Tins work introduced a numerical modelling of bending analysis of plates, continuous in two perpendicular directions. The model involved a finite element analysis including the design of mesh to determine the responses (deflection and stresses) of a typical two way plate These w ere synthesized in a finite element MATLAB program for the model The aim of the model was to develop coefficients for the internal stresses and displacements that could be used for two-way design of plates. By using a unit value of distributed load, moment coefficients for SSCS. CSCS. SSSC, CCCC, CCSS and CCSC were obtained for various aspect ratios ranging from 10 to 2.0. The analytical results of the two-way typical plate were compared to the British standard (BS8110) and the Indian standard (IS456) codes as well as that of the exact solution by Timoshenko and Woinowsky-Kxieger (1959). Finally, comparison was done for the coefficients of deflection and moments at various aspect ratios for the typical two-way plates and the ideal plates for different support conditions. From the comparison with the BSXI10 and IS456 it was observed that the maximum center moment (Mx) percentage difference recorded for SSSC and SCCS were 7 2% and 13.3% respectively; their center moment (My) vary above 25%. However, SSCS, CSCS, CCCC and CCSC had maximum center moment difference greater than 25%. These differences is due to the fact that the BSX110 and IS456 moment coefficients have been modified by factors or safety for design purposes while the coefficients obtained in this research work are results of pure and unmodified elastic analysis. Comparing the analytical results of the typical plate with the exact solution by Timoshenko and Woinowsky it was seen that SSCS, CSCS and CCSC recorded maximum center moment difference of 8 4%, 10.9% and 15.7% respectively. However. SSSC and CCCC showed maximum center moment variation of 84.6% and 24.5% respectively These differences are likely to disappear on refining the mesh The analytical results of the typical plate showed good agreement with the results obtained for the ideal plate in the deflection coefficients but the moment coefficients varied Based on the results of analysis, bending moment coefficients were proposed for the design of two-way plates in Nigeria.
TABLE OF CONTENTS
Front Cover i
Title page ii
Certification iii
Declaration iv
Dedication v
Acknowledgement vi
Table of
Contents vii
List of Tables ix
List of Figures xi
Abstract xii
CHAPTER 1: BACKGROUND OF STUDY
1.1
Statement of the Problem 3
1.2 Aim and Objectives 4
1.3 Scope of the Study 5
1.4 Significance of Study 5
1.5 Limitations 6
CHAPTER 2: LITERATURE REVIEW
2.1
Finite Strip Method 7
2.2 Finite Difference Method 8
2.3
The Finite Element Method 9
2.4 Why finite element method 14
2.5
Deductions from Literature Review 15
CHAPTER 3: METHODOLOGY
3.1 Classification of Panels 18
3.2 Development of the Numerical Method 19
3.2.1 Idealization of the continuous
surface 19
3.2.2 Selection of displacement
models 20
3.2.3
Relating the generalized displacement within an element to its nodal
displacement. 20
3.2.4 Strain –
displacement relationship 20
3.2.5 Relating the
internal stresses to the strains and to nodal displacements 21
3.3
Application of the Numerical Method to known Plate Problems 21
3.3.1 Discretization or mesh
generation 21
3.4 The Finite Element Formulation 21
3.4.1 Selection of displacement
models 23
3.4.2 Relating the generalized
displacement within an element to its nodal displacement 24
3.4.3 Strain – displacement
relationship 25
3.4.4 Relating the internal stresses
to the strains and to nodal displacements 25
3.5 Program Development 26
3.6 Response Points 27
3.7 Application of the Numerical Solutions to
known Plate Problems 28
3.7.1
Properties of the rectangular Isotropic Plate 28
CHAPTER 4: RESULTS AND DISCUSSIONS
4.1 Result
of the Typical Irregular Plate 30
4.2 Comparison of Numerical Result with Exact
Solution
37
4.3 Comparison of BS8110, IS456 and Exact
Solution 98
4.4 Comparison of Numerical Solution with
BS8110 and IS456:2000 51
4.5 Discussion of Results 75
4.5.1 Comparison of numerical results of typical
continuous
Plate
with the Exact Solution 76
4.5.2 Comparison of numerical results with BS8110
and IS456 76
4.5.3 Comparison of center deflection of typical
continuous plate,
Ideal
and Exact Solution 77
4.5.4 Derivation of moment coefficient for the
Nigerian use 78
CHAPTER 5: CONCLUSIONS AND RECOMMENDATIONS
5.1 Conclusions 81
5.2 Recommendations 83
REFERRENCES
APPENDIX 1
LIST
OF TABLES PAGES
4.1: Results for a plate perpendicular in two
directions 31
4.2 Result
of the numerical analysis of the ideal plates 34
