103,99 €
Mehr erfahren.
- Herausgeber: John Wiley & Sons
- Kategorie: Fachliteratur
- Sprache: Englisch
Plate and Shell Structures:
Selected Analytical and Finite Element Solutions
Maria Radwañska, Anna Stankiewicz, Adam Wosatko, Jerzy Pamin
Cracow University of Technology, Poland
Comprehensively covers the fundamental theory and analytical and numerical solutions for different types of plate and shell structures
Plate and Shell Structures: Selected Analytical and Finite Element Solutions not only provides the theoretical formulation of fundamental problems of mechanics of plates and shells, but also several examples of analytical and numerical solutions for different types of shell structures. The book contains advanced aspects related to stability analysis and a brief description of modern finite element formulations for plates and shells, including the discussion of mixed/hybrid models and locking phenomena.
Key features:
- 52 example problems solved and illustrated by more than 200 figures, including 30 plots of finite element simulation results.
- Contents based on many years of research and teaching the mechanics of plates and shells to students of civil engineering and professional engineers.
- Provides the basis of an intermediate-level course on computational mechanics of shell structures.
The book is essential reading for engineering students, university teachers, practitioners and researchers interested in the mechanics of plates and shells, as well as developers testing new simulation software.
Sie lesen das E-Book in den Legimi-Apps auf:
Seitenzahl: 551
Veröffentlichungsjahr: 2016
Ähnliche
Table of Contents
Cover
Title Page
Copyright
Dedication
Preface
Notation
Detailed notation for theoretical analysis
Conversions between imperial and metric system units
Part 1: Fundamentals: Theory and Modelling
Chapter 1: General Information
1.1 Introduction
1.2 Review of Theories Describing Elastic Plates and Shells
1.3 Description of Geometry for 2D Formulation
1.4 Definitions and Assumptions for 2D Formulation
1.5 Classification of Shell Structures
References
Chapter 2: Equations for Theory of Elasticity for 3D Problems
Reference
Chapter 3: Equations of Thin Shells According to the Three-Parameter Kirchhoff–Love Theory
3.1 General Equations for Thin Shells
3.2 Specification of Lame Parameters and Principal Curvature Radii for Typical Surfaces
3.3 Transition from General Shell Equations to Particular Cases of Plates and Shells
3.4 Displacement Equations for Multi-Parameter Plate and Shell Theories
3.5 Remarks
References
Chapter 4: General Information about Models and Computational Aspects
4.1 Analytical Approach to Statics, Buckling and Free Vibrations
4.2 Approximate Approach According to the Finite Difference Method
4.3 Computational Analysis by Finite Element Method
4.4 Computational Models – Summary
Reference
Chapter 5: Description of Finite Elements for Analysis of Plates and Shells
5.1 General Information on Finite Elements
5.2 Description of Selected FEs
5.3 Remarks on Displacement-based FE Formulation
References
Part 2: Plates
Chapter 6: Flat Rectangular Membranes
6.1 Introduction
6.2 Governing Equations
6.3 Square Membrane under Unidirectional Tension
6.4 Square Membrane under Uniform Shear
6.5 Pure In-Plane Bending of a Square Membrane
6.6 Cantilever Beam with a Load on the Free Side
6.7 Rectangular Deep Beams
6.8 Membrane with Variable Thicknesses or Material Parameters
References
Chapter 7: Circular and Annular Membranes
7.1 Equations of Membranes – Local and Global Formulation
7.2 Equations for the Axisymmetric Membrane State
7.3 Annular Membrane
References
Chapter 8: Rectangular Plates under Bending
8.1 Introduction
8.2 Equations for the Classical Kirchhoff–Love Thin Plate Theory
8.3 Derivation of Displacement Equation for a Thin Plate from the Principle of Minimum Potential Energy
8.4 Equation for a Plate under Bending Resting on a Winkler Elastic Foundation
8.5 Equations of Mindlin–Reissner Moderately Thick Plate Theory
8.6 Analytical Solution of a Sinusoidally Loaded Rectangular Plate
8.7 Analysis of Plates under Bending Using Expansions in Double or Single Trigonometric Series
8.8 Simply Supported or Clamped Square Plate with Uniform Load
8.9 Rectangular Plate with a Uniform Load and Various Boundary Conditions – Comparison of STSM and FEM Results
8.10 Uniformly Loaded Rectangular Plate with Clamped and Free Boundary Lines – Comparison of STSM and FEM Results
8.11 Approximate Solution to a Plate Bending Problem using FDM
8.12 Approximate Solution to a Bending Plate Problem using the Ritz Method
8.13 Plate with Variable Thickness
8.14 Analysis of Thin and Moderately Thick Plates in Bending
References
Chapter 9: Circular and Annular Plates under Bending
9.1 General State
9.2 Axisymmetric State
9.3 Analytical Solution using a Trigonometric Series Expansion
9.4 Clamped Circular Plate with a Uniformly Distributed Load
9.5 Simply Supported Circular Plate with a Concentrated Central Force
9.6 Simply Supported Circular Plate with an Asymmetric Distributed Load
9.7 Uniformly Loaded Annular Plate with Static and Kinematic Boundary Conditions
References
Part 3: Shells
Chapter 10: Shells in the Membrane State
10.1 Introduction
10.2 General Membrane State in Shells of Revolution
10.3 Axisymmetric Membrane State
10.4 Hemispherical Shell
10.5 Open Conical Shell under Self Weight
10.6 Cylindrical Shell
10.7 Hemispherical Shell with an Asymmetric Wind Action
References
Chapter 11: Shells in the Membrane-Bending State
11.1 Cylindrical Shells
11.2 Spherical Shells
11.3 Cylindrical and Spherical Shells Loaded by a Uniformly Distributed Boundary Moment and Horizontal Force
11.4 Cylindrical Shell with a Spherical Cap – Analytical and Numerical Solution
11.5 General Case of Deformation of Cylindrical Shells
11.6 Cylindrical Shell with a Semicircular Cross Section under Self Weight – Analytical Solution of Membrane State
11.7 Cylindrical Scordelis-Lo Roof in the Membrane-Bending State – Analytical and Numerical Solution
11.8 Single-Span Clamped Horizontal Cylindrical Shell under Self Weight
References
Chapter 12: Shallow Shells
12.1 Equations for Shallow Shells
12.2 Pucher's Equations for Shallow Shells in the Membrane State
12.3 Hyperbolic Paraboloid with Rectangular Projection
12.4 Remarks on Engineering Applications
References
Chapter 13: Thermal Loading of Selected Membranes, Plates and Shells
13.1 Introduction
13.2 Uniform Temperature Change along the Thickness
13.3 Linear Temperature Change along the Thickness – Analytical Solutions
References
Part 4: Stability and Free Vibrations
Chapter 14: Stability of Plates and Shells
14.1 Overview of Plate and Shell Stability Problems
14.2 Basis of Linear Buckling Theory, Assumptions and Computational Models
14.3 Description of Physical Phenomena and Nonlinear Simulations in Stability Analysis
