Reduced Stiffness Method Research Articles

Robust design of imperfection sensitive thin-walled shells under axial compression, bending or external pressure

Thin-walled shells like cylinders, cones and spheres are primary structures in launch-vehicle systems. When subjected to axial loading, bending or external pressure, these thin-walled shells are prone to buckling. The corresponding critical load heavily depends on deviations from the ideal shell shape. In general, these deviations are defined as geometric imperfections, and although imperfections exhibit comparatively low amplitudes, they can significantly reduce the critical load.Considering the influence of geometric imperfections adequately into the design process of thin-walled shells poses major challenges for structural design. The most common procedure to take into account the influence of imperfections is based on classical buckling loads obtained by a linear analysis which are then corrected by a knockdown factor. The knockdown factor represents a statistical lower-bound with respect to data obtained experimentally for different types of thin-walled shells.This article presents a versatile and simple numerical design approach for buckling of critical shell structures. The new design procedure is based on the reduced stiffness method and leads to significantly improved critical load estimations in comparison to lower-bounds obtained empirically. An analysis example is given which is based on the launch-vehicle stage adapter (LVSA) of NASAs Space Launch-system (SLS).

Design of laterally restrained web-tapered steel structures through a stiffness reduction method

A stiffness reduction method for the design of laterally restrained web-tapered steel structures fabricated through the welding of individual steel plates is presented in this paper. Stiffness reduction functions for welded members, accounting fully for the deleterious influence of the spread of plasticity and imperfections on the structural resistance, are developed. The method is implemented through (i) dividing tapered members into prismatic segments along their lengths, (ii) reducing the flexural stiffness of each segment by means of the developed stiffness reduction functions considering the first-order forces and cross-section properties of each segment, (iii) performing Geometrically Nonlinear Analysis and (iv) making cross-section strength checks. Essentially, it is proposed to replace the current typical approach to structural design of conducting a simple elastic (with nominal stiffness) structural analysis followed by elaborate member checks with an integrated process utilising more sophisticated second-order analysis (with stiffness reduction) but very simple design checks. The distribution of internal forces within the structure is captured more accurately due to the allowance for imperfections, residual stresses and plasticity through stiffness reduction and the allowance for frame and member instability effects through the use of second-order analysis. The need for determining effective lengths and for conducting member buckling checks is also eliminated. Verification of the proposed approach against the results obtained from nonlinear shell finite element modelling is presented for various tapering geometries, slenderness values and loading conditions. Assessment of the proposed method against the European and North American steel design codes for tapered steel structures is also provided.

05.06: Design of web‐tapered steel members through a stiffness reduction method

ABSTRACTStiffness reduction provides a practical means for the instability assessment of structural steel members. In this paper, a stiffness reduction approach for the design of web‐tapered steel members fabricated through the welding of individual plates is presented. Stiffness reduction functions for welded members, which fully take into account the adverse influence of plasticity and imperfections on the strength and stability, are described. The proposed approach is implemented by (i) dividing the tapered member into prismatic segments along its length, (ii) reducing the flexural stiffness of each segment through the developed stiffness reduction functions considering the withstood forces and the cross‐section properties of each segment, (iii) performing Geometrically Nonlinear Analysis using beam elements and (iv) making cross‐section strength checks. The accuracy of the proposed approach is assessed against the results of nonlinear shell finite element modelling for a large number of tapered members, with various slenderness values, tapering ratios (i.e. the ratios between the cross‐section depths of deep and shallow ends) and loading conditions. It is observed that the proposed stiffness reduction method provides a very accurate and efficient way to design web‐tapered steel members.

