Linear Stress Distribution Research Articles

Point Load Weight Functions for Semi-Elliptical Cracks in Finite Thickness Plate

Abstract This paper presents the application of the weight function method for the calculation of stress intensity factors for surface semi-elliptical cracks in finite thickness plates subjected to arbitrary two-dimensional stress fields. A new general mathematical form of point load weight function has been formulated by taking advantage of the knowledge of a few specific weight functions for two-dimensional planar cracks available in the literature and certain properties of the weight function in general. The existence of the generalized form of the weight function simplifies the determination of a specific weight function for specific crack configurations. The determination of a specific weight function is reduced to the determination of the parameters of the generalized weight function expression. These unknown parameters can be determined from reference stress intensity factor solutions. This method is used to derive the weight functions for semi-elliptical surface cracks in finite thickness plates. The derived weight functions are then validated against stress intensity factor solutions for several linear and non-linear two-dimensional stress distributions. The derived weight functions are particularly useful for the fatigue crack growth analysis of planar surface cracks subjected to fluctuating nonlinear stress fields resulting from surface treatment (shot peening), stress concentration, or welding (residual stress)

Elastic sectional stress analysis of variable section piers subjected to three-dimensional loads

The elastic stress analysis of beam-column structures of uniaxial symmetrical variable cross-sections is developed using an extension of Euler–Bernoulli beam theory. The applied loads are general considering axial (P), flexure (Mx–My), shear (Vx–Vy), and torsion (T). Three-dimensional analytical solutions for normal and shear stresses are derived using sectional analysis with warping functions for variable boundaries in elevation. The differential equations of equilibrium and deformations are accounting for the variations in the geometrical properties of the cross-section and related boundary conditions. The strong form solution is then written in a weak form that is implemented in a 2D sectional finite element (FE) code assuming linear normal stress distribution. Three application examples are presented to validate the proposed sectional approach and illustrate its accuracy by comparing with results from full 3D FE analyses: (a) a slender rectangular section pier with a sloped boundary, (b) a bulk rectangular section buttress with unsymmetrical slopes, and (c) a pier (squat wall) for a hydraulic structure. When the assumption of linear distribution for normal flexural stress is satisfied, the proposed sectional approach produces results within 1% of 3D FE with much reduced computational efforts. For bulk and squat walls the stress field distribution are very similar to 3D FE while the stress intensity shows some variations. This is of major practical significance because the proposed approach allows performing first a series of simplified yet acceptable sectional analyses in safety assessment of the three-dimensional type of structures considered herein.

Generalised stress intensity factors for rounded notches in plates under in-plane shear loading

The Notch Stress Intensity Factors (NSIFs) quantify the intensities of the asymptotic linear elastic stress distributions of sharp (zero radius) V-shaped notches. When the notch tip radius is different from zero, the singular sharp-notch field diverges from the rounded-notch solution in the close neighborhood of the notch tip. Nevertheless the NSIFs might continue to be parameters governing fracture if the notch root radius is small enough. Otherwise they can be seen simply as stress field parameters useful in quantifying the stress distributions ahead of the specific notch. Taking advantage of some analytical formulations which are able to describe stress distributions ahead of parabolic, hyperbolic and V-shaped notches with end holes, the paper discusses the form and the significance of the NSIFs with reference to in-plane shear loading, considering explicitly the role played by the notch opening angle and the notch tip radius. These parameters quantify the stress redistribution due to the root radius with respect to the sharp notch case to which they should naturally tend for decreasing values of the notch radius.

Design Lower Arm Using Optimum Approach

In the automotive industry, the riding comfort and handling qualities of an automobile are greatly affected by the suspension system. This paper presents the robust design of vehicle suspension arm using stochastic design improvement (SDI) technique based on Monte Carlo simulation. The main goal of this study is to determine the optimum design for the suspension arm. The structural model of the suspension arm was utilizing the Solid works and aluminum alloys (AA7075-T6) are selected as a suspension arm materials. The linear static stress distribution is investigated using the commercial Finite element analysis package, and dynamic analysis was performed using NASTRAN software. SDI has been performed to the design. A target output behavior is selected from the output variables available in the analysis. The result shows that the lower arm design has a higher capability to stand higher pressure as 9.18 MPa with the stress acted on lower arm is 41 MPa. The new parameter of material can be chose as optimum result for the lower suspension arm.