14.4 Analytical and Numerical Buckling Analysis for Selected Plates and Shells
14.5 Snap-Through and Snap-Back Phenomena Observed for Elastic Shallow Cylindrical Shells in Geometrically Nonlinear Analysis
References
Chapter 15: Free Vibrations of Plates and Shells
15.1 Introduction
15.2 Natural Transverse Vibrations of a Thin Rectangular Plate
15.3 Parametric Analysis of Free Vibrations of Rectangular Plates
15.4 Natural Vibrations of Cylindrical Shells
15.5 Remarks
References
Part 5: Aspects of FE Analysis
Chapter 16: Modelling Process
16.1 Advantages of Numerical Simulations
16.2 Complexity of Shell Structures Affecting FEM
16.3 Particular Requirements for FEs in Plate and Shell Discretization
References
Chapter 17: Quality of FEs and Accuracy of Solutions in Linear Analysis
17.1 Order of Approximation Function versus Order of Numerical Integration Quadrature
17.2 Assessment of Element Quality via Spectral Analysis
17.3 Numerical Effects of Shear Locking and Membrane Locking
17.4 Examination of Element Quality – One-Element and Patch Tests
17.5 Benchmarks for Membranes and Plates
17.6 Benchmarks for Shells
17.7 Comparison of Analytical and Numerical Solutions, Application of Various FE Formulations
References
Chapter 18: Advanced FE Formulations
18.1 Introduction
18.2 Link between Variational Formulations and FE Models
18.3 Advanced FEs
References
Appendix A: List of Boxes with Equations
Appendix B: List of Boxes with Data and Results for Examples
Index
End User License Agreement
Pages
xvii
xviii
xix
xx
xxi
xxii
xxiii
xxiv
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
73
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
179
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
xxiv
387
388
389
390
391
392
393
394
395
396
397
398
Guide
Cover
Table of Contents
Preface
Part 1: Fundamentals: Theory and Modelling
Begin Reading
List of Illustrations
Chapter 1: General Information
Figure 1.1 Structures: (a) bar (1D), (b) surface (2D) and (c) solid (3D)
Figure 1.2 (a) A middle surface with curvilinear coordinates and local base vectors at point , (b) straight fibre – intersection of planes , .
Figure 1.3 Three surfaces: (a) spherical, (b) cylindrical and (c) shallow hyperbolic, corresponding to appropriate coordinate systems
Figure 1.4 Description of objects: (a) on middle surface and in plane , (b) in plane .
Figure 1.5 Middle surface and equidistant surface .
Figure 1.6 Spherical surface
Figure 1.7 Cylindrical surface
Figure 1.8 Hyperbolic paraboloid
Figure 1.9 (a) Middle surfaces and (before and after deformation), (b) graphical interpretation of kinematic K–L hypothesis for the special case of a flat shell (plate) on plane ; analogical section can be shown for plane .
Figure 1.10 Description of rotations of vector normal to middle surface.
Figure 1.11 Position of a straight fibre before and after deformation of a plate according to the kinematic M–R hypothesis; analogical section can be shown for plane
Figure 1.12 (a) Section of a shell with characteristic stresses and vectors of: force and moment , (b) and (c) elementary surface segments with all resultant forces revealed on section lines in the membrane-bending state.
Figure 1.13 Surfaces , , defined for description of deformation.
Figure 1.14 Middle surfaces of structures with different shapes: (a) curved, (b) shallow and (c) flat
Chapter 2: Equations for Theory of Elasticity for 3D Problems
Figure 2.1 Notation and sign convention for stresses in a 3D body
Chapter 3: Equations of Thin Shells According to the Three-Parameter Kirchhoff–Love Theory
Figure 3.1 (a) Surface loading and (b) generalized displacements
Figure 3.2 (a) Forces in membrane state and (b) forces and moments in bending state
Figure 3.3 Boundary quantities: (a) kinematic and (b) static
Figure 3.4 Elementary segments of axisymmetric shells described in a coordinate system: (a) spherical and (b) cylindrical
Figure 3.5 Geometry of a spherical shell
Figure 3.6 (a) Cylindrical shell and (b) conical shell
Figure 3.7 Shallow shells: (a) paraboloid and (b) hyperbolic paraboloid
Figure 3.8 Segments from middle surface with two types of coordinates: (a) Cartesian and (b) polar
Figure 3.9 (a) Rectangular flat membrane and (b) its geometric 2D model
Figure 3.10 (a) Rectangular plate and (b) its geometric 2D model
Figure 3.11 Geometry of cylindrical shell
Chapter 4: General Information about Models and Computational Aspects
Figure 4.1 Loading configuration for: (a) statics and (b) buckling
Figure 4.2 Results for simply supported square plate: (a) static deflection mode, (b) first buckling mode and (c) first free vibration mode
Figure 4.3 FDM discretization for simply supported square plate
Chapter 5: Description of Finite Elements for Analysis of Plates and Shells
Figure 5.1 Flat rectangular four-node membrane FE
Figure 5.2 Bilinear shape functions of a four-node FE
Figure 5.3 (a) Conforming rectangular four-node plate bending FE and (b) reference FE
Figure 5.4 Hermitian (bicubic) shape functions , , , related to node number 1 in conforming plate bending element
Figure 5.5 Flat shell three- and four-node elements composed of membrane and plate bending FEs
Figure 5.6 Curved shell FEs with
Figure 5.7 Idea of transition from 3D FE: (a) FE20, (b) FE16 to (c) eight-noded 2D degenerated FE8
Figure 5.8 Solid 3D FEs: (a) eight-node and (b) 20-node
Figure 5.9 One-dimensional shell FEs: (a) straight and (b) curved
Figure 5.10 Transition from: (a) conical shell FE to (b) membrane, (c) plate and (d) cylindrical shell FEs
Chapter 6: Flat Rectangular Membranes
Figure 6.1 (a) Membrane as a 3D body, (b) membrane as a set of layers and (c) middle plane as a 2D model of a membrane
Figure 6.3 Sign convention of membrane forces for: (a) elementary segment and (b) segment with sides of unit length
Figure 6.2 Graphical interpretation of displacements and strains for membranes
Figure 6.4 Notation of boundary quantities related to line : (a) static and (b) kinematic; for rectangular domain angle has the following values: , , and
Figure 6.5 Examples of boundary constraints and loads applied to a rectangular flat membrane
Figure 6.6 Square membrane under unidirectional tension: (a) whole configuration, (b) quarter of the domain with relevant boundary conditions and (c) subdomain before and after deformation
Figure 6.7 Square membrane under uniform shear: (a) configuration and (b) deformation
Figure 6.8 Square membrane under pure in-plane bending: (a) geometry and tractions with equivalent moments and (b) configuration before and after deformation
Figure 6.9 Cantilever beam modelled as a flat rectangular membrane
Figure 6.10 Cantilever beam – two cases of boundary conditions on a clamped beam end
Figure 6.11 Cantilever beam – analytical results: (a) deformation, diagrams of (b) normal stresses and (c) in-plane shear stresses and along sections .
