2.2. Parameter Analysis for Determining the Importance Coefficient of Elements
An engineering case was analysed to explore the effects of the external load, the initial prestress, and cross-sectional areas of elements on the importance coefficients of elements. This case involved a flexible cable–strut tensile structure with a Geiger-type cable dome. The span of this structure had a length of 71.2 m and a rise of 5.5 m. The aforementioned structure was composed of 16 symmetrical parts, and each part included an outer ridge cable (denoted as RC1), inner ridge cable (denoted as RC2), outer diagonal cable (denoted as DC1), inner diagonal cable (denoted as DC2), outer strut (denoted as WG1), outer hoop cable (denoted as HC1), inner top hoop cable (denoted as THC2), inner lower hoop cable (denoted as LHC2), and inner strut (denoted as WG2). The profile of the considered structure and the corresponding numbers of elements are displayed in
Figure 1. The material parameters of the elements and the initial prestress are presented in
Table 1. The inner lower hoop cable and inner upper hoop cable constituted the inner hoop cable. For simplicity, the areas of the element sections of the aforementioned two types of elements were consistent in the parameter analysis and optimisation process. The elastic modulus values of the cables and struts were 160 and 206 GPa respectively. The structure was fixed on a hinged support and bore a design load of 0.4 kN/m
2.
The effects of the external load, the initial prestress, and the cross-sectional areas of elements on the importance coefficients of the elements are presented in
Table 2,
Table 3 and
Table 4, respectively. On the basis of these tables, the following inferences are obtained:
- (1)
Different elements had different importance coefficients. The outer hoop cable had the highest importance coefficient, followed by the inner lower hoop cable and inner upper hoop cable. The inner strut exhibited the smallest importance coefficient.
- (2)
The hoop cables exhibited the highest importance coefficients, followed by the ridge cables and diagonal cables. The struts exhibited the smallest importance coefficients.
- (3)
In terms of the positions of the elements, the importance coefficients of the outer elements were higher than those of the inner elements.
Table 2.
Importance coefficients of the elements under different initial prestress levels.
Table 2.
Importance coefficients of the elements under different initial prestress levels.
Element | Importance Coefficients | Prestress |
---|
0.5 P | 1.0 P | 1.5 P | 2.0 P |
---|
RC1 | | 0.070163 | 0.058055 | 0.053604 | 0.051754 |
RC2 | | 0.020178 | 0.016488 | 0.013876 | 0.01325 |
DC1 | | 0.235073 | 0.131665 | 0.067967 | 0.036867 |
DC2 | | 0.040101 | 0.027865 | 0.023377 | 0.021531 |
WG1 | | 0.032925 | 0.017571 | 0.011399 | 0.008266 |
WG2 | | 0.000098 | 0.000162 | 0.000168 | 0.000155 |
HC1 | | 0.415335 | 0.392424 | 0.384673 | 0.379775 |
THC2 | | 0.510334 | 0.318913 | 0.182745 | 0.203077 |
LHC2 | | 0.319583 | 0.309017 | 0.200168 | 0.118609 |
Table 3.
Importance coefficients of the elements under different external loads.
Table 3.
Importance coefficients of the elements under different external loads.
Element | Importance Coefficients | External Load |
---|
0.5 F | 1.0 F | 1.5 F | 2.0 F |
---|
RC1 | | 0.058936 | 0.058055 | 0.057238 | 0.056325 |
RC2 | | 0.017094 | 0.016488 | 0.015873 | 0.015250 |
DC1 | | 0.129883 | 0.131665 | 0.133305 | 0.134860 |
DC2 | | 0.027908 | 0.027865 | 0.027772 | 0.027710 |
WG1 | | 0.018979 | 0.017571 | 0.015920 | 0.014031 |
WG2 | | 0.000130 | 0.000162 | 0.000192 | 0.000222 |
HC1 | | 0.381889 | 0.392424 | 0.402959 | 0.413494 |
THC2 | | 0.321704 | 0.318913 | 0.316017 | 0.312898 |
LHC2 | | 0.311291 | 0.309017 | 0.305002 | 0.300766 |
Table 4.
