Christos C. Chamis

Glenn Research Center, Cleveland, Ohio

Probabilistic Methods for 

Structural Reliability and Risk

NASA/TM—2010-216078

January 2010

ECCM Paper No. 2413
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Christos C. Chamis

Glenn Research Center, Cleveland, Ohio

Probabilistic Methods for 

Structural Reliability and Risk

NASA/TM—2010-216078

January 2010

ECCM Paper No. 2413

National Aeronautics and

Space Administration

Glenn Research Center

Cleveland, Ohio 44135

Prepared for the

13th European Conference on Composite Materials

sponsored by the European Society for Composite Materials

Stockholm, Sweden, June 2–5, 2008
Acknowledgments

The author appreciates the diligent review by Fred Holland which improved the quality of the paper.

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Level of Review: This material has been technically reviewed by technical management. 

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NASA/TM—2010-216078 1 

Probabilistic Methods for Structural Reliability and Risk 

Christos C. Chamis 

National Aeronautics and Space Administration 

Glenn Research Center 

Cleveland, Ohio 44135 

Abstract 

A probabilistic method is used to evaluate the structural reliability and risk of select metallic and 

composite structures. The method is a multiscale, multifunctional and it is based on the most elemental 

level. A multifactor interaction model is used to describe the material properties which are subsequently 

evaluated probabilistically. The metallic structure is a two rotor aircraft engine, while the composite 

structures consist of laminated plies (multiscale) and the properties of each ply are the multifunctional 

representation. The structural component is modeled by finite element. The solution method for structural 

responses is obtained by an updated simulation scheme. The results show that the risk for the two rotor 

engine is about 0.0001 and the composite built-up structure is also 0.0001. 

Introduction 

The pursuit of achieving and retaining competitive advantages in world markets necessarily leads to 

proactive drives for better-cheaper-faster products to market. This becomes even more important in the high 

tech sector which includes aerospace vehicles. The awareness for conservation of natural resources also 

leads proactively to the reliable cost-effective useful-life extension of existing products. In the first case, it 

requires very effective use of available resources. In the second case, it requires formal methods to quantify 

the current strength of a specific structure/component and subsequent reliable evaluation of respective 

remaining strength. Both cases include a multitude of uncertainties. The first case includes uncertainties 

associated with lack of sufficient data of new potential material such as composites. The second case is full 

of uncertainties from (1) unknown assumptions and conditions of the initial design, (2) records of 

environmental exposure, and (3) material degradation of various factors, etc. In scenarios of multiple 

uncertainties described above, probabilistic methods offer formal approaches to quantify those uncertainties 

and their subsequent effects on material behavior, on service, and on reliabilities and risks. Therefore, the 

objective of this article is to describe a probabilistic method for predicting structural reliability and risk from 

material behavior to service life. That probabilistic method (1) is based on formulations that describe the 

physics in terms of primitive variables and (2) respective scatter ranges, at the lowest engineering 

manageable scale by a Multi-Factor Interaction Model (MFIM) (Ref. 1), and (3) relies on computational 

simulation methods to propagate those uncertainties from that elementary scale through all intermediate 

scales where metrics for structural reliability and risk are specified. The method has evolved over the past 

three decades and has matured sufficiently to evaluate structural reliability and risk under various scenarios. 

The schematic in Figure 1 illustrates the concept. The method has several unique features. The two that are 

most useful to present results are (1) quantifiable reliability in terms of Cumulative Distributions Functions 

(CDF) and (2) sensitivity factors of the primitive variables that affect reliability at every point in the CDF 

curve. The method was illustrated by first applying it to a metallic engine rotor (Ref. 2), and then to three 

different composite structures with progressive complexity (Ref. 3). 

