Structural Mechanics of Negative Stiffness Honeycomb Metamaterials

2021 ◽  
Vol 88 (5) ◽  
Author(s):  
Navid Mehreganian ◽  
Arash S. Fallah ◽  
Pooya Sareh
Keyword(s):  
Stiffness Matrix ◽  
Closed Form Solution ◽  
Form Solution ◽  
Negative Stiffness ◽  
Fe Model ◽  
Curved Beams ◽  
Analytical Treatment ◽  
Tangent Stiffness Matrix ◽  
Numerical Finite Element ◽  
Protective Structures

Abstract The development of multi-stable structural forms has attracted considerable attention in the design of architected multi-materials, metamaterials, and morphing structures, as a result of some unusual properties such as negative stiffness and, possibly, negative Poisson's ratio. Multi-stability is achieved through a morphological change of shape upon loading, and in doing so multi-stable structures undergo transitions from one equilibrium state to another. This paper investigates the structural performance of the negative stiffness honeycomb (NSH) metamaterials made of double curved beams which are emerging in various applications such as sensors, actuators, and lightweight impact protective structures with structural tunability and recoverability. An analytical treatment is pursued using the Euler–Lagrange theorem and the stability of the honeycomb has been studied. Based on a static analysis of the nonlinear elastic system, the developed tangent stiffness matrix and ensuing deformation curve were assessed through multiple phases of deformation. The closed-form solution was in good agreement with the numerical finite element (FE) model at different bistability ratios. It was shown that the bistability ratio had a pronounced effect on the overall response of the honeycomb and the desired negativity in the stiffness matrix could be achieved with high bistability ratios.

Stability of Thin Web Composite Cantilever Beams of Random Lamination

2020 ◽  
Vol 20 (13) ◽  
pp. 2041016
Author(s):  
Hayder A. Rasheed ◽  
Habiburrahman Ahmadi ◽  
Abdul H. Halim
Keyword(s):  
Shear Strain ◽  
Closed Form ◽  
Closed Form Solution ◽  
Form Solution ◽  
Cantilever Beams ◽  
Analytical Treatment ◽  
Lateral Curvature ◽  
Coupling Stiffness ◽  
Torsional Stability ◽  
Twisting Angle

This study addresses the analytical treatment of a closed-form buckling equation for lateral-torsional stability of thin web composite cantilever beams under mid-height tip force. The beam is composed of random ply fiber orientations. Classical lamination theory is embedded into the Vlasov plate formulation to make up the framework of the analytical treatment. A closed-form solution is realized when an innovative dimensional reduction is extended to the 3D constitutive stiffness matrix. This was made possible through a two-step process in which the shear strain, lateral curvature, and twisting curvature are retained first. By condensing the shear strain variable, effective lateral, torsional, and coupling stiffness terms were formulated. Applying the equilibrium conditions in the deformed configuration, two differential equations are obtained in terms of the lateral curvature and twisting angle. Eliminating the lateral curvature, the twisting angle differential equation with nonconstant coefficients is generated. This equation is solved using a hybrid numerical-analytical approach yielding an analytical buckling expression. Finite element results are generated to verify the accuracy of the buckling load predictions indicating very good correlation with the buckling equation results regardless of the random lamination applied.

Plastic Limit Pressure Solutions for Elbows With Slant Through-Wall Cracks

2020 ◽  
Vol 142 (5) ◽  
Author(s):  
Minkyu Kim ◽  
Jaehee Kim ◽  
Moon Ki Kim ◽  
Jae-Boong Choi ◽  
Nam-Su Huh ◽  
...  
Keyword(s):  
Three Dimensional ◽  
Closed Form Solution ◽  
Limit Load ◽  
Form Solution ◽  
Fe Model ◽  
Plastic Limit ◽  
Critical Crack ◽  
Critical Crack Length ◽  
Limit Pressure ◽  
Plastic Limit Pressure

