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Appendix C - Commentary on Notional Factors of Safety

C Notional Factors of Safety

C.1 Introduction

Bridge design standards changed several times during the latter half of the 20th Century. These changes affected load specifications and safety margins.

Safety of civil engineering structures is normally assured by applying safety factors to provide a margin between theoretical strengths and theoretical loading effects. Neither strengths nor loads can normally be exactly established, so they are represented by mathematical models based on observation and experiment. Some aspects of designs (such as structural weights) can often be assessed more accurately than others (such as road traffic load effects). More modern UK design standards cater for such variations by providing higher safety factors to those parts of the calculations where uncertainty is greatest.

This section of the document discusses the derivation of appropriate safety margins for the main cable of the Forth Bridge

C.2 BS153 Assessment

The Forth Bridge was originally designed to the current British Standard, BS153. According to the original designer's published information in the ICE paper (1967), the theoretical strength of the wires was 100 tons/sq.in, or 1550 N/mm2. The working stress in direct tension was defined as 40 tons/sq.in., or 620 N/mm2. The design target factor of safety was thus 2.50.

The cross section area was 350 sq. ins, (225800 mm2), and the 0.2% proof stress was taken to be 75 tons/sq.in., giving the relationships between loading and resistance in Table 1.

Table 1: Original Basis of Design Relationships

Therefore actual theoretical safety factor between the theoretical load and breaking strength was 2.58, and the safety factor between load and the permanent extension was 1.9.

At that time, the safety factor between load and yield capacity for steel structures was usually approximately 1.7.

C.3 Cable Angle Effect

The maximum cable loads presented in Table 1 appear at the top of the side span cables, where the slope is steepest. The cable slope at this point is close to 25°. Therefore we can deduce that the theoretical cable forces at the middle of the main span (where the cable is horizontal) are reduced to 91% of the Table 1 values.

The theoretical factor of safety at mid span was thus originally 2.83.

Since that time, the theoretical live load model has become heavier, thus reducing the theoretical safety factors. A 50% increase in live load would lead to a theoretical safety factor of total load of 2.45 in the side span near the tower tops, and 2.69 at mid span.

C.4 Review of Other Bridges

C.4.1 Severn Bridge

At the Severn Bridge (built shortly afterwards) the overall theoretical factor of safety that was used to prepare the design for the main cable was reduced from 2.5 to 2.25. This implies that, at the centre of the main span, the factor of safety was close to 2.45. (The cable geometry is slightly different).

This implies that the cable at Severn is some 10% less strong relative to the theoretical loads used in design than the cable at Forth Bridge.

C.4.2 Messina Crossing

The design specification for the projected Messina Crossing bridge differs from UK practice. A summary of the main cable design requirements follows.

Strength factors are:

• At SLS, main cable load <= Ultimate stress / 2.10
• At ULS, main cable load <= Ultimate stress / 1.67

Load factors are:

• 1.15 for steel elements
• 1.25 for concrete elements
• 1.50 for non-structural elements
• 1.50 for "dense variable load" = traffic load, deterministic model used that is not perhaps too different from HA

Therefore the "factor of safety" against ultimate strength is somewhere between:

• 1.15x1.67 = 1.92 for steel weight
• 1.50x1.67 = 2.50 for superimposed load and live load

If we assume (without actual figures being available) that the superimposed plus live load portion is some 20% of the total, this implies a factor of safety of close to 2.1.

If the total of superimposed load plus live load was 15% of total (which is probably more realistic), the factor of safety is then only 0.15x2.5 + 0.85x1.92 = 2.007, or close to 2.0.

It is notable that, at Messina, the partial factor on material strength has been reduced to a value that is close to the BS153 overall safety factor for steel bridges. However, the BS153 factor at that time included safety margins on live load etc. so the Messina criteria for cable design are still considerably more onerous than traditional Mid-20th Century design requirements for structural steel elements in bridge works.

C.4.3 Eurocode Recommendations

The Structural Eurocodes present another set of values. It is again difficult to make direct comparisons. In particular, prEN 1993-1-1, "Design of structures with tension components" specifically excludes aerial spun parallel wire suspension bridge cables from its scope, although it does state that some of its provisions may be applicable. It recommends that the limiting strength of parallel wire strands be the lesser of the proof strength (with a factor of 1.0) and the theoretical breaking strength divided by 1.50. The proposed Eurocode partial factors are different from current UK practice, and the National Annex values have not yet been ratified and are not in the public domain. However, the proposed strength factors on normal structural steelwork are currently equal to 1.00. If we assume that the reliability currently implied by UK structural steel design codes will not be significantly altered by implementation of Eurocodes, we see that the Eurocode system still requires parallel wire cables to have factors of safety 50% greater that those for structural steelwork.

Since the current UK partial factor system provides similar structural economy overall to that obtained using BS153, we see that the Eurocodes, as implemented in the UK, are likely to provide a factor of safety for parallel wire cables close to 1.7x1.5 = 2.55. This is as we saw with the Forth Bridge in the ICE paper of 1967.