4.3: Deflection coefficients for typical plate,
ideal plate and exact solution 35
4.4: Torsional moment coefficient from the
numerical analysis 36
4.5: Comparison of numerical results with exact solution for aspect
ratio 1.0. 37
4.5.1: Comparison of numerical results with exact solution for aspect
ratio 1.1. 38
4.5.2: Comparison of numerical results with exact solution for aspect
ratio 1.2. 39
4.5.3: Comparison of numerical results with exact solution for aspect
ratio 1.3. 40
4.5.4: Comparison of numerical results with exact solution for aspect
ratio 1.4. 41
4.5.5: Comparison of numerical results with exact solution for aspect
ratio 1.5. 42
4.5.6: Comparison of numerical results with exact solution for aspect
ratio 1.6. 43
4.5.7: Comparison of numerical results with exact solution for aspect
ratio 1.7 44
4.5.8: Comparison of numerical results with exact solution for aspect
ratio 1.8 45
4.5.9: Comparison of numerical results with exact solution for aspect
ratio 1.9 46
4.5.10: Comparison of numerical
results with exact solution for aspect ratio 2.0 47
4.6: Comparison of BS8110 and IS456 code values with
exact
Solution for various aspect ratios 48
4.7: Comparison of the numerical results with BS
8110 and IS 456 in
percentage
(%) for aspect ratios 1.0 (Positive centre moments) 51
4.7.1: Comparison of the numerical results with BS
8110 and IS 456 in
percentage
(%) for aspect ratios 1.0 (Edge centre moments) 52
4.7.2: Comparison of the numerical results with BS
8110 and IS 456 in
percentage
(%) for aspect ratios 1.1 (Positive centre moments) 53
4.7.3: Comparison of the numerical results with BS
8110 and IS 456 in
percentage
(%) for aspect ratios 1.1 (Edge centre moments) 54
4.7.4: Comparison of the numerical results with BS
8110 and IS 456 in
percentage
(%) for aspect ratios 1.2 (Positive centre moments) 55
4.7.5: Comparison of the numerical results with BS
8110 and IS 456 in
percentage
(%) for aspect ratios 1.2 (Edge centre moments) 56
4.7.6: Comparison of the numerical results with BS
8110 and IS 456 in
percentage
(%) for aspect ratios 1.3 (Positive centre moments) 57
4.7.7: Comparison of the numerical results with BS
8110 and IS 456 in
percentage
(%) for aspect ratios 1.3 (Edge centre moments) 58
4.7.8: Comparison of the numerical results with BS
8110 and IS 456 in
percentage
(%) for aspect ratios 1.4 (Positive centre moments) 59
4.7.9: Comparison of the numerical results with BS
8110 and IS 456 in
percentage
(%) for aspect ratios 1.4 (Edge centre moments) 60
4.7.10:
Comparison of the numerical results with BS 8110 and IS 456 in
percentage
(%) for aspect ratios 1.5 (Positive centre moments) 61
4.7.11:
Comparison of the numerical results with BS 8110 and IS 456 in
percentage
(%) for aspect ratios 1.5 (Edge centre moments) 62
4.7.12:
Comparison of the numerical results with BS 8110 and IS 456
in
percentage (%) for aspect ratios 1.75 (Positive centre moments) 63
4.7.13:
Comparison of the numerical results with BS 8110 and IS 456
in
percentage (%) for aspect ratios 1.75 (Edge centre moments) 64
4.7.14:
Comparison of the numerical results with BS 8110 and IS 456
in
percentage (%) for aspect ratios 2.0 (Positive centre moments) 65
4.7.15:
Comparison of the numerical results with BS 8110 and IS 456
in
percentage (%) for aspect ratios 2.0 (Edge centre moments) 66
4.8: Variation of the numerical results with BS
8110 and IS 456: 2000 67
4.9: Summary of the comparison of results of the
typical plate with the
results
of ideal plate in percentages 68
4.10: The proposed Bending Moment Coefficients of
Rectangular
Plates
for the Nigerian use 79
4.11: The Proposed Bending Moment Coefficients of
Rectangular
Plates for the Nigerian use 80
LIST
OF FIGURES
3.1: Numbering of the rectangular plates 16
3.2 Typical plate continuous in two
perpendicular
direction
*Referenced from TIMOSHENKO’S work
(1959) 17
3.3:
Boundary
Conditions Present in the Plate Model of Fig. 3.2 17
3.4: Discretized typical plate continuous in two Perpendicular
Direction 22
4.1 Graph of
typical and ideal solution for SSCS 69
4.2 Graph of
typical and ideal solution for CSCS 69
4.3
Graph of typical and ideal solution
for SSSC 69
4.4 Graph of
typical and ideal solution for CCCC 70
4.5 Graph of
typical and ideal solution for SCCS 70
4.6 Graph of
typical and ideal solution for CCSC 70
4.7 Graph
of typical, ideal and exact solution for SSCS 71
4.8 Graph
of typical, ideal and exact solution for CSCS 71
4.9 Graph of
typical, ideal and exact solution for SSSC 71
4.10 Graph of
typical, ideal and exact solution for CCCC 72
4.11 Graph of
typical, ideal and exact solution for SCCS 72
4.12 Graph of
typical, ideal and exact solution for CCSC 72
