SELECTED PROBLEMS OF MODELLING OF CONSTRUCTION ELEMENTS WITH VARIABLE MATERIAL AND GEOMETRICAL PROPERTIES

1.  Introduction   ................................................................................................ 5

1.1.  Overview on the issue of the modelling of construction elements with variable properties   ................ 5

1.2.  Scope of the monograph   ................................................................................. 9

2.  Dynamical stability of variable cross section bars - first natural frequencies and buckling critical forces   ... 13

2.1.  Introduction   ........................................................................................... 13

2.2.  The finite element method in the problems of dynamical buckling   ....................................... 17

2.3.  Bar CC   .................................................................................................. 20

2.3.1.   Bending stiffness variability of the type f(x⁴)   .................................................... 20

2.3.2.   Bending stiffness variability of the type f(x³)   ..................................................... 22

2.3.3.   Bending stiffness variability of the type f(x)   ..................................................... 26

2.4.  Bar PC   .................................................................................................. 29

2.4.1.   Bending stiffness variability of the type f(x⁴)   .................................................... 30

2.4.2.   Bending stiffness variability of the type f(x³)   .................................................... 31

2.4.3.   Bending stiffness variability of the type f(x)   ..................................................... 34

2.5.  Bar CP   .................................................................................................. 36

2.5.1.   Bending stiffness variability of the type f(x⁴)   .................................................... 36

2.5.2.   Bending stiffness variability of the type f(x³)   .................................................... 38

2.5.3.   Bending stiffness variability of the type f(x)   ..................................................... 40

2.6.  Curves l-s for various types of bars   ..................................................................... 43

2.7.  Comparison of results with the literature data  ........................................................... 44

2.8.  Convergence analysis   ................................................................................... 59

2.9.  Conclusions   ............................................................................................ 60

3.  Stability and vibrations of microheterogeneous plate band resting on elastic subsoil   ....................... 63

3.1.  Introduction   ............................................................................................ 63

3.2.  Basic assumptions   ...................................................................................... 66

3.3.  Transformation formulas of a band element   ............................................................. 67

3.4.  Element (cell) stiffness matrix   .......................................................................... 68

3.5.  Band stiffness matrix   ................................................................................... 74

3.6.  Calculation of the values of natural frequency and critical buckling force – examples   ..................... 77

3.6.1.   Example 1   ..................................................................................... 78

3.6.2.   Example 2   ..................................................................................... 79

3.6.3.   Example 3   ..................................................................................... 80

3.6.4.   Example 4   ..................................................................................... 81

3.6.5.   Comparative diagrams   ......................................................................... 82

3.7.  Conclusions   .......................................................................................... 83

4.  Free vibrations of non-homogeneous plates – application of the tolerance averaging technique   ........... 85

4.1.  Introduction   .......................................................................................... 85

4.2.  Description of system   ................................................................................. 86

4.3.  Equations of motion. Averaging tolerance technique   ................................................... 87

4.4.  Shape function   ........................................................................................ 90

4.5.  Calculational example   ................................................................................. 95

4.5.1.   Natural frequencies   ........................................................................... 95

4.5.2.   Fluctuations and total deflection   ............................................................. 100

4.6.  Conclusions   .......................................................................................... 102

5.  Method of calculation of non-homogeneous, rotationally symmetric circular plates supported on piles  . .. 103

5.1.  Introduction   .......................................................................................... 103

5.2.  Description of system, methods and equations   ........................................................ 105

5.3.  Calculational examples   ............................................................................... 112

5.3.1.   Plate with discontinuous, alternately variable stiffness   ........................................ 113

5.3.2.   Plate with discontinuously variable stiffness, increasing towards the circumference   ........... 120

5.3.3.   Comparative examples – plates with continuously variable stiffnesses   ........................... 122