Originally posted by little_chicken
mrowkoob .. this is probably what you are looking for ..
Usually the comparison and choice of the materials begins with the juxtaposition of their tensile strength and/or 0.2% proof stress (óTS or óPS). But, as a rule, these parameters have a very indirect relationship to the reliability of the majority of real constructions, especially welded constructions. There are many reasons for this. Note: the majority of examples described here are based upon our experience in the manufacture of welded bicycle frames.
In practice, constructions are extremely rarely broken due to the load exceeding the static material stress limit.
For example, high quality steel bicycle frames made of chrome-moly steel with óTS > 90 kg/mm2 have seat tubes typically of 28.6 x 1.0 mm (external diameter x wall thickness). This means that a force > 8 tons is necessary to break such a tube by static tension. The same situation is observed in
bicycle frames made of magnesium alloy (óTS > 32 kg/mm2) when a tensile force > 7 tons is necessary for static breaking of the standard seat tube 34,8 x 2,0 mm. Obviously the weight of even the heaviest cyclist is many times less. However, we see that in practice the seat tube is sometimes cracked and broken.
In fact, the primary cause of construction collapse is connected with fatigue - remember that it's very difficult to break iron wire by one tension or bending. But it's easy to do by repeatedly bending from side to side a few times). That is the reason why it is more correct to compare the materials by their fatigue limit (ó-1) for the corresponding amount of loading cycles. Usually the stand fatigue tests of bicycle frames demand from 100 000 to 1 million loading cycles. For other constructions the amount of reiterations may sufficiently differ: from 1-10 thousand to hundred million and more. It's rather curious that often the choice between two materials depends on the necessary amount of working cycles.
For the majority of real constructions it is even more correct to compare
the materials considering fatigue limit with stress concentration (fatigue
limit with notch ó-1n) which usually takes place. In welded constructions stress concentration is always just around the welds. Again, one material can be better when comparing ó-1 parameter and the other in case of ó-1n comparison when the first one is more sensitive to stress concentrators.
Thus we see that the choice of different material based simply on the comparison of tensile or proof strength may be absolutely false. This is due to the fact that fatigue breaking depends not only on the static strength parameters but also on possible elongation, visco-elastic properties including fracture toughness, fatigue crack growth, etc. For example, a quite good aluminum alloy of the Al-Zn-Cu-Mg system with the density ñ = 2.85 g/cm3 has the following mechanical properties óTS = 57 kg/mm2 , ó-1 = 16 kg/mm2, ó-1n = 9 kg/mm2 (for 20 million cycles). And one of the magnesium alloys of the Al-Zn-Cu-Mg system with the density ñ = 1.8 g/cm3 has the following properties: óTS = 32 kg/mm2 , ó-1 = 13 kg/mm2, ó-1n = 10 kg/mm2 (and that is for 50 million cycles). Thus, the specific mechanical characteristics (given in relative units) of these alloys are as follows:
Table 1
We can conclude that a "worse" magnesium alloy from the point of view of its static strength is much better by its fatigue parameters. Furthermore, the type of destruction (sudden brittle or steady viscous fracture) is a very important characteristic for any consumer. And namely the absence of instant brittle fracture can guarantee your safety.
Moreover, in case of welded constructions it’s not quite correct to use even ó-1 or ó-1n parameters for comparison, which are usually given for materials in the reference books. The fact is that a strong heat in the weld area and surrounds changes (and usually for the worse) the material’s structure and mechanical properties. As a rule, the welded construction breaks “in the weld” or near the weld. It should be noted that usually reference books give the data related to static tensile strength of the weld and rarely give information on fatigue properties of the weld.
2. Trusses and frames are two principally different construction types studied in the theory. The truss elements work mostly under tension-compression loads. The frame parts work under bend and torsion actions. It's quite logical for trusses to compare the materials by the ó/ñ (any necessary ó) parameter because both the strength and weight are linearly proportional to their cross-section. However, the same approach is incorrect for frames when the weight and strength (stiffness) are in different dependence on material properties. Thus, for frames in particular, the advantages of light alloys became clear and obvious.
Let us illustrate the above by a simple (but aposite) numerical example.
