national precedence. Drilling for oil was the fifth. It was offshore in deep in the waters. This was together with the creation of the pipeline of Alaska which demanded for fresh expertise for steel equipment. This was developed by Professors Popov, Bouwkamp and Powell.
At last, Professors Scordelis, Popov and Lin were still used as consultants during the formation of nuclear reactors together with nuclear towers which were required advanced skills of analysis as well as materials. They specifically were of aid on designing and construction of relevant shell structures of long span. In support of this research, Davis Hall-a structural engineering building, and a shaking table which was to arouse earthquake motions were constructed at Berkeley and Richmond Field Station respectively. This analysis and experimental researches were complemented by the Finite Element Method.
After Clough from sabbatical in Norway in 1957 at Berkeley not only applied for but he also received an NSF grant to support a research on computerized analysis of several structures. He also started a course known as Matrix Analysis of Structures. In the last semester of the course in 1957, he outlined some research areas of a graduate student which included topics on Finite Element, Analysis of the same on Plane Stress Structure and Plates.
Shells analysis
In the previous year, a digital computer, IBM 701 type which had a memory of 16 bit and 4k had been set up in the Engineering College. The computer could solve equations to a maximum of 40. To ensure that the students were not immediately required to study programming so as to tackle any finite element problems, Clough came up with a program on matrix algebra while operating with the computer set. It was thus made possible to solve larger systems using sub-matrix techniques and tape storage. Ed Wilson who was a graduate student shared an office with Adini was dissatisfied with the enormous work involved in solving the Finite Element problems. This was specifically while using the matrix algebra program. As a result, he, under Clough’s direction in 1958 began the construction of a finite element program that was automated. On this, he based it on the rectangular plane stress finite element which was developed at Boeing. It was after a number of months of studying how to program the computer that Wilson produced a semi-automated program. It had a limited capacity and operated on the force method. It was then followed with an MS research report which has been overridden with a rough title of ‘Computer Analysis of Plane Stress Structures.’
Installation of an IBM 704 computer in the Campus of Berkeley was done in 1959. Then, it had a memory capacity of 32 bit with 32K. In addition to that, it also had a balancing arithmetic unit point which was then approximated to be 100 times faster compared to the other computer. This enhanced the solving of practical measures by use of fine meshes. Still working under Clough’s direction, on NSF project, he, Wilson wrote an analysis program that was on a double dimensional frame. The program was with a nonlinear and a moment-curvature relationship which was under classical Ramburg’s definition: Osgood equation. A pushover kind of analysis was created after incremental application of the loads.
All types of finite elements systems could use the approach of incremental load because it was highly generalized. Using the matrix algebra program, Adini carried on with his research on finite element in solving plate-bending problems. This was using the rectangular finite problems as well as demonstrated. This was to ensure accurate modeling of that class of structures. Using the finite element method, it was possible to solve some plate bending problems. This was according to some demonstrations. With the acceptance of two papers from Berkeley, it was then acknowledged for presentation. This was at the ASCE Conference in1960. Using the approach of matrix algebra, together with more orders to form both membrane and bending stiffness matrices, Adini solved a number of simple shell structures. The commands were for rectangular elements. He was done with his PhD thesis on ‘the finite element analysis of shell structures’ in 1962. Wilson on the other hand came up with an automated finite element in 1960. In it was the storage of node equilibrium equations. This was in compact form and was solved by use of Gauss-Seidel iteration together with a factor of over-relaxation. It was thus possible for structural engineers then to solve plane stress structures without a strong background of mathematics in continuum mechanics. This was for structural structures of the arbitrary geometry which were constructed by use of several materials. Preparation of the computer input data was required was not a demanding task. Most structures are therefore completed within few hours. Incremental loading as well as nonlinear material capability were later added to this program by Wilson with which he then wrote his thesis on the same topic.
The Norfork dam project
At Berkeley, The University of California, researched on not only earth dams but rockfill ones also together with their material testing. This was just before the growth of the finite element method. Professor R. E. Davis in the nineteen twenties carried out studies for material for Hoover Dam. Professors J. Raphael, H. Eberhart and D. Pirtz, accomplished ventures on studies for material for the Oroville Dam. This was in the late nineteen fifties. Then, relevant investigation on the designation as well as building of dam structures was carried out in the Department of Civil Engineering. This was by a greater part of that faculty. The initial real appliance of the plane finite element program was definitely a dam structure. It was thus not a bombshell.
A proposal to conduct an analysis of the finite element of the Norfork Dam was presented by Clough. This was on approval of Dr. Roy Carlson who was a mentor of the unit of engineers. He was specific to the Little Rock District. The Norfork Dam was a gravity one. Close to the center of the section was a heat stimulated vertical crack. The proposal had a coarse mesh solution section the dam was produced the new program and clearly indicated the ability the method model structures arbitrary different the demand foundation. The group of Engineers agreed to proposal of the finite element analysis by Clough. This was because of the proposal of the analogue computer by Professor Richard MacNeal of Cal-Tech state-of-the-art method for responding to such problems.
