viii Contents 1.7 Long Uniformly Loaded Rectangular Plates Having a Small Initial Cylindrical Curvature...... 27 1.8 Cylindrical Bending of a Plate on an Elastic Foundation.… 30 2. Pure Bending of Plates .............. 33 2.9 Slope and Curvature of Slightly Bent Plates....… 33 2.10 Relations between Bending Moments and Curvature in Pure Bending of Plates ........... 37 2.11 Particular Cases of Pure Bending 42 2.12 Strain Energy in Pure Bending of Plates...... 46 2.13 Limitations on the application of the Derived Formulas 47 2.14 Thermal Stresses in Plates with Clamped Edges. 49 3. Symmetrical Bending of Circular Plates 51 3.15 Differential Equation for Symmetrical Bending of Laterally Loaded Circular Plates ........ 51 3.16 Uniformly Loaded Circular Plates ............ 54 3.17 Circular Plate with a Circular Hole at the Center ............ 58 3.18 Circular Plate Concentrically Loaded … 63 3.19 Circular Plate Loaded at the Center 67 3.20 Corrections to the Elementary Theory of Symmetrical Bending of Circular Plates ...... 72 This page has been reformatted by Knovel to provide easier navigation
viii Contents This page has been reformatted by Knovel to provide easier navigation. 1.7 Long Uniformly Loaded Rectangular Plates Having a Small Initial Cylindrical Curvature ..................................................... 27 1.8 Cylindrical Bending of a Plate on an Elastic Foundation ........................................ 30 2. Pure Bending of Plates ................................... 33 2.9 Slope and Curvature of Slightly Bent Plates ........................................................... 33 2.10 Relations between Bending Moments and Curvature in Pure Bending of Plates ............ 37 2.11 Particular Cases of Pure Bending ................ 42 2.12 Strain Energy in Pure Bending of Plates ...... 46 2.13 Limitations on the Application of the Derived Formulas ......................................... 47 2.14 Thermal Stresses in Plates with Clamped Edges ........................................................... 49 3. Symmetrical Bending of Circular Plates ........ 51 3.15 Differential Equation for Symmetrical Bending of Laterally Loaded Circular Plates ........................................................... 51 3.16 Uniformly Loaded Circular Plates ................. 54 3.17 Circular Plate with a Circular Hole at the Center .......................................................... 58 3.18 Circular Plate Concentrically Loaded ........... 63 3.19 Circular Plate Loaded at the Center ............. 67 3.20 Corrections to the Elementary Theory of Symmetrical Bending of Circular Plates ....... 72
Contents iⅸ 4. Small Deflections of Laterally Loaded Plates 79 4.21 The Differential Equation of the Deflection Surface............ 79 4.22 Boundary Conditions........ 83 4.23 Alternativme Method of Derivation of the Boundary Conditions........... 88 4.24 Reduction of the Problem of Bending of a Plate to That of Deflection of a Membrane........... 92 4.25 Effect of Elastic Constants on the Magnitude of Bending Moments 97 4.26 Exact Theory of Plates ............ 98 5. Simply Supported Rectangular Plates .......... 105 5.27 Simply Supported Rectangular Plates under Sinusoidal Load 105 5.28 Navier Solution for Simply Supported Rectangular Plates ......... 108 5.29 Further Applications of the Navier Solution ....... 111 5.30 Alternate Solution for Simply Supported and Uniformly Loaded Rectangular Plates ............. 113 5.31 Simply Supported Rectangular Plates under Hydrostatic Pressure ..... 124 5.32 Simply Supported Rectangular Plate under a Load in the Form of a Triangular Prism .... 130 This page has been reformatted by Knovel to provide easier navigation