Importance coefficients of the elements under different areas of the element sections.
Table 4.
Importance coefficients of the elements under different areas of the element sections.
|
Importance Coefficients | k1 | k2 | k3 | k4 | k5 | k6 | k7 | k8 | k9 |
---|
Section Sizes | |
---|
RC1 | 0.8 A | 0.05887 | 0.015314 | 0.12173 | 0.026334 | 0.015616 | 0.000153 | 0.364302 | 0.296647 | 0.297071 |
1.0 A | 0.058055 | 0.016488 | 0.131665 | 0.027865 | 0.017571 | 0.000162 | 0.392424 | 0.318913 | 0.309017 |
1.2 A | 0.057804 | 0.017396 | 0.139213 | 0.028982 | 0.01902 | 0.000169 | 0.413341 | 0.328802 | 0.316454 |
RC2 | 0.8 A | 0.05802 | 0.018158 | 0.12705 | 0.026886 | 0.01696 | 0.000218 | 0.379508 | 0.300345 | 0.302016 |
1.0 A | 0.058055 | 0.016488 | 0.131665 | 0.027865 | 0.017571 | 0.000162 | 0.392424 | 0.318913 | 0.309017 |
1.2 A | 0.058258 | 0.015652 | 0.134901 | 0.028905 | 0.018032 | 0.000251 | 0.401221 | 0.291511 | 0.301105 |
DC1 | 0.8 A | 0.055947 | 0.015901 | 0.131898 | 0.026784 | 0.016602 | 0.000157 | 0.414111 | 0.309294 | 0.298302 |
1.0 A | 0.058055 | 0.016488 | 0.131665 | 0.027865 | 0.017571 | 0.000162 | 0.392424 | 0.318913 | 0.309017 |
1.2 A | 0.0596 | 0.016933 | 0.131173 | 0.028698 | 0.018359 | 0.000165 | 0.376069 | 0.325345 | 0.316647 |
DC2 | 0.8 A | 0.058754 | 0.016018 | 0.126122 | 0.029171 | 0.017425 | 0.000187 | 0.377422 | 0.259439 | 0.293367 |
1.0 A | 0.058055 | 0.016488 | 0.131665 | 0.027865 | 0.017571 | 0.000162 | 0.392424 | 0.318913 | 0.309017 |
1.2 A | 0.057525 | 0.016819 | 0.135547 | 0.026818 | 0.017648 | 0.000168 | 0.402469 | 0.360229 | 0.320426 |
WG1 | 0.8 A | 0.057569 | 0.016287 | 0.127649 | 0.027517 | 0.017575 | 0.000159 | 0.397635 | 0.315285 | 0.307203 |
1.0 A | 0.058055 | 0.016488 | 0.131665 | 0.027865 | 0.017571 | 0.000162 | 0.392424 | 0.318913 | 0.309017 |
1.2 A | 0.058399 | 0.016637 | 0.134634 | 0.028202 | 0.017574 | 0.00017 | 0.387521 | 0.319196 | 0.307949 |
WG2 | 0.8 A | 0.057499 | 0.016179 | 0.131715 | 0.027839 | 0.017671 | 0.000176 | 0.39246 | 0.298214 | 0.305824 |
1.0 A | 0.058055 | 0.016488 | 0.131665 | 0.027865 | 0.017571 | 0.000162 | 0.392424 | 0.318913 | 0.309017 |
1.2 A | 0.058563 | 0.016524 | 0.131621 | 0.028003 | 0.017487 | 0.000149 | 0.392389 | 0.332275 | 0.308337 |
HL1 | 0.8 A | 0.0559 | 0.01576 | 0.115023 | 0.026894 | 0.016284 | 0.000161 | 0.420049 | 0.309944 | 0.297148 |
1.0 A | 0.058055 | 0.016488 | 0.131665 | 0.027865 | 0.017571 | 0.000162 | 0.392424 | 0.318913 | 0.309017 |
1.2 A | 0.059647 | 0.017043 | 0.140861 | 0.028555 | 0.018584 | 0.000162 | 0.370816 | 0.324687 | 0.317635 |
LHC2 | 0.8 A | 0.057737 | 0.016635 | 0.131025 | 0.027884 | 0.01751 | 0.000167 | 0.390575 | 0.305499 | 0.305819 |
1.0 A | 0.058055 | 0.016488 | 0.131665 | 0.027865 | 0.017571 | 0.000162 | 0.392424 | 0.318913 | 0.309017 |
1.2 A | 0.058158 | 0.016422 | 0.132089 | 0.027596 | 0.017603 | 0.000178 | 0.393661 | 0.325046 | 0.311724 |
- (1)
Effects of the initial prestress on the importance coefficients of the elements
The importance coefficients of the elements were analysed under different initial prestress levels (
Table 2). In
Table 2, 0.5
P, 1.5
P, and 2.0
P represented 0.5, 1.5, and 2 times the initial prestress, respectively. The results presented in