Figure 6.12 Cantilever beam – numerical results: (a) and (b) beam before and after deformation for a coarse FE mesh (), contour plots of translations [cm]: (c) , (d) , stresses [MPa]: (e) , (f) for fine FE mesh (); (g) directions and magnitudes of principal membrane stresses for coarse mesh; FEM results from ROBOT.
Figure 6.13 Model of: (a) beam and (b) deep beam – with characteristic distribution of stress along the vertical line
Figure 6.14 Square deep beam – diagrams of membrane forces: (a) normal in the left-hand part of the domain, tangential in the right-hand part of the domain and (b) normal in whole domain; results obtained using FDM. Source: Andermann (1966).
Figure 6.15 Square deep beam: (a) deformation, contour plots of membrane forces: (b) , (c) , (d) , (e) diagram of distribution of [MN/m] along central vertical line with maximum value of [MN/m] 1.5 and (f) vizualization of directions and magnitudes of principal membrane forces; FEM results from ROBOT
Figure 6.16 Example of membrane with thickness discontinuity
Figure 6.17 Membrane with thickness discontinuities – contour plots of: (a) force [kN/m] and (b) stress [MPa], with diagrams of distributions along vertical line (denoted as A-A1); FEM results from ROBOT
Figure 6.18 Membrane with different elastic constants in subdomains: (a) configuration and (b) contour plot of stress [MPa] with diagram of distribution along central vertical line (denoted as A-A1); FEM results from ROBOT
Chapter 7: Circular and Annular Membranes
Figure 7.1 Membranes described in a polar coordinate system: (a) circular, (b) annular and (c) segment
Figure 7.2 Membrane forces in a general case of deformation described in a polar coordinate system, shown on the boundary of an elementary surface segment
Figure 7.3 Membrane forces in the axisymmetric state
Figure 7.4 Configuration of axisymmetric annular membrane: dimensions, boundary load and constraints
Figure 7.5 Annular membrane: functions obtained from analytical solution: (a) , (b) , (c) and (d) , (e)
Figure 7.6 Annular membrane: (a) radial cross section (for .) and (b) configuration discretized with SRK FEs
Chapter 8: Rectangular Plates under Bending
Figure 8.1 Plate under bending: (a) 3D geometrical model and (b) middle plane () as a 2D geometrical model
Figure 8.2 Description of plate behaviour following K–L constraints – respective segments and before and after deformation: (a) in the plane and (b) in the plane
Figure 8.3 (a) Stress distribution along the thickness of plate under bending and (b) the sign convention of surface load and generalized resultant forces
Figure 8.4 Example of rectangular plate under bending with various boundary constraints and loads
Figure 8.5 Description of plate behaviour following M–R constraints – respective segments and before and after deformation: (a) in the plane and (b) in the plane
Figure 8.6 Rectangular, simply supported plate under sinusoidal load: (a) configuration and (b) criterion of notation and sign convention for moments and transverse shear forces – repeated from Figure 8.3
Figure 8.7 Deflection for square simply supported plate under sinusoidal load: (a) diagram and (b) contour plot
Figure 8.8 Bending moments , for a square simply supported plate under sinusoidal load: (a) diagram and (b) contour plot (the same for and )
Figure 8.9 Twisting moment for a square simply supported plate under sinusoidal load: (a) diagram and (b) contour plot
Figure 8.10 Transverse shear force for a square simply supported plate under sinusoidal load: (a) diagram and (b) contour plot
Figure 8.11 Transverse shear force for a square simply supported plate under sinusoidal load: (a) diagram and (b) contour plot
Figure 8.12 Rectangular, simply supported plate under sinusoidal load: diagrams of distributions (along plate edges) of: (a) effective transverse shear forces, (b) linear boundary and corner reactions and (c) presentation of all actions in the analysed plate as concentrated forces, see also Equation ((8.53))
Figure 8.13 Rectangular plates with two types of boundary conditions analysed using two different methods: (a) double and (b) single trigonometric series
Figure 8.14 Simply supported square plate with uniformly distributed load: (a) configuration, (b) middle surface of plate after deformation and (c) distribution of bending moments
Figure 8.15 Two benchmarks for bending analysis – square plate with uniform load and different boundary conditions: (a) simply supported plate (case A) with principal bending moments at corner K and (b) clamped plate (case B)
Figure 8.16 Simply supported square plate with uniform load: (a) deflected surface , contour maps of moments [kNm/m]: (b) , (c) , (d) ; vizualization of directions and magnitudes of principal stresses (for FE mesh with ) on two surfaces: (e) bottom and (f) top; FEM results from ROBOT.
