Safety and Efficiency

The safety factor for the Eiffel Tower is the ratio of the ultimate stress and the actual stress.  The ultimate stress for iron is about 45 ksi.  This is the maximum it can withstand in both tension or compression before it fails or breaks.  Unlike masonry where compression and tension failure occur at different stress levels, iron failure occurs at the same stress for both types of force.  The value of actual stress, -9.9 ksi, is used with the ultimate stress, 45 ksi, in the safety factor formula:

The Tower could withstand four and a half times as much load as it ordinarily carries without danger of collapse.  It may seem as if Eiffel over designed the Tower because it will never be subjected to this much loading.  However, structures are usually designed with at least a safety factor of two, i.e. only half a building material’s maximum strength is used.  This is due to the fact that long before a material will fail, it will start to stretch or shorten and its dimensions will either increase or decrease slightly.  While not critically dangerous, these size changes  deformations  are unsightly and can cause connection problems and high local stresses.  Therefore, in design, one wants a structure not only to be strong, but stiff, so that these deformations cannot occur.

These considerations are accounted for in structural design by using an allowable stress.  This value is a set percentage of the ultimate stress and is used as the basis for design, i.e., ideally the designer of a structure will proportion its elements so that they are stressed to this value.  Higher stresses can be dangerous and cause deformation, and lower stresses do not use the full strength of the material.  Designing with the allowable stress is essentially building a safety factor into the design.  If a structure is stressed to the allowable limit, the safety factor is found as usual:

The ultimate stress is a property of the material, but the allowable stress is a set standard based on deformation calculations, i.e., a minimum safety factor is chosen for a material and the allowable stress is found as

The safety factor for wrought iron is about three, so the allowable stress is:

This value is a standard used in building codes at the time of the Tower’s design.  As mentioned, the elements of an ideal design will be stressed to the value.  This rarely occurs, so a measure of a design’s efficiency is

This value is the percentage of allowable stress that is actually used.  An ideal structure has an efficiency of 1.0 or 100% where actual stress equals the allowable.  The Eiffel Tower, with a maximum stress of 9.9 ksi, has an efficiency of

This means that 66% of the iron’s allowable stress is utilized in the design.  The Eiffel Tower is a moderately efficient and conservative design.  This conclusion is reinforced by a comparison of the safety factors.  The actual structure has a factor of four and a half while the material could withstand a factor of only three.

The explanations used and the estimates found in this analysis are all valid, and accurate, but still the analysis must be viewed as a simplification of the actual situation.  A brief review of the assumptions used in this analysis will emphasize its simplicity; the actual Tower’s geometry has been idealized in a number of critical ways by using solid columns and a two-dimensional model, and the loads, both dead and wind, were simplified.  Although the efficiency and safety factor are reasonable for such a structure, they are only estimates, and cannot be seen as precise measures.  They should be interpreted as indicative of a relatively safe and efficient structure.

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