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Anchoring the masts and vertical antennas - 2

Martin Huml, OL5Y/OK1FUA, JLIB_HTML_CLOAKING

In the first part we talked about forces and the issues regarding anchoring in general; today we will talk about the mast itself. Before I begin I would like to thank for all your feedback, questions and other topics. I am glad that you were interested in the previous article and I will try to keep it that way.

In this sequel we will focus on the most basic version – the tube mast anchored in one level below the antenna. This situation is illustrated under figure 1. To simplify the calculations, we assume that the entire mast is the same tube diameter and has the same properties throughout its length. We will also assume that the wind velocity along the length of the mast is the same (in reality it is lower just above the ground).

mastr sily en

  When analyzing quantities and properties that affect the behavior of the system we get to this list:

  • the total height of the mast, height of the attached rope, distance of the anchor from the mast bottom (to determine forces affecting the system)
  • external and internal diameters of the tube (to determine the strength and the weight of the mast)
  • physical properties of the material from which the tube is made from: density, elastic modulus, strength limit, proportional limit (to determine the strength and weight of the mast)
  • area and weight of the antenna (to determine the wind resistance force)
  • coefficient of the resistance of the mast and the antenna (to determine the wind resistance force)
  • properties of the environment (air): kinetic viscosity, gravitation acceleration, air density
  • wind velocity

Outputs of the calculations we wish to obtain:

  • force in the axis of the mast bottom (action on the point of placement of the mast bottom)
  • force in the axis of the rope (for the selection of the suitable rope)

However, we will be interested particularly in safety - if the mast will survive and to what degree of safety.

But how to assess and compare safety if it has no unit and its expression in words is quite difficult and above all subjective? We will be probably unable to measure it. Construction sectors use a unit called safety coefficient. It is calculated differently for each type of structure, but its interpretation (sense) is always the same: If it is greater than 1, “there is a theoretical guarantee that the structure will survive”. The recommended minimum value is 1.4. If the safety of the structure involves several factors, the coefficient is calculated for each factor separately and the total safety of the structure is the smallest one of them. In our case, there are two critical factors: the strength of the material of which the mast is made (i.e. the tension in it), and the mast buckling (so the mast will not bend). Our considerations will result in the assessment of the total safety of the system.

From the foregoing, it is clear that there are a large number of quantities that are different for specific situations. Everybody has a different antenna, different mast, different mast height ... For illustration I have chosen several situations, that I find appropriate for demonstration and for which I calculated different outputs. In each case I chose the height of the rope attachment so that the total safety is the greatest. Individual variations are as follows:

  • The mast height of 13 m, on which an ECO antenna is placed (3el. tribander for 10/15/20m). This version is calculated for 3 different masts: tube diameter 80 mm with 3 mm thick wall from an average quality duralumin (ver. A), tube from the same material 100/4 mm (B) and steel tube 60/3 mm (C).
  • The mast height of 13 m with 11el. antenna for a 2 m band in two versions: average duralumin 60 mm in diameter with a 2 mm thick wall (D) and fiberglass 60 mm in diameter with a 5 mm thick wall (E).
  • The last version is a 23 m mast with a bulky antenna TH7DX (7el. tribander for 10/15/20m) again in 2 versions: high quality duralumin 100 mm in diameter with a 10 mm thick wall (F) and steel 100 mm in diameter with a 5 mm thick wall (G).

Other parameters used for calculations are: air density = 1.2 kg/m3, gravity acceleration = 9.82 m/s2, wind velocity = 36 m/s = 130 km/h, coefficient of mast and antenna resistance C = 1.2. The results are shown in table no. 1.

Quantity Symbol A B C D E F G Unit
11m duralumin ECO 11m duralumin ECO

11m

steel ECO

11m duralumin 11el. 2m 11m fibreglass11el. 2m 23m duralumin TH7DX

23m

steelTH7DX

mast - tube                  
total height h 13 13 13 13 13 23 23 m
height of rope attachment h_ki 12 12 12 11 9 17 20 m
anchor distance r_ki 10 10 10 10 10 15 15 m
external diameter D_o 80 100 60 60 60 100 100 mm
internal diameter D_i 74 92 54 56 50 80 90 mm
mast density ro_s 2700 2700 7850 2800 1200 2800 7850 kg/m3
elastic modulus E_s 60000 60000 200000 60000 18000 60000 200000 MPa
strength limit sigma_t 300 300 320 300 220 350 320 MPa
proportional limit sigma_tu 200 200 120 200 200 200 120 MPa
rope reaction F_ropex 1355 1486 1223 629 768 2446 2079 N
reaction in mast bottom F_forces in the axis of the mast 2023 2346 2153 856 858 4951 5808 N
reaction in mast bottom perpendicular F_axis -381 -492 -270 -267 -128 -376 -743 N
force in rope axis F_rope 2116 2321 1911 934 1034 3697 3466 N
antenna                  
area of antenna S_ant 0,82 0,82 0,82 0,18 0,18 0,9 0,9 m2
weight of antenna m_ant 15 15 15 3,5 3,5 40 40 kg
assesing the safety                
tension in the mast k_t 4,01 6,21 2,86 2,37 2,14 3,16 3,34  
buckling k_b 2,47 5,84 3,62 2,01 1,89 3,27 4,75  
total safety k 2,47 5,84 2,86 2,01 1,89 3,16 3,34  

In version (A) I wanted to show that although a relatively thick tube is used the total safety is not as perfect as some might expect based on their experience. This is because the arrangement of the system with a single anchor height is definitely not optimal and places high demands on the strength of the mast material. We will talk about other versions next time, but I can disclose that the strength of the system in dual anchoring is four times greater and even nine times greater in triple anchoring levels (of course, if they are placed in optimal heights). I have also included version (E) because I have seen similar masts being used by several radio amateurs.

material density elastic modulus strength limit proportional limit
  kg/m3 MPa MPa MPa
duralumin 2800 60000 180-450 x
aluminum 2700 60000 60-150 x
steel 7850 200000 320-835 120-290
fiberglass 1200 18000 220 x

Table no. 2: physical properties of the materials

In addition to its own safety system, it is also interesting to look at the distribution of some quantities along the length of the mast. This is shown in figure 2 (for version C) and 3 (E). If the anchoring height is chosen at a height to maximize safety, then the curve shapes are very similar - that is why I show only 2 typical examples.

dopl01

dopl02

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