Figure 8.17 Clamped square plate with a uniform load: (a) deflected surface , contour maps of moments [kNm/m]: (b) , (c) , (d) ; vizualization of directions and magnitudes of principal stresses (for FEs mesh with ) on two surfaces: (e) bottom and (f) top; FEM results from ROBOT.
Figure 8.18 Plate with various boundary conditions: (a) configuration and (b) actual senses of bending moments at points (1–5, K, L)
Figure 8.19 Plate with various boundary conditions along edges, (a) numerical model and (b) deformed middle surface; FEM results from ROBOT
Figure 8.20 Plate with various boundary conditions, contour plots of: (a) deflected surface [m], moments: (b) , (c) , (d) [kNm/m]; vizualization of directions and magnitudes of principal stresses (for FEs mesh with ) on two surfaces: (e) bottom and (f) top; FEM results from ROBOT
Figure 8.21 Plate with protruding cantilever part: (a) configuration and (b) engineering sketch of trajectory lines of principal moments (with tension lines at bottom and top)
Figure 8.22 Plate with protruding cantilever part: (a) numerical model and (b) deformed middle surface; FEM results from ROBOT
Figure 8.23 Plate with protruding cantilever part – contour plots of: (a) bending moment , (b) bending moment , (c) twisting moment [kNm/m] and (d) deflection ; FEM results from ROBOT
Figure 8.24 Plate with protruding cantilever part, vizualization of directions and magnitudes of principal stresses (for FE mesh with ) on limiting surfaces: (a) bottom and (b) top; FEM results from ROBOT
Figure 8.25 Discrete representation of function of one variable
Figure 8.26 Rectangular plate under bending with various boundary constraints
Figure 8.29 Graphical representation of difference formulae for moments: (a) , (b) and (c)
Figure 8.30 Graphical representation of difference formulae for: (a) transverse shear force and (b) effective transverse shear boundary force
Figure 8.27 Graphical representation of a fourth-order differential equation for a plate under bending, adopted in the finite difference method
Figure 8.28 Grid points in vicinity of two types of plate edge: (a) clamped and (b) simply supported
Figure 8.31 Simply supported square plate with uniform load: (a) configuration, (b) set of grid points for FDM analysis and (c) visualization of deflection function on the basis of a finite set of deflection values
Figure 8.32 Simply supported square plate under bending resting on elastic foundation
Figure 8.33 Simply supported rectangular plate with uniform load
Figure 8.34 Two kinds of thickness jump in a plate
Figure 8.35 Plate under bending with a jump of thickness: (a) configuration of the plate, (b) axonometric view of plate deflection, (c) contour map of plate deflection field, (d) directions of principal moments; contour plots of bending moments: (e) and (f) [kNm/m]; FEM results from ROBOT
Figure 8.36 Simply supported, square plate with point load: (a) configuration of the plate and (b) discretization of quarter of the domain using eight-node FEs
Chapter 9: Circular and Annular Plates under Bending
Figure 9.1 Elementary segment of circular plate: surface load and generalized resultant forces for general bending state
Figure 9.2 Configuration of clamped circular plate: dimensions, load and boundary constraints
Figure 9.3 Clamped circular plate with uniformly distributed load – functions obtained from analytical solution: (a) , (b) , (c) , (d)
Figure 9.4 Different meshes for a quarter of circular plate: (a) discretization with improper FEs around the centre, (b) three-node triangular FEs, (c) four-node quadrilateral FEs and (d) eight-node isoparametric FEs
Figure 9.5 Simply supported circular plate with concentrated central force: (a), (b) configurations for two different approaches and (c) 3D model of the loaded central part of the domain
Figure 9.6 Simply supported circular plate with concentrated central force – functions obtained from analytical solution: (a) , (b) , (c) and (d)
Figure 9.7 Simply supported circular plate with asymmetric load: (a) plate configuration and (b) transverse load distribution
Figure 9.8 Simply supported circular plate with asymmetric load – functions: (a) , (b) and (c) for , and for ,
Figure 9.9 Annular plate: (a) configuration with loads and boundary constraints and (b) FE model discretized with SRK FEs
Figure 9.10 Annular plate – results of analytical solution: (a) deflection , (b) bending moment , (c) bending moment and (d) transverse shear force
Figure 9.11 Annular plate – two elementary segments with bending moments and transverse shear forces in the vicinity of the: (a) internal edge and (b) external edge
Chapter 10: Shells in the Membrane State
Figure 10.1 Shells of revolution in axisymmetric membrane state – analysis of equilibrium of shell segment described in: (a) spherical and (b) cylindrical coordinate system
Figure 10.2 Hemispherical shell under self weight: (a) configuration, (b) segment of a shell in equilibrium; diagrams of forces [kN/m]: (c) meridional and (d) circumferential
Figure 10.3 Hemispherical shell under normal pressure: (a) configuration, (b) segment of shell limited by running angle , analysed in equilibrium state; diagrams of membrane forces [kN/m]: (c) meridional and (d) circumferential
Figure 10.4 Hemispherical tank: (a) configuration; diagrams of membrane forces [kN/m]: (b) meridional and (c) circumferential
Figure 10.5 Hemispherical tank supported along the intermediate parallel: (a) configuration; diagrams of membrane forces [kN/m]: (b) meridional and (c) circumferential
Figure 10.6 Hemispherical shell supported at the intermediate parallel, subjected to hydrostatic pressure – contour maps of membrane forces: (a) meridional, (b) circumferential and (c) diagram of meridional force along the section described by , with extreme ordinates kN/m and kN/m; FEM results from ANSYS.
Figure 10.7 Conical shell: (a) configuration, diagrams of membrane forces [kN/m]: (b) meridional and (c) circumferential
Figure 10.8 Cylindrical shell – configuration and loads
Figure 10.9 Cylindrical shell – diagrams of characteristic functions: (a) , (b) , (c) , (d) , (e) and (f)
Figure 10.10 Hemispherical shell under wind action: (a) configuration and (b) distribution of normal wind pressure
Figure 10.11 Hemispherical shell under wind action – analytical solution, diagrams of three membrane forces: (a) , (b) and (c)
Figure 10.12 Hemispherical shell under wind action: (a) vector vizualization of wind load, contour maps of membrane forces [N/m]: (b) meridional , (c) circumferential and (d) tangent ; FEM results from ANSYS.