Determining the double‐K fracture parameters for three‐point bending notched concrete beams using weight function

ABSTRACTParameters of universal form of weight functions having four terms and five terms are derived for edge cracks in finite width of plate. The standard Tada Green's function is taken as the basis for the derivation. The shape of universal form of weight functions considered enables closed form expressions for cohesive toughness of three‐point bending test geometry of notched concrete beams due to linear cohesive stress distribution in the fictitious fracture zone. This solution provides a viable method to determine the double‐K fracture parameters: the initiation toughness, and the unstable toughness for mode I fracture of concrete beam. A comparison with existing analytical method shows that the weight function method for determination of the double‐K fracture parameters yields results without any appreciable error. The use of weight function will not only simplify the calculation to obtain the double‐K fracture parameters, and but also it will avoid the need of skilled numerical integration technique due to singularity problem at the integral boundary.

Bimodular Analysis of Hot-Mix Asphalt

ABSTRACT In this paper, the bimodular (different tensile and compressive moduli) stress distribution in the IDT specimen is determined for different compressive to tensile modulus ratios. The difference between the modulus calculated from the IDT test assuming Hondros' linear elastic stress distribution, and the compressive and tensile moduli is then obtained. The results show that this difference is significantly smaller for the case of the compressive modulus compared to the case of the tensile modulus. Even for a compressive to tensile modulus ratio of 2, the difference from the compressive modulus is small enough (5 to 10%) to be indistinguishable from experimental testing variation. This would explain why IDT experimental results tend to agree with uniaxial compression test results. It however contradicts the assumption that the IDT creep compliance test can approximate the uniaxial tension creep compliance.

On the failure of a brittle material by high velocity gas jet impact

Failure of brittle solid bodies due to the impingement of a high velocity air jet on the body surface is studied, experimentally and theoretically. Using the linear elastic theory and stress distribution analysis, a general criterion for the failure of brittle materials impacted by a gas jet is derived. Several special cases of jet–solid body interaction including failure of thin and thick layers and cylindrical objects immersed in a crossflow gas stream are investigated and proper material failure criteria are developed. These criteria correlate the minimum jet peak impact pressure (PIP) required to break the material to the material's tensile strength and Poisson's ratio. A series of experiments were performed using a laboratory-scale apparatus. Gypsum cast on steel tubes forming cylindrical samples was used as the model brittle material. Experimental data and high-speed breakup movies are employed to understand the gas jet–solid body interaction and to validate the theoretical criteria developed for the material failure. It is deduced that the failure of cylindrical samples impacted by a gas jet is by the formation and propagation of cracks. However, when the impact jet diameter is small, the cracks cannot propagate, and the material is failed due to localized surface pitting. One of the practical applications of this research is in Kraft recovery boilers, where high velocity supersonic steam jets are employed to remove deposits accumulated on the outer surfaces of the steam tubes.

Determination of approximate point load weight functions for embedded elliptical cracks

This paper presents the application of weight function method for the calculation of stress intensity factors in embedded elliptical cracks under complex two-dimensional loading conditions. A new general mathematical form of point load weight function is proposed based on the properties of weight functions and the available weight functions for two-dimensional cracks. The existence of this general weight function form has simplified the determination of point load weight functions significantly. For an embedded elliptical crack of any aspect ratio, the unknown parameters in the general form can be determined from one reference stress intensity factor solution. This method was used to derive the weight functions for embedded elliptical cracks in an infinite body and in a semi-infinite body. The derived weight functions are then validated against available stress intensity factor solutions for several linear and non-linear stress distributions. The derived weight functions are particularly useful for the fatigue crack growth analysis of planer embedded cracks subjected to fluctuating non-linear stress fields resulting from surface treatment (shot peening), stress concentration or welding (residual stress).

