Suggested replacement of steel ropes by Mastrant synthetic ropes - based on strength only!
Suggested replacement of steel ropes by Mastrant synthetic ropes - based on strength only!
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).
When analyzing quantities and properties that affect the behavior of the system we get to this list:
Outputs of the calculations we wish to obtain:
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:
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.
|11m duralumin ECO||11m duralumin ECO||
|11m duralumin 11el. 2m||11m fibreglass11el. 2m||23m duralumin TH7DX||
|mast - tube|
|height of rope attachment||h_ki||12||12||12||11||9||17||20||m|
|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|
|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|
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|
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.
(note: all tables are available in Excel format)
During my radio amateur activities, I dedicate most of my time (probably like any one of us) to the question of antennas. And one of the most important and maybe the most challenging task is to get the antenna “into the air” and keep it there. Apparently this applies to all antennas, perhaps with the exception of beverage antennas... I wrote about this subject in the article “How to build and anchor simple antenna masts” (magazine “Radioamatér” 2 & 3/2004). At that time, I already felt that this issue is so interesting, complicated and extensive that it would be appropriate to return to it.
Another motive for writing the following article was to include the questions asked either by my friends or emerging on their own, e.g. “Why are you anchoring the vertical so low down?”, “Will this rope be strong enough?”, “Will this mast be able to carry the antenna?” and so forth. Most of the time I couldn't find a better answer than “because I think it's enough” or “because I saw it somewhere like that”. This doesn't sound very scientific. Practice and experience are great and irreplaceable, but too much of it is guesswork, so when my activities brought me to “anchoring” I decided to look into it from the theoretical perspective as well. First, I would like to state that I am not an engineer; so, after studying some time, I consulted Engineer Richard Beber, who, unlike me, has studied this issue. Here, I would like to thank him - the article wouldn't have been written if it weren't for him.
Hence, as indicated by the name of this article, its main subject is focused on how to anchor antennas. This doesn't mean that owners of the non-anchored masts won't find something interesting. For instance, calculations of forces acting on the antenna in the wind or other matters might be useful.
Used terms and simplification
I ask the experts and linguists for leniency - I use these terms as I know them from amateur practice:
A vertical antenna is actually a mast without the antenna. Therefore, where it is not appropriate in the following text, I will not distinguish between these two types of antennas. In other words, e.g. the phrase “anchoring in the middle of the mast” will have to be understood as “anchoring in the middle of the vertical”.
Unless stated otherwise, we assume that the mast is built on a horizontal surface; therefore, the anchoring points and the mast bottom are at the same level, perpendicular to the axis of the mast. This is for simplicity - the reality definitely tends to be different. Therefore, the following text will indicate how to deal with the reality.
In our considerations, we also do not deal with the behaviour of the antenna itself - we assume that the antenna on the mast does not change.
And, finally, for those who are not friends with physics - we will talk a lot about force, whose unit is 1 N (Newton). For instance, if you lift a weight of 1 kg, the acting force on you is approximately 10N.
The Antenna and the Mast
If we look at the simplified model of the anchored mast and the antenna at its top (fig. 1), the following forces will affect the system:
This set of forces will produce a reaction so that the resulting forces will be balanced. The reaction will appear in the gripping of the mast bottom and in the places of mounting of the anchoring ropes. There will also be a flexible deformation of the mast and the ropes. In this respect, we do not consider the irreversible deformation or destruction of the material - it is these cases that we want to avoid and therefore we will focus on the identification of all acting forces.