Analysis of the influence of stiffness reduction on the load carrying capacity of ring-stiffened cylindrical shell

Load carrying capacity is of importance in the evaluation of the stability of submarine pressure shell structure. During the buckling process of cylindrical shell under hydrostatic pressure, the total potential energy of ring-stiffened cylindrical shell can be divided into constant component, linear component and square nonlinear component. Based on the derivations of buckling loads of smooth cylindrical shell and ring-stiffened cylindrical shell, the reduced stiffness method (RSM) is applied to assess the role of different stiffness reductions in the evaluation of stable load carrying capacity of cylindrical shell. This study shows: (1) the circumferential membrane stiffness is influence the stability of cylindrical shell largest; (2) the ring ribs have a significant effect on the overall stability of ring-stiffened cylindrical shells under hydrostatic pressure; (3) the circumferential buckling wave number of ring-stiffened cylindrical shell may not depend on the membrane stiffness reduction of shell, but it increases with the bending stiffness reduction of ring ribs.

Micromechanical prediction of damage due to transverse ply cracking under fatigue loading in composite laminates

To predict the matrix microcracking of laminated composites under fatigue loading, a novel energy based model is presented in the framework of micromechanics. For this purpose, strain energy release rate (SERR) of microcracks which had been derived previously for the whole laminate, is developed for a lamina, and then is calculated using a stress transfer-based stiffness reduction method. The advantages of the proposed method include its capability to predict the matrix cracking of general lay-ups based on the local stresses and stiffnesses of each plies separately and not being limited to a special stacking sequence. In order to predict micro-cracking propagation of composites under cyclic loading, the coefficients of the modified Paris law are extracted using the available experimental data of crack density–cycle curves. Then using multi-scale modelling and continuum damage mechanics concept, the proposed algorithm is implemented in ANSYS finite element software, as a new user defined material (Usermat). The static progress of failure on [45/−45]s laminate is simulated and the obtained results are compared with the existing experimental data in a good agreement. Finally, the results of implemented fatigue algorithm for different cross-ply laminates under different stress levels are obtained and compared with the available experimental data.

Fatigue Performance of Recycled Hot Mix Asphalt: A Laboratory Study

The paper introduces and analyses the results of an experimental trial on the fatigue resistance of recycled hot mix asphalt for road pavements. Based on the gyratory compaction and the indirect tensile strength test, the mix design procedure has optimized nine different mixes, considering both conventional limestone and Reclaimed Asphalt Pavement (RAP), the latter used at different quantities, up to 40% by weight of the aggregate. A standard bitumen and two polymer modified binders were used for the production of the mixes. The fatigue study was carried out with four-point bending tests, each one performed at 20°C and 10 Hz. The empirical stiffness reduction method, along with the energy ratio approach, based on the dissipated energy concept, was adopted to elaborate the experimental data. Unaged and aged specimens were checked, to analyse the ageing effects on the fatigue performance. In comparison with the control mixes, produced only with limestone, improved fatigue performance was noticed for the mixtures prepared with RAP, especially when made with polymer modified binders, under both aged and unaged conditions. Both the approaches adopted for the experimental data analysis have outlined the same ranking of the mixes.

Lower bound buckling loads for design of laminate composite cylinders

ABSTRACTOver a period of more than 45 years, an extensive research program has allowed a series of very simple propositions, relating to the safe design of shells experiencing imperfection sensitive buckling, to be recast in the form of a series of lemmas. These are briefly summarized and their practical use is illustrated in relation to the prediction of safe lower bounds to the imperfection sensitive buckling of axially loaded, fiber reinforced polymeric, laminated cylinders. With a fundamental aspect of the approach, sometimes referred to as the reduced stiffness method, being the delineation of the various shell membrane and bending stiffness (or perhaps more appropriately energy) components contributing to the buckling resistance, the method will be shown to also provide a powerful way of making rational design decisions to optimize the use of fiber reinforcement.

Identification of cracks in thick beams with a cracked beam element model

The effect of a crack on the vibration of a beam is a classical problem, and various models have been proposed, ranging from the basic stiffness reduction method to the more sophisticated model involving formulation based on the additional flexibility due to a crack. However, in the damage identification or finite element model updating applications, it is still common practice to employ a simple stiffness reduction factor to represent a crack in the identification process, whereas the use of a more realistic crack model is rather limited. In this paper, the issues with the simple stiffness reduction method, particularly concerning thick beams, are highlighted along with a review of several other crack models. A robust finite element model updating procedure is then presented for the detection of cracks in beams. The description of the crack parameters is based on the cracked beam flexibility formulated by means of the fracture mechanics, and it takes into consideration of shear deformation and coupling between translational and longitudinal vibrations, and thus is particularly suitable for thick beams. The identification procedure employs a global searching technique using Genetic Algorithms, and there is no restriction on the location, severity and the number of cracks to be identified. The procedure is verified to yield satisfactory identification for practically any configurations of cracks in a beam.

