Content

1. Introduction

2. Tensile test

2.1 Test principle

2.2. External load

2.3. Variations in temperature and gravity

2.4. Applicability

2.5. Strength - Density diagram

3. Bending test

3.1. Test principle

3.2. External load

3.3. Derivation

3.4. Temperature

3.5. Applicability

3.6. Modulus - Density diagram

4. Ashby diagrammes

5. References

Supplement.

This paper is giving some considerations concerning minimizing the weight of a construction. Two different loading modes are addressed.

b [m] width

d [m] diameter

E [N / m²] modulus of elasticity

F [N] force

F_w own weight

F_ext weight of an external load

g [m / s²] acceleration

h [m] height (thickness)

I [m⁴] quadratic surface moment

k [m³ / s²] constant

l [m] length

l_b length at which a cable breaks under its own weight

l_max maximum cable length given an external load

m [kg] mass

Rm [N /m² ] tensile strength (material dependant)

V [m³] volume

W [N] weight

r [m³ / kg] density

s [N / m²] stress

The Elsevier Materials Selector [1] is giving guide lines for the selection of materials best fullfilling predescribed functions. An important tool in the method used is the Ashby diagram [1,2]. An Ashby diagram presents materials in a way that allows a quick first assesment of the materials fitness to meet a certain design objective, e.g. minimum weight or fracture save design.

When designing for minimum weight the selection criterion is depending on the mode of loading, such as tensile, compression, bending, torsion, shear or combinations thereoff. Reference [1] is not explicitely giving the line of reasoning in establishing the selection criteria. To get an idea how such a line develops the following is giving it for simple tensile and bending. The reasoning is based on natural material tests.

2.1. Test principle.

In the tensile case the test is to determine at which length a cable breaks under its own weight. (Imagine the case that a cable is lowered into a very deep mine shaft).

The weight of the cable is given by

F_w= m * g = V * r * g = (p /4) * d² * l * r * g (1)

The tensile stress at any given cross section of the cable is

s = F_w / ((p /4)*d²) = l * r * g (2)

So, the length l_b , at which the cable breaks under its own weight equals

l_b = Rm / (r * g) (3)

The length l_b and the tensile strength Rm can be considered as material characteristics together with the density r . Besides depending on Rm and r which are to be considered as internal material quantities , l_b is also depending on gravity, which is to be considered as an external quantity. In relation to light weight materials the beauty of this test is that l_b in fact is the strength to specific weight ratio (see formula 3).

Note that l_b is not depending on the diameter d of the cable. The external load which can be carried by the cable however is. In that case the tensile stress in any given cross section of the cable is

s = (F_w+F_ext) / ((p /4)*d²) = l * r * g + F_ext / ((p /4)*d²) (4)

and

l_max = (Rm - F_ext / ((p /4)*d²)) / (r * g). (5)

Formula (5) implies that the cable diameter can be smaller or the maximum length can be greater when using a material with a higher strength to specific weight ratio. However when a material has a given strength to specific weight ratio, at low values of Rm and r the diameter of the cable needed to carry a given load is larger than that of a material that has the same strength to weight ratio at higher values of Rm and r . It can be shown that maximum length and weight of the cable are always the same at a given strength to specific weight ratio, see supplement.

Note that Rm and r usually are temperature dependent and that g can vary along the length of a very long cable. Also such a cable may lengthen and contract under its own weight.

The test is of course not a practical one, but it sheds some light upon the matter of strength related to weight. For practical purposes the standardized tensile test is used.

When a diagram is drawn of Rm against r , as an Ashby diagam in reference [1], straight lines through the origin represent equal strength to specific weight ratio’s at different absolute value’s of Rm and r . (Rm / r = l_b / g ).

For light weight constructions materials positioned in the upper left part of the diagram are preferred. This means that the highest Rm at the lowest r is preferred. This is however not always the choice given in practice.

Figure 1. s / r diagram

Figure 1 represents different possibilities for changes in materials strength and density. These changes, and the consequences for cable length and weight in the mine lift application, are summarized in table 1 with material A as the reference. For example:

changing from material A to material B the strength to specific weight ratio remains the same at lower values of strength and density. Cable diameter increases and the maximum cable length and weight stay the same.