Contents ix This page has been reformatted by Knovel to provide easier navigation. 4. Small Deflections of Laterally Loaded Plates ................................................................ 79 4.21 The Differential Equation of the Deflection Surface ......................................................... 79 4.22 Boundary Conditions .................................... 83 4.23 Alternativme Method of Derivation of the Boundary Conditions .................................... 88 4.24 Reduction of the Problem of Bending of a Plate to That of Deflection of a Membrane .................................................... 92 4.25 Effect of Elastic Constants on the Magnitude of Bending Moments .................. 97 4.26 Exact Theory of Plates ................................. 98 5. Simply Supported Rectangular Plates ........... 105 5.27 Simply Supported Rectangular Plates under Sinusoidal Load ................................. 105 5.28 Navier Solution for Simply Supported Rectangular Plates ....................................... 108 5.29 Further Applications of the Navier Solution ........................................................ 111 5.30 Alternate Solution for Simply Supported and Uniformly Loaded Rectangular Plates ........................................................... 113 5.31 Simply Supported Rectangular Plates under Hydrostatic Pressure .......................... 124 5.32 Simply Supported Rectangular Plate under a Load in the Form of a Triangular Prism ..... 130
X Contents 5.33 Partially Loaded Simply Supported Rectangular Plate 135 5.34 Concentrated Load on a Simply Supported Rectangular Plate ........ 141 5.35 Bending Moments in a Simply Supported Rectangular Plate with a Concentrated Load ........... 143 5.36 Rectangular Plates of Infinite Length with Simply Supported Edges ........... 149 5.37 Bending Moments in Simply Supported Rectangular Plates under a Load Uniformly Distributed over the Area of a Rectangle..… .144 158 5.38 Thermal Stresses in Simply Supported Rectangular Plates ........... 162 5.39 The Effect of Transverse Shear Deformation on the Bending of Thin Plates ..................... 165 5.40 Rectangular Plates of Variable Thickness… 173 6. Rectangular Plates with Various Edge Conditions.......... 180 6.41 Bending of Rectangular Plates by Moments Distributed along the Edges ........ 180 6.42 Rectangular Plates with Two opposite Edges Simply Supported and the Other Two Edges Clamped ........ 185 6.43 Rectangular Plates with Three Edges Simply Supported and One Edge Built In .... 192 This page has been reformatted by Knovel to provide easier navigation
x Contents This page has been reformatted by Knovel to provide easier navigation. 5.33 Partially Loaded Simply Supported Rectangular Plate ......................................... 135 5.34 Concentrated Load on a Simply Supported Rectangular Plate ....................... 141 5.35 Bending Moments in a Simply Supported Rectangular Plate with a Concentrated Load ............................................................. 143 5.36 Rectangular Plates of Infinite Length with Simply Supported Edges .............................. 149 5.37 Bending Moments in Simply Supported Rectangular Plates under a Load Uniformly Distributed over the Area of a Rectangle ..................................................... 158 5.38 Thermal Stresses in Simply Supported Rectangular Plates ....................................... 162 5.39 The Effect of Transverse Shear Deformation on the Bending of Thin Plates ...................... 165 5.40 Rectangular Plates of Variable Thickness ..................................................... 173 6. Rectangular Plates with Various Edge Conditions ........................................................ 180 6.41 Bending of Rectangular Plates by Moments Distributed along the Edges ......... 180 6.42 Rectangular Plates with Two opposite Edges Simply Supported and the Other Two Edges Clamped .................................... 185 6.43 Rectangular Plates with Three Edges Simply Supported and One Edge Built In ..... 192
Contents xi 6.44 Rectangular Plates with All Edges Built In ..197 6.45 Rectangular Plates with One Edge or Two Adjacent Edges Simply Supported and the Other Edges Built In ................ 205 6.46 Rectangular Plates with Two Opposite Edges Simply Supported,the Third Edge Free,and the Fourth Edge Built In or Simply Supported ............... 208 6.47 Rectangular Plates with Three Edges Built In and the Fourth Edge Free ....... 211 6.48 Rectangular Plates with Two Opposite Edges Simply Supported and the Other Two Edges Free or Supported Elastically ...214 6.49 Rectangular Plates Having Four Edges Supported Elastically or Resting on Corner Points with All Edges Free 218 6.50 Semi-infinite Rectangular Plates under Uniform Pressure........... 221 6.51 Semi-infinite Rectangular Plates under Concentrated Loads ....... 225 7. Continuous Rectangular Plates 229 7.52 Simply Supported Continuous Plates .......... 229 7.53 Approximate Design of Continuous Plates with Equal Spans ......... 236 7.54 Bending of Plates Supported by Rows of Equidistant Columns (Flat Slabs)............... 245 7.55 Flat Slab Having Nine Panels and Slab with Two Edges Free ....................... 253 This page has been reformatted by Knovel to provide easier navigation