The project provided opportunity improve the numerical methods used the program and extend the finite element method the solution the crack due loading Wilson and a new graduate student. Ian King then performed thorough analyses that the study needed. Alongside the report submitted by the Engineering corps on the analysis of the same dam was a paper which was prepared and later in 1962, it was presented at Lisbon in Portugal. This was during a Symposium on ‘The Use of Computers in Civil Engineering’.
Reaction from the continuum mechanics community
During nor fork Dam Berkeley colleague Karl of Clough, it then came to Pister’s awareness that the approach of finite element was a distinctive expression of the Ritz Method. In it, the three functions were independent of one another within each element, and they were also similar at the edges of the elements. He also conducted a vigorous investigation within a brief time limit. However, examiners on continuum mechanics without the group of Structural Engineering and Mechanics at Berkeley where not willing to give into that method. It is appealing to note that the assumption constants train fields the elements essentially equivalent the concept regional had been proposed many years earlier and however, these ideas were practiced then due to the low speed computers. Clough and Boeing were uninformed of those mathematical citations while carrying out their advancement projects. It was until 1964 that the approach of finite element was accepted as one of the methods of solving problems related to continuum mechanics. This was after continuum mechanics investigators found out those initial mathematical papers. The approach of finite element was also used in 1965 to solve problems of heat transfer. This changed the appropriateness of the name direct stiffness.
Isoparametric elements
Irons and Zienkiewicz in 1968 presented the finite matrices. That work had an immediate and significant impact on the finite element research which was being conducted. Professor Taylor was not only the first Professor to program that new approach, but he also demonstrated its power. This took place at Berkeley. In 1969, William Doherty who was working under Taylor’s direction came up with the original program for the analysis of the trio-dimensional, steady flow of fluids in porous media. This was with the aid of the eight node isoparametric element. Hitherto, several firms are using the same program in the prediction of hydrostatic pressures under dams. Around that time, Kenneth Kavanagh who was working under Clough’s supervision used the eight node solid element for the structural analysis of the trio-dimensional solids. To eliminate that problem, Taylor and Wilson on the other hand experimented using the incompatible displacement modes as well as reduced integration.
The SMIS and SAP programs
Wilson in association with Clough constructed a Symbolic Matrix Interpretive System [SMIS]. This was to enhance the teaching of the static as well as dynamic analysis of several structures. That program was to specifically narrow the disparity of structural analysis that was between the traditional manual calculation techniques and the matrix ones. This program [FORTMAN] was not only distributed several times at different advanced learning institutions, but it was also modified. Its current modification is the CAL 91 which is being used for the teaching of modern structural analysis. In the initial years of the research at Berkeley on finite element, each learner came up with their own computer program if not modified that of another colleague. This was to solve a certain kind of structure. In most cases, those programs were and would not be used by any other person apart from the developer. Therefore, without extensive costs of development, most of the associates of the profession of engineering would not use it. It was because of this that Wilson came up with the expansion of the universal dynamic and structural expansion of Structural Analysis Program [SAP]
The SAP program then utilized the technology that was at that time and each joint could have a range of up to six displacement degrees-of-freedom. In it was the creation of an integer pointer array that was six times that number of the joints. That array therefore allowed every node a varying number of displacements. As a result, the equilibrium equations that were for the unknown displacements were the only ones formed during the compilation of the stiffness of the element. The program thus was just like special purpose programs. These used fixed rate of displacements per node. Within a year, the opening SAP program was brought up. This was by Wilson together with three other scholars. It was generously dispersed to all undergraduates as well as the professionals. It thus became the fundamental preparatory program for a series of projects on finite element. Dr. Jurgen Bathe in 1973 did not only revise the vibrant response options. He also came up with another documentation so as to produce SAP IV which became among the universal quickest and prevalent program of structural analysis. It was then liberally circulated so as to effectively transfer the finite element research that had progressed in Berkeley to other learning institutions as well as to the profession.
Application to creep and incremental construction
Assumption of the condition of stress in the concrete dam before applying the concept of hydrostatic loading was important while the Nor fork dam project was in progress. It was evident that since several factors like temperature changes resulting from the heat of hydration as well as the properties of creep and modulus and geometry of the concrete were all operational at the same time, they influenced the condition of the stress of the dam. Therefore, Ian King decided to work on the same topic in 1962 as his Ph.D. thesis. King worked under professors Clough and Raphael. To solve that problem, he made more adjustments to Wilson’s program. King research was there and was used the construction stresses the foot high. Two years were the approximated time for completion of the construction of a typical monolith. Just before its completion, the maximum temperature stresses also occurred.
It was therefore basic that he came up with a complex re-meshing system while working on the solution procedure. This was to ensure the problem was solved on the 1964 mainframe computers. That venture described the merits of the procedure and creep during the construction which according to the awareness this form of analysis has not been operated on another dam. It was later in the 1970s that professor Scordelis together with his graduate scholars performed a relevant research on incremental construction as well as the creep of bridges and shells among other concrete structures.
