Experimental Critical Loads Research Articles

A reliability-based design against post-buckling load drop in spherical shell cap with stochastic imperfections

Elastic buckling is a critical design consideration for deep spherical shell caps. Significant reduction is observed in the experimental critical load from the linearized eigen-buckling analysis, for reasons extensively discussed in the past and relooked recently through the energy barrier concepts. The Knock Down Factors (KDFs) were proposed to account for such a reduction. The highly conservative KDFs constitute the lower bound of a large experimental dataset; that can be reconciled by nonlinear stability analysis of the imperfect shell. This study models the KDFs as Uncertain-but-Bounded (UBB) variables to present a Reliability-Based Design (RBD) for spherical shell caps having random geometric imperfections in radius (R) and thickness (t). The RBD furnishes the pertinent design variables ((R/t) ratio and the dome height-to-base radius (H/r0) ratio) for a target reliability index (β) and the respective KDFs. A metamodelling approach is employed to reduce the computational burden of the proposed RBD for a thin spherical shell cap having zero-mean, stationary, homogeneous, gaussian stochastic geometric imperfections. The Riks algorithm is employed in the commercial code ABAQUS for the finite element-based nonlinear stability analysis of shells. The KDFs and the reliability bounds are presented for varying degrees of imperfections. Remarkably, the margin of conservatism in the KDFs is shown to be narrowed down by incorporating the dependency on (H/r0).

Fracture analysis of Brazilian circular hole disk under mixed mode loading

The paper presents an investigation into a Brazilian disk specimen featuring a concentric circular hole, accompanied by two internal edge radial cracks. By introducing rotations of the cracks relative to the vertical axis, a combination of fracture modes I and II is induced. Employing the finite element method, we compute stress intensity factors related to fracture modes I and II, alongside T-stresses, while varying geometric parameters such as inner and outer diameters, crack length-to-ring width ratio, and crack inclination angle. Experimental analyses of ebonite's fracture toughness in the mixed mode are conducted, subjecting samples to static loading until complete failure. Each test provides the angle at which crack initiation occurs and the corresponding critical load. Three distinct fracture criteria - the generalized maximum tangential stress criterion (GMTS), extended maximum tangential strain criterion (EMTSN), and generalized strain energy density (GSED) criterion - are employed to predict fracture direction and critical load. The outcomes exhibit good correspondence between experimental critical loads and theoretical predictions.

Predicting Critical Loads in Fused Deposition Modeling Graphene-Reinforced PLA Plates Containing Notches Using the Point Method.

The use of 3D-printed composites in structural applications beyond current prototyping applications requires the definition of safe and robust methodologies for the determination of critical loads. Taking into account that notches (corners, holes, grooves, etc.) are unavoidable in structural components, the presence of these types of stress risers affects the corresponding load-carrying capacity. This work applies the point method (PM) to the estimation of the critical (fracture) loads of graphene-reinforced polylactic acid (PLA-Gr) plates obtained via fused deposition modeling (FDM) with a fixed raster orientation at 45/-45. Additionally, the plates contain three different notch types (U-notches, V-notches, and circular holes) and comprise various thicknesses (from 5 mm up to 20 mm) and ratios of notch length to plate width (a/W= 0.25 and a/W = 0.50). The comparison between the obtained experimental critical loads and the corresponding estimations derived from the application of the PM reveals that this approach generates reasonable accuracy in this particular material that is comparable to the accuracy obtained in other structural materials obtained via traditional manufacturing processes.

Reliability analysis and design of randomly imperfect thin cylindrical shells against post-critical drops

Thin-walled cylindrical shells are employed for high specific bending stiffness and light design. Elastic buckling of shell is a critical design consideration. The Knock Down Factors (KDFs) are stipulated for accounting the markedly reduced experimental critical load by multiplying the linearized critical load by the KDF. Key to such reductions is the nonlinearity induced modal interactions among the closely spaced buckling modes, over which, the geometric imperfections act in synergy. Being the lower bound of an extensive experimental dataset; the stipulated design KDFs are largely over-conservative; which can be relaxed by accounting for imperfections. With this as the eventual goal, this study first demonstrate the feasibility of modeling the KDFs as uniformly distributed Uncertain-but-Bounded (UBB) type variable. The UBB description is employed to obtain the reliability bounds and the design variables (radius-to-thickness R/t and L/R ratio) for achieving a target reliability index through Reliability Based Design (RBD). Repeated computations of KDFs for reliability assessment are bypassed by fitting the KDFs vs. R/t relations for given L/R. The approach is demonstrated on a thin cylindrical shell with a zero-mean, stationary, gaussian, stochastic imperfection fields. The Riks method is employed for the nonlinear finite element analysis of the shell using the code ABAQUS. The KDFs and reliability bounds are provided for varying imperfections and are compared with an existing study on refining the KDFs. Furthermore, the RBD presents the desired R/t for a given L/R for achieving a target reliability index and the respective KDF for a given imperfection level. The L/R dependency is shown to reduce the over-conservatism in the design stipulation.

