Mass of Extremely Tall Truss Towers

Gerry Flanagan

Alna Space Program

May 6, 2012

Introduction

This study was conducted with the goal of determining whether it would be possible to free-standing towers to the fringes of the atmosphere. Such compressive structures have been proposed [1,2,3] as a means of reducing the cost of space launches. Other possible applications include placing wind turbines in the jet stream, or permanent observation platforms. It is not clear that any application could justify the enormous cost of such a structure, but it is an interesting engineering exercise to determine some limits present-day materials and passive structure are used. Only conventional truss structures will be considered. The tower must be naturally stable (no actively controlled structures) from collapse and able to withstand hurricane force winds. We have also eschewed the use of guy wires for resisting wind and preventing buckling. Guys wires tens of kilometers long introduce more technical problems, and also introduce a significant nonlinear element into the analysis.

This study is for what will be termed “first-order” trusses. By this, we mean that the basic tower geometry is broken up into a system of bar elements. The bars will be analyzed as structural cylinders. For a structure of this scale, it is likely the bar elements would be further broken into truss structures. This would have the benefit of potentially lower weight for the same buckling performance along with much reduced area presented to the wind. The elements of these trusses could also be trusses, as in a fractal regression. Higher-order truss structures require much additional analysis effort and therefore first-order results are presented as a preliminary effort to learn more about these structures and the optimization trends.

Bolonkin [mass_of_extreme_towers_1.gif] has presented the basic equations for a constant stress tower. However, his results do not include in discussion of stability or wind. As an alternative to solid structures, Bolonkin has also presented calculations for gas-filled structures [2]. Again, only the strength aspect of the problem is presented. Landis [3] discusses the advantages of a tall tower for space launch, but does not go into any substantive detail on the tower structure.

Analysis Approach

Automated Design Approach

The design approach being taken is not meant to be construed as a truely “optimal”, but rather a rational design process. The reason for a “rational” design instead of a true optimal method is simply the computational cost. Some of the design being considered use over a thousand truss elements. Any of the formal mathematical optimization procedures would require thousands of finite element analyses. This would not be practical for our purposes. The methods used typically converge within forty analysis cycles.

The automated design procedures only address the tube diameters and wall thickness. The nodal positions are not varied, so there is no geometric optimization of the tower. Several of the studies that will be presented look as some of the overall geometric parameters such as the base width. However, these studies are performed one-dimensionally. It is likely that better designs could be found if several parameters could be varied simultaneously using mathematical optimization, but these studies would again require more computational resources. However, these studies have been repeated multiple times with the findings feeding back into the parameters used to perform the one-dimensional searches. In a sense, an informal multi-parameter search has been performed.

There are three phases to the design process. For the first phase dimensions are assigned to the tubes using forces that are estimated by summing the total weight above any vertical location. By starting from the top and proceeding down the tower the forces can be determined without performing a global analysis. In the second phase the loads are determined by the finite element method. Each tube is then individually optimized for the minimum weight that will satisfy all of the tube criteria. A new load distribution is then determined and the iteration continues until no tube changes by more than a set tolerance. Global buckling is not considered during this phase and there is no guarantee that the buckling criterion will be meet. Because the tube resizing is done after the finite element analysis, there is also no guarantee that the local failure modes will meet the factor-of-safety goals after an additional finite element re-analysis, but the method will typically come close to meeting all the local failure modes criteria. The method then moves into a third phase. During the third phase, tubes are first resized based on the strain-energy-density distribution of the global buckling eigenvector. If the buckling criterion is not meet, the method increases the stiffness of the tubes in proportion to the normalized strain energy. After the buckling resize, the tubes are then checked and resized for the local criteria. The sizing method will satisfy all of the criteria for each tube, but is not necessarily a weight optimal solution. For both the buckling sweep and the local failure sweep, the tube area can only increase. The approach prevents the buckling and local strength operations from working against each other and going into a non-convergent loop. After the global buckling and local criteria sweeps, the load is recalculated from the finite element method and the process continues. The iteration loop will terminate when both the global buckling criterion and the local tube failure criteria are meet, or if the iteration count is exceeded. It is possible for the method not to converge. This may indicate that the tower geometry is inadequate. Buckling perform ace can be improved by making a wider tower for change other aspects of the geometry.

