Monday, November 21, 2011

Design Methods: Quality Robust Design and Limit State Design

Soft Conversion of Permissible Stress Design to Limit State Design.

Some time around 1989, Australia started to convert their structural codes from permissible/allowable stress design to limit state design. Such conversions had apparently been tried many times before in the past, but always eventually scrapped, and this was the general view again, that the idea would be shorted lived. However by 2000 all relevant structural codes had been converted to limit state methodology, and the permissible stress codes scrapped. The one exception I know of, being the aluminium structures code for which there is both a permissible stress version and limit state version: the limit state version does not conform with any of the other limit state version codes.

Limit state theory is based on the concept that every product has multiple states-of-nature in which it operates, for each state-of-nature there are unique acceptable criteria for acceptance. A car for example may be fuel efficient for city driving but extremely inefficient for long cross country trips, another car may be the exact opposite. The end-user selects the vehicle which is optimum for their most frequent type of journey and accepts the inefficiency for the other type of journey. Alternatively if frequently make both types of journey, a vehicle which is optimum across both types of journey, but less than optimum on any individual type of journey compared to those vehicles optimised for one type of journey only. As a vehicle ages it also becomes less fuel efficient. As a vehicle ages, the brakes also become less effective, and stopping distances increase. Another concept known as quality robust design (QRD) aims to ensure that whilst there is high variability in the operating conditions, there is minimum variability in the performance. This also takes into consideration the variability in the production process, and the capability to produce an end-product with the required performance.

For example in a region where there is a high demand for housing, and a scarcity of steel, and where clearly the steel is not required in the concrete to prevent the building collapsing during construction or day to day usage, then the steel is not going to be installed in the reinforced concrete structure no matter how much inspection is implemented. The steel will be seen as an unwarranted and unnecessary expense and if not available it cannot be installed. The need for the steel will not become apparent until its too late, during an earthquake or hurricane. Traditional quality control (QC) would place the responsibility with the builder for not following the specifications. Modern quality assurance (QA) places the responsibility with the designer, for not designing a product which can be constructed from the available resources: labour skills, materials and equipment.

A product experiences differing states-of-nature during its fabrication and construction, handling and transportation, in-service usage, maintenance and repair and during and after future modifications and renovations. Structurally these states-of-nature are classified into 3 large groups:

  1. Stability
  2. Strength
  3. Serviceability

{As is usual I got side stepped from my original train of thought}. Limit state design and quality robust design are both dependent on statistics and probability to allow for the variability in operating condition of the product and the environment. The ultimate strength limit state, can be a state of collapse of a member or assembly or the fracture of materials. Most traditional design has been based on keeping materials in the linear elastic range, so that when a load is removed the deflection caused by that load is also removed: elastic recovery. The end of the linear elastic range is typically marked by the yield strength of the material if the material has one. Above the yield strength materials deform plastically: that is when the load causing deformation is removed the deformation remains. This is important for manufacturing where a large slab of steel is to be rolled into a a thin sheet or the I-section of a universal beam, or coiled sheet steel roll formed into a c-section. In these cases once the section has been formed, it is not desired that the material spring back to its original condition. Once these formed sections are put to use in the structure of a building or machine then any further permanent deformation would typically be considered end-of-life for the structure. Collapse or fracture of the structure is thus an important limit state, and the operating and environmental conditions under which this is acceptable need to be determined.

Risk analysis, failure mode, effect and criticality analysis (FMECA) can involve some highly qualitative reasoning before anything is quantitified, if at all. Probability concepts can also be relatively complex. Design for collapse involving probabilities of events can also be fairly daunting and scary, coming from a tradition of designing for operation and concept of safety. However it is this latter concept of safety, that wish to remove. In my mechanical engineering studies we were explicitly advised against making reference to factors-of-safety, or margins of safety, we were reprimanded if we used such terms. The preference was design factor or factor of ignorance. These numbers can in general be fairly arbitrary and give a false sense of safety which is not present.

For example a cable may be broken a certain load (N), and we choose to use the cable only for situations where the operating load is 50% of the load (0.5N). It is a mistake to assume that the cable is twice as strong as it needs to be. The cable may well break at a load less than 0.5N, it just as a lower probability of occurring than breakage at a load N, or 0.9N or 0.75N or what ever higher load chosen. The higher the load to N, the higher the probability of the cable breaking, the further away the lower the probability of failure so 0.1N may be good choice, but there is still a probability of failure. It is not safe from breaking. Further more we cannot be certain that the operating environment will not exceed N, or the 0.5N, or the 0.1N that we choose. All we know is that the strength of the cable can vary and that the load applied can vary, and we need to accomodate this variability in design. Whilst permissible stress design hides the probability and reliability concepts behind the scenes in the derivation of a design factor, limit state design wants to make these risk concepts foremost in the designers mind. But tradition is in the way, so a soft conversion of the permissible stress codes was carried out, to set a path in place towards a more risk based probabalistic approach to design.