Table 1. Possible changes in Rm and r and consequences.

In the bending case the test is to determine the bending in the middle of a rectangular beam of a given length under its own weight in a two point bending test.

The maximum deflection in the middle of the beam is given by

f_max = (5 / 384) * (F * l³) / (E * I) (6)

With F_w = m * g = V * r * g = b * h* l * r * g and I = b * h³ / 12 this leads to

f_max = (5 / 32) * g * (r / E) * (l⁴ / h²) (7)

From here the problem can be treated in different ways.

If we take a standardized beam (l⁴ / h²) becomes a fixed value and (5 / 32) * g * (l⁴ / h²) can be treated as a constant value and the expression for the deflection becomes

f_max = k₁ * (r / E) (8)

This expression is in its form equivalent to (3). Note that the width b has disappeared from the equation.

Another possibility is taking the deflection f_max, and length l as fixed values and adjust the thickness h of the beam. The expression for the beam thickness then becomes

h = k_{2 *} (r / E)^1/2 (9)

Again b has disappeared from the equation.

With an over the beam evenly distributed external load expression (8) becomes

f_max = (k₁+k₂) * (r / E) (10)

Here the impact of an external load is lower the higher the value of the Young’s modulus E is. So again an absolute value comes into play. The Young’s modulus is considered to be a material characteristic.

The derivation of (6) isn't taken into into consideration here, so there may be other way's to arrive at (8) and (9) or similar formulas.

3.4. Temperature.

Value’s of r and E usually are temperature dependent

The test involving bending in the unloaded situation is not known to be standardized. It can however been done easily if sensitive equipment for displacement measurements is available. Standardized bending tests usually are done in dedicated testing machines applying a load to the test piece.

3.6. E / r diagram.

When a diagram is drawn of E against r , as an Ashby diagram in reference [1], straight lines through the origin represent equal Young’s modulus to weight ratio’s at different absolute value’s of E and r . (E / r = k / f_max). For light weight constructions materials positioned in the upper left part of the diagram are preferred.

Possible weight reduction for a construction is depending on design criteria, as for instance minimizing deflection and weight at given dimensions. With fixed values for l and h as in formulas (8) and (10) the volume is constant and therefor the weight of the beam is decreasing with decreasing r . The deflection is decreasing with decreasing r / E. The impact of an external load is decreasing with increasing E.

Here for a few materials an example is given how they are placed in an Ashby diagram. Comprehensive information is to be found in [1].

Figure 2. Strength - Density.

Figure 3. E-Modulus - Density

5. References.

[1] Elsevier Materials Selector. Vol 1.

Edited by Norman A. Waterman and Michael F. Ashby

[2] The development of steelite.

R. Boesenkool, A. Hurkmans, R.J. Jansen and M. Roukema.

Figure 4. Maximum cable length as function of diameter at a load of 125 kg. The Rm / r ratio is constant. The open dots represent value's in the steel range (Rm = 400 N/mm² , r = 7850 kg / m³) and the open triangles represent value's in the area of polymers foams and cork (Rm = 4 N/mm² , r = 78,5 kg / m³)

Figure 5. Maximum cable length as function of diameter. The Rm / r

ratio for the "+" is higher than for the open dots due to a higher strength. Value's are in the steel range.

Figure 6. Maximum cable length as function of diameter. The Rm / r ratio for the "*" is higher at lower values of Rm and r . Value's for the open dots are in the steel range and value's for the "*" are in the wood range (Rm = 44 MPa, r = 785 kg / m³)

The weight of a cable made of material 1 that can can carry a given load F_ext at a given diameter d₁ is

W₁ = p /4*r ₁*g*l*d₁² (11)

and the weight for material 2 is

W₂ = p /4*r ₂*g*l*d₂² (12)

If l = l_max2 = l_max1 then

d₂² = 4/p *(F_ex / (Rm₂-l_max1*r ₂*g)) (13)

And if the cables also have the same weight then because of formula (11) and (12)

r ₁*d₁² = r ₂*d₂² (14)

should hold.

Substituting (13) in (14) and applying formula (5) for l_max1 yields that (14) only holds if

Rm₁ / r ₁ = Rm₂ / r ₂, or in other words if the ratio Rm / r is constant.

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