Development and assessment of a practical stiffness reduction method for the in-plane design of steel frames

In this paper, the development and assessment of a stiffness reduction method for the in-plane design of steel frames is presented. The adopted stiffness reduction approach is implemented by reducing the flexural stiffnesses (EI) of the members of a steel frame by considering the first-order forces they are subjected to through the stiffness reduction functions and performing Geometrically Nonlinear Analysis (i.e. second-order elastic analysis). Since the presented approach uses stiffness reduction functions that fully take into account the deleterious influence of imperfections and the spread of plasticity on the structural response and member strengths, it obviates the need of using member design equations, and only requires cross-section strength checks. The accuracy of the presented approach is illustrated for individual steel members and non-redundant and redundant benchmark steel frames from the literature. In all the considered cases, the presented method is verified against the results obtained from nonlinear finite element modelling. A comparison of the presented approach against the notional load method of the European structural steel design code EN 1993-1-1 and the direct analysis method of the US structural steel design code AISC 360-10 is also provided.

Laboratory investigation of interlayer shear fatigue performance between open-graded friction course and underlying layer

The objective of the study was to evaluate the bonding fatigue performance between open-graded friction course (OGFC) and underlying layer through laboratory testing at different tack coat application rates. Direct shear fatigue test was performed to obtain the shear fatigue properties of the composite specimens composed of OGFC and its underlying layer. Two types of dense graded asphalt mixture (named BM and TLD in Tennessee) were selected as the underlying layer. In addition to the fatigue life determined according to the conventional 50% stiffness reduction method, energy approach was also employed to analyze the fatigue behavior of the composite specimens. Results showed that the OGFC-TLD structure gave a better shear fatigue performance than OGFC-BM. With the increase in tack coat application rate, the plateau value (PV) increased and the total cumulative dissipated energy decreased for both combined structures. The contact area between OGFC and the underlying layer was measured and correlated to the fatigue life. The contact area between OGFC and TLD was larger than that between OGFC and BM, leading to a better fatigue performance of OGFC-TLD.

Flexural–torsional buckling assessment of steel beam–columns through a stiffness reduction method

In this paper, a stiffness reduction method for the flexural–torsional buckling assessment of steel beam–columns subjected to major axis bending and axial compression is presented. The proposed method is applied by reducing the Young’s E and shear G moduli through the developed stiffness reduction functions and performing Linear Buckling Analysis. To account for second-order forces induced prior to buckling, the in-plane (in the plane of bending) and out-of-plane analyses of a member are separated and stiffness reduction for the out-of-plane instability assessment is applied on the basis of member forces determined from the in-plane analysis. Since the developed stiffness reduction functions fully take into account the detrimental influence of imperfections and spread of plasticity, the proposed method does not require the use of member design equations, thus leading to practical design. For the purpose of verifying this approach, the strength predictions determined through the proposed stiffness reduction method are compared against those obtained from nonlinear finite element modelling for a large number of regular, irregular, single and multi-span beam–columns.

Plateau lower-bounds to the imperfection sensitive buckling of composite shells

The buckling of thin FRP laminated shells are sensitive to initial geometric imperfections. A large number of geometric and material variables prohibit the traditional experiment-based lower-bound design method for metallic shells from being extended to composite ones. As an alternative, the so-called reduced stiffness method (RSM) has been applied to the lower bound buckling of FRP laminated shells. It has been shown for shorter shells the method predicted excellent lower bounds to the nonlinear buckling loads. This paper aims to extend the study to longer shells. It is shown for longer composite shells the lower-bound buckling modes generally occur in the long axial wave mode having one axial halfwave but require imperfection amplitudes of impractical largeness. Further studies of the geometric parameters identify the existence of intermediate and significant plateaus to the lower bounds. These plateau lower-bounds demonstrate the importance of the short axial wave modes having more than one axial halfwave, associated with the imperfection amplitudes of practical smallness. Using the plateau values and modes to predict lower bounds is suggested to provide an important alternative for improving shell buckling design.

