Keywords

1 Introduction

The Space Needle in Seattle, Washington, is one of the most popular tourist spots around the world. Approximately 1.3 million guests visit the Space Needle per year, and nearly 60 million visitors have visited the tower since it opened in 1962. The tower was designed by John Graham & Company.

The structure of Space Needle comprises a steel tripod, with each of the three legs pinched just above the middle of their height and topped by a multi-level tophouse reminiscent of a flying saucer. This tophouse consists of five stacked layers: a revolving restaurant, a mezzanine level, an observation deck, a mechanical equipment level, and at the tower’s pinnacle, an elevator penthouse. The structure was also originally crowned by a 15 m natural gas torch. On the basis of its structural features, the tower is 42 m wide, weighs 8660 metric tons, and is built to withstand winds of up to 32 0 km/h and earthquakes of up to 9.0 magnitude, as strong as the 1700 Cascadia earthquake [1].

Earlier studies on the Space Needle tower mainly focus on the analysis of its acoustical design and commercial values. However, the natural frequencies and deformation type of the tower are merely studied. Thus, this study focus on calculation of natural frequencies and analysis of the tower’s weakness [2].

2 Method

2.1 Model Establishment

For the purpose of simplification, the model constructed in the ANSYS WORKBENCH is composed with 11 components [3].

Component 1 is a triangle base with a total height of 154 m, which can be further divided into three platforms. The length of the six support bars on platform one (the lower one) is 11.3 m, and the thickness is 2.3 m. The length of the support bars on platform two (the middle one) is 7.1 m, and the thickness is 1.2 m. The highest platform is composed with two cylinders whose edges are linearly connected. The diameter of the upper cylinder is 33 m, and the diameter of the lower cylinder is 21 m (Fig. 1).

Fig. 1.
figure 1

Component 1

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As shown in Fig. 2, the curvature of the upper part of the tripod changes from 0.011 \({\text{m}}^{-1}\) to 0.008 \({\text{m}}^{-1}\) from top to bottom.

Fig. 2.
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Component 1 Upper Part Dimension

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The curvature of the middle part ranges from 0.0002 \({\text{m}}^{-1}\) to 0.0078 \({\text{m}}^{-1}\), see Fig. 3.

Fig. 3.
figure 3

Component 1 Middle Part Dimension

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As shown in Fig. 4, the curvature of the lower part is constant, which is 0.0003 \({\text{m}}^{-1}\). And each hollow cuboid has a length of 9 m, a width of 3.3 m, and a thickness of 1.2 m.

Fig. 4.
figure 4

Component 1 Lower Part Dimension

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In addition to Component 1, component 2 is another support for the Space Needle Tower (See Fig. 5). It was constructed by creating a circle and then using the circular pattern button to make it a hexagon with an edge length of 3.4 m and finally extruded it to a height of 151.5 m.

Fig. 5.
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Component 2

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Component 3 is composed with two cylinders (See Fig. 6). A big circle with a diameter of 21.1 m is first created and then it is stretched to a thickness of 0.6 m. The smaller one is created by repeating the same process, its diameter is 15.1 m and its thickness is 1.6 m.

Fig. 6.
figure 6

Component 3

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As shown in Fig. 7, component 4, 5, 6 have the same shape but are different in size. Three types of edges are named Edge # 1, 2, and 3 based on their length, from shortest to longest. The dimensions of each component are recorded in Table 1.

Fig. 7.
figure 7

Component 4, 5, 6

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Table 1. Dimensions of Component 4, 5, 6
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Figure 8 shows component 7, the top of the tower, which was composed with four cylinders. The cylinders are named Cylinder #1, 2, 3, 4 from top to bottom. The curvature range of the connecting surface between cylinders 3 and 4 is 0.27 \({\text{m}}^{-1}\) to 0.47 \({\text{m}}^{-1}\). The dimensions of each cylinder are recorded in Table 2.

Fig. 8.
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Component 7

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Table 2. Dimensions of the Cylinders in Component 7
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Figure 9 shows component 8&9, which are two cylinders with the same diameter of 30.2 m, the height of component 8 is 3.1 m and the height of component 9 is 2.8 m.

Fig. 9.
figure 9

Component 8, 9

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As shown in Fig. 10, the longer edges of component 10 have a length of 11.7 m and the shorter edges are 4.5 m. The angle between two adjacent short edges is \(120^\circ \).

Fig. 10.
figure 10

Component 10

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Component 11 is the frustum of a cone at the top of the model, which has a diameter of 0.6 m at the upper bottom, 2.1 m at the lower bottom, and a height of 11.7 m.

Fig. 11.
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Component 11

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All components are combined into a whole after being established by using the combine and joint button. The final modeling result is shown in Fig. 12.