Failure Modes and Criteria

A feasible tower is designed to meet a global buckling requirement, and to meet a selected number of local failure criteria for the individual tubes that make up the truss structure.

The tube sizing (radius and thickness) is determined by simultaneously satisfying strength (load divided by area is less than the material failure stress), the Euler buckling is given by mass_of_extreme_towers_2.gif where e is the Young’s modulus. Wall buckling stress of the tube is estimated using a semi-empirical equation

mass_of_extreme_towers_3.gif

These are equations for isotropic thin wall buckling with ν=0.3, taken from NASA SP-8007, "Buckling of Thin Walled Circular Cylinders".  Some of the massive tubes that result from this analysis are probably well out of the range of data used to create these equations, but we will use them until a better method comes to light. In addition, the wall buckling equation is derived for an isotropic material. We will consider the case of a highly orthotropic composite material which certainly falls outside the reliable range of the equation. However, we are assuming a monolithic wall for the tubes. In reality, if wall buckling was a governing factor the performance could be improved in a weight efficient manner by using stiffeners or a sandwich construction.

The various failure criteria must not only be meet, but the structural capability should exceed any failure mode by a specified factor-of-safety (FS). In this analysis, the same FS is applied to all the modes. For the present study, a FS of 1.5 is applied.

The Finite Element Method

The load distribution and global buckling analysis are performed using a standard finite-element formulation for an assemblage of bar elements. If x is the local axial direction of the bar, then the element stiffness matrix is

mass_of_extreme_towers_4.gif

where EA is the product of the material Young’s modulus and the cross-sectional area of the bar, and L is the length. The subscripts 1 and 2 refer to the two nodes. F is the nodal force, and u is a nodal displacement. The local stiffness matrix is transformed into the global coordinate system, and the total matrix is assembled as in the standard finite element method. For buckling, a geometric stiffness matrix is formed using

mass_of_extreme_towers_5.gif

where N is the bar force, obtained from a previous linear displacement analysis. If the assembled linear stiffness matrix is designated as K, and the assembled geometric stiffness as mass_of_extreme_towers_6.gif, then the buckling factor comes from the solution of the eigenvalue problem

mass_of_extreme_towers_7.gif

where lowest value of the eigenvalue λ multiplies the current loads to give a buckling value. Thus, λ is the buckling factor-of-safety on load that we desire.

It is possible for the lowest eigenvalue of the system to be negative, which implies that buckling would occur if the sign of the loads was reversed. These cases are not of interest and the algorithm searches for the lowest positive eigenvalue in order to find the gravity driven buckling mode. The buckling analysis is performed with the wind loads applied. Studies have shown that the wind loads can have a significant impact on the tower stability.

A limitation of the analysis is that the bar elements of the truss do not represent bending. When the tube is not vertical there will be a gravity load that tends to bend the tube. This additional bending stress is not part of the analysis. Bending of the tube is part of the Euler buckling calculation for one of the local failure modes.

Assumed Material Properties

A goal of this study was to use existing engineering materials rather than speculate on future developments of advanced materials. The four selected materials are shown in Table 1. The steel properties are for AISI 4130, taken from MIL-HDBK-5 [4]. The aluminum properties are for 7075 4” plate [5]. This is a typical aircraft quality aluminum. The material labeled “GRP Laminate” is graphite reinforced plastic in a quasi-isotropic laminate form. The properties are typical values for laminates using low-cost manufacturing methods. Although the analysis only uses the material axial properties, practical structures usually end up being multi-directional laminates. “GRP Uni.” is a unidirectional graphite reinforced plastic, specifically HITEX 33/E7K8 from MIL-HDBK-17-2F [6].  This represents a high-end aircraft quality material, but a material that is in current applications. The majority of initial geometric studies and trade-offs will be made using the GRP Laminate properties. The strength checks used in the analysis code do not distinguish between tension and compression stress. The structure will be dominated by compression, and therefore the quoted strength values are for compression. For metals, yield  is considered failure.