4.13 Graph of
typical, ideal, BS8110 and IS456 solution for SSCS 73
4.14 Graph of
typical, ideal, BS8110 and IS456 solution for CSCS 73
4.15 Graph of
typical, ideal, BS8110 and IS456 solution for SSSC 73
4.16 Graph of
typical, ideal, BS8110 and IS456 solution for CCCC 74
4.17 Graph
of typical, ideal, BS8110 and IS456 solution for SCCS 74
4.18
Graph of typical, ideal, BS8110 and
IS456 solution for CCSC 74
CHAPTER 1
BACKGROUND
OF STUDY
One
of the common structural members which is widely employed in many fields of
engineering are plate structures. Plates are flat structural members which are
always bounded by two parallel planes referred to as faces and edge (Vantsel,
E. and Krauthemmer, T. 2001). Plates are used to model retaining walls, bridge
decks and floor slabs (osadebe et al.,
2016). Plates may be visualized as intersecting, closely spaced, grid beams and
thus, are considered to be highly indeterminate. In fact, plate structures
always have complex boundary conditions and loading patterns that are often
difficult to find for the exact solution. Numerical approaches for these cases
therefore serve as a good alternative approach. Therefore, the numerical
modelling of the bending analysis of plates continuous in two perpendicular
directions is a practically important case.
Analytically, the plate is a complex
problem when it is continuous in two perpendicular directions. (Bari et al., 2004). Remarkable authors have
proposed tentative solutions. Timoshenko and Woinowsky-Kreiger (1959) developed
classic thin-plate solutions for an isotropic linear elastic thin plate. In the
past years, plate problems have been treated by the use of Fourier series or
trigonometric series as the shape function of the deformed plate. However, no
matter the approach used, the use of trigonometric series (double Fourier
series and single Fourier series) has been predominant. Most times, when it is
becoming intractable to use the trigonometric series, trial and error means of
getting a shape function that would approximate the deformed shape of the plate
would be used (Ibearugbulem et al.,
2011). Osadebe et al., (2016) worked
on application of the galerkin-vlasov method to the flexural analysis of simply
supported rectangular kirchhoff plates under uniform loads. Here, the Galerkin
- Vlasov variational method was used to present a general formulation of the
Kirchhoff plate problem with simply supported edges and under distributed
loads. The problem was then solved to obtain the displacements, and the bending
moments in a Kirchhoff plate with simply supported edges and under uniform
load. Maximum values of the displacement and the bending moments were found to
occur at the plate center. Several other methods were explored by researchers
like Emmanuel et al., (2018), Okafor
and Udeh (2015), Taylor & Govindgee (2004). These are generally accepted as
approximate methods.
This research work introduces the
numerical modelling of bending analysis of plates, continuous in two
perpendicular directions. A finite element MATLAB program was developed for
this model. Because of the irregular shape the data was designed for automatic
mesh generation. The developed program was used to determine design
coefficients as obtained in BS8110 and IS456. The aim is to compare the
theoretical coefficients obtained in this work with that of the exact solution,
BS8110 and Indian code, and hence, make a proposal for such coefficients that
could be used in Nigeria. BS 8110 is the official code of practice used in the
design of structures in Nigeria.
Accordingly, design tables and charts from
BS 8110 are used to analyze and design structural elements such as columns,
bases, slabs, beams, etc. This was the official practice to date before and
after the independence of Nigeria. All the same, Nigeria's frequent building
collapse is at a disturbing rate, with a moderately large impact.
Despite all the investigations, no serious
and justifiable development is recorded to avert this happenings (Mansor et al., 2017). Findings from researchers
(Tanko et al., 2013; Ede, 2013;
Ayodeji, 2011; Abubakar et al., 2014)
concluded that substandard construction materials are the real causes of
structural failure and collapse in Nigeria.
The Nigerian Engineer has not taken any
bold step to evaluate the suitability of this code (BS 8110) to the Nigerian
construction industry particularly at this time, when there are several
structural failure cases in Nigeria. India, a commonwealth nation, has taken
the bold step to pull away from the British code (BS 8110) and established the
Indian code of practice (IS456:2000) which serves her purpose better.