Lateral–torsional buckling assessment of steel beams through a stiffness reduction method

This paper presents a stiffness reduction approach utilising Linear Buckling Analysis (LBA) with developed stiffness reduction functions for the lateral–torsional buckling (LTB) assessment of steel beams. A stiffness reduction expression is developed for the LTB assessment of beams subjected to uniform bending and modified for the consideration of moment gradient effects on the development of plasticity. The proposed stiffness reduction method considers the influence of imperfections and plasticity on the response through the reduction of the Young's modulus E and shear modulus G and obviates the need of using LTB buckling curves in design. The accuracy and practicality of the method are illustrated for regular, irregular, single and multi-span beams. In all of the considered cases, the proposed method is verified against the results obtained through nonlinear finite element modelling.

An optimised stiffness reduction method for simulating infinite elastic space using commercial Finite Elements codes

A common goal when using Finite Element (FE) modelling in time domain wave scattering problems is to minimise model size by only considering a region immediately surrounding a scatterer or feature of interest. The model boundaries must simulate infinite space by minimising the reflection of incident waves. This is a significant and long-standing challenge that has only achieved partial success. Industrial companies wishing to perform such modelling are keen to use established commercial FE packages that offer a thorough history of validation and testing. Unfortunately, this limits the flexibility available to modellers preventing the use of popular research tools such as Perfectly Matched Layers (PML). Unlike PML, Absorbing Layers by Increasing Damping (ALID) have proven successful offering practical implementation into any solver that has representation of material damping. Despite good performance further improvements are desirable. Here, a Stiffness Reduction Method (SRM) has been developed and optimised to operate within a significantly reduced spatial domain. The technique is applied by altering damping and stiffness matrices, inducing decay of incident waves. Variables are expressed as a function of known model constants, easing implementation for generic problems. Analytical and numerical solutions have shown that SRM out performs ALID, with results approaching those of PML.

A stiffness reduction method for the in-plane design of structural steel elements

Stiffness reduction offers a practical means of considering the detrimental influence of geometrical imperfections, residual stresses and the spread of plasticity in the analysis and design of steel structures. In this paper, a stiffness reduction approach is presented, which utilises Linear Buckling Analysis (LBA) and Geometrically Nonlinear Analysis (GNA) in conjunction with developed stiffness reduction functions for the design of columns and beam-columns in steel frames. This approach eliminates the need for modelling geometrical imperfections and requires no member buckling checks. For columns, inelastic flexural buckling loads can be obtained using LBA with appropriate stiffness reduction, while GNA with stiffness reduction is required to determine an accurate prediction of beam-column failure. The accuracy and practicality of the proposed method is shown in several examples, including regular and irregular members. For the latter case in particular, it is found that the proposed approach provides more accurate capacity predictions than traditional design methods, when compared to results generated by means of nonlinear finite element modelling.

Reduced Stiffness Criteria for the Safe Buckling Design of Thin Shells and its Numerical Validation

Thin metallic shells have long been adopted as major structural components in weight-sensitive applications, especially in marine engineering. Imperfection sensitive buckling is a critical design factor when these structures are loaded in compression. Traditional experimental design method depends on deriving lower bounds to the scatter of large sets of test results. This paper aims to present an analytical approach, the so-called reduced stiffness method (RSM) to the lower-bound buckling of thin-metallic shells. The validity of the RSM for the prediction of the safe lower bounds to the buckling of thin shells is verified through carefully controlled finite element analysis and the comparative studies confirm the reliability of the RSM.