The mass of optimized structures is usually driven by either specific stiffness (Young’s modulus divided by density), or specific strength (strength divided by density). For convenience, these properties are given in Table 2.

Table 1. Assumed Material Properties

Stiffness, GPa Strength, MPa mass_of_extreme_towers_8.gif
Steel 200. 480. 7833
Aluminum 71. 360. 2975
GRP Laminate 55. 350. 1570
GRP Uni. 117. 1160. 1570

Table 2.  Stiffness/Density and Strength/Density

mass_of_extreme_towers_9.gif mass_of_extreme_towers_10.gif
Steel 2.5533*10^^7 61279.2
Aluminum 2.38655*10^^7 121008.
Quasi GRP 3.50318*10^^7 222930.
Uni GRP 7.45223*10^^7 738854.

Examples for Three-Leg Towers

Repeating Unit

The basis repeating unit for a three leg tower will consists of (near) vertical legs, three pairs of crossing diagonal members, and a horizontal ring of three elements at each node elevation, as shown below. The diagonal elements cross, but it is assumed that they move freely at the crossing point. The bar-element analysis assumes that the bars can rotate freely about the connecting nodes.

mass_of_extreme_towers_11.gif

Figure 1. Repeating Unit for 3-Leg Tower

Optimum Geometric Parameters

The first question to be address is the appropriate base radius for the tower. This could also be called a slenderness ratio. By “radius” we mean the radius of a circle that fits the footprint of the three legs. In order to perform a one dimensional parametric study, additional parameters need to be selected and fixed. We assume a tower shape that follows the exponential relation

mass_of_extreme_towers_12.gif

where x (the height coordiante) has the range of 0 to 1, or it’s equivalent

mass_of_extreme_towers_13.gif

where mass_of_extreme_towers_14.gif is the radius at the bottom, and mass_of_extreme_towers_15.gif is the radius at the top. We will further assumed that mass_of_extreme_towers_16.gif The validity of this assumption will be reviewed below. The material is the quasi-isotropic GRP.  The modeling function  keeps the local radius to height ratio of a repeating unit constant. The unit height ratio is set to 1.5 (another parameter to be examined later in this report).

The results in Figure 2 show that a minimum mass tower results when the base radius is 5% of the tower height. Of course, the optimal radius may be a more complex function of height instead of a simple ratio, but this possibility was not explored.

mass_of_extreme_towers_17.gif

Figure 2 Tower Mass versus Base Radius to Height Ratio.  mass_of_extreme_towers_18.gif, mass_of_extreme_towers_19.gif.

The next parametric study examines the ratio of the top radius to the bottom radius. For this study, we assume that the bottom radius is 5% of the tower height. The results, shown in Figure 3 indicate a fairly flat minimum for mass_of_extreme_towers_20.gif>10.

mass_of_extreme_towers_21.gif

Figure 3 Tower Mass versus Base Radius to Top Radius Ratio. mass_of_extreme_towers_22.gif, mass_of_extreme_towers_23.gif

The next study examine the height-to-radius ratio mass_of_extreme_towers_24.gifof the individual repeating units. For this study, we keep the base radius ratio at 5% of the height, and the top radius at 1/10 of the base. mass_of_extreme_towers_25.gifis the local radius at the base of the repeating unit.

mass_of_extreme_towers_26.gif

Figure 4. Tower Mass Versus Unit Cell Height to Local Radius Ratio.  mass_of_extreme_towers_27.gif, mass_of_extreme_towers_28.gif.

All of the examples have used the exponential radius function. The following example compares the exponential shape to a linearly tapering tower with all of the other design parameters held constant. The results show that the power-law shape is more efficient.

Table 1. Tower Mass for Different Radius Distribution Functions

Exponential shape 1.49442*10^^9
Linear shape 2.09269*10^^9

Mass versus Height Study

Putting these results together, we chose a tower for further study with mass_of_extreme_towers_29.gif, mass_of_extreme_towers_30.gif. Furthermore, we assume the tower supports mass_of_extreme_towers_31.gifkg at the top. We can then compute the tower mass as a function of height. The results are shown in Figure 5. Two curves are shown. One is for a cylinder coefficient of drag mass_of_extreme_towers_32.gif) equal 1. This is the expected value for a cylinder with a high Reynolds number. The second curve (red) is for a reduced mass_of_extreme_towers_33.gif of 0.25. The second curve is a quick evaluation of the impact of wind on the tower mass, and might be indicative of the benefits that could be gained with the cylinder truss elements were replaced with additional trusses with less area presented to the wind.