The theory of critical distances applied to fracture of rocks with circular cavities

This work presents experimental and theoretical analyses of the fracture behavior of different rocks containing circular holes subjected to uniaxial compressive uniform loads. The experimental critical loads are compared with those derived from the Theory of Critical Distances (TCD) and finite element simulations. The rocks being analyzed are Floresta sandstone, Moleanos limestone, Macael marble and Carrara marble. They were previously characterized under compressive and tensile conditions, obtaining their corresponding Poisson's ratios, Young's moduli and ultimate tensile and compressive strengths. Besides, their fracture properties, specifically their fracture toughnesses and critical distances, were also available from a previous experimental campaign on Single Edge Notched Beam specimens with different notch radii and tested under 4-point-bending conditions. In this paper, prismatic specimens (9 per each type of rock) containing a cylindrical cavity at their center were uniaxially compressed until failure. The fracture pattern was revealed and agrees with previously published cases. The TCD, specifically the Point Method, was applied with the aim of comparing the corresponding fracture load predictions with the experimental ones. Three-dimensional finite element analyses were used to calculate the stress fields. The application of the TCD on the Floresta sandstone and the Moleanos limestone provides reasonable predictions of fracture loads, given their linear-elastic behavior. For the two studied marbles, the resulting predictions are overly conservative despite their quasi-brittle behavior. This may be attributed to complex process zones with intense microcracking that are revealed as white patches prior to failure and generate dust and small fragments after failure.

Modelling mode I failure at crack tip with verifications using digital image correlation

Background: Linear elastic fracture mechanics (LEFM) applies to sharp cracks, although crack sites such as holes and slots are often blunt cracks with nonzero widths. The advancement in finite element method (FEM) has enabled the calculation of stress intensity factor (SIF) for unstable crack growth prediction, regardless of crack shape and size, but the calculations often differ from contour to contour. Hence, the purpose of this work is to determine if SIFs computed using commercial FEM have experimentally verifiable advantages over the traditional stress-based modelling approaches in predicting Mode I brittle failure at blunt crack tip. Methods: Experiments and simulations were conducted on brittle Poly(methyl methacrylate) (PMMA) plastic to compare the actual and predicted strain fields and SIFs at crack tip, and the critical force at which unstable crack growth initiates. A centrally straight cracked Brazilian disc made of PMMA was subjected to purely Mode I fracture. Its strain fields were measured from deformed speckle patterns using digital image correlation software. The same disc was modelled using plane stress model in FEM. By applying the critical force, SIFs were then computed using ANSYS pre-meshed crack method at different contours away from the crack tip. The effects of element type, mesh size and crack width on the simulated SIFs were investigated. Results: It was found that the experimental critical load agreed well with LEFM prediction based on PMMA fracture toughness in published literature. Disc failure happened at the first sign of tensile yield at the crack tip in the finite element model with a triangle mesh. Digital image correlation clearly shows the occurrence of unstable crack growth at critical force. It also shows comparable far field strain responses to the FEM model. Conclusions: The computed SIFs were inconsistent, and their usefulness in predicting unstable crack growth requires further investigation.

Experimental Studies of the Cylindrical Mesh Shell Model

Experimental studies of the effect of full and one-sided external load on a metal cylindrical mesh shell have been conducted. Regularities of stress and displacement distribution over the surface are recorded. The regions with the largest internal force factors and deformation parameters were identified. The values of the experimental critical load and the number of half wave of the form of stability loss are obtained. The risk of unilateral application of the load was established when the critical load was reduced and the maximum displacement exceeded.

Notch Fracture in Polymeric Specimens under Compressive Stresses: The Role of the Equivalent Material Concept in Estimating the Critical Stress of Polymers

In this paper, the fracture of notched polymeric specimens under compressive stresses was investigated both experimentally and theoretically. In the experimental section, to determine the load-carrying capacity (LCC) of U-notched specimens made of general-purpose polystyrene (GPPS) and polymethyl-methacrylate (PMMA) polymers, tests were performed on notched square samples under compression, i.e., negative mode I loading. In the observation of the nonlinear behavior of the two polymers in the standard compressive tests, for the first time, the equivalent material concept (EMC) was used under compressive loading to theoretically estimate the critical stresses of the two polymers, which were shown to be significantly different from the ultimate strengths obtained from the standard compression tests. By linking the EMC to the maximum tangential stress (MTS) and mean stress (MS) criteria, the LCC of the notched specimens was predicted. The outcomes are twofold: First, MTS, MS, EMC–MTS, and EMC–MS criteria provide accurate predictions of the experimental critical loads observed in the U-notched polymeric specimens; second, the combination of the EMC with the MTS and MS criteria, allow such predictions to be obtained without any need for experimental calibration.