C.5 Target Reliability and Safety Factors

The main cable of a suspension bridge comprises typically several thousand parallel wires, each of which has been created by a process that requires it to withstand the tensions of the wire drawing process whilst it is being manufactured. The large number of parallel elements allows a very reliable strength model to be developed for a new cable. Therefore, a new main cable is one of the most reliable structural elements that an engineer can specify. The obvious question is why they are then provided with higher safety factors than any other structural element.

There appear to be a number of reasons, some of which are more easily justified than others. They include:

• Individual wires are relatively fragile, and might be damaged on assembly. This may be so, but main cables are assembled using carefully designed machines and methods, and significant numbers of wires are not likely to be mis-handled.
• Wires might not share equal loads. However, the spinning process ensures they all hang in the same catenary to begin with, and their initial loads are a consequence of their geometry and weight, both of which are essentially fixed.
• Wires cross over and cut each other, so aggregate strength is not the same as true strength. This has some effect in wire ropes, but even there its effect can be determined by testing. It is not relevant to suspension bridge cables.
• Wire ropes used in lightweight structures such as guyed masts may suffer from all manner of unexpected duty cycles, including galloping vibration, vortex excitation effects, and uncertainties in end-connection strengths. None of these applies to large suspension bridge cables.
• Ropes may corrode more quickly than other steel elements, because they are thinner and more difficult to protect. This is a real issue, and will be discussed below.
• Wire ropes used for general engineering purposes are frequently mishandled in use, by being pulled around unsatisfactory radii, dragged through dirty ground, allowed to lie in rain water, and generally exposed to dirt and grit. They wear when they are repeatedly pulled through guides, and suffer from fatigue when they are repeatedly bent around sheaves. None of these effects is relevant to suspension bridge cables.
• Tradition dictates to the engineering profession that wire ropes shall have large factors of safety. Engineers are familiar with the fact that ropes used in lifting equipment typically have safety factors of about 6 or 7, and some engineers have difficulty in accepting safety factors as low as 2.50 for cables that are as important as those used for suspension bridge cables.
• Finally, safety factors are employed to provide enough margin between theoretical loads and theoretical strengths for structures to remain safe and serviceable even if the theoretical loads and strengths do not precisely model the actual.

The main justification for retaining a safety factor as high as 2.5 for suspension bridges would appear to be to ensure that they withstand the real problem of long term corrosion and to cater for uncertainty in loading and resistance. We would not advocate constructing a new bridge with significantly reduced cable safety factors unless the problem of corrosion could be reliably solved.

However, where we are looking at an existing structure the arguments are very different, always bearing in mind that the objective is to protect public safety.

C.5.1 Strength Reduction

We have seen that the main justification for the relatively high safety factors in suspension bridge cables is to cater for the future prospect that the strength will be reduced. Therefore we should not be unduly concerned by the fact that cables do indeed behave as expected, since we had already designed to cater for this very effect.

Now that the partial factor format for safety factors has become widely accepted, it would be logical to redefine the strength factors for suspension bridge cables. Thus: if we use the BS5400 arrangement of factors, an aggregate factor of 2.50 could be interpreted as follows:

• Permanent loads (maybe 80% of all loads): Factor = 1.15
• Live loads (maybe 20% of all loads): Factor = 1.50
• Aggregate load factor therefore = .80x1.15 + .20x1.50 = 1.22
• Uncertainty in analysis: 1.10
• Main cable strength factor: 1.86

Thus the effective factor of safety = 1.22 x 1.10 x 1.86 = 2.50

However, we have just seen that the high cable factor ( i.e. the '1.86' value) caters for strength reduction. However, if 'strength reduction' is a 'design action' (in Eurocode parlance), it ought to be explicitly included in the model. That would allow the wire strength itself to provided with the same factor as any other steel element, which in BS5400 is typically between 1.05 and 1.20. If we decide to retain the '1.20' value, we still see that our cable is as safe as any other steel element even after it has lost 45% of its strength.

These values can be discussed and re-calculated, but the principle seems to be clear. If we can obtain reliable strength models, we should use them as the basis for cable strength: but we should not expect to strengthen a cable to restore it to provide a high that was originally provided in order to allow strength to fall safely in the first place.

C.5.2 Rational Partial Factor Format for Main Cable

The logical process of determining the acceptable overall factor of safety on the cable ought to use the same partial factors as those used for normal steel elements for which we have a similar degree of uncertainty in our strength predictions.

Partial factors should be applied to 'Actions' and "Resistances' individually. Factors should not be employed to model physical processes (such as corrosion). They should only be used to protect structures from uncertainty. The strength model for a cable should comprise the following:

• Initial cross section area. This is probably close to being deterministic, with a factor of 1.0.
• Material properties. For steel, the factor in BS5400 format is typically 1.05
• Strength reduction model. This is derived for the Forth Bridge from the theoretical analysis of cable strength reduction. It effectively applies a factor onto the wire area.
• Partial factor on the strength reduction model.
• Corrosion model. This provides a predictive model for the future progress of corrosion.
• Partial factor on the corrosion model. This may be greater than the factor on the strength reduction model.

C.6 Conclusion

A rational safety assessment can only be made if the origins of loads and capacities are rationally considered.

Partial factors should not be used to model physical processes.

It is irrational to be concerned when a structure which has been designed to be safe after it has deteriorated has indeed deteriorated - unless the deterioration has progressed to such an extent that its reliability is becoming unacceptably low.

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