The mass_of_extreme_towers_34.gif curve (blue) initially shows a near linear relation in log-log space until a height of about 10 km. At that point, the algorithm begins to struggle to find a feasible design. This behavior may indicate that the particular geometry and material combination is reaching an inherent limit. The red cruve is discontinuous; it stop as 10 km and restarts at 25 km. The break is because the algorithm failed to find feasible designs for the intermediate points. Later, it will be shown that a six-leg tower geometry converges more readily over the entire range.

mass_of_extreme_towers_35.gif

Figure 5. Thee-Leg Tower Mass Versus Maximum Height. mass_of_extreme_towers_36.gif.

Total Mass versus Top Mass

The various geometric studies were performed assumed that the dead mass at the top of the tower was mass_of_extreme_towers_37.gif kg.  If thinking in aerospace terms, top mass is in a sense the “payload”. We will no explore the relation between the top mass and the tower mass. In a tension structure where stability is not a concern, one would expect the total mass to be proportional to the top mass. For these solutions, the total mass is nearly independent of the top mass as shown in the figure below. The slight dip in mass going from mass_of_extreme_towers_38.gif to mass_of_extreme_towers_39.gif kg of top mass is most likely numerical noise due to the limited precision of the iteration loops. The result is surprising at first, but upon further reflection may be expected. The sizing of upper most units are influenced by the top mass, but the sizing of lower units is quickly overwhelmed by stability requirements and the mass of the tower above a given position. A lesson would be that once you build an extreme tower, you might as well take advantage of a large payload capacity.

mass_of_extreme_towers_40.gif

Figure 6  Tower Mass Versus Mass at Top.

Examples for Six-Leg Towers

As a second example, consider a basic unit with 6 vertical legs. Figure 7 shows the repeating unit that will be used. Figure 8 is a typical six-leg tower with an exponential taper.

mass_of_extreme_towers_41.gif

Figure 7  Repeating Unit for 6-Leg Structure

mass_of_extreme_towers_42.gif

Figure 8  Example of 6-Leg Truss Tower.

Geometric Studies

The same geometric parameters as considered for the 3-leg case will be repeated for the 6-leg case. Figure 9 shows that the base radius to height ratio minimizes at 4%, a bit lower than the 5% for the 3-leg tower. Figure 10 shows the ratio of the bottom radius to top radius, and Figure 11 gives the mass versus repeating unit height. From these results, we will select mass_of_extreme_towers_43.gif, mass_of_extreme_towers_44.gif=10, mass_of_extreme_towers_45.gif=1.5 for further studies.

mass_of_extreme_towers_46.gif

Figure 9 Tower Mass versus Base Radius to Height Ratio.  mass_of_extreme_towers_47.gif, mass_of_extreme_towers_48.gif.

mass_of_extreme_towers_49.gif

Figure 10 Tower Mass versus Base Radius to Top Radius Ratio. mass_of_extreme_towers_50.gif, mass_of_extreme_towers_51.gif

mass_of_extreme_towers_52.gif

Figure 11. Tower Mass Versus Unit Cell Height to Local Radius Ratio. . mass_of_extreme_towers_53.gif, mass_of_extreme_towers_54.gif.

Figure 12 gives the mass of the tower as a function of the height. The mass_of_extreme_towers_55.gifcase was performed to indicate the benefit of have a reduce wind area (using higher order trusses to replace the cylindrical tubes). The results are similar to the 3-leg case up to 15 km in height, as shown in Figure 13. The sudden change in slope for the 3-leg case may be an anomaly of the algorithm, or it may show that the more stable 6-leg case can be used for higher towers before the method begins to struggle finding feasible designs.

mass_of_extreme_towers_56.gif

Figure 12. Six-Leg Tower Mass Versus Maximum Height. mass_of_extreme_towers_57.gif.

