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FEM shakedown analysis of structures under random strength with chance constrained programming
(2022)
Direct methods, comprising limit and shakedown analysis, are a branch of computational mechanics. They play a significant role in mechanical and civil engineering design. The concept of direct methods aims to determine the ultimate load carrying capacity of structures beyond the elastic range. In practical problems, the direct methods lead to nonlinear convex optimization problems with a large number of variables and constraints. If strength and loading are random quantities, the shakedown analysis can be formulated as stochastic programming problem. In this paper, a method called chance constrained programming is presented, which is an effective method of stochastic programming to solve shakedown analysis problems under random conditions of strength. In this study, the loading is deterministic, and the strength is a normally or lognormally distributed variable.
A new formulation to calculate the shakedown limit load of Kirchhoff plates under stochastic conditions of strength is developed. Direct structural reliability design by chance con-strained programming is based on the prescribed failure probabilities, which is an effective approach of stochastic programming if it can be formulated as an equivalent deterministic optimization problem. We restrict uncertainty to strength, the loading is still deterministic. A new formulation is derived in case of random strength with lognormal distribution. Upper bound and lower bound shakedown load factors are calculated simultaneously by a dual algorithm.
Background:
Additional stabilization of the “comma sign” in anterosuperior rotator cuff repair has been proposed to provide biomechanical benefits regarding stability of the repair.
Purpose:
This in vitro investigation aimed to investigate the influence of a comma sign–directed reconstruction technique for anterosuperior rotator cuff tears on the primary stability of the subscapularis tendon repair.
Study Design:
Controlled laboratory study.
Methods:
A total of 18 fresh-frozen cadaveric shoulders were used in this study. Anterosuperior rotator cuff tears (complete full-thickness tear of the supraspinatus and subscapularis tendons) were created, and supraspinatus repair was performed with a standard suture bridge technique. The subscapularis was repaired with either a (1) single-row or (2) comma sign technique. A high-resolution 3D camera system was used to analyze 3-mm and 5-mm gap formation at the subscapularis tendon-bone interface upon incremental cyclic loading. Moreover, the ultimate failure load of the repair was recorded. A Mann-Whitney test was used to assess significant differences between the 2 groups.
Results:
The comma sign repair withstood significantly more loading cycles than the single-row repair until 3-mm and 5-mm gap formation occurred (P≤ .047). The ultimate failure load did not reveal any significant differences when the 2 techniques were compared (P = .596).
Conclusion:
The results of this study show that additional stabilization of the comma sign enhanced the primary stability of subscapularis tendon repair in anterosuperior rotator cuff tears. Although this stabilization did not seem to influence the ultimate failure load, it effectively decreased the micromotion at the tendon-bone interface during cyclic loading.
Clinical Relevance:
The proposed technique for stabilization of the comma sign has shown superior biomechanical properties in comparison with a single-row repair and might thus improve tendon healing. Further clinical research will be necessary to determine its influence on the functional outcome.
Extension fractures are typical for the deformation under low or no confining pressure. They can be explained by a phenomenological extension strain failure criterion. In the past, a simple empirical criterion for fracture initiation in brittle rock has been developed. In this article, it is shown that the simple extension strain criterion makes unrealistic strength predictions in biaxial compression and tension. To overcome this major limitation, a new extension strain criterion is proposed by adding a weighted principal shear component to the simple criterion. The shear weight is chosen, such that the enriched extension strain criterion represents the same failure surface as the Mohr–Coulomb (MC) criterion. Thus, the MC criterion has been derived as an extension strain criterion predicting extension failure modes, which are unexpected in the classical understanding of the failure of cohesive-frictional materials. In progressive damage of rock, the most likely fracture direction is orthogonal to the maximum extension strain leading to dilatancy. The enriched extension strain criterion is proposed as a threshold surface for crack initiation CI and crack damage CD and as a failure surface at peak stress CP. Different from compressive loading, tensile loading requires only a limited number of critical cracks to cause failure. Therefore, for tensile stresses, the failure criteria must be modified somehow, possibly by a cut-off corresponding to the CI stress. Examples show that the enriched extension strain criterion predicts much lower volumes of damaged rock mass compared to the simple extension strain criterion.
We propose the so-called chance constrained programming model of stochastic programming theory to analyze limit and shakedown loads of structures under random strength with a lognormal distribution. A dual chance constrained programming algorithm is developed to calculate simultaneously both the upper and lower bounds of the plastic collapse limit and the shakedown limit. The edge-based smoothed finite element method (ES-FEM) is used with three-node linear triangular elements.
Background
Osteoporosis is associated with the risk of fractures near the hip. Age and comorbidities increase the perioperative risk. Due to the ageing population, fracture of the proximal femur also proves to be a socio-economic problem. Preventive surgical measures have hardly been used so far.
Methods
10 pairs of human femora from fresh cadavers were divided into control and low-volume femoroplasty groups and subjected to a Hayes fall-loading fracture test. The results of the respective localization and classification of the fracture site, the Singh index determined by computed tomography (CT) examination and the parameters in terms of fracture force, work to fracture and stiffness were evaluated statistically and with the finite element method. In addition, a finite element parametric study with different position angles and variants of the tubular geometry of the femoroplasty was performed.
Findings
Compared to the control group, the work to fracture could be increased by 33.2%. The fracture force increased by 19.9%. The used technique and instrumentation proved to be standardized and reproducible with an average poly(methyl methacrylate) volume of 10.5 ml. The parametric study showed the best results for the selected angle and geometry.
Interpretation
The cadaver studies demonstrated the biomechanical efficacy of the low-volume tubular femoroplasty. The numerical calculations confirmed the optimal choice of positioning as well as the inner and outer diameter of the tube in this setting. The standardized minimally invasive technique with the instruments developed for it could be used in further comparative studies to confirm the measured biomechanical results.