With respect to bending the permissible stress formula for hot-rolled steel design was something of the form:

M <= 0.6.fy.Z

M = applied bending moment
fy = yield strength
Z = elastic section modulus {S=plastic section modulus. Though some countries may reverse these notations.}

{NB: actually not the exact form since code was based on permissible stresses, however much of the steel was designed using the Australian Institute of Steel Construction (AISC) safe load tables. Now the Australian Steel Institute (ASI) and design capacity tables (DCT's)}

The problem with the permissible stress equation is that it gets rearranged:

1.67M <= fy.Z

Thus inferring that the structure is 67% stronger than it needs to be. This however is incorrect, for it fails to allow for variability in the strength of the material (fy), and variability in the dimensions of the section used to calculate elastic section modulus (Z). It also fails to allow for variability in the actual design action-effect (M). Given that we can have fairly tight control on the strength of materials and the dimensions of the sections, the strength or resistance (fy.Z) has little variability (or small standard deviation), whilst the magnitude of the loading has significantly greater variability, the design factor (1.67) can be split into two parts to accommodate variability on both sides of the expression and remain calibrated against the old code and provide a step towards a new philosophy. Thus the expression becomes:

1.5M <= 0.9fy.Z

This can be expressed more generally as:

psi.M <= phi.fy.Z


psi = partial load factor
phi = capacity reduction factor

The fundamental requirement of the building code of Australia (BCA) is that the resistance in this case (phi.fy.Z) is the 5th percentile resistance of the part. The value of phi can therefore be adjusted to suit the origins of the values of fy and Z. In general fy should be the 5th percentile yield strength of the material, so that phi mostly applies to the variability in Z. So that phi is a simple way to allow for variability present in the dimensions of the section which go into calculating the elastic section modulus. The derivation of the value of phi=0.9 is something hidden behind the scenes of the code, but it is something which can be questioned and brought more into the open. Anycase it reflects an expected low variability in the resistance of the structural member. None the less there is variability and there is a 5% probability that this strength will not be achieved in practice.

Our code has no explicit reference on the probability of exceedance for the design load (psi.M), however in the 1989 version of the wind loading code, the 1000 year mean return period used back then was derived from a 5% probability of exceedance for a 50 year life expectancy. Currently wind loading is based on wind speed with a 1/500 annual probability of exceedance for buildings of normal importance, this relates to a 500 year mean return period. So unless otherwise noted the basic principle is that the design action and/or design action-effect should have a 5% probability of exceedance for a given life expectancy: or is otherwise the 95th percentile load.

So when we work with wind loads we do not use the psi=1.5, instead we use the design action (psi.M) which has the required probability of exceedance. So that the most generic version of limit state structural design is:

95th percentile action-effect <= 5th percentile member resistance

There is no safety as such, there is always some probability of failure. We could choose the 99th percentile action-effect, but there would still be some probability of it being exceeded. We cannot choose the 100th percentile action-effect because we don't know what it is: everything we measure has variability. Whilst statistical assessment of manufacturing output can control resistance fairly tightly, the statistical estimates of loading is fairly crude and in some instances possibly highly unreliable.

Engineers Australia in 1990 issued to its members a booklet titled :"Are you at risk! Managing expectations". Part of the exercise was to get engineers and other technical professionals away from declarring things to be safe. When something is declared as "safe" the public tends not to perceive that it will fail no matter what the conditions. Little seems to have changed: there are still engineers declaring buildings to be earthquake resistant, hurricane resistant and flood proof: and as to be expected they continue to fail. The response is make the design load bigger will make it safer. Keep making design loads bigger, just makes more expensive, uses more materials, and limits supply to fewer and fewer people.

The magnitude of the design load is not the issue. The real issue is the qualitative consideration of the modes of failure and the consequences, consideration of the full continuous spectrum of limit states.

Some of the poorest regions of the world, are also prone to seismic activity, and they have been making use of steel reinforced concrete. Whilst the concrete is obviously abundant, the investigations after destructive earthquakes, indicates the steel is obviously not so abundant.

Since the design load can always be exceeded, making the design load bigger is of little real benefit, it just makes the structures less affordable. Reading about the 2008 Sichuan earthquake, it reinforced the perspective that consequences of failure are the issue. Traditional Mongolian yurts may not have resisted the earthquake, but their collapse would have caused fewer deaths, less severe injuries, and further more could have been replaced rapidly. Far from being a disaster, would have been more like an inconvenience. Our ancestors were mobile, that is the benefit of being an animal rather than a plant. Plants are stuck in the paths of earthquakes, hurricanes and floods, animals are not. Architects and civil/structural engineers are turning us into plants, and many of our modern world problems are associated with us being more like static plants than mobile animals: not least of which is concentration of pollutants and waste of fuel. Sure there is an issue of travel: work/home/work/home etc... which is a major waste. But brought about because the city is a giant plant with massive global root system. If going to use concrete in a structure in these regions because it is abundant then make it a compression only structure, so that doesn't require the steel: get creative. However, if the design load is exceeded still going to get crushed. Tension membranes and cable-nets can certainly cover large areas: but back to issues of availability of suitable materials. Also what is housing for, protection from the environment or privacy? A large membrane structure could protect a village, but not provide privacy to the individuals within.

If we can get back to the qualitative issues instead of thinking we are smart because we can do some complex mathematics, then we can find better design solutions to the problems that we encounter. There are different states-of-nature, a great deal of variability and uncertainty to be accounted for, and differing criteria for acceptable performance.

At the moment the community has little to say on the performance requirements imposed in the built environment, yet it is the people who have to pay one way of another. The BCA talks about loss of amenity: at the present point in time the primary loss of amenity is not getting it in the first place. Further more most houses do not comply with current code requirements.

So first here is an opportunity to knock down the price of an established house because it is not compliant with current codes. Second an opportunity to assess when these established houses will fail, what the consequences of failure are, and then use this as a basis for BCA alternate-solutions, which will provide more affordable housing and less hazard to life when they actually collapse. Note the building doesn't have to be made of cotton wool so that when it collapses it cause minimum injury: rather the structure should provide adequate warning of its impending collapse. A warning from a government department not adequate because that may relate to current codes of practice, not the capability of existing structures. So need an early warning system which alerts occupants before they hear the loud cracking sound of the members of the structure failing.