Figure 10.13 Hemispherical shell under wind action – vector visualization of displacement field: (a) top view, (b) side view; the Figure is continued on next page and presents directions of principal stress at bottom limiting surface: (c) top view and (d) side view; FEM results from ANSYS.
Chapter 11: Shells in the Membrane-Bending State
Figure 11.1 Configuration of circular cylindrical shell: (a) cylindrical segment and (b) shell of revolution
Figure 11.2 Cylindrical shell – distributions of exponential-trigonometric functions in interval : (a) , (b) , (c) and (d) , for 1/m
Figure 11.3 Cylindrical shell: (a) description of geometry and loads; boundary conditions: (b) in stage I for membrane state and (c) in stage II for the membrane-bending state
Figure 11.4 Long cylindrical shell (analytical solution) – characteristic functions: (a) normal displacement , (b) circumferential force , (c) meridional moment and (d) meridional transverse shear force
Figure 11.5 Long cylindrical shell – results of calculations with 2D FEs – distributions of functions: (a) meridional force , (b) circumferential force , (c) meridional moment , (d) meridional transverse shear force , [kN], [m]; FEM results from ANSYS.
See Plate section for color representation of this figure.
Figure 11.6 Short cylindrical shell (analytical solution) – graphs of four characteristic functions: (a) normal displacement , (b) circumferential force , (c) meridional moment and (d) meridional transverse shear force
Figure 11.7 Angular coordinates used in a spherical shell
Figure 11.8 Spherical shell – description of geometry and boundary conditions: (a) clamped support (case A), (b) hinged support (case B) and (c) support used for membrane state analysis in stage I
Figure 11.9 Spherical shells under self weight: left column – results for case A (shell clamped along bottom edge), right column – case B (simply supported shell); graphs of following functions: (a), (b) meridional membrane forces , (c), (d) meridional moment and (e), (f) meridional transverse shear forces , [N],[m]; FEM results from ANSYS.
See Plate section for color representation of this figure.
Figure 11.10 Cylindrical shell with two uniformly distributed boundary loads on contour : (a) normal moment and (b) horizontal force
Figure 11.11 Cylindrical shell with boundary load kNm/m (see Figure 11.10a) – diagrams of: (a) normal displacement , (b) meridional moment – see ordinate kNm/m and (c) meridional transverse shear force
Figure 11.12 Cylindrical shell with boundary load kN/m (see Figure 11.10b) – diagrams of: (a) normal displacement , (b) meridional moment and (c) meridional transverse shear force with ordinate kN/m
Figure 11.13 (a) Spherical shell with uniformly distributed boundary loads and ; shell deformation caused by: (b) moment and (c) horizontal force
Figure 11.14 Cylindrical shell with spherical cap: (a) configuration with two coordinate systems: spherical with and cylindrical with and (b) interactions and introduced in interface of two parts of reservoir
Figure 11.15 Cylindrical shell with spherical cap – distribution of: (a) circumferential force and (b) meridional moment in spherical shell in vicinity of connection of two shells
Figure 11.16 Cylindrical shell with spherical cap – distribution of: (a) circumferential force and (b) meridional moment in cylindrical shell in the vicinity of connection of two shells
Figure 11.17 Cylindrical shell with spherical cap – distribution of (a) circumferential force caused by hydrostatic pressure and (b) meridional moment in section (extreme moment kNm/m); FEM results from ROBOT.
SubFigure (a) shown in Plate section for color representation of this figure.
Figure 11.18 Configuration of cylindrical segment
Figure 11.19 Cylindrical shell with semicircular cross section: (a) configuration and (b) peripheral of cylindrical shell as domain with and
Figure 11.20 Cylindrical shell with semicircular cross section under self weight – contour maps of: (a)–(c) three membrane forces , , and (d)–(f) three translations , , ; drawn on the basis of analytical solution
Figure 11.21 Cylindrical shell with marked characteristic points
Figure 11.22 Scordelis-Lo roof: (a) vector presentation of displacement field [cm] – FEM results from ANSYS; graphs of displacements on four cross lines [m]: (b) UX and (c) UZ; FEM results from ROBOT.
Figure 11.23 Scordelis-Lo roof – contour maps for: membrane forces [kN/m]: (a) , (b) , (c) and (d) twisting moment [kNm/m] ; FEM results from ANSYS.
See Plate section for color representation of this figure.
Figure 11.24 Scordelis-Lo roof – contour maps for bending moments [kNm/m]: (a) , (c) , transverse shear forces [kN/m]: (b) , (d) ; FEM results from ANSYS.
Figure 11.25 Scordelis-Lo roof – visualization of the directions and magnitudes of the principal stresses on surfaces: (a) bottom and (b) top; FEM results from ANSYS.
Figure 11.26 Horizontal cylindrical tube: (a) configuration, description of loads: (b) self weight and (c) hydrostatic pressure
Figure 11.27 Horizontal cylindrical tube under self weight – graphs of three functions for two lines with coordinates : (a) meridional membrane force [kN/m], (b) meridional bending moment [kNm/m] and (c) final meridional force [kN/m]
Figure 11.28 Horizontal tube under self weight – distribution of meridional bending moment – graphs on lines: (a) , (b) (characteristic value Nm/m kNm/m) and (c) contour map (characteristic value Nm/m kNm/m); FEM results from ANSYS.
Figure 11.29 Horizontal tube under self weight – contour maps for membrane forces: (a) , (b) and for transverse shear forces: (c) , (d) [N,m]; FEM results from ANSYS.
See Plate section for color representation of this figure.
