THEORY OF PLATES AND SHELLS S.TIMOSHENKO Professor Emeritus of Engineering Mechanics Stanford Universily S.WOINOWSKY-KRIEGER Professor of Engineering Mechanics Laval Universily SECOND EDITION MONOGRAPHS MCGRAW-HILL BOOK COMPANY Auckland Bogota Guatemala Hamburg Lisbon London Madrid Mexico New Delhi Panama Paris San Juan Sao Paulo Singapore Sydney Tokyo
THEORY OF PLATES AND SHELLS S. TIMOSHENKO Professor Emeritus of Engineering Mechanics Stanford University S. WOINOWSKY-KRIEGER Professor of Engineering Mechanics Laval University SECOND EDITION MCGRAW-HILL BOOK COMPANY Auckland Bogota Guatemala Hamburg Lisbon London Madrid Mexico New Delhi Panama Paris San Juan SSo Paulo Singapore Sydney Tokyo
PREFACE Since the publication of the first edition of this book,the application of the theory of plates and shells in practice has widened considerably, and some new methods have been introduced into the theory.To take these facts into consideration,we have had to make many changes and additions.The principal additions are (1)an article on deflection of plates due to transverse shear,(2)an article on stress concentrations around a circular hole in a bent plate,(3)a chapter on bending of plates resting on an elastic foundation,(4)a chapter on bending of anisotropic plates,and (5)a chapter reviewing certain special and approximate methods used in plate analysis.We have also expanded the chapter on large deflections of plates,adding several new cases of plates of variable thickness and some numerical tables facilitating plate analysis. In the part of the book dealing with the theory of shells,we limited ourselves to the addition of the stress-function method in the membrane theory of shells and some minor additions in the flexural theory of shells. The theory of shells has been developing rapidly in recent years,and several new books have appeared in this field.Since it was not feasible for us to discuss these new developments in detail,we have merely referred to the new bibliography,in which persons specially interested in this field will find the necessary information. S.Timoshenko S.Woinowsky-Krieger
PREFACE Since the publication of the first edition of this book, the application of the theory of plates and shells in practice has widened considerably, and some new methods have been introduced into the theory. To take these facts into consideration, we have had to make many changes and additions. The principal additions are (1) an article on deflection of plates due to transverse shear, (2) an article on stress concentrations around a circular hole in a bent plate, (3) a chapter on bending of plates resting on an elastic foundation, (4) a chapter on bending of anisotropic plates, and (5) a chapter reviewing certain special and approximate methods used in plate analysis. We have also expanded the chapter on large deflections of plates, adding several new cases of plates of variable thickness and some numerical tables facilitating plate analysis. In the part of the book dealing with the theory of shells, we limited ourselves to the addition of the stress-function method in the membrane theory of shells and some minor additions in the flexural theory of shells. The theory of shells has been developing rapidly in recent years, and several new books have appeared in this field. Since it was not feasible for us to discuss these new developments in detail, we have merely referred to the new bibliography, in which persons specially interested in this field will find the necessary information. S. Timoshenko S. Woinowsky-Krieger
NOTATION x,y,a Rectangular coordinates r,Polar coordinates ra,ry Radii of curvature of the middle surface of a plate in xz and yz planes, respectively h Thickness of a plate or a shell g Intensity of a continuously distributed load p Pressure P Single load Weight per unit volume ,Normal components of stress parallel to y,and z axes Normal component of stress parallel to n direction Ur Radial stress in polar coordinates ,o Tangential stress in polar coordinates Shearing stress Tav:Tan Tuz Shearing stress components in rectangular coordinates u,v,w Components of displacements eUnit elongation Unit elongations in z,y,and z directions Er Radial unit elongation in polar coordinates a,se Tangential unit elongation in polar coordinates Unit elongations of a shell in meridional direction and in the direetion of parallel circle,respectively Y,Y,Yy Shearing strain components in rectangular coordinates r Shearing strain in polar coordinates E Modulus of elasticity in tension and compression G Modulus of elasticity in shear y Poisson's ratio V Strain energy D Flexural rigidity of a plate.or shell M,M Bending moments per unit length of sections of a plate perpendicular to x and y axes,respectively M Twisting moment per unit length of section of a plate perpendicular to x axis Ma,M Bending and twisting moments per unit length of a section of a plate perpendicular to n direction Q,Q Shearing forces parallel to z axis per unit length of sections of a plate perpendicular to x and y axes,respectively Q Shearing force parallel to a axis per unit length of section of a plate perpendicular to n direction Na,N Normal forces per unit length of sections of a plate perpendicular to x and y directions,respectively xiii