Multifactor Interaction Model 

The complex material behavior is generally affected by several factors. These factors herein are 

represented by a Multi-Factor Interaction Model (MFIM) of multiplicative form as is illustrated in Figure 

2. The concept of this model is that the complex behavior of any material is assumed to be a surface in 

space. This surface is represented by an N dimensional vector from some origin. For illustration purposes, 
NASA/TM—2010-216078 2 
NASA/TM—2010-216078 3 

this figure indicates 6-vectors that define the shape of the assumed surface. The figure also describes the 

product form of the MFIM. Note the definitions of A. An expanded form of the MFIM is illustrated in 

Figure 3. Note in Figure 3 that each property is defined by Po at the origin and by an unsubscripted P at its 

evolution as it is affected by the various factors. There are two conditions associated with the MFIM 

expanded form: (1) the exponents must be positive and that their value is unity at the origins and has 

nanotonic behavior from the origin to their failure value, and (2) all the factors within the parentheses 

have absolute values. 

The MFIM describes behavior as is illustrated in Figure 4. Note that when the exponent is less than 

one the exponent describes “sudden death” type of function. When it is one it describes a straight line 

between the origin and the final value. When it is greater than one it describes an inverse death function 

(infant mortality), rapid degradation early on and then progressively decreasing values as the curve 

approaches its final value. 

Simulated Two Stage Rotor 

A simulated two-stage engine rotor is illustrated in Figure 5 top left. At the right are the cumulative 

distribution functions for the four conditions noted in the table at the bottom. These results are noted for 

each of the three conditions and a combined one which coincides with the fracture at the rim. In the table 

down below is a list of the components that were evaluated. The last column in the table shows the 

resistance associated with each component failure. Risk is not appropriate when the CDF is available. The 

reason is that for any 0 < probability < 1 the values are known. For example, take the probability of 0.6, 

which corresponds to a resistance lost of 0.9. The remaining resistance is only 0.1 or 10 percent. 

Therefore, the margin of safety prior to failure is only 10 percent. The remaining strength (resistance) is a 

better indicator compared to risk. The probabilistic sensitivity factors for the two-stage engine rotor are 

listed in Figure 6 in hierarchical order. The largest factor (value-wise) is that of the rotor speed. The 

second largest is the rotor density. The third largest is the rotor temperature. The rest of them vary 

between 0.01 and “0”. The magnitude of the factors is indicative of the effect that the factor will have in 

the CDF of the response. The sum of the squares of the factors equals unity which means that the rotor 

speed has about four times the effect of the density and about twenty-five times the temperature effect. 

Other points to be noted in Figure 6 are the variables associated with fracture mechanics K1c, Ao, Nl, Kt, 

A-LCF which have negligible affects on the reliability and risk of the fracture probability. 
NASA/TM—2010-216078 4 
NASA/TM—2010-216078 5 

Probability Evaluation of Composites 

Composites really constitute a multiscale structure with multifunctional material. These two features 

of composites are illustrated in the computational simulation cycle denoted in Figure 7. As can be seen in 

this figure, the multifunctional aspect starts at the material behavior space at the lower right where the 

local properties are described probabilistically (A). Next there is a schematic of a fiber with two matrix 

slabs indicating the probabilistic simulation of micromechanics (B). The next scale is the probabilistic 

description of a mono-fiber ply indicated by (C). Next is the probabilistic description of a laminate which 

ends the composite probabilistic description. Next is the probabilistic description of a finite element (D), 

the beginning of the structural analysis with the probabilistic loads, structural geometry and boundary 

conditions. A to D on the left side of the chart represent an upward synthesis. After the finite element 

structural analysis, the downward decomposition starts with the probabilistic description of 

displacements, loads, stresses and strains in each finite element or node. With these probabilistic values as 

inputs, the probabilistic decomposition continues to F, G and to the probabilistic micromechanics 

description of the material behavior space. The probabilistic cycle continues until all the loads have been 

finished and structural fracture has occurred. This describes an incremental approach where any change 

can be included in the upward synthesis or the downward decomposition. As can be imagined, the input 

probabilistic material properties can be quite numerous. These are shown in the next Figure 8. This table 

has five columns; the first column has a description of each property; the second has reference values of 

each property, the third has corresponding mean properties, the fourth has the assumed scatter for each 

property, and the fifth has the distribution type of each property. Note that the distribution type can be 

different for each property. Only two distributions are shown in this table. This is strictly for convenience 

because after the simulation is completed changes in the scatter and in the distribution type can be 

introduced. Now the comparison can be made between the two probabilistic results of the CDF and of the 

sensitivities. If the two CDF’s are about the same, and if the order of the sensitivities is about the same, 

then the assumed scatter and their respective distributions do not influence the probabilistic results and 

the conclusion is that no additional testing is required at the materials quality control level, which, in 
NASA/TM—2010-216078 6 

essence, is a considerable cost saver. Also, at the same time it illustrates how the probabilistic simulation 

is “self corrective.” The results described and their significance constitute a multifunctional and a 

concurrent multiscale probabilistic simulation. 

Probabilistic Fatigue Life 

Now after the three essential parts (MFIM, multifunctional and multiscale) have been described, we 

are ready to implement them in specific problems. A composite panel is subject to in-plane cyclic loads as 

is illustrated in Figure 9. As is seen in the figure, the panel is subjected to four different frequencies. The 

schematic of the panel with a cyclic load is at the top of Figure 9 while results are in the middle and the 

loading conditions are at the bottom. The small table at the right is mixed because it presents inputs in the 

first line and outputs in the next three lines. The first line indicates the frequency inputs while the other 

three lines represent respective frequency outputs. For example, at 

25 cps the mean life is about 0.91; the scatter percent is about 24.85 and the panel relative life is 0.21 if 

the maximum fatigue life is assumed to be unity (1). The corresponding CDF is shown in the left 

schematic which has four different CDF’s corresponding to each one of the frequencies. Each CDF is 

identified with the frequency that produced it. The 25 cps CDF is the solid curve to the right. Below the 

figure the dominant failure modes are listed as transverse tension in the 90° plies at low frequencies and 

90° plies compression at higher frequencies. It is seen in Figure 9 that the 200 cps curve is substantially to 

the left of the other three curves. This effect is also reflected in the small table which shows the 

corresponding degradation to be 0.6 mean life, 37.65 percent scatter and relative life 0.073. The other 

significant information in Figure 9 is the probabilistic sensitivity factors which are plotted horizontally in 

the middle right. Two major observations are important in this figure: (1) The frequencies have almost no 

effect on these factors. (2) The three dominant sensitivities are (in decreasing order of magnitude): panel 

thickness, matrix tensile strength, and fiber modulus. The next dominant factor is the matrix modulus and 

the last two factors are the fiber volume ratios and the matrix shear strength. The other 11 properties listed 

in Figure 8 had less than 0.1 effects and hardly worth testing to quantify their influence. 

This problem demonstrates the benefits of probabilistic simulation in multiscale multifunctional 

composite structure. The wealth of information obtained is definitely worth the effort required to generate it. 
NASA/TM—2010-216078 7 

Probabilistic Evaluation of Ceramic Hot Component 

In order to demonstrate the versatility of probabilistic methods, we discuss its application to a 

ceramic high temperature aircraft engine liner. The liner is schematically illustrated in Figure 10 with 

loads, geometry and finite element model. 

The multiscale simulation is comparable to Figure 7 and the input properties are the same as in 

Figure 8 with the exceptions that the fiber properties and the matrix are replaced with high 

temperature ceramic materials and the processing temperature is much higher. It is about 3000 °F. 

The use temperature is about 2800 °F. The high temperature effects of the ceramic materials were 

simulated by the use of the MFIM. Probabilistic results obtained for strength and stress at two 

different probabilities are shown in Figure 11 as functions of normalized cycle numbers. The most 

significant point to be observed in this figure is the greater spread between strength and stress at 0.01 

to 0.001 probability. This effect illustrates the point that probabilistic evaluations must be used 

judiciously. The spread between the strength and stress at the 0.01 probability is relatively large and 

will lead to an erroneous conclusion especially at very low fatigue cycles. 

The probability of survival which is comparable to that of Figure 8 is plotted in Figure 12 as a 

function of the normalized cycle numbers. This figure illustrates a very interesting point which is the 

nearly constant variation from about 0.25 to 0.65 of the normalized cycles. It indicates a relatively 

rapid degradation at low fatigue cycles to about 0.25 and then again at relatively high cycles at cycles 

greater than 0.65. Note that the survival probability scale varies from about 0.9999 down to about 

0.9986. This is a relatively small change of about 4/10,000. This small variation is very interesting 

indeed. It leads to conclusion that ceramic matrix composites operating at high temperatures are not 

fatigue prone. 

Probabilistic Evaluation of a Built-Up Composite Structure 

The last select example to illustrate the probabilistic evaluation of a multiscale multifunctional composite 

structure is the built-up structure depicted schematically in Figure 13. Figure 13 illustrates the structural 

geometry, the finite element mesh, and the laminate configuration. Note that all dimensions are in inches. The 

results from the two types of analyses are shown in Figure 14. The deterministic results are in the left figure 

and the probabilistic in the right where the two CDF curves are shown, 1) just before fracture and 2) at fracture 

(dashed curve). What is interesting in this figure is that the two CDF curves are parallel and that the shape of 

the energy curve is also almost parallel to the probabilistic curves. The reason for this is that both curves are 

based on the same constitutive values. The probabilistic sensitivity factors before and after fracture are shown 

in Figure 15 for two probabilities: 0.001 and 0.999. The dominant sensitivities remain about the same. It is 

seen in the figure that the pressure dominates followed by the fiber volume ratio, the fiber longitudinal 

modulus and the ply thickness respectively. The reason that these two sets of probabilistic sensitivities are 

about the same is that they both represent the same physical system. 
NASA/TM—2010-216078 8 
NASA/TM—2010-216078 9 
NASA/TM—2010-216078 10 

Concluding Remarks 

The results and discussions of probabilistic methods for structural reliability and risk lead to the 

following concluding remarks: The failure probability in metallic and composite structures can be 

computationally simulated. Probabilistic evaluation includes or can include some or all factors that 

influence component/system reliability. One important aspect is that the probability can be evaluated by 

the cumulative distribution function of the system response. Then the risk can be assessed by the 

remaining strength at some probability in the cumulative distribution function. Another important aspect 

of probabilistic evaluations is the probability sensitivities of all the variables that constitute the system 

design. The sensitivities can be used to fine-tune the design for more weight savings. 

References 

1. Chamis, C.C. and Minnetyan, L., “A Multi-Factor Interaction Model (MFIM) for Damage Initiation 

and Progression in Composites,” 2001 ASME International Mechanical Engineering Congress and 

Exposition, (IMECE), New York, NY, November 11–16, 2001. 

2. Chamis, C.C., “Probabilistic Structural Analysis Methods for Space System Components,” 

Probabilistic Engineering Mechanics, 1987 vol. 2, no. 2, pp. 100–110. 

3. Shiao, M.C., Abumeri, G.H. and Chamis, C.C., “Probabilistic Assessment of Composite Structures,” 

AIAA/ASME/ASCE/AHS/ASC, 34th Structures, Structural Dynamics and Materials Conference, 

1993, Proceedings pp. 1174–1186. 
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14. ABSTRACT

A probabilistic method is used to evaluate the structural reliability and risk of select metallic and composite structures. The method is a 

multiscale, multifunctional and it is based on the most elemental level. A multifactor interaction model is used to describe the material 

properties which are subsequently evaluated probabilistically. The metallic structure is a two rotor aircraft engine, while the composite 

structures consist of laminated plies (multiscale) and the properties of each ply are the multifunctional representation. The structural 

component is modeled by finite element. The solution method for structural responses is obtained by an updated simulation scheme. The 

results show that the risk for the two rotor engine is about 0.0001 and the composite built-up structure is also 0.0001. 

15. SUBJECT TERMS

Probability; Multi-factor; Fatigue; Composite fatigue; Ceramic composite fatigue 

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