Contents xi This page has been reformatted by Knovel to provide easier navigation. 6.44 Rectangular Plates with All Edges Built In ... 197 6.45 Rectangular Plates with One Edge or Two Adjacent Edges Simply Supported and the Other Edges Built In ..................................... 205 6.46 Rectangular Plates with Two Opposite Edges Simply Supported, the Third Edge Free, and the Fourth Edge Built In or Simply Supported ......................................... 208 6.47 Rectangular Plates with Three Edges Built In and the Fourth Edge Free ........................ 211 6.48 Rectangular Plates with Two Opposite Edges Simply Supported and the Other Two Edges Free or Supported Elastically .... 214 6.49 Rectangular Plates Having Four Edges Supported Elastically or Resting on Corner Points with All Edges Free ............... 218 6.50 Semi-infinite Rectangular Plates under Uniform Pressure ......................................... 221 6.51 Semi-infinite Rectangular Plates under Concentrated Loads ..................................... 225 7. Continuous Rectangular Plates ...................... 229 7.52 Simply Supported Continuous Plates ........... 229 7.53 Approximate Design of Continuous Plates with Equal Spans ......................................... 236 7.54 Bending of Plates Supported by Rows of Equidistant Columns (Flat Slabs) ................. 245 7.55 Flat Slab Having Nine Panels and Slab with Two Edges Free ................................... 253
xii Contents 7.56 Effect of a Rigid Connection with Column on Moments of the Flat Slab 257 8. Plates on Elastic Foundation ............. 259 8.57 Bending Symmetrical with Respect to a Center ........... 259 8.58 Application of Bessel Functions to the Problem of the Circular Plate ....... 265 8.59 Rectangular and Continuous Plates on Elastic Foundation ......... 269 8.60 Plate Carrying Rows of Equidistant Columns...... 276 8.61 Bending of Plates Resting on a Semi- infinite Elastic Solid 278 9. Plates of Various Shapes 282 9.62 Equations of Bending of Plates in Polar Coordinates....... 282 9.63 Circular Plates under a Linearly Varying L0ad… 285 9.64 Circular Plates under a Concentrated Load 290 9.65 Circular Plates Supported at Several Points along the Boundary 293 9.66 Plates in the Form of a Sector ................... 295 9.67 Circular Plates of Nonuniform Thickness .... 298 9.68 Annular Plates with Linearly Varying Thickness .......... 303 This page has been reformatted by Knovel to provide easier navigation
xii Contents This page has been reformatted by Knovel to provide easier navigation. 7.56 Effect of a Rigid Connection with Column on Moments of the Flat Slab ........................ 257 8. Plates on Elastic Foundation .......................... 259 8.57 Bending Symmetrical with Respect to a Center .......................................................... 259 8.58 Application of Bessel Functions to the Problem of the Circular Plate ....................... 265 8.59 Rectangular and Continuous Plates on Elastic Foundation ........................................ 269 8.60 Plate Carrying Rows of Equidistant Columns ....................................................... 276 8.61 Bending of Plates Resting on a Semiinfinite Elastic Solid ...................................... 278 9. Plates of Various Shapes ................................ 282 9.62 Equations of Bending of Plates in Polar Coordinates .................................................. 282 9.63 Circular Plates under a Linearly Varying Load ............................................................. 285 9.64 Circular Plates under a Concentrated Load ............................................................. 290 9.65 Circular Plates Supported at Several Points along the Boundary ........................... 293 9.66 Plates in the Form of a Sector ...................... 295 9.67 Circular Plates of Nonuniform Thickness ..... 298 9.68 Annular Plates with Linearly Varying Thickness ..................................................... 303