Chapter 12: Shallow Shells
Figure 12.1 Model of shallow shell with rectangular projection
Figure 12.2 (a) Surface segment of doubly curved shallow shell and (b) ruled segment with membrane forces and their projections on horizontal plane
Figure 12.3 Hyperbolic paraboloid – configuration
Figure 12.4 Hyperbolic paraboloid – contour plots of: (a) middle surface and analytically determined distributions of projections of membrane forces: (b) , (c) and (d)
Figure 12.6 Complex roofs: (a) cross-roof and (b) umbrella-roof
Figure 12.5 Two cases of ruled shells with point supports as well as visualization of actions along edge beams, and their projections on horizontal plane
Chapter 13: Thermal Loading of Selected Membranes, Plates and Shells
Figure 13.1 Distribution of temperature changes along the thickness: (a) uniform and (b) linear
Figure 13.2 Configurations of three structures and their thermal deformations under uniform distribution of temperature change along the thickness : (a) circular membrane, (b) cylindrical shell and (c) hemispherical shell
Figure 13.3 Hemispherical shell – idea of superposition of states induced sequentially by: (a) thermal load, distributed boundary loads (b)
and (c)
; (d) configuration of the shell with boundary loads corresponding to clamped edge
Figure 13.4 Hemisphere clamped on bottom edge and uniformly heated – diagrams of: (a) meridional membrane force , extreme value N/m, (b) circumferential membrane force , extreme value N/m, (c) meridional bending moment , extreme value Nm/m, (d) circumferential bending moment , extreme value Nm/m and (e) meridional transverse shear force , extreme value N/m; FEM results from ANSYS
Figure 13.5 Notation for linear distribution of temperature change along thickness for: (a) flat plate and (b) curved shell
Figure 13.6 Influence of thermal loads for two analysed circular plates with: (a) simply supported edge and (b) clamped edge
Figure 13.7 Cylindrical shell in stage I – subdomain far from edges: (a) description of linear change of temperature along thickness and (b) diagrams of constant meridional moment along meridian and linear distribution of stress along thickness
Figure 13.8 Cylindrical shell (subdomain near bottom edge) – superposition of effects resulting from: (a) thermal load , (b) boundary moment and (c) two functions of meridional moment and normal displacement in the neighbourhood of simply supported moveable contour
Chapter 14: Stability of Plates and Shells
Figure 14.1 Example equilibrium paths with either bifurcation point B or limit point L, obtained in geometrically nonlinear analysis; denotes a load factor and is a selected representative displacement
Figure 14.2 Examples of membranes susceptible to buckling under various loads
Figure 14.3 Example equilibrium paths for a plate: pre- and post-buckling paths for a perfect plate and membrane-bending path for an imperfect plate with initial deflection ; denotes the active postbuckling deflection
Figure 14.4 Rectangular membrane under unidirectional compression: (a) configuration and (b) diagram of function
Figure 14.5 Square membrane under unidirectional compression – FDM discretization
Figure 14.6 (a) Shallow cylindrical shell and (b) its projection onto the horizontal plane – geometry, load and load imperfection
Figure 14.7 Shallow cylindrical shell () and flat membrane () – relations for load imperfection cases
Figure 14.8 Three types of bifurcation points: (a) – asymmetric, (b) – symmetric stable and (c) – symmetric unstable. Source: Radwańska (1990). Reproduced with permission of Cracow University of Technology.
Figure 14.9 (a) Membranes subjected to three types of load (, , ), (b) prebuckling deformations in middle plane and (c) primary buckling modes. Source: Waszczyszyn et al. (1994). Reproduced with permission from Elsevier.
Figure 14.10 Buckling modes for unidirectionally compressed square plate; FEM results from ANKA. Source: Waszczyszyn et al. (1994). Reproduced with permission from Elsevier.
Figure 14.11 Prebuckling deformation and buckling modes for square plate under uniform shear; FEM results from ANKA. Source: Waszczyszyn et al. (1994). Reproduced with permission from Elsevier.
Figure 14.12 Prebuckling deformation and buckling modes for square plate under in-plane bending; FEM results from ANKA. Source: Waszczyszyn et al. (1994). Reproduced with permission from Elsevier.
Figure 14.13 Membrane under unidirectional compression – spurious modes resulting from the use of (a) S8 and (b) and (c) L9 finite elements with reduced integration (RI). Source: Radwańska (1990). Reproduced with permission of Cracow University of Technology.
Figure 14.14 Circular plate under radial compression: (a) configuration and deformation for case A, (b) Bessel function , see Equation (14.35) and (c) function for case B, see Equation (14.40)
Figure 14.15 Cylindrical shells: (a) under axial compression and (b) under external pressure
Figure 14.16 Relation between dimensionless pressure and the Batdorf's parameter for cylindrical shell under axial compression. Source: Batdorf (1947), NASA.
Figure 14.17 (a) Distribution of experimental buckling test data for cylindrical shells subjected to axial compression; (b) characteristic equilibrium paths of perfect and imperfect axially compressed cylindrical shells. Source: Harris et al. (1957). SubFigure (a) reproduced with permission of AIAA.
Figure 14.18 Diagram of relation between dimensionless critical pressure and Batdorf's parameter for cylindrical shells with external lateral pressure. Source: Batdorf (1947), NASA.
Figure 14.19 Cylindrical shell under external pressure: (a) configuration and (b) postbuckling mode for and
Figure 14.20 Axisymmetric shells with various shapes of meridian (, , ) and two types of loads and
Figure 14.21 Cylindrical shell – configuration with load and boundary conditions; FEM mesh for shell quarter
Figure 14.22 Cylindrical shell with concentrated central force – equilibrium paths for five thickness values. FEM results from ANKA. Source: Radwańska (1990). Reproduced with permission of Cracow University of Technology.
Chapter 15: Free Vibrations of Plates and Shells
Figure 15.1 Configuration of a square plate with all simply supported edges
Figure 15.2 Simply supported square plate – three first modes of transverse natural vibrations: (a), (b) for frequency Hz, (c)–(f) for equal frequencies Hz; two techniques of mode presentation, ROBOT (2006) and ANSYS.
Figure 15.3 Three natural vibration modes related to three initial frequencies for a long shell with
(first column) and for a short shell with
(second column); FEM results from ANSYS.
Chapter 16: Modelling Process
Figure 16.1 Independent displacement modes of bilinear membrane element
Figure 16.2 Disadvantageous finite element shapes
Chapter 17: Quality of FEs and Accuracy of Solutions in Linear Analysis
Figure 17.1 Positions of Gauss points in 2D FEs: (a) Q4 with FI (), ; (b) Q4 with RI (), ; (c) Q8 with FI (), and (d) Q8 with RI ()
Figure 17.2 Initial eight deformation modes of single nine-node Lagrangian membrane FE for different NI quadratures: (a) FI – 3/3, (b) SI – 3/2 and (c) URI – 2/2; some eigenmodes are associated with zero eigenvalues
Figure 17.3 Rectangular membrane in pure in-plane bending: (a) proper deformation and (b) deformation mode
Figure 17.4 Selected examples of single-element models in typical states: (a) membrane in-plane bending state, (b) plate in bending, (c) plate twisted by two forces, (d) plate twisted by two moments and (e) pure bending of infinitely long cylindrical shell
Figure 17.5 (a) Patch with one internal node and (b) patch with one internal element; material data, dimensions and values of coordinates of all nodes are given
Figure 17.6 Patch tests with loading (PT-L) for examination of the following states: (a) constant curvatures and (b) constant twist (warping)
Figure 17.7 Cantilever beam
Figure 17.8 Swept panel – dimensions, load, boundary conditions and different discretization.