Therefore, the present research work
examines the bending moment coefficients from BS 8110 used to analyze and
design two-way slab continuous in two perpendicular directions, using a
numerical approach (FEM)
The finite element analysis method is
generally the most powerful, versatile and precise analytical method of all
available methods and has quickly become a well-known technique for the
computer solution of complex engineering problems. It is very effective in
analyzing complex structures such as modeling a plate, continuous in two
perpendicular directions; with complex geometrical properties, material
properties and conditions of support and subject to a number of conditions of
loading.
Finite element modelling involves the
description of the domain, discretization of the continuum into sub-domain of
appropriate shape called finite elements and selection of the interpolation
function that will be used to reduce complex plate bending equations into
simple differential and integral equation which are solved simultaneously so as
to get the stiffness matrix, Daniel (2015).
Finite element analysis has several
advantages in comparison to other methodologies for numerical analysis. It is
very powerful and applicable in many engineering issues such as structural
system displacements, stress-strain analysis, etc.
A few applications of finite element
modelling that have been used for engineering research are seen in the research
works of Mustafa et al.,2013; kanber
and bozkurt 2005; bari et al.,2004;Agrawal
et al., 2016 etc
1.2 STATEMENT OF THE PROBLEM
Given a typical plate and a number of
ideal plates, all of which are continuous in two perpendicular directions; it
is required to carry out structural analysis of such plates. Over the years,
dividing the typical model into a number of ideal plates has been the normal
practice for such analysis. Again, the BS8110 has been the official code of
practice used in the concrete plates (slabs) as a structural element.
Based on the knowledge of numerical
analysis, the present work developed a numerical model that will carry out
structural analysis of such plate wholistically. The formulation of the model
shall be synthesized in a finite element based MATLAB program.
The
developed MATLAB program will be used to study the structural response (bending
moment coefficients) when the analytical structures (plates) are subjected to
unit loads. These results will be used for a proposal of bending moment
coefficients for the Nigerian use. It is expected that this proposal will form
the bed rock in future in case Nigeria decides to have her own code.
1.3
AIM AND OBJECTIVES
The aim of this work is to develop a
numerical model for the structural analysis of plates, continuous in two
perpendicular directions using a MATLAB computer program for the determination
of the static response of a two-way slab using the finite element method. The
objectives of this research work are:
- To develop a finite element MATLAB
program for the numerical analysis of the two way plate model.
- To
validate the model by statistically comparing analytical results
(responses) obtained with the program to other results in published
literature.
- Application
of the finite element program to the model in other to determine the
responses of the two way plate model.
- To
determine bending moment coefficients using the developed program and
compare the results with exact solution, BS8110, IS456. The analytical
structures here will be a set of ideal plates, all of which are continuous
in two perpendicular directions.
- Based
on the comparison in 4 above, make a proposal for bending moment
coefficients for use in Nigeria. (Note that Nigeria has not developed any
such coefficients, BS8110 has been in use in Nigeria).
1.3 SCOPE OF THE STUDY
This study is limited to the analysis of
two-way slab. There are other types of slabs which are not of interest to this
work. The MATLAB program to be developed for this work is for the elastic
analysis of the static response of two-way slab. Plastic analysis is,
therefore, not within the scope of this work. Also, dynamic and stability analysis
will not be considered.
1.4 SIGNIFICANCE OF STUDY
The present work introduced the numerical
modelling of bending analysis of plates continuous in two perpendicular
directions. The following are the significance of the study:
i.
The result of this study
will be used as a standard for the construction of two-way plates taking into
consideration the fact that local requirements are often different compared to
the provisions of BS8110 which has been the official code of practice.
ii.
Serpell et al., (2002) observed that the
development process of standard is difficult, cumbersome and unstable. This
research will be a contribution to the body of literature in this area of
construction standard and regulations in Nigeria, thereby constituting the
empirical literature for future research in this subject area.
iii. Finite
element analysis of irregularly shaped rectangular plates continuous in two
perpendicular directions will obviously involve a mesh design which is very
crucial and challenging. The mesh design in this research will therefore
present a knowledge based approach to other researchers, which is necessary to
experience in this aspect of finite element analysis.
iv. The
bending moment coefficients that will be obtained from this work, after being
appropriately enhanced by a factor of safety, could serve as a proposal for a
possible Nigerian code of practice in this regard.
1.5
LIMITATIONS.
One
way slab is not considered in this study. Again, the response obtained in this
research are linearly elastic. Plastic analysis is not considered.
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