Extension of the Gravity Method for 3D Cracking Analysis of Spillway Piers Including Uplift Pressures

The classical structural stability evaluation of concrete hydraulic structures, such as spillways, is based on the application of rigid body equilibrium to compute force and moment resultants, and in most instances beam theory assuming linear stress distribution. The computations are done supposing a symmetric structure subjected to load acting in the upstream/downstream direction leading to a unidimensional axial force (P) -bending moment (M) interaction problem. However, several load conditions (ice floes, earthquakes) could apply transverse force components leading to a nonlinear biaxial flexural problem (P- Mx - My ) taking into account the potential cracking of the sections, as well as pressurized water penetration in cracks. A three-dimensional extension of the gravity method is presented herein for arbitrary sections such as those of spillway piers. Robust algorithms based on the strength of materials have been developed to compute the kernel of arbitrary sections, locate the neutral axis (NA), and...

The T-stress solutions for through-wall circumferential cracks in cylinders subjected to general loading conditions

The elastic T-stress is a parameter used to define the level of constraint at a crack tip. It is important to provide T-stress solutions for practical geometries to apply the constraint-based fracture mechanics methodology. In the present work, T-stress solutions are provided for circumferential through-wall cracks in thin-walled cylinders. First, cylinders with a circumferential through-wall crack were analyzed using the finite element method. Three cylinder geometries were considered; defined by the mean radius of the cylinder ( R) to wall thickness ( t) ratios: R/ t = 5, 10, and 20. The T-stress was obtained at eight crack lengths ( θ/π = 0.0625, 0.1250, 0.1875, 0.2500, 0.3125, 0.3750, 0.4375, and 0.5000, θ is the crack half angle). Both crack face loading and remote loading conditions were considered including constant, linear, parabolic and cubic crack face pressures and remote tension and bending. The results for constant and linear crack face pressure were used to derive weight functions for T-stress for the corresponding cracked geometries. The weight functions were validated against several linear and non-linear stress distributions. The derived weight functions are suitable for T-stress calculations for circumferential cracks in cylinders under complex stress fields.

Comparison of stress intensity factor solutions for cylinders with axial and circumferential cracks

Numerous stress intensity factor solutions have been proposed so far depending on the objects of evaluation including the variations of structures, cracks, and applied loads. In applying the flaw evaluation methodology to components of nuclear power plants, the use of reliable stress intensity factor solutions is essential. In this study, cracked cylinders were focused on as one of the typical configurations in actual plants. Existing stress intensity factor solutions for cracked cylinders were reviewed, and the accuracy of these solutions was investigated thorough the comparison with each other. Specific solutions were then recommended for cylindrical structures. Approximate expressions were newly derived for axially through-wall cracked cylinder subjected to linear stress distribution and for circumferentially through-wall cracked cylinder subjected to bending to realize simple evaluation of stress intensity factor. Considering that the cylindrical structures are often replaced with flat plates in the evaluation of actual components, the propriety of the replacement was also studied.

Residual Stress Distribution in Pure Bending Beam Subjected to Tensile Failure on One Side

The stress distribution on the midsection of a pure bending beam where tensile strain localization band initiates on the tensile side of the beam and propagates within the beam is analyzed. Using the static equilibrium condition on the section of the midspan of the beam and the assumption of plane section as well as the linear softening constitutive relation beyond the tensile strength, the expressions for the length of tensile strain localization band and the distance from the tip of the band to the neutral axis are derived. After superimposing a linear unloading stress distribution over the initial stress distribution, the residual stress distribution on the midsection of the beam is investigated. In the process of strain localization band’s propagation, strain-softening behavior of the band occurs and neutral axis will shift. When the unloading moment is lower, the length of tensile strain localization band remains a constant since the stress on the base side of the beam is tensile stress. While, for larger unloading moment, with an increase of unloading moment, the length of tensile strain localization band decreases and the distance from the initial neutral axis to the unloading neutral axis increases. The neutral axis of midsection of the beam will shift in the unloading process. The present analysis is applicable to some metal materials and many quasi-brittle geomaterials (rocks and concrete, etc) in which tensile strength is lower than compressive strength. The present investigation is limited to the case of no real crack. Moreover, the present investigation is limited to the case that the length of strain localization band before unloading is less than half of depth of the beam. Otherwise, the residual tensile stress above the elastic neutral axis will be greater than the tensile strength, leading to the further development of tensile strain localization band in the unloading process.

