Analysis of deep beam using Finite Element Method

Research article ISSN 2277 – 9442

Analysis of deep beam using Finite Element Method

1

Enem, J.I.,1Ezeh, J.C.,2Mbagiorgu, M.S.W.,1Onwuka, D.O.

14

Department of Civil Engineering, Federal University of Technology, Owerri

2

Department of Civil Engineering, Enugu State University of science and technology, Enugu

doi: 10.6088/ijaser.0020101035

Abstract: There are several analytical tools available for analyzing deep beams. Among all the available

analytical methods, finite element analysis (FEA) offers a better option. This method discretizes the continuum into finer elements, making it somewhat tedious and complex to handle manually. Computer– assisted programmes make the use of finite element method quite easier. The availability of advanced analysis tools based on finite elements and matrix structural analysis concepts has enabled engineers to model, analyze and design innovative complex and unusual structures. This paper focuses on development of novel software (ADBUFEM) for analysis of deep beams. The software is simple, modular and governed by fundamental principles and aspects of finite element methodology. The results from this study were compared with results from renowned finite element software (ADINA) and the results obtained were shown to reasonably agree.

Key Words: Finite Element, Concrete Idealization, Development of Software, Discretization, Deep Beam.

1. Introduction

Existing methods of predicting Deep Beam behaviour involve either elastic theory or semi–empirical equation, neither of which is entirely satisfactory (Yoo, et. al, 2004; Kong and Chemrouk, 2002). The basic assumption that plane sections remain plane after loading and that the material is homogeneous and elastic do not hold for deep beams. Finite element method (FEM) offers a powerful and general analytical tool for studying the behaviour of reinforced concrete deep beams (Sciarmmarella, 1963; Singh, et. al. 1980 and Tan, et. al. 2003). Finite element method as a tool can provide realistic and satisfactory solutions for linear and nonlinear behaviour of deep beam structural elements (Quanfeny and Hoogenboom, 2004; Samir and Chris, 2005).

Finite element method uses many elements in analysing any continuum which makes it cumbersome for manual analysis. As the number of elements used increases, the manpower and effort required to prepare the relevant and necessary data and interpret the results increase. Therefore computer based programmes help to reduce the effect and rigour involved by analyzing a continuum using manual analysis. The limitations of some programmes may prevent the use of a large number of finite elements to idealize the continuum. The effectiveness of a programme depends essentially on the following factors: Firstly, the use of efficient finite elements; Secondly, efficient programming methods and effective use of the available computer hardware and software. Thirdly, a very important aspect in the development of a finite element programme is the use of appropriate numerical techniques (Klus, 1990). In this research work a novel software was developed in the light of making the analysis of deep beams using finite element method simple and easy. The name of the software developed is Analysis of Deep Beam Using Finite Element

2. Finite Element Idealization

The aim of this study is to develop a simple finite element model aided with computer programme which leads to a satisfactory discretization of any continuum into finer elements, analysis and prediction of the behavior of concrete deep beams. The concrete was idealized using 2–D plane stress elements. A host of finite element programmes are presently available for use in plane stress analysis. The mostly used are isoparametric elements as a result of their versatility and simplicity (Khalaf, 2007; Zienkiewicz and Taylor, 2000). There are several isoparametric elements applied in finite element analysis ranging from four-node to higher order isoparametric elements.

3. Development of Adbufem

Finite element modeling in engineering without computer codes has little practical value. Programmes are an invaluable resource with which to perform a wide variety of computing tasks. Hence, high premium is placed on the development of a functional programme packages for the finite element analysis.

Employing developed programme in finite element analysis, the stiffness and load matrices are accomplished automatically; it only requires the input of information concerning the physical characteristics of the members, joints and boundary conditions of the structure. In writing this programme some factors and considerations were taken into cognizance which include the following:

1. Accuracy and Correctness: The important objective in developing this software is that it

should meet its specification in solving both small and large problems. ADINA Finite Element Software was used to validate the programme’s accuracy and capability.

2. Clarity: A programme is necessary as complex as the algorithm which describes it. Programme

clarity is important to the programmer himself, in the design and debugging of the programmes and to others who may have to read and amend the programme at later stage.

3. Efficiency and Cost: The actual cost of FEA software is relative and this cost does not really

depend on the programmes capabilities or the efficiency of the software. Some programmes have limited capability and are expensive. The cost of executing a computer programme is normally measured in terms of

i) Time taken by the computer to carry out the sequence of operations involved. ii) The amount of computer storage memory used in doing so.

4. Problem Size and Mesh Generation: What makes software capable is its ability to handle a

sizeable problem. The programme developed in this work contains an excellent means of mesh generation and can handle a large problem.

Essentially, there are three basic tasks/steps that were involved in the development of this software:

1. Specifying the task that the software is to carry out in terms of the input data to be supplied and the output data or results to be produced.

2. Devising an algorithm or sequence of steps, by which the software can produce the required output from the available input.