A) We have a plate of cross-section S being extended with the force F. In this case the specific load (stress) in the material is equal to F/S and reliability condition is
F/S < ómax (1)
The precise ómax value (óTS, ó-1, ó-1 n) depends on the loading type (static, periodic, pulsed), presence and type of concentrators, etc., is not important now. The condition (1) defines minimal admissible area of the plate:
S > F/ómax (2)
and for constant length l of the plate its weight M has the lower limit:
M = ñlS > ñlF/ómax = lF(ómax/ñ) -1 (3)
This means that the bigger is ómax/ñ ratio, the lighter the plate which can withstand the given external force. It is the same combination of material characteristics that we have already considered above.
B) Now we have the same plate affixed in one end, for example welded to a powerful support. The force is applied to the other end of the plate creating the bending moment Fl. In accordance with the theory stress in the material reaches maximum near the support and its value is proportional to such ratio:
ó ~ Fl/h2 (4)
where h is the plate thickness and here we do not indicate all the numerical coefficients. If the maximal admissible stress is ómax, then we again calculate the minimal thickness of the plate can withstand the given external moment
hmin ~ (Fl/ómax) 1/2 (5)
and, correspondingly, the minimal weight of the plate (assuming constant width) is defined as
Mmin ~ ñhmin ~ ñ (Fl/ómax )1/2 ~ (ñ2 / ómax ) 1/2 (6)
This means that a smaller weight plate can be produced out of the material with the smallest ratio (ñ2 /ómax )1/2 or with the bigger combination of ómax /ñ2 parameters (it is the second power of previous ratio inverse value). If we rewrite this combination as (ómax /ñ)/ñ the increase of smaller density influence becomes obvious.
Now we shall take the same two alloys (aluminum-based and magnesium-based) considered above and make a table, which can help us to compare the weight of the constructions giving the same reliability depending on the type of the load. In the case of tension-compression actions (hereinafter indicated as T-C) the materials are compared by the ó/ñ parameter, and in the case of bends and torsions (hereinafter indicated as B-T) they are compared by ó1/2/ñ parameter (if ó is measured in kg/mm2 and ñ is measured in g/cm3 the conventional units are used):
Table 2
Thus now we see that in the case of bending the advantage of using low density materials increases and it is much more desirable to use magnesium even for an occasional static bend. In comparison with heavier alloys the weight economy rises considerably. A titanium plate having the same length, width and weight as a magnesium one will be able to stand the destruction by the same bending moment only if it is 6.25 times stronger (titanium is 2.5 times heavier than magnesium). And you will need a titanium alloy having tensile strength equal to 200 kg/mm2 to replace a magnesium alloy with óTS = 32 kg/mm2 ! Taking into account that real strength of the titanium alloys is 2-2.5 lower, the magnesium plate will be as much lighter.
As a conclusion to this paragraph we would like to point out the following: the plate rigidity is proportional to the Eh3 product where E is the Young modulus. The specific stiffness E/ñ is almost equal for the majority of alloys (except for the beryllium ones). The difference does not exceed 4-5%. As a result, a magnesium plate having the same weight as a titanium one (2.5 times thicker) can not only withstand almost a double load but is also approximately 6 (2.52 ) times stiffer. Although the precise dependencies change for tubes and other profiles, the principle of
substantial increase of efficiency of light alloys application for the frames totally remains.
3. There is a minimum thickness for plates, tubes, etc. In practice you cannot use very thin profiles in constructions even if the conditions of sufficient stiffness permit to do so in theory. For example, when producing magnesium bicycle frames the tube wall thickness ä is normally equal to 1.5-2.2 mm at the diameter D of 40-60 mm. A steel tube of the same diameter and weight will have the wall as thick as 0.35-0.5 mm (steel is 4.4 times heavier). It is rather hard to process such tubes mechanically, weld them, etc. The last problem is partially resolved by using tubes having variable thickness where the weld area is thicker (butted tubes). Yet we can use the same approach to the lighter alloys. In other places (except the weld zone) the minimum thickness of the tube wall is limited by the condition of rigidity towards the side loads (incidences, impacts, etc.). In such cases the stability of the tube drops sharply with the decrease of the ratio of the wall thickness to the tube diameter ä/D. As a result thin steel tubes having the same diameter as magnesium ones crumple easily like beer cans.