A simplified method for nonlinear failure analysis of stiffened plates

This paper shows that the application of pre-determined failure equations, derived from nonlinear finite element analyses, is effective in determining failure of structural components in a simpler linear finite element analysis. An analysis method is presented which is called simplified failure analysis. The first step of this method is the nonlinear determination of a component's failure limit. Next, a linear coarse-meshed finite element model of the component is analyzed under the failure load determined in the previous step. The resulting linear stress distribution is a ‘representative failure stress’ for the component because it is in equilibrium with the applied failure load. This ‘failure stress’ is then used in simpler linear analysis to provide a representative failure limit. This method is verified by an analysis of a structural grillage.

A generalized stress intensity factor to be applied to rounded V-shaped notches

In the presence of sharp (zero radius) V-shaped notches the notch stress intensity factors (N-SIFs) quantify the intensities of the asymptotic linear elastic stress distributions. They are proportional to the limit of the mode I or II stress components multiplied by the distance powered 1 − λ i from the notch tip, λ i being Williams’ eigenvalues. When the notch tip radius is different from zero, the definition is no longer valid from a theoretical point of view and the characteristic, singular, sharp-notch field diverges from the rounded-notch solution very next to the notch. Nevertheless, N-SIFs continue to be used as parameters governing fracture if the notch root radius is sufficiently small with respect to the notch depth. Taking advantage of a recent analytical formulation able to describe stress distributions ahead of rounded V-notches, the paper gives a generalized form for the notch stress intensity factors, in which not only the opening angle but also the tip radius dimension is explicitly involved. Such parameters quantify the stress redistribution due to the root radius with respect to the sharp notch case.

Technical solutions to reduce the risk of subsoil compaction: effects of dual wheels, tandem wheels and tyre inflation pressure on stress propagation in soil

The use of heavy machinery is increasing in agriculture, which induces increased risks of subsoil compaction. Hence, there is a need for technical solutions that reduce the compaction risk at high total machine loads. Three field experiments were performed in order to study the effects of dual wheels, tandem wheels and tyre inflation pressure on stress propagation in soil. Vertical soil stress was measured at three different depths by installing probes into the soil horizontally from a dug pit. In one experiment, also the stress distribution below the tyre was measured. Beneath the dual wheels, vertical stresses at 0.15 and 0.3 m depth were lower between the two wheels than under the centre of each wheel, despite the gap between the wheels being small (0.1 m). At 0.5 m depth, vertical stress beneath the wheels was the same as between the two wheels. The stress interaction from the two wheels was weak, even in the subsoil. Accordingly, measured stresses at 0.3, 0.5 and 0.7 m depth were highest under the centre of each axle centre line of tandem wheels, and much lower between the axles. For a wheel load of 86 kN, tyre inflation pressure significantly affected stress at 0.3 m depth, but not at greater depths. Stress directly below the tyre, measured at 0.1 m depth, was unevenly distributed, both in driving direction and perpendicular to driving direction, and maximum stress was considerably higher than tyre inflation pressure. Calculations of vertical stress based on Boussinesq's equation for elastic materials agreed well with measurements. A parabolic or linear contact stress distribution (stress declines from the centre to the edge of the contact area) was a better approximation of the contact stress than a uniform stress distribution. The results demonstrate that stress in the soil at different depths is a function of the stress on the surface and the contact area, which in turn are functions of wheel load, wheel arrangement, tyre inflation pressure, contact stress distribution and soil conditions. Soil stress and soil compaction are a function of neither axle load nor total vehicle load. This is of great importance for practical purposes. Reducing wheel load, e.g. by using dual or tandem wheels, also allows tyre inflation pressure to be reduced. This reduces the risk of subsoil compaction.

Weight functions for the determination of stress intensity factor and T-stress for edge-cracked plates with built-in ends

This paper presents the weight functions for the determination of the stress intensity factor and T-stress solutions for edge-cracked plates with built-in ends under complex stress distributions. First, a compliance analysis approach is used to calculate stress intensity factor and T-stress for edge cracks in finite width plates with built-in ends with uniform or linear stress distributions acting on the crack face. The results serve as the reference solutions for the next step in which the approaches of deriving weight functions from reference stress intensity factor and T-stress solutions developed for stress boundary conditions are extended to obtain the corresponding weight functions for edge-cracked plates with built-in ends. Finite element analysis is conducted to validate the derived solutions. The weight functions derived are suitable for obtaining stress intensity factors and T-stress solutions under any complex stress field.