Figure 17.9 Plane square cantilever – results of an adaptive solution in a plane region using linear strain triangles.
Figure 17.10 Three shell configurations, input data, verification results: (a) cylindrical shell roof, (b) pinched cylinder and (c) hemisphere
Figure 17.11 Simply supported square plate with force at the centre – comparison of numerical results using different types of rectangular FEs.
Chapter 18: Advanced FE Formulations
Figure 18.1 One-field four-node rectangular membrane FE
Figure 18.2 Two-field triangular membrane FE
Figure 18.3 Two-field four-node membrane FE
Figure 18.4 Deep beam: (a) configuration, contour maps of membrane force obtained using two FEs based on the following functionals: (b) and (c) ()
Figure 18.5 Two-field triangular plate FE
Figure 18.6 Hybrid displacement triangular plate FE
Figure 18.7 Hybrid stress three-node plate FE
Figure 18.8 Displacement-based three-node membrane FE with three dofs per node
Figure 18.9 The initial (a) and final (b) DKT finite element with reduction of dofs
Figure 18.10 Semiloof shell FE with three types of nodes; crossed dofs are eliminated in the FE formulation
Figure 18.11 (a) FE with curvilinear internal coordinate system; positions of points used for interpolation of: (b) displacements and rotations, (c) in-surface strains and (d), (e) two transverse shear strains, respectively
Figure 18.12 Silo structure: (a) fine -mesh; contour plots of: (b) radial displacements , (c) azimuthal stresses and (d) vertical stresses . Source: Tews and Rachowicz (2009). Reproduced with permission of Elsevier
Figure 18.13 Spherical container: (a) fine -mesh; contour plots of: (b) radial displacements in cylindrical coordinates, (c) and (d) radial and meridional , (e) azimuthal stresses in spherical coordinates, (f) meridional stresses . Source: Tews and Rachowicz (2009). Reproduced with permission of Elsevier
List of Tables
Chapter 4: General Information about Models and Computational Aspects
Table 4.1 Input data for the simply supported square plate in Figure 4.1
Table 4.2 Equations of mathematical and numerical models
Chapter 8: Rectangular Plates under Bending
Table 8.1 Effect of , mesh density and NI on dimensionless centre deflection
Chapter 15: Free Vibrations of Plates and Shells
Table 15.1 Values of relative errors for natural frequencies for eight cases
Chapter 17: Quality of FEs and Accuracy of Solutions in Linear Analysis
Table 17.1 Numbers of zero eigenvalues for plate FEs based on M–R theory: L4, S8, L9, S12, L16
Table 17.2 Number of spurious modes for various FEs: S8, H8/9, L9 and NI: RI, SI
Table 17.3 Values of normalized critical loads for plate under unidirectional compression for two ratios and three types of NI
Plate and Shell Structures
Selected Analytical and Finite Element Solutions
Maria Radwańska
Anna Stankiewicz
Adam Wosatko
Jerzy Pamin
This edition first published 2017
© 2017 John Wiley & Sons Ltd
Registered office
John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, United Kingdom
For details of our global editorial offices, for customer services and for information about how to apply for permission to reuse the copyright material in this book please see our website at www.wiley.com.
The right of Maria Radwańska, Anna Stankiewicz, Adam Wosatko and Jerzy Pamin to be identified as the authors of this work has been asserted in accordance with the Copyright, Designs and Patents Act 1988.
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by the UK Copyright, Designs and Patents Act 1988, without the prior permission of the publisher.
Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books.
Designations used by companies to distinguish their products are often claimed as trademarks. All brand names and product names used in this book are trade names, service marks, trademarks or registered trademarks of their respective owners. The publisher is not associated with any product or vendor mentioned in this book.
Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. It is sold on the understanding that the publisher is not engaged in rendering professional services and neither the publisher nor the author shall be liable for damages arising herefrom. If professional advice or other expert assistance is required, the services of a competent professional should be sought
Library of Congress Cataloging-in-Publication data applied for
ISBN: 9781118934548
A catalogue record for this book is available from the British Library.
Cover image: Jasmin Kämmerer/EyeEm/Gettyimages
To our families
Preface
This book deals with the mechanics and numerical simulations of plates and shells, which are flat and curved thin-walled structures, respectively (called shell structures for short in this book). They have very important applications as complete structures or structural elements in many branches of engineering. Examples of shell structures in civil and mechanical engineering include slabs, vaults, roofs, domes, chimneys, cooling towers, pipes, tanks, containers and pressure vessels; in shipbuilding – ship and submarine hulls, in the vehicle and aerospace industries – automobile bodies and tyres and the wings and fuselages of aeroplanes.
The scope of the book is limited to the presentation of the theory of elastic plates and shells undergoing small deformation (thus assuming linear constitutive and kinematic equations).
The book is aimed at the large international community of engineering students, university teachers, professional engineers and researchers interested in the mechanics of shell structures, as well as developers testing new simulation software. The book can be the basis of an intermediate-level course on (computational) mechanics of shell structures at the level of doctoral, graduate and undergraduate studies. The reader should have the basic knowledge of the strength of materials, theory of elasticity, structural mechanics and FEM technology; basic information in these areas is not repeated in the book.
The strength of the book results from the fact that it not only provides the theoretical formulation of fundamental problems of mechanics of plates and shells, but also several examples of analytical and numerical solutions for different types of shell structures. The book also contains some advanced aspects related to the stability analysis and a brief description of classical and modern finite element formulations for plates and shells, including the discussion of mixed/hybrid models and so-called locking phenomena.