It might have seemed possible to decrease the steel tube diameter, for example, by 1.5 times and to raise the wall thickness accordingly. This could potentially lead to substantial rise of tube stability towards the side shocks and crumpling without changing its weight. But this also leads to reduction of ordinary stiffness according to the decrease of the product äD3 that drops in 1,52 = 2.25! It explains why lighter magnesium frames are stiffer than those made of heavier alloys.
4. Until now we discussed issues of stiffness and reliability in the case of cyclic loads on some abstract elements and welded constructions, but not real ones. The analysis was useful though substantially incomplete. We considered neither frequency (frequencies) of the loads nor their character (smooth loads, impacts, etc.). In reality, the type and character of the external load plays a great role not only for the reliability (resource) of the product but also for its consumer qualities. To be more precise we will take a bicycle frame as an example for our analysis.
A bicycle frame is a construction having its own oscillation frequencies (resonances), to which external forces are applied within a wide range of frequencies. These are impacts (up to 1000 Hz), riding on a rough road (up to 200 Hz), pedaling (up to 3-5 Hz) with all their various harmonics and other aspects. You will know what happens when the frequency of a small load coincides with the resonance frequency of a very rigid construction. This can be illustrated by the classic example of a bridge's collapse caused simply by an infantry battalion marching across hen that same bridge had easily withstood tanks driving over it.
This means that the most important reliability characteristic of the construction in real exploitation conditions is the ability to oppose resonances. That is the ability to absorb and disperse the energy of the impacts and cyclic loads by "smoothening" the resonance. The importance of this property is due to the fact that destruction of the constructions working in a wide range of the loading frequencies is mostly explained by the growth of oscillations (vibrations) on the resonance frequencies.
Oscillations can be suppressed by appropriate measures undertaken in the two directions. The first one is the optimizing of the construction, tuning its resonance frequencies away from the typical frequencies of the external influence, exclusion of coincidence of one resonance frequency and the harmonic of the other resonance frequency, etc. The second one is the use of materials having high damping characteristics. This leads to rapid fading of free oscillations in the construction, decrease of the amplitudes of forced resonance oscillations and abrupt decrease of stresses caused by impacts. As a result it is often more appropriate to give preference to the material having higher damping qualities with formally smaller fatigue parameters when working with impact loads. The logarithmic fading decrement ä (showing reduction of the amplitude of free oscillations within a period) or the damping index Ø (showing the part of energy being dispersed in the material within one oscillation period) are used for description of damping properties of the material, where Ø = 2ä.
Let us assume: the static load Fst (force, moment, etc.) causes the deformation in the element of the construction equal to Ast. It is easy to show that the cyclic load Fstcosùt at the resonance frequency ù leads to deformation equal to (ð/ä)Ast. The tension in the element in its linear approach grows as much. For example, if ä = 0.5% then the deformation and the stress at the resonance frequency will increase by 630 times!!! Surely, often only a small part of the external influence is applied at the resonance frequency but it leads to colossal upraise! It means that in many cases the materials should be compared by the product Øó-1 /ñ and not by the ó-1 or ó-1 /ñ parameters.
Good alloys usually may have little difference in ó/ñ characteristic but they differ for numerical orders by their damping qualities. Moreover, a slight change of the alloying element concentration may result in a serious change of the damping capacity of the alloy.
It should be noted that the Ø value seriously depends on the amplitude of the applied load. It is clear that when a load near the elastic limit is applied a substantial part of the mechanical energy is dispersed in the material and the Ø coefficient is quite high. In practice it is desirable that Ø value is to be high under stress of the ó-1 level when the material really can stand multiple cyclic loads. When the tension is approximately equal to (ó-1)/2 the Ø value (it should be mentioned that the exact determination of the Ø value is more difficult than that of the standard mechanic characteristics) of the most widespread materials is as follows:
Quality steels: Ø ~ (0,2-0,5-1)%;
Titanium alloys: Ø ~ (0,03-0,05-0,08)%;
Aluminum alloys: Ø ~ (0,05-0,1-0,2)%;
Magnesium alloys: Ø ~ (0,5-2-10)%.
Let look now at magnesium alloys. They include some high-proof alloys having poor damping qualities of the aluminum kind. But the most important thing is that there are quality magnesium alloys having great damping characteristics in comparison to the usually used materials. If an automobile wheel or a bicycle frame are made of such an alloy then even ordinary consumer will easily feel the difference without putting them through the specials tests.