But forces do not represent all factors that will influence the behaviour of the system. We must not forget about the structure of the mast (tube, lattice structure etc.) and the material of which it is made, especially its physical properties such as density (specific gravity), flexibility and strength. Similarly, we need to know the properties of the anchoring ropes - their strength and elongation. Let's summarise what we need to know:
And the parameters with which we will be dealing are as follows:
The distance of the anchoring
The first question that we will address is the influence of the distance of anchoring on the size of the forces (acting on the mast and the anchoring ropes). We will divide this task into the following marginal situations - in the first case, the wind is blowing from the anchoring direction; in the second case, the wind is blowing “in between the anchors”. (fig. 2)
If the wind is blowing from the point of anchoring direction, it is a simple composition of forces - the case of a right triangle where one of the legs is the mast (h), the second one the distance between the mast bottom and the point of anchoring (r), where the anchoring rope is the hypotenuse (l). The proportion in which the individual legs of this triangle are is the proportion of forces acting in individual directions. We know (can calculate) the force Fant caused by the wind acting on the antenna. (Its specific value is not important at this moment; we will deal with it later - as we are now assessing the influence of the distance of the point of anchoring.) So the force on the mast (in its axis) is Fst = Fant * h / r, and the force acting on the anchoring rope is Fko = Fant * l / r. We will calculate the length of the rope by the Pythagorean Theorem: l = √(h2 + r2).
When the wind is blowing from the direction of the axis between the anchors the situation is somewhat more complicated, because we have to bear in mind the angle between the anchors - in other words, to how many directions the mast is anchored.
For this purpose we look at the mast from the top and introduce a total of 3 points - the mast and 2 anchoring points. Further we will introduce the direction from which the wind is blowing - it is the axis between the anchoring points going through the mast. The place where this axis intersects the connecting line between both of our anchoring points is a point that represents the virtual anchoring point for the calculation of acting forces. We see that it is much closer to the mast bottom than the anchoring distance - the closer it is, the larger the angle between the anchors.
This is again a case of triangles, but not necessarily right angle triangles. If we apply the basic goniometric functions we get to these relations:
For 4 anchors in 90°:
Fst = Fant * (h / r) * √2 = Fst = Fant * (h / r) * 1,414
Fko = Fant * (l / r) * (√2) / 2 = Fant * (l / )r * 0,707
for 3 anchors 120° each:
Fst = Fant * (h / r) / cos(60/180π) = Fant * (h / r) / 0,5
Fko = Fant * (l / r) / cos(60/180π) = Fant * (l / )r .
At a first glance, an interesting feature might not be seen - when anchoring in 3 directions the force acting on the anchoring rope is the same as with wind “from the anchor” as well as “in between the anchors”.
How does this look in practice
From the theoretical perspective, which is probably boring for most of us, we move on to practical effects. (I promise that the following text will be without formulas - they would be much more complicated ...)
For illustration, I chose a simple example - a mast 10 m high with a tribander (10/15/20m) on top. I repeat again – it is an example to show the impact of anchoring distance and the number of directions to which the mast is anchored.
Therefore we are not interested in the mast or rope properties. So - we let the wind blow on the tribander at 130 km/h. It can be approximately calculated that the wind's acting force is approximately 775N. How I got to this result will be explained in the next chapter; for now this result will do.
I will however mention a very important fact, i.e. that the force is proportional to square (squared) velocity (e.g. half velocity = quarter force; the force in the case of 80 km/hour will thus be approx. 290 N). However, the force grows at a similar pace – so you can then understand what a tornado with its wind velocity reaching over 300 km/h in its centre can do, so please do not think that cars flying the air are a mere invention of the American filmmakers.
But now back to anchoring. In the following tables you will see the force calculated for both versions of anchoring (3 and 4 directions) and for the anchoring of 10 and 5 m. I think it is obvious that bringing the anchoring points closer to the mast results in unnecessary increase of the acting forces. This is similar in case of anchoring into 3 directions, which increases the load on the mast.
|High of the fixation||10 m|
|Distance of anchoring points||10 m|
|Speed of the wind||130 km/h|
|Direction of the wind||number of directions||Force [N] effecting:|
|from direction of the guy-wire||4||1 095||775|
|between of the guy-wire||4||775||1 095|
|from direction of the guy-wire||3||1 095||775|
|between of the guy-wire||3||1 095||1 245|
|High of the fixation||10 m|
|Distance of anchoring points||5 m|
|Speed of the wind||130 km/h|
|Direction of the wind||number of directions||Force [N] effecting:|
|from direction of the guy-wire||4||1 732||1 549|
|between of the guy-wire||4||1 224||2 191|
|from direction of the guy-wire||3||1 732||1 549|
|between of the guy-wire||3||1 732||3 098|
As you can see, given a sensible layout, forces caused by such high wind are not so huge. In other words - we do not need any extreme ropes for anchoring such an antenna. As can be found in a lot of places, even a relatively weak rope will endure if it’s designated for this purpose. The weakest places are all joints ... And in our conditions, we must not forget situations where the antenna is encased with frost... But this is really a distraction here and we will return to it.