Analysis of underground Structures
With the guidance of Professor Eberhart, a part of photo elastic analysis of the Oroville Dam which was started in 1957 was carried out. This was on analyzation of underground structures. by use of photo elasticity, there was an exploration of a number of the possible models. This experimental approach however happened to consume much time and was also quite expensive. As a result of this, it was probable for a prompt research on the stress concentration in not only underground concrete structures but the rock ones also. This was after the advancement of the automated finite element program which was use for the project of the Norfork Dam. Clough and Raphael later in 1962 were approved to use their approach of finite element. This was in a class of problems by the Department of Resources of the State of California. Joe Rashid who was a fresh graduate student was appointed to work on that project.
Apart from him working on the project, under Pister’s direction on ‘Finite Element Analysis of Ax symmetric solids’, in 1964 he compete his Ph.D. work. He also advanced Wilson’s program, so it was able to solve ax symmetric structures that were subjected to axis metric loads.
The attention of Professor O. C. Zienkiewicz was drawn to approach of finite element analysis. He then was at Norway University. This was in 1960 after Clough produced his primary paper on the use of the same. Few weeks down the line, Clough received Zienkiewicz’s invitation on a seminar to specifically present that approach [finite element] to his students. Among the universal specialists on the appliance of finite difference method was Zienkiewicz. This was specifically in relation to solving problems of continuum mechanics in Civil Engineering. Because of this, Clough prepared for a presentation of the relative advantages of both methods. Zienkiewicz almost concerted to Clough’s approach after hi handling of several queries on his finite element method.
In the 1964 and 1965 academic year, Clough was at Cambridge University working as a Visiting Professor. He still continued to oversee a couple of students via mail. He also met other intercontinental professionals in the same field. He thus wrote a series of research papers on not only the finite element method but on earthquake engineering, as well. Within that period, Dr. Zienkiewicz who then worked in Swansea at the University of Wales as a professor asked Clough among other experts in the same field to partake of a conference at the same university on the Stress Analysis. Their lectures were amassed in the book ‘Stress Analysis’ with each expert writing an independent chapter. For instance, Clough and Zienkiewicz who associated with B. Fraeijs de Veubeke who did not give any numerical presentations but he came up with a triangular element of the six node pane as well as alluded to the performance of Pager and Synge [5]of 1947 wrote on chapters dealing with the finite element method.
The Aero jet experience
In August 1963, Wilson agreed to the position of a senior research engineer. This was in the Solid Rocket Plant at Aero jet general which was specifically in California at Sacramento located at an approximated distance of 80 miles from Berkeley. It was at this place that he went on with both the research on computerization of the finite element method and the program on computer development. At that place, he joined efforts with Stan Dong who is currently professor Emeritus of UCLA. In that collaboration were also Len Herrmann who is lately Professor of UC Davis and Professor Pister’s Ph.D. students. This group consequently was directly an extension of their research programs [Clough and Pister]. In the following couple of years, at Aerojet, Berkeley and Swansea, the two classical fields, continuum mechanics together with the structural analysis would be unified. Since then, the new field was named computational mechanics.
Stress analysis of a grain which was a side view of a solid rocket propellant was among the first difficulties at Aerojet with which Wilson put his program of double-dimensional finite element into use to solve. The grain was induces to internal pressure Figure 9. These stresses were put in to comparison with a photo elastic analysis that had been achieved before. He was startled when the finite element stress was higher than that of the photo elastic one with around 5% while at its maximum stress. This was especially because it was expected that it was less than that of the exact result. Therefore, an original analysis of the photo elastic was done. The fresh outcome was closer to the computer one. The photo elastic results still turned out lesser than those of the finite element stresses, owing to the trio-dimensional effects close to the stress concentration. Wilson thus came up with a definite purpose edition with mesh generation to his program. This was to routinely perform stress analyses for solid motors that were of arbitrary geometry. In the following few years, the size of the photo elastic group at Aerojet was decreased. Most photo elasticity engineers were either writing or using the finite element programs.
In 2000, SAP modeled and also analyzed three distinct bridges. These were known as Culpeper, Welland River Bridge and Door Creek Bridge. Several analyses were performed on these bridges including linear and non-linear analyses. The Door Creek Bridge underwent both of them while the other two only went through the linear elastic ones. In 1998 however, the Culpeper as well as the Welland River Bridge were modeled and also analyzed by Issa. The re-analysis therefore was only for the verification of the SAP way of modeling. It was also for the identification of the least requisite pre-stressing level. This was across the intersections under AASHTO LRFD in 2007 service loads. It also included impact. At that time, analysis of The Door Creek Bridge was for the assessment of the least needed pre-stressing level. However, this was across longitudinal linkages as well as the transverse ones. It was also for the prediction of the bridge’s structural behavior when overloaded.
References.
International Modal Analysis Conference, & Proulx, T. (2011). Dynamics of bridges: proceedings of the 28th IMAC, a conference on structural dynamics, 2010. New York, Springer.
National Cooperative Highway Research Program (Etats-Unis). (2008). Development of design specifications and commentary for horizontally curved concrete box-girder bridges.
Suthar, K. (2007). The effect of dead, live and blast loads on a suspension bridge. Thesis (M.S.)–University of Maryland, College Park, 2007.