Table 2 indicated, as the prestress increased, that the importance coefficients of all the elements decreased by different extents. The importance coefficients of the outer diagonal cable, outer strut, and inner top hoop cable changed by −84.3%, −74.9%, and −62.9%, respectively. However, the importance coefficients of the outer hoop cables changed marginally by an average of −8.6%.
- (2)
Effects of the external load on the importance coefficients of the elements
The importance coefficients of the elements were analysed under different external loads, and the results were presented in
Table 3, where 0.5
F, 1.5
F, and 2.0
F represented 0.5, 1.5, and 2 times the initial external load, respectively. As the external load increased, different elements exhibited different trends in their importance coefficients. The importance coefficients of the outer ridge cable, inner ridge cable, inner diagonal cable, outer strut, inner lower hoop cable, and inner top hoop cable decreased with an increase in the external load. By contrast, the importance coefficients of the outer diagonal cable, inner strut, and outer hoop increased with an increase in the external load. When the load increased from 0.5
F to 2.0
F, the importance coefficients of the inner strut and outer strut changed by 70.8% and −26.1%, respectively. For the aforementioned increase in the load, the importance coefficients of the inner diagonal cable, inner lower hoop cable, and inner top hoop cable changed marginally by −0.7%, −2.7%, and −3.4%, respectively.
- (3)
Effects of the areas of the element sections on the importance coefficients of the elements
The importance coefficients of the elements were analysed under three cross-sectional areas of the elements, namely 0.8 times the initial cross-sectional area (0.8 A), the initial cross-sectional area (1.0 A), and 1.2 times the initial cross-sectional area (1.2 A). The obtained results were presented in
Table 4. The importance coefficients of different elements exhibited different trends with changes in the cross-sectional areas of the elements. For example, when the cross-sectional area of the outer ridge cable increased from 0.8 A to 1.2 A, its importance coefficient decreased gradually, whereas the importance coefficients of the other elements increased gradually. The areas of different elements had different effects on the importance coefficients of the other elements. For instance, when the cross-sectional areas of the outer ridge cable, inner ridge cable, outer diagonal cable, inner diagonal cable, outer strut, inner strut, outer hoop cable, and inner hoop cable increased from 1.0 A to 1.2 A, the importance coefficients of these elements changed by 5.3%, 2.2%, −4.3%, 2.6%, −1.3%, −0.01%, −5.8%, and 0.3%, respectively. The outer hoop cable exhibited the highest importance coefficient. Its importance coefficient decreased with decreases in the cross-sectional areas of the outer ridge cable, inner ridge cable, inner diagonal cable, inner lower hoop cable, and inner upper hoop cable and increases in the cross-sectional areas of the outer diagonal cable, outer strut, inner strut, and outer hoop cable.
In summary, different elements had different importance coefficients, and the outer hoop cable had the highest importance coefficient in this study. Moreover, the importance coefficients of different elements exhibited different trends with variations in the external load, the initial prestress, and areas of element sections.