Buckling of cylindrical shells under axial compression with loading imperfections: An experimental and numerical campaign on low knockdown factors

Abstract Thin-walled cylindrical shells are primary structures in aerospace, marine and civil engineering. A major loading scenario for these imperfection sensitive shells is axial compression. For this load case, there is a critical disagreement between the theoretical and experimental critical load. This difference is mainly contributed due to shape deviations of the shell middle plane which are commonly described as geometric imperfections. However, some deviations between theoretical and experimental buckling loads are sometimes so severe that other imperfection types are possibly to blame. This article describes experimental and numerical studies on the influence of loading imperfections on the buckling load of thin-walled cylinders. A global loading imperfection was applied by mistake during a buckling test of a thin-walled cylinder which led to a severe buckling load reduction and the corresponding load level was similar to the post-buckling load. Also, a series of experimental studies with localized loading imperfections is described which significantly reduced the buckling load of CFRP cylinders. The experimental results of this article represent an example for some of the very low buckling knockdown factors from early experimental campaigns and give insights in how to avoid these critical imperfections in experimental buckling campaigns.

Стійкість циліндричних оболонок

Мета: Отримати більш точні формули теоретичної осьової критичного навантаження шарнірно закріпленої циліндричної оболонки; знайти причину великих відмінностей між розрахунковими і експериментальними критичними навантаженнями. Метод: Використовується енергетичний критерій стійкості і співвідношення загальної лінійної теорії тонкостінних оболонок. Результати: Отримано нові формули залежності критичного навантаження від механічних і геометричних характеристик оболонки і параметрів хвилеутворення. Значення критичних навантажень, розраховані за цими формулами, близькі до експериментальних даних. Виявлено велика залежність критичних навантажень від: відносини радіуса до товщини оболонки; відношення довжини до радіусу; граничних умов. Обговорення: Циліндричні оболонки широко використовуються в інженерних конструкціях. Втрата стійкості оболонки може привести до руйнування конструкції. Для оцінки несучої здатності інженерних споруд, необхідні точні формули для обчислення критичних навантажень оболонок при осьовому стисненні. Такі формули поки не отримані. Причину великих розбіжностей між теоретичними і експериментальними значеннями осьових критичних навантажень циліндричних оболонок не знайдено. У даній роботі зроблена спроба вирішити цю проблему. На відміну від традиційного підходу, тут передбачається, що при випинанні оболонки, відстань між її кінцями не змінюється. Такий підхід дозволив отримати формули осьових критичних навантажень які більш точно описують процес втрати стійкості циліндричної оболонки при осьовому стисненні. Аналіз отриманих результатів дозволяє зробити висновок, що отримані формули можна використовувати для реальних розрахунків критичних навантажень циліндричних оболонок, а запропонований підхід може бути використаний для продовження досліджень стійкості тонкостінних конструкцій.

Experimental investigation on the flexural-torsional buckling behavior of pultruded GFRP angle columns

In this paper, the findings from an experimental investigation on the flexural-torsional buckling behavior of pultruded glass-fiber reinforced polymer (GFRP) angle columns are reported and discussed. The program included the study of two sizes of equal-leg angles made with different resins (polyester and vinylester). Prior to testing, a detailed material characterization was carried out and signature curves (critical load x length) were obtained using a generalized beam theory (GBT) software for predicting critical loads. Lengths were selected in order to ensure ‘pure’ flexural-torsional buckling, in a range of slenderness not studied in previous works. Twenty-two members with fixed ends and clamped walls were tested in concentric compression and had their motions measured during loading. Load-deflection curves are presented and the influences of post-buckling reserve of strength, damage and differential rotation of legs are discussed. Experimental critical loads obtained using Koiter's method are reported and shown to be in good agreement with GBT predictions. Finally, a design recommendation through the use of a Winter-type equation accounting for the plate-like behavior is made.