The Equivalent Strain Energy Density approach re‐formulated and applied to sharp V‐shaped notches under localized and generalized plasticity

ABSTRACTIn the case of a rounded notch, the stress and strain at the notch tip can be determined by the traditional Neuber rule or by the Equivalent Strain Energy Density (ESED) approach, as formulated by Glinka and Molski. In the latter case the elastoplastic strain energy density at the notch tip is thought of as coincident with that determined under purely elastic conditions. For sharply V‐shaped notches this approach is not directly applicable, since the strain energy density at the notch tip tends toward infinity both for a material obeying an elastic law and a material obeying a power hardening law. By using the notch stress intensity factors, the present paper suggests a re‐formulation of the ESED approach which is applied no longer at the notch tip but to a finite size circular sector surrounding the notch tip. In particular we have adopted the hypothesis that, under plane strain conditions, the value of the energy concentration due to the notch is constant and independent of the two constitutive laws. When small scale yielding conditions are present, such a hypothesis immediately results in the constancy of the strain energy averaged over the process volume. As a consequence, plastic notch stress intensity factors valid for sharp V‐shaped notches can be predicted on the basis of the linear elastic stress distributions alone.

Non-linear static–dynamic finite element formulation for composite shells

Three non-linear finite element formulations for a composite shell are discussed. They are the simplified large rotation (SLR), the large displacement large rotation (LDLR), and the Jaumann analysis of general shells (JAGS). The SLR and the LDLR theories are based on total Lagrangian approach, and the JAGS is based on a co-rotational approach. Both the SLR and LDLR theories represent the in-plane strains exactly the same as Green's strain–displacement relations, whereas, only linear displacement terms are used to represent the transverse shear strain. However, a higher order kinematic through the thickness assumption is used in the SLR theory, which leads to parabolic transverse shear stress distribution compared to a first order kinematic through the thickness relationship used in the LDLR theory that leads to linear transverse shear stress distribution. Furthermore, the LDLR theory uses an Euler-like angle in the kinematics to account for the large displacement and rotation. The JAGS theory decomposes the deformation into stretches and rigid body rotations, where an orthogonal coordinate system translates and rotates with the deformed infinitesimal volume element. The Jaumann stresses and strains are used. Layer-wise stretching and shear warping through the thickness functions are used to model the three-dimensional behavior of the shell, where displacement and stress continuities are enforced along the ply interfaces. The kinematic behavior is related to the original undeformed coordinate system using the global displacements and their derivatives. Numerical analyses of composite shells are performed to compare the three theories. The commercial code ABAQUS is also used in this investigation as a comparison.

Stress intensity factors and weight functions for deep semi‐elliptical surface cracks in finite‐thickness plates

ABSTRACT Three‐dimensional finite element analyses have been conducted to calculate the stress intensity factors for deep semi‐elliptical cracks in flat plates. The stress intensity factors are presented for the deepest and surface points on semi‐elliptic cracks with a/t‐values of 0.9 and 0.95 and aspect ratios (a/c) from 0.05 to 2. Uniform, linear, parabolic or cubic stress distributions were applied to the crack face. The results for uniform and linear stress distributions were combined with corresponding results for surface cracks with a/t= 0.6 and 0.8 to derive weight functions over the range 0.05 ≤ a/c ≤ 2.0 and 0.6 ≤ a/t ≤ 0.95. The weight functions were then verified against finite element data for parabolic or cubic stress distributions. Excellent agreements are achieved for both the deepest and surface points. The present results complement stress intensity factors and weight functions for surface cracks in finite thickness plate developed previously.

Measurement of through-thickness stresses using small holes

Previously the deep-hole method was developed to measure residual stresses in thick-section engineering components. In this paper it is demonstrated that stresses arising from external applied loading can also be measured. A series of calibration studies using aluminium and steel samples, with thicknesses varying from 5 to 50 mm, are presented. The samples were subjected to tensions, bending and torsion. It is shown that the deep-hole method can measure linear and non-linear stress distributions.