mass_of_extreme_towers_58.gif

Figure 13 Tower Mass Versus Height for Three-Leg (Red) and Six-Leg (Blue) Trusses

Taking the six-leg tower and the best geometric parameters, we can now compare the performance of different materials. They results are summarized in Figure 14. Each of the materials shows the same near-linear relation for mass versus height when plotted on a log-log scale. I was surprised that the slopes of the different curves are nearly equal. The change in material performance mostly shifts the curve vertically. The linear curve fit parameters are shown in Table 4. To obtain the curve fits, the last points (beyond 20 km) are dropped for the steel and aluminum structures because the curves appear to start diverging from the linear behavior at that point. I suspect that the change in slope indicates that a height limit is being reached for that material in the particular geometric arrangement, but this hypothesis was not explored further. From the parameters in Table 4, mass can be estimated from the relation

mass_of_extreme_towers_59.gif

where M is the mass (in kg) and x is the tower height (in km). Figure 15 shows the original results as solid lines, and the curve-fits as dashed lines.

mass_of_extreme_towers_60.gif

Figure 14  Tower Mass Versus Height for Six-Leg Truss with Different Materials

Table 4  Curve-Fit Parameters

a b
Steel 1.46758*10^^7 2.76333
Aluminum 7.99259*10^^6 2.63114
GRP Laminate 4.82841*10^^6 2.51545
GRP Uni 1.99588*10^^6 2.54892

mass_of_extreme_towers_61.gif

Figure 15  Tower Mass Versus Height for Six-Leg Truss with Different Materials. Solid lines are original results, dashed lines are curve fits

Observations for Typical Tower Design

Buckling Modes

This section demonstrates some of the buckling modes that appear in the designs. Small changes in the input parameters can make significant changes in the mode. In the array of buckling shown in Figure 16, modes (a) and (b)  involve local distortions of many individual repeating units. Mode (c) is a local “bulge” in the structure. Mode (d) is a more classic bending shape. For our purposes I can see no preference in the buckling mode as long as the eigenvalue satisfies the factor-of-safety.

mass_of_extreme_towers_62.gif

Figure 16 Buckling Modes for Series of Towers

Tube Factors-of-safety for “Optimized” Design

We will now take a look at the tube failure modes and factors-of-safety (FS) for an optimized design. For the next series of results we will assume a 10K tall tower with mass_of_extreme_towers_63.gif=0.04, mass_of_extreme_towers_64.gif0.15, mass_of_extreme_towers_65.gif The material is the multidirectional GRP. This is the same tower as case (d) in Figure 16. Figure 17 shows the minimum FS for each tube as a color plot. The plot is limited to a maximum FS of 10 in order to concentrate the color levels to a range of interest. Because bending due to wind the elements on one side of the tower are closer to the target FS of 1.5 than those on the other side. The algorithm tends to overshoot the desired FS by a factor of 1.1. The 1.1 factor comes from a built-in over-design that accelerates the algorithm convergence.

In Figure 18 the minimum FS within a repeating unit and for a given element type (leg, diagonal, or ring) is determined and applied to all the elements of that type within the unit. Recall that the algorithm assigns the same dimensions to all the elements of a type within a unit. The figure shows that the design algorithm effectively sizes the elements so that nearly all of the elements are at the target FS. The base ring is unloaded because of the fixed boundary conditions.

mass_of_extreme_towers_66.gif

Figure 17 Minimum Factor-of-Safety for Individual Tubes. 70 m/sec Wind

mass_of_extreme_towers_67.gif

Figure 18 Minimum Factor-of-Safety for Tubes Within a Set (70 m/sec Wind)

Figures 17 and 18  show the minimum FS for a tube, selected from the three failure modes considered. We can further examine the individual failure modes. Figure 19 plots each FS for all the failure modes. The figure shows that the algorithm tends to bring all of the failure modes close to the goal for many elements. It should be noted that the plot range cuts of elements in which the FS is greater than 3. It is interesting that strength is only critical for the lower elements (element number roughly corresponds to height). We will see in a moment that the strength failure mode is driven by the wind loads.

mass_of_extreme_towers_68.gif

Figure 19 Factors-of-Safety for All the Local Tube Failure Modes. All Elements. 70 m/sec Wind