Characterization of Delamination Crack in Multidirectional E-glass/epoxy Composite under Mode I Loading

In this study, mode I critical strain energy release rate ( G Ic ) of unidirectional E-glass/epoxy was determined by double cantilever beam (DCB) test. Calculated G Ic was used to initiate delamination in numerical models. The effect of stacking sequence and fiber orientation of the sublaminates and the effect of thickness on mode I delamination crack length and G Ic distribution along the specimen width in 0 o //0 o interface have been studied using ANSYS ® . 3-D 8-node linear interface element INTER 205 is used to create a predefined crack path. To investigate the fiber orientation effect , composites with [+θ 2 , 90 2 , -θ 2 , 0 2 ] s stacking sequences were modeled. Interlaminar fracture analyses were performed by Virtual Crack Closure Technique (VCCT). Experimental and numerical critical loads ( P cr ) showed a good agreement. According to the results, while fiber orientation affects significantly the extended crack length and the G Ic distribution along the specimen width, stacking sequence only affects the G Ic distribution.

ASTER Shell: a simple concept to significantly increase the plastic buckling strength of short cylinders subjected to combined external pressure and axial compression

This paper proposes a new type of shell, similar to a cylindrical shell, which has significantly higher buckling strength when subjected to an arbitrary combination of uniform external pressure and axial compression. The underlying principle consists in a slight modification of the perfect cylinder in order to counteract the natural deformations which get larger and larger and lead to the collapse of the structure. Such shells are called ASTER shells. The concept has been validated through experiments, then analyzed numerically in order to explain what was observed and to propose avenues for improvements. The shells were made of electrodeposited nickel. The material was characterized. The chosen specimens were carefully measured to characterize their thickness and initial imperfections, then tested under the various types of loading. Then they were analyzed using finite elements. Thus, we were able to compare the finite element predictions with the experimental results. This comparison shows that plasticity has a decisive influence on the critical load and that linear elastic dimensioning leads to a serious overestimation of the experimental critical load. Contrary to perfect cylindrical shells, this type of shell is not significantly affected by geometric imperfections: this is another advantage of this type of design. Finally, we propose a numerical analysis in order to optimize the choice of the shape and propose shapes which resist buckling much better than a smooth cylinder when subjected to uniform external pressure, axial compression or a combination of both.

Numerical and experimental studies of compressed composite columns with complex open cross-sections

The article presents results of experimental investigations of thin-walled beams made with carbon fiber composite. Experimental studies were conducted to confirm results obtained from numerical calculations, which was performed using two different software based on finite element method and analytical–numerical method based on Koiter’s asymptotic stability theory of conservative systems modified by Byskov and Hutchinson. The studies consisted of axially compressed thin-walled columns with channel and top-hat cross section.The specimen for the tests were made of unidirectional composite – epoxy resin matrix with carbon fibers reinforcement. The specimen was manufactured using autoclave technique. Thin-walled channel columns with dimensions of 80×40×1.048mm and length of 300mm and with four different symmetrical ply arrangements were prepared.The resulting axial force, longitudinal shortening and lateral displacement were recorded during experimental tests. Additionally in chosen points the strain were registered using strain gauges measurement technique.The experimental critical loads using the different method were determined. The experimental results was compared with these obtained from finite element method (ANSYS and ABAQUS) and analytical–numerical method. The postbuckling equilibrium paths obtained from experimental tests and numerical calculation for all considered columns were compared.

Characterization of the Tenacity of the Concrete Mortars Exposed to Elevated Temperatures

The main objective of this research was the characterisation of the fracture toughness of concrete mortars exposed to high temperatures by using notched specimens in three point bending tests. The specimens have different lengths and were heated at different temperatures ranging from room temperature up to 900°C. The experimental test results obtained are closed with those reported by the literature, it shows that beyond approximately 300°C, the fracture loads decrease strongly with the rise in heating temperature. However, between the room temperature (25°C) and 300°C, we recorded an increase of the fracture load. Two methods using the experimental critical load were applied to calculate the fracture toughness of concrete mortars: an analytical method based on the mechanics of continuous medium theory and a numerical approach using finite elements method. The numerical modelling has then been validated, discussed and compared with analytical solution. Comparison between the experimental results and the theory predictions was quite good.

Characterization of the Tenacity of the Concrete Mortars Exposed to Elevated Temperatures

The main objective of this research was the charac terisation of the fracture toughness of concrete mortars exposed to high temperatures by using notched specimens in three point bending tests. The specimens have different lengths and were heated at different temperatures ranging from room temperature up to 900°C. The experimental test results obtained are closed with those reported by the literature, it shows tha t beyond approximately 300°C, the fracture loads decrease strongly with the rise in heating temperat ure. However, between the room temperature (25°C) and 300°C, we recorded an increase of the fr acture load. Two methods using the experimental critical load were applied to calculat e the fracture toughness of concrete mortars: an analytical method based on the mechanics of continu ous medium theory and a numerical approach using finite elements method. The numerical modelli ng has then been validated, discussed and compared with analytical solution. Comparison between the experimental results and the theory predictions was quite good.