Just to complete, one more paragraph - observant readers have definitely noticed that it is not about absolute height and distance - we will get the same results with mast of 20m and the distance of 20 and 10m. It is the angle formed between the anchoring rope and the mast. This should be acknowledged especially in situations where the anchoring point cannot be placed at a level perpendicular to the mast (it is on a slope). In such case, for example, if you had to place the anchoring point 3m lower than the mast bottom and you want to keep the angle between the rope and the mast 45° you have to place the anchoring point 13m from the axis of the mast (be careful, not from the mast bottom!). This can be hardly calculated at times, that is why it is possible to calculate and measure the length of the anchoring rope - in this case being 13*1,41 = 18.3m (1,41 = √2).
If we place an object in the fluid stream, in our case the air, it will cause resistance by the friction and pressure elements. This resistance is determined experimentally in aerodynamic wind tunnels and is expressed in relation to the resistance force:
where FO is the resistance force [N], C is the resistance coefficient of the object (-), A is the area of the object perpendicular to the wind direction [m2], ρvz is the air density [kg/m3], &vvz is the wind velocity [m/s].
The value of the resistance coefficient is not constant, but depends on the so-called Reynold's number - non-dimensional criterion, expressing the ratio of inertial and viscous
Re = vvz * d / v ,
where d is the characteristic dimension [m] a v is the kinematic viscosity of the air.
If we simplify the antenna (in our case a tribander) to several cylinders, then the characteristic dimension is the diameter and Re equals roughly 104 in normal conditions while the resistance coefficient of the object C is approximately 1.2. By the resistance force acting on the antenna (tube diameter d = 35 mm, length l = 23.6 m, wind velocity vvz = 36 m/s = 130 km/h; air density 1.2 kg/m3) we can determine:
This force acts as a continuous load on the whole antenna (if we simplify the actual situation on the same velocity profile for the whole surface of the antenna). We can determine the resistance force acting on the mast in a similar way.
The following table illustrates forces calculated for several typical antennas (I apologize to those who can’t find their exact antenna in the table, please just extrapolate).
|Antenna||Wind area [m2]||Force of the wind [N] at speed as [km/h]|
|HF - 3el. tribander ECO||0,82||116||765||1476|
|HF - 3el. tribander A3S||0,40||56||373||720|
|HF - 3el. tribander TH3JRS||0,32||45||299||576|
|HF - 7el. tribander TH7DX||0,88||124||821||1584|
|HF - 11el. 5-bander TH11DX||1,17||165||1092||2106|
|HF - 10el. LP 10-30m LP1010||1,49||210||1390||2682|
|HF - 5el. for 6m F9FT||0,13||18||121||234|
|HF - 5el. for 10m LJ105CA||0,37||52||345||666|
|HF - 5el. for 15m LJ155CA||0,49||69||457||882|
|HF - 5el. for 20m LJ205CA||0,84||119||784||1512|
|HF - 6el. for 20m HD OWA||1,80||254||1680||3240|
|HF - vertical 6-20m R6000||0,14||20||131||252|
|HF - vertical 6-40m R8||0,24||34||224||432|
|HF - Inv.V full-size 20-160m||0,90||127||840||1620|
|2m - 9el. 2M9||0,12||17||112||216|
|2m - 11el. F9FT||0,18||25||168||324|
|2m - 12el. 2M12||0,14||20||131||252|
|2m - 17el. F9FT||0,29||41||271||522|
|2m - 18el. 2M18XXX||0,30||42||280||540|
|70cm - 18el. 440-18||0,08||11||75||144|
|70cm - 21el. 440-21ATV||0,12||17||112||216|
|70cm - 38el. 432-13WLA||0,24||34||224||432|
|23cm - 35el. 23CM35EZ||0,06||8||56||108|
What can be caused by frost?
