Structural models of reliability

Machines and engineering structures consist of a large number of structural elements that can fail. The way elements are connected and their reliability define the reliability of the whole structure. There are examples with series connection A, parallel connection B and a combination C.

For a series system, failure of an elements results in failure of the whole structure. Reliability of the structure RS is the product of the reliabilities of its elements. The failure rate of the series structure is the sum of failure rates of its elements. For a parallel system the failure rate is smaller than failure rate of its elements.

The failure rate decreases if the number of redundant elements increases. The failure rate decreases over all time except the initial and final stages.

An example of an inverse problem: If the reliability of the entire oil pipeline is known the reliability of a single butt-weld can be predicted.

The figure shows examples of structures with parallel connections. The analysis shows that the reliability of structures with a larger number of redundant elements is higher. This does not mean that parallel connection is always economically effective.

Among structural elements there are parts that don't carry the load and failure of these parts has no significant effect on the load-carrying ability of the structure. For example, in a 7200 volt electric power line ACSR (aluminum conductor, steel reinforced) the aluminum wire does not accept enough of the load. The failure of the steel wire results in failure of the line. A single element (steel wire) reliability model can be proposed to judge the carrying ability of the electric line.

LIMITING STATE

If the probability density functions for strength and stress of a structure are described by normal functions with known means mi and standard deviation D_i the probability density function for difference in strength and stress is also a normal distribution.

The reliability is equal to 0.5 if the means of the distributions coincide. It is larger than 0.5 if the mean for stress is smaller than for strength.

The summed area under a density function is equal to 1.
The probability of failure is equal to the integral over the range where both functions f(s) and F'(s) are nonzero.

The summation is over three intervals. The probability of failure of a structure is 0.09 and its reliability is 0.91.

Engineering changes can alter the positions of the probability density functions.

A. Cross section area increase results in a stress decrease.
B. Stress concentration results in a stress increase.
C. Service in low temperatures results in brittle fracture (strength decrease).
D. Heat treatment of welded joints results in residual stress decrease (strength increase).

The reliability index is a measure of the reliability of a structure. The minimum reliability index of structural components corresponds to the reliability index of the whole structure. The smaller the reliability index bi, the more probable the failure.

DISPERSION

The histogram shows how many specimens have strength in the corresponding intervals. The experimental data and corresponding analytical distributions can be characterized by mean value m_s and standard deviation D_s. The standard deviation is expressed in terms of the same units of measurement as the variable.
_m

The standard deviation does not depend on number of tests N if the number is large. The shapes of the two histograms are similar.
The standard deviations are approximately equal for both examples.

All distributions (exponential A, uniform B, normal C and D) correspond to the mean ms=200 MPa.
The areas under the curves are equal.
The larger part of the area is placed far from the mean, the bigger standard deviation D_s:

D_s(A) > D_s(B) > D_s(C) > D_s(D)

The mean of the safety margin is equal to ratio of the means for strength and stress:
mg = 40 / 20 = 2.0

There is probability of failure for the structure

P(g) > 0 if g < 1 .

The reliability index helps to compare two variants. If the acting stress is constant (s = 180 MPa) the reliability index depends on variation parameters of strength. The bigger the reliability index, the bigger the reliability of the structure.

As result of damage growth in a heat exchanger, the strength of the material decreases by 10%. To keep the initial reliability the engineers decrease the stress due to vapor pressure by 20% after 10 years of service. In this case the reliability index is equal for both time t=0 and t=10 years.

t = 0 year : Dm = 200 - 100 = 100 MPa.
t = 10 years : Dm = 180 - 80 = 100 MPa.

A carbon-fiber reinforced plastic element of an airplane has microdamages, its rigidity and strength decrease with time. The maximum stress increases due to smaller rigidity of the elements. There is a correlation between stress and strength. The correlation coefficient affects the reliability index of the structural elements.
The negative correlation coefficient decreases the reliability index. Positive correlation coefficient between stress and strength is preferential.

DURABILTY

Durability of structures is connected with degradation processes such as fatigue, creep and others. "Stress - Number ofциклы на неудачу» (S-N) диаграмма описывает усталость металлов. Усталость прочности s-1 соответствует горизонтальный сегмент кривой. Если нет горизонтального сегмента, значение можно предположить при N = 106 или N = 107. На высоких напряжений усталость жизни и его дисперсия малы. При низком давлении функция плотности вероятности не симметрично и его дисперсия больше.

Initiation and growth of fatigue cracks occur under cyclic loading. The figure shows an experimental diagram "fatigue crack growth rate vs stress intensity factor" for two butt-welds made from different steels. The smaller the fatigue crack growth rate, the better. The smaller the fatigue crack growth rate at small values of stress intensity factor, A, the longer the fatigue macrocrack growth is in the initial stage.

The dispersion increases with time for fatigue cracks. Short cracks have no significant increase, but the longer cracks grow faster. Theoretical distributions for fatigue crack go from zero to infinity. This means that the theoretical reliability measure cannot be equal to 1.0.

A fatigue crack grows in a structural element from initial size a_o to its critical size a_c. The reliability is approximately equal to 1 for a small crack. The fatigue crack reaches its critical size at failure and the reliability decrease. When the means coincide, the reliability is equal to 0.5.

Airplane fuselages are periodically inspected for fatigue crack identification. Probability of crack detection depends on the method of nondestructive testing (NDT) and the inspection interval. Probability of crack detection is approximately equal to 1 for short intervals and decreases as the intervals increase. The ultrasonic NDT is the most effective technique.

The figure shows the dependence of maximum stress on the number of cycles sustained by an element of a pressure vessel. The engineers can suppose service at different stress, for example:

1000 MPa - 100 cycles
500 MPa - 500 cycles
100 MPa - 50000 cycles

The element fails if the sum damage parameter exceeds 1.
In our case:

w = 100/200 + 500/1000 + 50000/100000 = 0.5 + 0.5 + 0.5 = 1.5 > 1.0

The element fails.