Postbuckling behaviour of stainless steel (SUS304) columns under loading–unloading cycles

Based on the equilibrium configuration paths, this study investigates the postbuckling behaviour, including the shape recovery capability, of stainless steel (SUS304) columns after a complete loading–unloading cycle. Experiments were conducted and later numerical simulation was carried out in order to analyze the observed buckling and postbuckling behaviour. Large deflection analysis is necessary for slender columns. Moreover, comparison of the experimental critical load with analytical predictions verifies the fact that physical nonlinearity should be taken into account to analyze the column’s buckling behaviour comprehensively. Thus, based on large deflection theory as well as the nonlinear stress–strain relations, the load–deformation curves of the columns during a complete loading–unloading cycle have been predicted by using the finite element modelling (FEM) code ANSYS. Precise and quantitative analyses of the results verify the fact that the material’s stress–strain properties, both in tension and compression, attribute to the column’s buckling behaviour.

Buckling and Postbuckling Characteristics of the Superelastic SMA Columns – Numerical Simulation

Numerical simulation has been carried out to analyze several unique buckling and postbuckling behaviors of superelastic shape memory alloy (SMA) columns observed from experimental evidences. Comparison of the experimental critical load with analytical predictions verifies the fact that physical nonlinearity should be taken into account to comprehensively analyze the buckling behaviors of SMA columns. Thus, based on the nonlinear stress–strain relations as well as the large deformation theory, the load–deformation curves of the columns during a complete loading–unloading cycle have been predicted by using the FEM code ANSYS. Simulation has also been performed for the stainless steel (SUS304) columns. The total strain distributions in the column materials have been calculated at different states of loading. Precise and quantitative analyses of the results verify that the unique behaviors of the SMA columns could be attributed to the special nature of the stress–strain curves. It is found that the slender SMA column buckles in the austenite phase. Due to SMA's high strength for large strains, the slender columns can sustain the load with a very small change in magnitude for the postbuckling compression. A special method has been devised to simulate the unique unloading path of the slender superelastic SMA columns and its usefulness has been demonstrated. It has been found that defining a few intermediate unloading curves between the actual loading and unloading curves can help to minimize the stress discontinuity at the beginning of unloading. Simulation results also verify that because of the elastic shape recovery of the buckled slender SMA columns, the load increases when they are unloaded.

Lumped damping and stability of Beck column with a tip mass

Lumped external damping, acting on the concentrated mass at the tip of Beck’s column, is taken into account. This additional external damping removes the prolonged inconsistency between theory and experiment for a column with a tip mass, subjected to follower force. In Sugiyama et al. [J. Aerospace Eng. 8 (1995) 9] and Wood et al. [Proc. Roy. Soc. Lond. Ser. A. 313 (1969) 239] the experimental critical loads for a beam with a tip mass agree well with theoretical values for an undamped system. On the other hand, the critical loads, calculated in Sugiyama et al. with account for the internal and distributed external damping, were 50% lower than the experimental values. Since the actual experimental system is subject to damping, the experiment and theory come into conflict. This conflict is resolved in this paper by observing that the experimental system was subjected not only to the distributed external damping (due to distributed mass), but also to the lumped damping, acting on the tip mass. When the lumped external damping included in the analysis, analytical critical forces are in good agreement with experimental ones.

Tapered beam on elastic foundation model for compliance rate change of TDCB specimen

A modified beam theory is developed to predict compliance rate change of tapered double cantilever beam (TDCB) specimens for mode-I fracture of hybrid interface bonds, such as polymer composites bonded to wood. The analytical model treats the uncracked region of the specimen as a tapered beam on generalized elastic foundation (TBEF), and the effect of crack tip deformation is incorporated in the formulation. A closed-form solution is obtained to compute the compliance and compliance vs. crack length rate change. The present TBEF model is verified with finite element analyses and experimental calibration data of compliance for wood–wood and wood–composite bonded interfaces. The compliance rate change can be used with experimental critical fracture loads to determine the respective critical strain energy release rates or fracture toughness of interface bonds. The present analytical model, which accounts for the crack tip deformation, can be efficiently and accurately used for compliance and compliance rate-change predictions of TDCB specimens and reduce the experimental calibration effort that is often necessary in fracture studies. Moreover, the constant compliance rate change obtained for linear-slope TDCB specimens can be applied with confidence in mode-I fracture tests of hybrid material interface bonds.