The frost, which is a very important element in our calculations, affecting the function (as well as survival) of the antenna, causes
Because I do not have any personal experience with the frost I have consulted this issue with a few friends who have their antennas on a variety of problematic places including those where the frost lasts several months. Their experience indicates that frost on the components adds up to 50-100 % and in extreme cases up to 200 % to their original diameter (the component then increases its diameter). Let’s see how much this increases wind loading.
The previous chapter shows that the resistance force of the wind is directly proportional to the surface of the antenna perpendicular to the wind direction and, therefore, to the diameter of its components. So if the diameter of the components increases 2x, it doubles the wind force. These are very easy calculations - if you wish to design an antenna system for extreme frost, multiply the forces by 3.
The mass issue is more complicated. For its calculation, we need to know not only the thickness of the frost but also its density. Although the density of ice is 917 kg/m3, the density of frost stated in literature and standards is considered 400-500 kg/m3 - let 's calculate preferably 500 kg/m3. Since we are interested in the amount of increase of the mass of the standing antenna, we also need to know the density of the material, which it is made from. This is usually some aluminium alloy whose density is around 2800 kg/m3.
A brief remark about the calculation: for the sake of simplicity, we would again consider an antenna consisting of tubes or rods. The mass of the object m = V * ρ, where V is the volume and ρ the density. The volume of the cylinder is V = π * r2 * l, where r is the radius and l the length. If we substitute the known values we will calculate the original mass and the mass of the frost. This is demonstrated below.
|Design of element /boom||Multiplication of weight with icing:|
|Spar (all diameters)||1,2||1,5||2,4|
|Pipe ø 8 mm, wall 1 mm||1,5||2,2||4,3|
|Pipe ø 10 mm, wall 1 mm||1,6||2,5||5,0|
|Pipe ø 14 mm, wall 1 mm||1,8||3,0||6,4|
|Pipe ø 24 mm, wall 1 mm||2,4||4,4||9,9|
|Pipe ø 24 mm, wall 2 mm||1,7||2,8||5,7|
|Pipe ø 30 mm, wall 1 mm||2,7||5,2||12,1|
|Pipe ø 30 mm, wall 2 mm||1,9||3,2||6,7|
|Pipe ø 50 mm, wall 1 mm||3,8||7,8||19,2|
|Pipe ø 50 mm, wall 2 mm||2,5||4,5||10,3|
The facts shown in the table may be shocking for some people. It is not easy to admit that the antenna consisting, for example, of tubes with 24mm in diameter with 1mm wall increases its mass in a 100% frost by 4.4 times! On the other hand, it is necessary to consider that the conditions for creating continuous frost on the entire perimeter of the tube, especially on those with larger diameters, are relatively rare in ordinary QTHs.
As a matter of interest, Wikipedia states the following about frost:
The frost is an atmospheric phenomenon that develops by creating ice crystals on the surface of the object by inflicting the following effects:
• freezing tiny droplets of the air's humidity (clouds, fog etc.) in it's contact with the surface of the ground, object or other subjects at the temperature of 0°C and below;
• precipitation (sublimation) of air's humidity on a sufficiently cold surface of the ground or subjects and that even without the presence of fog or clouds.
The highest probability of frost creation is with contact at a temperature (0 to -4 °C) between the surface of the object and the moist airflow. With temperature below -4 °C the possibility of frost creation decreases and at temperatures below -12 °C frost does not occur or is very weak.
The above data is interesting especially in the consideration of structure & design of the antenna - to what extent the components and the boom should be dimensioned and reinforced to avoid bending. That is again a completely different topic.
Next time, we will focus on anchoring a real mast, including considerations about its strength and other influencing elements.
Beim Entwurf eines „Verankerungssystems“ (Verankerungsseil mit Endstücken, Kupplungen und Seilklemmen) müssen folgende Faktoren in Betracht gezogen werden: