Keywords

1 Introduction

Current methods for risk and reliability assessment are mostly deterministic methods (i.e. based on constant inputs and outputs). But in the real world, the typical engineering values as loads, dimensions, material properties etc. are not constant but variable. The developments of computers and reliability theories allow to applied stochastic/probabilistic approaches for a qualitative improvements of the safety and serviceability assessment of structural components and systems especially in the engineering. In structural reliability assessment, the concept of a limit state separating a multidimensional domain of random (stochastic, probabilistic) variables into “safe” and “unsafe” domains has been generally accepted. For more general information about the history and development of stochastic methods in the branch of engineering, see Refs. [1–5]. There are solved structures on elastic foundations, medical problems, movement of Earth plates, design of machines and its parts, experiments etc.

Hence, the SBRA Method is connected with new innovative engineering. Therefore, the probabilistic applications in the engineering are new scientific trends.

This paper presents some computational stochastic and probabilistic solutions (i.e. SBRA—Simulation-Based Reliability Assessment Method) applied in the branch of engineering. The origins and developments of SBRA Method are described in [2]. The main attention is focused on the “fully” probabilistic approach to reliability assessment using SBRA Method as a suitable tool. Let the resistance of the structure be expressed by random variable R and load effect by random variable S. The failure is defined by RF < 0 that is

$$ RF = R - S < 0 . $$
(1)

In SBRA Method, the reliability function (1) is analysed using direct Monte Carlo simulation and Anthill computer programme, see [2]. Let N f be the number of simulation steps when RF < 0 and let N be the total number of simulation steps. Then, the probability of failure can be expressed as \( P_{f} = P_{RF\, < 0} = N_{f} /N \).

At first, the 2D unbraced steel frames are solved (i.e. buckling; mechanics; elastic transformation models are used in order to express the response of the structural components including the 2nd order theory effects; probabilistic reliability assessment). At second, the femoral screw (made of Ti6Al4V and stainless steel) for treatment of proximal femoral neck fractures of humans are solved (i.e. bending; biomechanics, implant rested in bones is solved as a planar beam on elastic foundation including the 2nd order theory effects; probabilistic reliability assessment). At third, other applications are mentioned.

2 Unbraced Planar Frame (Buckling)

The common approach to the stability (buckling) assessment of structural steel components and systems is based so far mainly on the conventional models using buckling length, buckling coefficients, compressive strength and other characteristics, see [6]. However, the computer technology allows for qualitative improvement using probabilistic approach and advanced transformation models based on 2nd order theory.

The unbraced planar steel frame shown in Fig. 1a and Table 1 consists of two cantilevered columns (1, 2), two leaning columns (3, 4) and three crossbars. The supports of cantilevered columns are not rigid, but elastic with stiffness k j . The frame is loaded by forces F i , H, and by forced deformation caused by temperature differences ∆T i (with reference to the temperature during erection), see Fig. 1b. Initial curvatures with amplitudes f j , were considered. Unavoidable eccentricities of forces F i are e i . Imperfections a i represent the initial deviations.

Fig. 1
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Planar frame a undeformed situation, b deformed situation

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Table 1 Nomenclature in Sect. 2 (Planar frame)
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Analytical transformation model of the frame is expressed by following equations

$$ \begin{aligned} \delta_{1} & = \frac{{H + \sum\limits_{i = 3}^{4} {\frac{{F_{i} a_{i} }}{{l_{i} }} + \sum\limits_{i = 1}^{2} {\frac{{F_{i} V_{i} }}{{G_{i} l_{i} }} + \frac{{F_{3} }}{{l_{3} }}d_{1} \alpha_{1}\Delta T_{1} - \frac{{F_{2} }}{{G_{2} l_{2} }}\sum\limits_{i = 1}^{2} {d_{i} \alpha_{i}\Delta T_{i} + \frac{{F_{4} }}{{l_{4} }}} \sum\limits_{i = 1}^{3} {d_{i} \alpha_{i}\Delta T_{i} } } } }}{{\frac{{F_{1} }}{{G_{1} l_{1} }} + \frac{{F_{2} }}{{G_{2} l_{2} }} - \sum\limits_{i = 3}^{4} {\frac{{F_{i} }}{{l_{i} }}} }};\omega_{j} = \sqrt {\frac{{F_{j} }}{{E_{j} J_{j} }}} ; \\ \delta_{2} & = \delta_{1} + \sum\limits_{i = 1}^{2} {d_{i} \alpha_{i}\Delta T_{i} } ;M_{j} = F_{j} [ (1 + \frac{1}{{G_{j} }} )\delta_{j} - \frac{{V_{j} }}{{G_{j} }} + e_{j} + f_{j} ];P_{j} = 1 - \frac{{F_{j} l_{j} }}{{k_{j} }}\frac{{\tan (\omega_{j} l_{j} )}}{{\omega_{j} l_{j} }}; \\ G_{j} & = \frac{{\tan (\omega_{j} l_{j} )}}{{P_{j} \omega_{j} l_{j} }} - 1{\kern 1pt} ;V_{j} = e_{j} [\frac{1}{{P_{j} \cos (\omega_{j} l_{j} )}} - 1 ]+ f_{j} \left\{ {{\kern 1pt} \left( {P_{j} [1 - ({{2\omega_{j} l_{j} } \mathord{\left/ {\vphantom {{2\omega_{j} l_{j} } \pi }} \right. \kern-0pt} \pi } )^{2} ]} \right)^{{{\kern 1pt} - 1}} - 1} \right\}. \\ \end{aligned} $$
(2)

Input random variables are summarized in Table 2. Names of histograms are the same as in [2] (i.e. Anthill software). Except EQ, all input random variables are mutually uncorrelated. Force EQ is correlated with forces F i . Force H is the sum W and EQ, both of them may act to the left or to the right. The coefficient of temperature expansion is α i  = 12 × 10−6 K−1. Dimensions l 1  = 6 m, l 2  = 7.6 m, l 3  = 5.4 m, l 4  = 6 m and d 1  = d 2  = d 3  = 10 m. Numerical results are summarized in Table 3 and [6]. Two million simulation steps were used for both of the safety and serviceability assessment.

Table 2 Stochastic input values in Sect. 2 Planar frame
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Table 3 Calculated probabilities of failure P f —safety (carrying capacity, Planar frame)
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Load carrying capacity is evaluated with reference to the onset of yielding (most exposed fibers of columns 1, 2 are considered, see Fig. 2). Serviceability assessment of the structure refers to the lateral displacement limit value of the upper end of column 1. Computed probabilities of failure (probability of exceeding the lateral displacement value, see Fig. 2) are summarized in [6].

Fig. 2
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Histograms of the safety function a Carrying capacity, b serviceability—Planar frame, Anthill software

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To assess the safety and serviceability of this unbraced planar frame using current codes (i.e. different methods LRFD or ASD approach, see [7]) and considering its complexity would not be an easy task, see, e.g. the discussion on the assessment of such system in [8]. However, in this article, the authors offer better stochastic/probabilistic solution which is based on own analytical model applied in Anthill software (SBRA Method).

3 Probabilistic Reliability Assessment of Femoral Screws Intended for Treatment of “Collum Femoris” Fractures (Bending)

Proximal femoral neck fractures, see Fig. 3a, remain a vexing clinical problem in traumatology and orthopaedics (i.e. common type of trauma), see [9–11].

Fig. 3
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a Femoral neck fracture. b Femoral screws. c Treatment of fracture—X-ray snapshot. d Beams on elastic foundation

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One possible treatment method for femoral neck fractures, is the application of femoral screws made up from stainless steel or Ti6Al4V material, see Fig. 3b, c.

The analytical model for strength analyses of femoral screws is based on the theory of beams on an elastic foundation, where the bone is approximated by the elastic foundation prescribed by stiffness k/Pa/, see [1, 4, 10]. Quite large and complex review about the theory and practice of elastic foundation is performed in author’s book [1].

Three screws are applied in parallel positions on the elastic foundation (i.e. in the bone). The force F/N/ acting in one screw can be defined via total loading force Fm/N/, see Fig. 3d, by the equation \( {\text{F}} = {\text{F}}_{\text{m}} /{\text{n}} = {\text{m k}}_{\text{m}} {\text{k}}_{\text{dyn}} {\text{g}}/{\text{n}} \). The variables are as follows: m/kg/ is the entire mass of a patient; km/1/ is the coefficient of mass reduction; kdyn/1/ is the dynamic force coefficient; g/ms−2/ is the gravitational acceleration; and n/1/ is the coefficient of inequality in the division of force \( {\text{F}}_{\text{m}} \) into three screws. These variables are defined via truncated histograms. The force F can be decomposed into forces \( {\text{F}}_{1} = {\text{F cos}} \propto \) and \( {\text{F}}_{2} = {\text{F sin}} \propto \), see Fig. 4a. The femoral screw angle α/deg/, which is defined by the limits of adduction and abduction, see Fig. 4b, and the yield limit Re/MPa/ for material, are likewise defined via truncated histograms, see Fig. 5a. According to the 2nd order theory and the theory of beams on an elastic foundation, three linear differential equations for the intervals \( x_{1, 2, 3} \), can be written as \( EJ_{\text{ZT}} \frac{{d^{4} v_{\text{i}} }}{{dx_{\text{i}}^{4} }} + {\text{F}}_{2} \frac{{d^{2} v_{\text{i}} }}{{dx_{\text{i}}^{2} }} + kv_{\text{i}} = 0 \) together with 12 boundary conditions. \( EJ_{\text{ZT}} \)/Nm2/ is flexural stiffness, \( v_{\text{i}} \)/m/ is displacement and x i/m/ are coordinates. Hence, bending moments \( M_{{o{\text{i}}}} \)/Nm/, and maximum stresses \( \sigma_{\text{MAX}} \)/MPa/, see e.g. Fig. 4a, can be calculated.

Fig. 4
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a Beam on elastic foundation. b Histogram of femoral angle

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Fig. 5
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a Histogram of yield limit for Ti6Al4V material (Anthill software); b Probabilistic assessment of femoral screw (i.e. result of SBRA Method, Anthill software)

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Probabilistic reliability assessment can be carried out via SBRA Method by means of the reliability function (1), depending on load capacity, compared to the extreme stress values with yield limit. The probability that plastic deformation will occur in the beam is \( P_{f} = 2.51 \times 10^{ - 5} \) i.e. \( P_{f\% } = 0.00251\,\% \) (calculated for 5 × 106 Monte Carlo simulations, see Fig. 5b). Hence, the femoral screws are safe and suitable for patient treatment and the surgeons can use them for treatment.

4 Conclusion

The transition from the pre-computer era reliability assessment to modern computers technology leads to re-engineering of the entire assessment procedures. The probabilistic SBRA Method is connected directly with this transition. Methods such as SBRA can be considered and applied in order to evaluate the safety and serviceability of structural components and systems taking into the account variabilities of inputs. One of the main prerequisites, related to the application of qualitatively higher reliability assessment methods using the potential of modern computers, is the transition of thinking of designers from deterministic to probabilistic. In case of the stability (buckling) problems (i.e. unbraced planar frame), the current assessment criteria based on buckling length and compressive strength, buckling coefficients, etc. can be gradually replaced by a fully probabilistic approach, using transformation models and 2nd order theory. In case of the design of femoral screws intended for treatment of “collum femoris” fractures in traumatology/orthopaedics (i.e. bending of planar beam rested in femur), the probabilistic assessment criteria can be based on fully probabilistic approach, using transformation models and 2nd order theory. Other examples are mentioned too.