3. Expressing this algorithm as a computer programme in a programming language. The software package is composed of various subroutines, each representing a particular step of the finite element analytical procedure. The programme is designed in a manner that the programme calls for various subroutines in a predefined sequence. The algorithm of this programme is outlined in the flowcharts shown in figure1.

COMPUTATION OF EFFECTIVE DISPLACEMENT AND SUPPORT REACTIONS SUBROUTINE

START INPUT SUBROUTINE REDUCTION OF [K] MATRIX SUBROUTINE COMPUTATION OF DISPLACEMENT SUBROUTINE

COMPUTATION OF STRAIN, STRESS AND NODAL FORCES SUBROUTINE PRINT STOP IF ANY FORCES IS APPLY

GENERATE A MESH FOR THE BEAM SUBROUTINE

GENERATE NODAL COORDINATE SUBROUTINE

GENERATE ELEMENT CONNECTIVITIES

SUBROUTINE

GENERATE GRADIENT MATRIX [B]

SUBROUTINE

GENERATE STIFFNESS MATRIX [K] SUBROUTINE

4. Structural Modeling

Figure 2: Model for continuous deep beam under two–point loading.

Model of the beam that is analyzed in this study is well articulated in figure 2. The beam is modeled as a continuous deep beam of total length 12m and 1.8m depth, with fixity at the edges and at the centre of the beam. Two–point loading is used in the analysis. The magnitude of the loads are equal to 50KN, however, they are located at 4m from the edges of the beam.

The analysis method used in this research is the displacement method. Approximate displacement functions were assumed and the functions expressed in terms of the approximate models. Displacement compatibility conditions are satisfied and the governing equations that are generated are approximate equilibrium equations. These are solved to yield the unknown nodal displacement. The displacement formulations are summarized as:

1. The structure is discretized into a number of finite elements separated by imaginary lines or surfaces. .

2. The elements are assumed to be interconnected only at a discrete number of nodal points situated on their boundaries.

3. A set of functions is chosen to define uniquely the state of displacement within each “finite element” in terms of its nodal displacements.

4. The displacement function now defines uniquely the state of strain within an element in terms of the nodal displacements. These strains, together with any initial strains and the constitutive properties of the material will define the state of stress throughout the element and also on its boundaries.

5. A system of forces concentrated at the nodes and equilibrating the boundary stresses and any distributed loads is determined, resulting in a stiffness relationship.

The choice of the shape or configuration of the element to use is most often governed by the geometry of the body, the number of independent space coordinate necessary to describe the problem and engineering judgment. For the course of this work element configuration that was used is rectangular element. The finite element mesh consists of 36 elements and is shown in figure 3. It is imperative to adopt a systematic method for numbering the element and nodes of the discretized structure. The proper numbering will minimize the band width of the final system of algebraic equations.

LEGEND E2 = Element two = node number two 2 E1 2m 2m 2m 2m 2m 2m 10 1 2 3 4 5 6 7 15 16 17 18 19 20 21 8 9 11 12 13 14 24 29 30 31 32 33 34 35 22 23 25 26 27 28 38 43 44 45 46 47 48 49 36 37 39 40 41 42 E2 E3 E4 E5 E6 E12 E18 E24 E30 E36 E11 E17 E23 E29 E35 E10 E16 E22 E28 E34 E9 E15 E21 E27 E33 E8 E14 E20 E26 E₃₂ E7 E13 E19 E25 E31 0 . 3 m 0 . 3 m 0 .3 m 0 .3 m 0 .3 m 0 .3 m 50KN 50KN

Figure 3: Rectangular discretization of the structure into finite elements with node and element

numbering.

4.1 Choice of displacement function

The choice of an appropriate displacement function is a very important step in the analysis of a structure using the finite element method. The assumption that elements are interconnected only at their nodes implies that the continuity requirements are satisfied only at the nodal points. See figure 4

5 0 K N 5 0 K N U3 6 U35 U3 0 U2 9 U32 U3 1 U3 4 U33 U38 U3 7 U4 0 U3 9 U4 8 U4 7 U42 U4 1 U4 4 U43 U4 6 U4 5 U5 0 U4 9 U5 2 U5 1 U6 0 U5 9 U54 U5 3 U5 6 U55 U5 8 U5 7 U6 2 U6 1 U6 4 U6 3 U7 2 U7 1 U6 6 U6 5 U68 U6 7 U7 0 U6 9 2 m 2 m 2 m 2 m 2 m 2 m U8 8 U8 7 U9 0 U8 9 U9 2 U9 1 U9 6 U9 5 U8 0 U7 9 U82 U8 1 0 .3 m 0 .3 m 0 .3 m 0 .3 m 0 .3 m 0 .3 m L E G E N D UI = H o r iz o n ta l d is p la c e m e n t U2 = V e r tic a l d is p la c e m e n t U2 U1 U4 U3 U12 U1 1 U6 U5 U8 U7 U1 0 U9 U1 4 U1 3 U1 6 U1 5 U2 4 U2 3 U1 8 U1 7 U2 0 U1 9 U2 2 U2 1 U2 6 U2 5 U2 8 U2 7 U8 6 U8 5 U98 U9 7 U9 4 U9 3 U7 6 U7 5 U7 8 U7 7 U8 4 U8 3 U7 4 U7 3