A Stiffness Reduction Method for efficient absorption of waves at boundaries for use in commercial Finite Element codes

Commercially available Finite Element packages are being used increasingly for modelling elastic wave propagation problems. Demand for improved capability has resulted in a drive to maximise the efficiency of the solver whilst maintaining a reliable solution. Modelling waves in unbound elastic media to high levels of accuracy presents a challenge for commercial packages, requiring the removal of unwanted reflections from model boundaries. For time domain explicit solvers, Absorbing Layers by Increasing Damping (ALID) have proven successful because they offer flexible application to modellers and, unlike the Perfectly Matched Layers (PMLs) approach, they are readily implemented in most commercial Finite Element software without requiring access to the source code. However, despite good overall performance, this technique requires the spatial model to extend significantly outside the domain of interest. Here, a Stiffness Reduction Method (SRM) has been developed that operates within a significantly reduced spatial domain. The technique is applied by altering the damping and stiffness matrices of the system, inducing decay of any incident wave. Absorbing region variables are expressed as a function of known model constants, helping to apply the technique to generic elastodynamic problems. The SRM has been shown to perform significantly better than ALID, with results confirmed by both numerical and analytical means.

Research on Flexural Stiffness Reduction Factor of Reinforced Concrete Column with Equiaxial + Shaped Section

The reduced stiffness method had been adopted to evaluate the material nonlinearity characteristics of reinforced concrete structures in compliance with concrete structure standards of the United States, New Zealand and Canada. Concrete structure design code in China also accepts the reduced stiffness method as a supplement of the second-order effects. However, the concrete structure with specially shaped columns code of China still use amplified coefficients of eccentricity to consider nonlinearity characteristics of reinforced concrete structure with special shaped columns. Based on the numerical integral method, a flexural stiffness reduction factor is proposed to consider characteristics of material nonlinearity and geometrical nonlinearity of reinforced concrete columns with equiaxial + shaped section

Geometric imperfections and lower-bound methods used to calculate knock-down factors for axially compressed composite cylindrical shells

The important role of geometric imperfections on the decrease of the buckling load for thin-walled cylinders had been recognized already by the first authors investigating the theoretical approaches on this topic. However, there are currently no closed-form solutions to take imperfections into account already during the early design phases, forcing the analysts to use lower-bound methods to calculate the required knock-down factors (KDF). Lower-bound methods such as the empirical NASA SP-8007 guideline are commonly used in the aerospace and space industries, while the approaches based on the Reduced Stiffness Method (RSM) have been used mostly in the civil engineering field. Since 1970s a considerable number of experimental and numerical investigations have been conducted to develop new stochastic and deterministic methods for calculating less conservative KDFs. Among the deterministic approaches, the single perturbation load approach (SPLA), proposed by Hühne, will be further investigated for axially compressed fiber composite cylindrical shells and compared with four other methods commonly used to create geometric imperfections: linear buckling mode-shaped, geometric dimples, axisymmetric imperfections and measured geometric imperfections from test articles. The finite element method using static analysis with artificial damping is used to simulate the displacement controlled compression tests up to the post-buckled range of loading. The implementation of each method is explained in details and the different KDFs obtained are compared. The study is part of the European Union (EU) project DESICOS, whose aim is to combine stochastic and deterministic approaches to develop less conservative guidelines for the design of imperfection sensitive structures.

Study on Flexural Stiffness Reduction Factor of Reinforced Concrete Column with Equiaxial T Shaped Section

The reduced stiffness method had been adopted to evaluate the material nonlinearity characteristics of reinforced concrete structures to be in compliance with concrete structure standards of the United States, New Zealand and Canada. Concrete structure design code in China also accepts the reduced stiffness method as a supplementary method of considering the second-order effects problem. However, the concrete structure with specially shaped columns code of China still use amplified coefficients of eccentricity to consider nonlinearity characteristics of reinforced concrete structure with special shaped columns. Based on the numerical integral method, a flexural stiffness reduction factor is proposed to consider characteristics of material nonlinearity and geometrical nonlinearity of reinforced concrete columns with equiaxial T shaped section.