The book contains a comprehensive presentation of the theory of elastic plates and shells, formulations and solutions of fundamental mechanical problems (statics, stability, free vibrations) for these structures using exact approaches and computational (approximate) methods, with emphasis on modern capabilities of the finite element (FE) technology. In the book we introduce a large number of examples that illustrate various physical phenomena associated with the behaviour of shell structures under external actions. Comparisons of analytical and numerical solutions are given for several benchmark tests. The book includes plenty of boxes and tables that contain sets of formulae or data and check values describing the examples. They help the reader to find and integrate the information provided and draw conclusions.
The authors are researchers and teachers from the Institute for Computational Civil Engineering of Cracow University of Technology. They have done research on structural mechanics for years, in particular on the theories and advanced computational methods for shell structures, and they also have a long history of teaching the subject to students and practitioners. The selection of the contents of the book is based on this experience. The motivation to write the present book has also come from the fact that there are no books that contain, in one volume, the foundation of the theory and solutions of selected problems using simultaneously analytical and numerical methods.
Following a sequence of subjects: mathematics, theoretical mechanics, strength of materials, structural mechanics, computer science, numerical methods and the finite element method – we have developed a comprehensive course on the mechanics of shell structures. This course contains: (i) discussion of the assumptions and limits of applicability of selected theories on which mathematical models are based, (ii) choice of a method to solve the problem efficiently, (iii) analytical and/or numerical calculations simulating physical phenomena or processes, (iv) confrontation of the results of theoretical and numerical analysis and (v) evaluation of the calculation methodology and results.
Maria Radwańska and Jerzy Pamin were members of Professor Zenon Waszczyszyn's research team, who implemented the finite element code ANKA for buckling and nonlinear analysis of structures at the end of the twentieth century. This resulted in the 1994 Elsevier book: Waszczyszyn, Z. and Cichoń, Cz. and Radwańska, M., Stability of Structures by Finite Element Method.
Next, we briefly describe the contents of the book, which is divided into five parts. Part 1 is the introductory part that gives a compact encyclopedic overview of the fundamentals of the theory and modelling of plates and shells in the linear elastic range. A description of static analysis of (plane) plates is contained in Part 2 and of (curved) shells in Part 3. Part 4 includes information on the selected problems of buckling and free vibrations of shell structures. In Part 5, the authors discuss the general aspects of finite element analysis, including the modelling process, evaluation of the quality of finite elements and accuracy of solutions, Part 5 also contains a brief presentation of advanced formulations of finite elements for plates and shells.
While working on the book, we felt special gratitude to two of our teachers: Professors Zenon Waszczyszyn and Michał Życzkowski, who we always thought of as scientific authorities in the field of structural mechanics. In particular, we are deeply indebted to Professor Zenon Waszczyszyn for his invaluable contribution to our knowledge, motivation to do research and to participate in high-level university education. Under his guidance we got to know the theory of plates and shells, computational mechanics applied in civil engineering and modern numerical methods; in particular, the finite element method.
The authors wish to express their appreciation to several colleagues fromthe Institute for Computational Civil Engineering for discussions and help during the preparation of the book, in particular to A. Matuszak, E. Pabisek, P. Pluciński, R. Putanowicz and T. Żebro. We also record our gratitude to our students who cooperated with us in the computation of numerous examples: M. Abramowicz, M. Bera, I. Bugaj, M. Florek, S. Janowiak, A. Kornaś and K. Kwinta.
Notation
Detailed notation for theoretical analysis
Indices
Greek indices (for curvature lines and surface coordinates)
Latin indices (for 3D space)
,
,
indices for membrane, bending, transverse shear states
number of a components of trigonometric series or number of circumferential wave (half-wave)
indices describing number of waves of deformation in two directions
Coefficients and variables
coefficient of thermal expansion of a material
:
,
Lame coefficients
prescribed body forces
components of II (second) metric tensor
coefficient in equation of local bending state in cylindrical shell
,
initial and current configuration of a body (shell)
,
matrices of local flexibility and stiffness in constitutive equations
cross-sectional stiffness in membrane state
cross-sectional stiffness in bending state
,
cross-sectional stiffness in transverse shear state
,
local base versors on middle surface and on equidistant surface from the middle surface in initial configuration
local base versors on middle surface in current configuration
generalized strain vector (membrane, bending and transverse shear components)
membrane strain vector
,
,
membrane strains: normal and shear in middle surface
,
,
membrane strains in cylindrical system
,
,
membrane strains in spherical system
bending strain vector
,
,
bending strains: changes of curvature and warping of middle surface
,
,
bending strains in cylindrical system
,
,
bending strains in spherical system
transverse shear strain vector
,
transverse shear strains in cylindrical system
,
transverse shear strains in spherical system
,
,
material constants: Young's modulus, Poisson's ratio, Kirchhoff's modulus
rise of shallow shell
Airy's stress function
base versors related to Cartesian coordinates
components of I (first) metric tensor
thickness of shell
Gaussian curvature of surface
length of half-wave for exponential-trigonometric function in local membrane-bending state of cylindrical shell
,
,
moments: bending and twisting in middle surface
,
,
moments in cylindrical system
,
,
moments in spherical system
,
,
membrane forces: normal and tangential in middle surface
,
,
membrane forces in cylindrical system
,
,
membrane forces in spherical system
,
,
,
principal membrane forces and bending moments
,
effective boundary forces (tangential membrane and transverse shear)
,
,
,
generalized boundary forces
,
,
,
prescribed generalized boundary loads
,
,
directions of boundary base vectors
vector of prescribed surface loads
,
vectors of prescribed generalized boundary loads and displacements
prescribed concentrated force in corner
,
middle and equidistant surfaces in initial configuration
,
middle and equidistant surfaces in current configuration
,
cross-sectional planes: normal and tangent to middle surface
two transverse cross-sectional planes normal to middle surface
,
,
total potential energy, internal energy, external load work
,
meridian equation for axisymmetric shell
:
,
principal curvature radii for middle surface of a shell
arch coordinate for a line on surface
vector of generalized resultant forces for membrane, bending and transverse shear states
vector of generalized boundary forces
vector of presribed boundary forces
vector of membrane forces
vector of bending and twisting moments
vector of transverse shear forces
coefficient in equation of local bending state in spherical shell
effective force in a corner used in static boundary conditions
vector of rotations
:
,
two rotations of normal to middle surface
rotation around normal to middle surface
,