5. Discussion of the results

The predicted behavior and the comparisons between the ADBUFEM and ADINA software are summarized thus:

5.1 Displacement Magnitude

Apparently, from results obtained it could be seen vividly that the difference in the displacement magnitude between the ADBUFEM and ADINA is insignificant and not quite appreciable. The values from ADBUFEM are comparatively smaller. The concentration of the displacement is higher within the central region of the beam, and propagates appreciably at the axis of loading from the top (points of loading). The fixity area has smaller value of displacement.

Again, the maximum and minimum effective displacement from the both software occurred at the same point of loading and fixity respectively. ADBUFEM predicted its own maximum and minimum displacements to be 1.15063E – 005 and 0.00 while, ADINA has its own to be 1.209E-005 and 6.355E-14 (approximately 0.000).

The analytical displacement magnitude– distance relationships of both bottom and top fibres are shown in figures 5 and 6 respectively. The representations clearly show that the result from both software (ADBUFEM and ADINA) follow the same pattern. The displacement magnitude (effective) – distance relationship of the bottom fibre shows a linear graph towards the support region. This linearity shows that the beam will be rigid around this area. In the same light the trajectory at the span is parabolic climaxing at the point of loading, which explains that the deformation of the beam will concentrate more on the span. The displacement magnitude (effective) – distance relationship of the top fibre shows that, the deformation or failure at the region of loading will be sudden which is indicated by the linearity of the graph towards that region. 0 2 4 6 8 10 12 0 0.2 0.4 0.6 0.8 1 1.2x 10 -5 Distance B o tt o m F ib re D is p la c e m e n t( m ) fo r T ri a n g u la r

Figure 5a: Displacement magnitude (Effective)–Distance Relationship for rectangular idealization for

Figure 5b: Displacement Magnitude (Effective)– Relationship for ADINA (bottom fibre)

Figure 6b: Displacement Magnitude (Effective)–Relationship for ADINA(top fiber)

5.2 Conclusion

It is well recognized that the exact analysis of concrete deep beams is a complex problem. The numerical method of analysis developed in the present study is capable of providing information about the responses of concrete deep beams with ease in contrast to other analysis techniques. This ease can be attributed to the use of computer programmes. In principle ADBUFEM has the capacity of idealizing any continuum into finer mesh which in turn enhances the results obtained, and with a high speed of operation. The results obtained from ADBUFEM compared favourably with those from ADINA. The development of ADBUFEM is geared towards empowering engineers to automate most of their analysis and design routines of deep beams.

6. References

1. Yoo, T. M; Doh, J. H., and Guan, H. (2004) Experimental work on Reinforced and Prestressed Concrete Deep Beans with Various Web Openings. Griffith school of Engineering, Griffith University Gold Coast Campus, Queensland, Australia.

2. Kong, F. K; and Chemrouk, M. (2002) Reinforced concrete deep beans. University of Newcastle Upon Yyne

3. Sciarmmarella, C. A. (1963). Effect of holes in deep beams with reinforced vertical edges”, Engineering progress, University of Fla, 17, No. 12.

4. Singh, R., Ray, S. P. and Reddy, C. S. (1980). Some tests on reinforced concrete deep beams with and without opening in the web, The Indian concrete journal, vol. 54, No. 7, Pp. 189 – 194. 5. Tan, K. H., Tong, K. and Tang, C. Y. (2003). Consistent strut – and – tie modeling of deep beams

with web openings, Magazine of concrete Research, 55(1), 572-582.

6. Yang, K. H., Eun, H.C. and Chung, H. S., The influence of web openings on the structural behavior of reinforced high – strength concrete deep beams, Engineering structures, in press,

available on online www.sciencedirect.com.

7. Quanfeny, W., Hoogenboom, P. C. J. (2004). Nonlinear Analysis of Reinforced Concrete Continuous Deep Beams using stringer – Panel Model. Asian Journal of Civil Engineering (Building and Housing). 5(1-2), 25.

8. Samir, M. O. H. D. and Chris, T. M. (2005). Nonlinear Finite Element Analysis of Reinforced Concrete Deep Beams. Department of Engineering, University of Cambridge, Cambridge, CB2 IPZ, England.

9. Klus–Jurgen, B. (1990). Finite Element Procedures in Engineering Analysis.

10. Khalaf, I. M. (2007). Prediction of Behaviour of Reinforced Concrete Deep Beams with Web Openings Using Finite Elements. Civil Engineering Department, University of Mosul.

11. Zienkiewicz, O. C. and Taylor, R. L. (2000). The finite element method, Butterworth – Heinemann, U. K.