The exercise can be repeated, but without a wind load. Figure 20 shows the individual minimum FS. In this case there is no need to look at the minimum FS within a unit because the symmetry of the load condition assures equal factors. The rings have a large FS, which may indicate that they are unnecessary. The algorithm has a minimum diameter and gage for the tubes, which can result in the algorithm leaving some elements with a large FS.  Figure 21 shows all the failure modes. For the no-wind case, strength does not appear to be a governing factor.

mass_of_extreme_towers_69.gif

Figure 20 Minimum Factor-of-Safety for Individual Tubes. No Wind

mass_of_extreme_towers_70.gif

Figure 21 Factors-of-Safety for All the Local Tube Failure Modes. All Elements. No Wind

Dimensions for Optimized Design

We next look at the final dimensions for a typical design. We will be using the 10km tower described in the section above, with wind loads applied. Figure 22 shows the individual tube dimensions (radius and wall thickness) for the leg elements. Figure 23 is for the rings, and Figure 24 is for the diagonal elements. The radii are enormous and this demonstrates that we should be considering the use of truss structures instead of cylindrical tubes. However, even for these large dimensions, but tower appears to be reasonable proportioned when considering it’s great height, as shown in Figure 25

mass_of_extreme_towers_71.gif

Figure 22 Dimensions of Leg Elements

mass_of_extreme_towers_72.gif

Figure 23 Dimensions of Ring Elements

mass_of_extreme_towers_73.gif

Figure 24 Dimensions of Diagonal Elements

mass_of_extreme_towers_74.gif

Figure 25 Portion of Tower with Tubes Drawn to Scale

Conclusions

In looking at the very large masses involved, one must ask what would be a reasonable limit for a large scale engineering project in the 21th century? One way to view the question would be to consider the mass of existing large buildings. The total mass of the World Trade Center Towers has been estimated as 9.3 10^8 kg. Burj Dubai is similar, involving mass_of_extreme_towers_75.gif kg of steel and concrete. From these numbers, a mass_of_extreme_towers_76.gif kg structure would appear feasible. From Figure 12, an 8-kilometer tall tower could be build for this mass using a modestly performing composite material. The tower could withstand winds equivalent to the top range of a category-4 hurricane. With a current high-performance unidirectional carbon composite, 10 km could be achieved (Figure 14). These are huge towers, bur far from reaching space. Such a tower would be high enough to place wind turbines into the jet stream.  If total mass is not limited to the realm of “reasonable”, then with existing materials a 40 km tower could be built.

The main goal of this study was to test out the automated design methodology. The methods appear to work in most cases and give reasonable dimensions and masses. The methods sometimes fail to converge, but this may highlight the physical limits; there may not be a feasible solution for a given set of inputs. The method is fast. For a 1200 tube model, the algorithm converges within 4 minutes.

It is clear that we need to consider converting the cylindrical tubes into truss structures in order to push the height limits further. Trusses have the major advantage of presenting less area to the wind, with a consequent reduction in wind induced bending loads.

Calculation Notes

All of the figures and calculations in the study were generated in the Mathematica Notebook “first order tower calculations.nb”, available on the Alna Space Program web site (alnaspaceprogram.org). The calculations require the package file ttas.m (Release 2, May 1 2012) also available on the website. ttas stands for Truss Tower Analysis System. The package defines all of the analysis and optimization functions used in the study, including the finite-element code.

References

1.      Alexander Bolonkin, Optimal Solid Space Tower, AIAA-2007-0367.

2.     Alexander Bolonkin, Optimal Inflatable Space Towers with 3- 100 km Height, JBIS Vol 56, pp.87-97,  2003.

3.     Geoffrey A. Landis and Vincent Denis, High Altitude Launch for a Practical SSTO, Space Technology & Applications International Forum, Albuquerque, NM, Feb. 2-6 2003. Available at http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/20030022661_2003025516.pdf

4.        MIL-HDBK-17-2F, Table 4.2.2(d)

5.        MIL-HDBK-5H, Table 2.3.1.0

6.        MIL-HDBK-5H, Table 3.7.4.0

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