NOTATION x, y, z Rectangular coordinates r, 0 Polar coordinates rx, ry Radii of curvature of the middle surface of a plate in xz and yz planes, respectively h Thickness of a plate or a shell q Intensity of a continuously distributed load p Pressure P Single load 7 Weight per unit volume (Tx, Radial stress in polar coordinates at, (re Tangential stress in polar coordinates r Shearing stress Txy, Txz, Tyz Shearing stress components in rectangular coordinates u, v, w Components of displacements e Unit elongation «*, «•/, fz Unit elongations in x, y, and z directions er Radial unit elongation in polar coordinates et , eo Tangential unit elongation in polar coordinates ttp, eo Unit elongations of a shell in meridional direction and in the direction of parallel circle, respectively yxy, Vxz, jyz Shearing strain components in rectangular coordinates 7r0 Shearing strain in polar coordinates E Modulus of elasticity in tension and compression G Modulus of elasticity in shear v Poisson's ratio V Strain energy D Flexural rigidity of a plate or shell Mx, My Bending moments per unit length of sections of a plate perpendicular to x and y axes, respectively Mxy Twisting moment per unit length of section of a plate perpendicular to x axis Mn, Mnt Bending and twisting moments per unit length of a section of a plate perpendicular to n direction Qx, Qy Shearing forces parallel to z axis per unit length of sections of a plate perpendicular to x and y axes, respectively Qn Shearing force parallel to z axis per unit length of section of a plate perpendicular to n direction JVx, Ny Normal forces per Unit length of sections of a plate perpendicular to x and y directions, respectively
xiv NOTATION N Shearing force in direction of y axis per unit length of section of a plate perpendicular to x axis M,M,M Radial,tangential,and twisting moments when using polar coordinates Q,Q.Radial and tangential shearing forces N,N:Normal forces per unit length in radial and tangential directions ri,r:Radii of curvature of a shell in the form of a surface of revolution in meridional plane and in the normal plane perpendicular to meridian, respectively xe,xe Changes of curvature of a shell in meridional plane and in the plane perpendicular to meridian,respectively XO Twist of a shell X,Y,2 Components of the intensity of the external load on a shell,parallel to x,y,and z axes,respectively Ne,Ne,Ne Membrane forces per unit length of principal normal sections of a shell Ma,M.Bending moments in a shell per unit length of meridional section and a section perpendicular to meridian,respectively x=,xe Changes of curvature of a cylindrical shell in axial plane and in a plane perpendicular to the axis,respectively Ne,N=,N Membrane forces per unit length of axial section and a section perpen- dicular to the axis of a cylindrical shell M,M:Bending moments per unit length of axial section and a section perpen- dicular to the axis of a cylindrical shell,respectively M Twisting moment per unit length of an axial section of a cylindrical shell Shearing forces parallel to z axis per unit length of an axial section and a section perpendicular to the axis of a cylindrical shell,respectively log Natural logarithm logio,Log Common logarithm
Nxu Shearing force in direction of y axis per unit length of section of a plate perpendicular to x axis Mr , Mt, MH Radial, tangential, and twisting moments when using polar coordinates Qr , Qt Radial and tangential shearing forces Nn Nt Normal forces per unit length in radial and tangential directions ri, r2 Radii of curvature of a shell in the form of a surface of revolution in meridional plane and in the normal plane perpendicular to meridian, respectively X<pi XQ Changes of curvature of a shell in meridional plane and in the plane perpendicular to meridian, respectively X6ip Twist of a shell X, Y, Z Components of the intensity of the external load on a shell, parallel to x, y, and z axes, respectively N<p, Ne, NfB Membrane forces per unit length of principal normal sections of a shell MBy M<p Bending moments in a shell per unit length of meridional section and a section perpendicular to meridian, respectively Xx, x<p Changes of curvature of a cylindrical shell in axial plane and in a plane perpendicular to the axis, respectively N<p, Nx, Nx<p Membrane forces per unit length of axial section and a section perpendicular to the axis of a cylindrical shell M9, Mx Bending moments per unit length of axial section and a section perpendicular to the axis of a cylindrical shell, respectively Mx<p Twisting moment per unit length of an axial section of a cylindrical shell QiP, Qx Shearing forces parallel to z axis per unit length of an axial section and a section perpendicular to the axis of a cylindrical shell, respectively log Natural logarithm log10, Log Common logarithm
Contents Preface Notation Introduction 7 1. Bending of Long Rectangular Plates to a Cylindrical Surface ........ 4 1.1 Differential Equation for Cylindrical Bending of Plates............ 4 1.2 Cylindrical Bending of Uniformly Loaded Rectangular Plates with Simply Supported Edges… 6 1.3 Cylindrical Bending of Uniformly Loaded Rectangular Plates with Built-in Edges ...... 13 1.4 Cylindrical Bending of Uniformly Loaded Rectangular Plates with Elastically Built-in Edges ......... 17 1.5 The Effect on Stresses and Deflections of Small Displacements of Longitudinal Edges in the Plane of the Plate ................ 20 1.6 An Approximate Method of Calculating the Parameter u 24 This page has been reformatted by Knovel to provide easier navigation. vii
vii This page has been reformatted by Knovel to provide easier navigation. Contents Preface ..................................................................... v Notation .................................................................... xiii Introduction ............................................................ 1 1. Bending of Long Rectangular Plates to a Cylindrical Surface .......................................... 4 1.1 Differential Equation for Cylindrical Bending of Plates ......................................... 4 1.2 Cylindrical Bending of Uniformly Loaded Rectangular Plates with Simply Supported Edges ........................................................... 6 1.3 Cylindrical Bending of Uniformly Loaded Rectangular Plates with Built-in Edges ........ 13 1.4 Cylindrical Bending of Uniformly Loaded Rectangular Plates with Elastically Built-in Edges ........................................................... 17 1.5 The Effect on Stresses and Deflections of Small Displacements of Longitudinal Edges in the Plane of the Plate .................... 20 1.6 An Approximate Method of Calculating the Parameter u ................................................. 24
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
