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Ribbed (Waffle) Rectangular Reinforced Concrete Slabs at Ultimate Limit State

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Ribbed (Waffle) Rectangular Reinforced Concrete Slabs at Ultimate Limit State AN INVESTIGATION INTO THE BEHAVIOUR OF HOLLOW RIBBED (WAFFLE) RECTANGULAR REINFORCED CONCRETE SLABS AT ULTIMATE LIMIT STATE by Sai Lung HO, B.Sc.(Hons) A Thesis Presented For the award of the Degree of Doctor of Philosophy of the Council for National Academic Awards Department of Civil Engineering, Surveying & Building Dundee Institute of Technology July, 1989. ACKNOWLEDGEMENTS The author would like to express his sincere gratitude to his Supervisor Dr. Susanta Sarkar, Head of Department of Civil Engineering, Surveying & Building for his encouragement, guidance and supervision given to the author throughout the course of this research which have been invaluable and have made this research possible. The author is indebted to the valuable advice and help received from his second Supervisor Dr. J.G.L. Munday, Senior Lecturer in the Department of Civil Engineering, Surveying & Building at Dundee Institute of Technology. Thanks are also due to my external supervisor Mr. C.M.C. Marshall, Senior Lecturer at Hatfield Polytechnic for his helpful discussion and advice. Dr. Thorpe is acknowledged for his assistance in the Finite Element Analysis. Thanks are also due to the staff of Computer Centre particularly Dr. J. Clayton and Mr. S. Gardner, for their help and advice. Grateful thanks are given to the entire technical staff of the Department of Civil Engineering, Surveying & Building for their practical assistance, preparing and testing the models and specially to Mr. J. Galloway, Mr. A. Thomson and Mr. J. Webster who were most closely associated with and coordinated much of the technical work. The author would like to record his thanks to the Dundee Institute of Technology for providing the research studentship which enabled him to carry out the work. Finally, the author would like to take this opportunity to express his greatest appreciation and gratitude to his wife, Susan, for her patience and support throughout this work, and especially for her exceptional job typing the manuscript. (i) AN INVESTIGATION INTO THE BEHAVIOUR OF HOLLOW RIBBED (WAFFLE) RECTANGULAR REINFORCED CONCRETE SLABS AT ULTIMATE LIMIT STATE by Sai Lung HO, B.Sc.(Hons) ABSTRACT The elastic to ultimate load behaviour of reinforced concrete waffle slabs subjected to uniformly distributed load and 4-point loads are reported in this thesis. A total of twenty 1/4- and one 1/2-scale reinforced concrete waffle model slabs, with different area of steel and rib depth to slab thickness ratio and various edge conditions were made and tested to destruction. The theoretical analysis of the slabs essentially consists of three phases: (i) elastic behaviour up to the appearance of first cracks; (ii) nonlinear behaviour due to progressive concrete cracking and yielding of reinforcements; and (iii) ultimate load behaviour. Two 2-dimensional elastic models viz., thin plate bending element and grillage element models, and one 3-dimensional elastic-fracture model using finite element package LUSAS/AN have been employed to investigate the applicability of these methods for the analysis of the slabs. In addition to these, three ultimate load methods viz., modified yield line method, equivalent open grillage analysis and modified strip method were developed and used to predict the ultimate load carrying capacity of the slabs. Based on the experimental and theoretical results, it is concluded that the behaviour of R.C. waffle slabs is very similar to that of R.C. solid slabs except that waffle slabs tend to attract more bending moments at the support. As a result, the span moments of the slab are reduced when compared to the span moments given by BS8110 for solid slabs. The combined bending and torsion interaction at the rib joints is also considered in this investiagtion. It is found that for slabs with restrained edges membrance action significantly increases the load carrying capacity. A method of predicting this enhancement has also been developed. The theoretical results obtained by the proposed methods of analysis for the three phases above show a good correlation with the experiemntal results. ( i i ) LIST OF TABLES TABLE PAGE 2.1 Summary of the Design methods for Reinforced Concrete 6 Slabs 2.2 Results of Analysis of Mid-span Deflection of Stiffened Plate Subjected to Uniformly Distributed Load Using Various Approaches 9 2.3 Distribution of Momentsin Internal Square Plate Panel 13 2.4 Moments in Rectangular Plain Internal Plate Panel 13 2.5 Moments in Rectangular Internal Ribbed Plate Panel 14 3.1 Coefficients Obtained by Filoneuko-Borodich (16) for Simply Supported Rectangular Plates Subjected to Uniformly Distributed Load 28 3.2 Coefficients Obtained by Timoshenko