Fig. 12.
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Final Model of the Space Needle

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2.2 Material Settings

The main material of the Space Needle Tower is structural steel. The actual total weight is 9550 tons (excluding the foundation), and the volume of the model is 25958.26 \({\text{m}}^{3}\). We chose structural steel as the only material, and as a simplification, we set the equivalent density of the model to the actual mass of the building divided by the model volume [4]:

$${\varvec{\rho}}=\frac{{\varvec{M}}}{{\varvec{V}}}=\frac{9550000{\varvec{k}}{\varvec{g}}}{25958{{\varvec{m}}}^{3}}=367.9{\varvec{k}}{\varvec{g}}/{{\varvec{m}}}^{3}$$
(1)

The corresponding material settings are shown in Table 3.

Table 3. Material Settings
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2.3 Boundary Conditions

The bottom of the space needle tower is actually connected to a foundation, as our model didn’t consider the foundation, the seven faces at the bottom of the model are considered fixed and supported (see Fig. 13) [5].

Fig. 13.
figure 13

Boundary Conditions

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2.4 Mesh Generation

Appropriate mesh generation is the key to accurate model analysis. To determine the appropriate grid size, an independence study was conducted to test the dependence of the result on mesh density. By changing the mesh size, the relationship between the number of nodes and the fundamental frequency is shown in Fig. 14 [6, 7].

Fig. 14.
figure 14

Mesh Independence Study

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According to Fig. 11, the fundamental frequency hardly varies with the number of nodes in the range of 80000 to 130000 nodes, so the weak dependence can be ignored. In order to obtain the most accurate results, a grid size of 0.767 m was used. The model was then divided into 126144 nodes and 66287 units (see Fig. 15).

Fig. 15.
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Mesh Generation Result

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3 Results

The modal analysis was carried out by using ANSYS Student Version. Six modes and the corresponding natural frequencies were obtained (see Fig. 16).

Fig. 16.
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Frequencies of Different Modes

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For six modes with different frequencies, deformation analysis was carried out separately, and the result was shown in Fig. 17, 18, 19, 20, 21 and 22.

Fig. 17.
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Deformation Diagram Mode 1

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Fig. 18.
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Deformation in Diagram in Mode 2

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Fig. 19.
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Deformation Diagram in Mode 3

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Fig. 20.
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Deformation in Diagram in Mode 4

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Fig. 21.
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Deformation Diagram Mode 5

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Fig. 22.
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Deformation Diagram in in Mode 6

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Figure 17, 18, 19, 20, 21 and 22 shows the deformation of six modes with different frequencies. According to the Figs, the area where the maximum normalized deformation occurs and the tower’s weakest points can be clearly identified (See Table 4).

Table 4. Deformation at Six Different Frequencies
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4 Conclusions

4.1 Summary

Modal analysis is performed on the simplified model of the Space Needle tower. Proper meshing and boundary conditions are applied during the simulation process. Six mode types with corresponding natural frequencies have been obtained. The natural frequencies range from 0.72 Hz to 7.49 Hz, the corresponding maximum normalized deformation is from 0.062 cm to 0.189 cm. Component 1, the triangle base has the greatest degree of deformation in all six modes. Component 4, 5, 6 (which together form the first platform) have the smallest degree of deformation. In mode 1&2, bending is the most obvious and primary form of deformation [7]. Within a component, the part at a higher altitude tends to show a larger degree of deformation. In mode 3, torsion centered on Y-axis appears and greatest deformation occurs at the highest platform (UFO shaped) throughout the entire tower. In mode 4, 5, 6, sway occurs and the location where the maximum deformation occurs is in the middle of the two connection points (component 1& platform 1, component 1 and platform 3).

4.2 Discussions

According to an empirical study, the natural frequency of tall buildings can be simply given by:

$$ f = \frac{46}{L}\,({\text{Hz, L is the height of the building in m}}$$
(2)

Thus, the empirical natural frequency should be around 0.3 Hz and the difference between the value of mode 1 and the empirical value is relatively small. As shown in the deformation diagrams, the triangular structure and the complex curves of the steel bars (component 1) effectively make the overall structure safer and more stable. It should be pointed out that for the purpose of simplification, the complicated surface structure and the foundation of the tower is not considered during the modeling process, and the weight of the tower is distributed uniformly throughout the whole tower. Actually, the designers took into account the characteristics of seismic waves and specially used a heavy and solid foundation (5850 ton, including 250 tons of refined steel) to frame the entire tower. Therefore, the real tower must exhibit a much more stable state than the model does since its center of gravity is much lower. These steps of simplification are the major causes of inaccuracies in the simulation results. As for future maintenance, one practical method is to install a dynamic monitoring system at the top of the Space Needle tower. Continuously recorded signals can automatically identify the natural frequencies of key vibration modes of the structure and provide engineers with latest information as references.