for Simply Supported Rectangular Plates Subjected to Uniformly Distributed Load 32 3.3 Coefficients Obtained by Timoshenko for Built-in Supported Rectangular Plates Subjected to Uniformly Distributed Load 33 3.4 Huber's Coefficients for Simply Supported Rectangular Orthotropic Plates with H = y(Dx.Dy) , Subjected to Uniformly Distributed Load Extracted from Timoshenko and Woinowisky-Krieger 36 3.5 Central Deflection of an Isotropic Square Plate using Different Meshes by Zienkiewicz (107) 66 5.1 Results of Bridge Model Tests by Reddy and Henry 86 5.2 Data of R.C. Waffle Model Slabs by Marshall 87 5.3 Theoretical and Test Results of R.C. Waffle Model Slabs by Marshall 88 5.4 Results of Bridge Models by Nacaraja and Lash 91 5.5 Results of Prestressed Concrete Grillage Models by Reynolds 93 5.6 Results of the Twenty-Four T-Beams at Cracking Load by Kirk 107 5.7 Results of the Eighteen T-Beams at Ultimate Load by Kirk 108 5.8 Results of the Fourteen Rectangular Beams with All Stirrups Subjected to Bending and Torsion by Ramakrishnan and Vijayarangan 110 (iii) TABLE PAGE 5.9 The Results of the Eleven Tests on Longitudinally Reinforced T-Beams Without Stirrups by Victor and Ferguson 112 5.10 The Cracking Torque of Reinforced T-Beams Subjected to Pure Torsion by Kirk 113 5.11 Results of R.C. Slab Tests with All Edges Fully Fixed 117 5.12 Results of Unreinforced Concrete Slab Tests with All Edges Fully Fixed 118 5.13 Results of R.C. Slab Tests with One Side Simply Supported and Three Sides Fully Fixed 118 6.1 R.C. Waffle Model Slab Schedule 124 6.2a Results of Concrete Trial Mixes 127 6.2b Results of Trial Mix at Age 7-days 127 6.2c Results of Trial Mix at Age 28-days 127 6.3 Concrete Strength of the Model Slabs 140 6.4 Yield Strengths of Reinforcing Bars 141 6.5 Summary of Principal Test Results at Cracking Loads 141 6.6 Summary of Principal Test Results at FailureLoads 142 7.1 Plate with All Edges Fully Fixed and Subjected to Uniformly Distributed Load 159 7.2 Sectional properties of 'QF4' elements 168 7.3 Comparison of the Maximum Deflection between the Experimental and the Analytical Results at Cracking Load 169 7.4 Torsional Constants For R.C. Waffle Model Slabs 186 7.5 Sectional Properties of 'GRIL' Elements 186 7.6 Moment coefficients for R.C. Waffle Slabs Using Finite Element Method 190 7.7 Moment Coefficients For R.C.Waffle Slabs Using Grillage Analysis 191 7.8 Moment Coefficients For Square R.C. Waffle Slabs Using Finite Element Method 191 7.9 Comparison Between Experimental Loads and Ultimate Loads Using Modified Yield Line Method for R.C. waffle slabs 208 (iv) PAGE Results Of The Load Capacity Due To Membrane Action For Unreinforced Concrete Slabs With Restrained Edges 214 Results of the Load Capacity Taking Account of Membrane Action for R.C. Slabs with Restrained Edges 214 Comparison Between The Analytical And Experimental Results For R.C. Waffle Slabs In Series A To F 215 The Analytical Results For R.C. Waffle Slabs In Series H 218 The Analytical Results For R.C. Waffle Slabs In Series I 220 The Analytical Results For R.C. Waffle Slabs In Series J 223 Edge Beam To Span Moment RatioFor R.C. Waffle Slabs In Series H To I 226 Analytical Results for R.C. Waffle Slabs In Series H To J 226 Analytical Values Obtained From Eqn.7.54a & Eqn.7.54b For R.C. Waffle Slabs In Series H To J 226 Comparison Of The Experimental And Theoretical Results Using Equivalent Open Grillage Analysis 234 Comparison Of The Experimental And Theoretical Results Using Modified Strip Method 244 Loads Of R.C. Waffle Slabs At Deflection Limit Of Span/250 248 Deflections of R.C. Waffle Slabs at the Initial Cracking Load 251 Comparison Between the Failure and Cracking Loads 260 Experimental and Theoretical Ultimate Load Capacities of R.C. Waffle Slabs 266 Comparison between the Theoretical and Experimental Ultimate Loads 271 Summary of the Various Parameters and the Failure Load of R.C waffle slabs. 273 Predicted Load Capacities for Various Areas of Steel and d/t Ratios 273 (v) LIST OF FIGURES FIGURE PAGE 2.1 Deflection Line of the Ribbed Slab 8 2.2 Distribution of Moment for the 45 Ribbed Slab by Testa and Levy (115) 12 3.1 Stresses Acting on a Thin Plate Element 19 3.2 Deflection of a Thin Plate Element 20 3.3 Distribution of Stresses on a Thin Plate Element 22 3.4 Free Body Diagram of a Thin Plate Element 23 3.5 Simply Supported Rectangular Plate 25 3.6 Simply Supported Plate with Infinite Width 29 3.7 Free Body Diagram of a Grillage Element 39 3.8 Distribution of Stresses and Strains of a Grillage Element 41 3.9 Notations of a Ribbed Plate Element 42 3.10 Distribution of Stresses and Strains of a Ribbed Plate Element 44 3.11 Notations of a One-way Ribbed Slab 47 3.12 Notations of a Two-way Ribbed Slab 48 3.13 Layout of a Grillage System 51 3.14 Displacement of a Grillage Element 51 3.15 Co-ordinate axes of a Grillage System 53 3.16 Rectangular
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