Abstract For leak-before-break (LBB) assessment, an idealized through-wall crack (TWC) is typically postulated to determine the critical crack length of cracked piping. However, such an idealization in terms of crack shape can lead to underestimations of plastic limit pressure. Although many studies have been performed to obtain accurate limit load solutions for cracked straight pipes by considering realistic crack geometries, there is still a lack of information regarding slant TWC at elbow. Therefore, three-dimensional finite element (FE) models of an elbow considering the effects of slant TWC on plastic limit pressure are developed. The proposed FE model and analysis procedure were verified through comparisons to the existing solutions for idealized TWCs in elbow. On this basis, the effect of slant TWC on the plastic limit pressure is analyzed, and a closed-form solution of the plastic limit pressure is proposed, for an elbow containing a longitudinal or a circumferential through-wall crack.

scholarly journals Developed Mathematical Model for Indeterminate Elements with Variable Inertia and Curved Elements with Constant Cross-Section

2019 ◽  
Vol 2019 ◽  
pp. 1-11
Author(s):  
Mohamed A. El Zareef ◽  
Mohamed E. El Madawy ◽  
Mohamed Ghannam
Keyword(s):  
Mathematical Models ◽  
Numerical Integration ◽  
Cross Section ◽  
Closed Form Solution ◽  
Form Solution ◽  
Optimum Number ◽  
Fe Model ◽  
Constant Cross Section ◽  
Structural Elements ◽  
Good Agreement

Issues such as analysis of indeterminate structural elements that have variable inertia as well as a curved shape still have no closed form solution and are considered one of the major problems faced by design engineers. One method to cope with these issues is by using suitable the finite element (FE) software for analyzing these types of elements. Although it saves time, utilization of FE programs still needs professional users and not all engineers are familiar with it. This paper has two main objectives; first, to develop simple mathematical models for analyzing indeterminate structural elements with variable inertia and that have a curved shape with constant cross section, this model is much easier to be used by engineers compared to the FE model. For simplicity and saving time, a MATLAB program is developed based on investigated mathematical models. The force method combined with numerical integration technique is used to develop these models. The developed mathematical models are verified using the suitable FE software; good agreement was observed between the mathematical and the FE model. The second objective is to introduce a mathematical formula to determine the accurate number of divisions that would be used in the mathematical models. The study proves that the accuracy of analysis depends on the number of divisions used in the numerical integration. The optimum number of divisions is obtained by comparing the output results for both FE and developed mathematical models. The developed mathematical models show a good agreement with FE results with faster processing time and easier usage.

Pure Bending of Curved Beams of Thin-Walled Rectangular Box Section

1991 ◽  
Vol 58 (1) ◽  
pp. 154-156 ◽  
Author(s):  
R. D. Cook
Keyword(s):  
Finite Element ◽  
Analytical Method ◽  
Circumferential Stress ◽  
Closed Form Solution ◽  
Form Solution ◽  
Finite Element Analyses ◽  
Curved Beams ◽  
Pure Bending ◽  
The Subject ◽  
Thin Walled

A closed-form solution of the subject problem is presented. The analytical method resembles that used by Bleich (1933) to study curved beams of I or T section. It is found that the circumferential stress may be smaller than a perpendicular stress that arises from flexing of parts of the box. Accuracy of the solution is verified by comparison with finite element analyses.

An Efficient Method to Generate Element Stiffness Matrix of Quadrilateral Elements in Closed Form on Application of Vehicle Analysis

2016 ◽  
Vol 852 ◽  
pp. 582-587
Author(s):  
P.V. Jeyakarthikeyan ◽  
R. Yogeshwaran ◽  
Karthikk Sridharan
Keyword(s):  
Closed Form ◽  
Stiffness Matrix ◽  
Outer Boundary ◽  
Closed Form Solution ◽  
Simple Calculation ◽  
Field Variable ◽  
Form Solution ◽  
Gauss Quadrature ◽  
Quadrilateral Elements ◽  
Hybrid Grid

This paper presents about generating elemental stiffness matrix for quadrilateral elements in closed form solution method for application on vehicle analysis which is convenient and simple as long as Jacobian is matrix of constant. The interpolation function of the field variable to be found can integrate explicitly once for all, which gives the constant universal matrices A, B and C. Therefore, stiffness matrix is no longer integration of the given functional, it is simple calculation of universal matrices and local co-ordinates of the element. So time taken for generation of element stiffness can be reduced considerably compared to Gauss numerical integration method. For effective use of quadrilateral elements hybrid grid generation is recommended that contains all interior element edges are parallel to each other (rectangle or square elements) and outer boundary elements are quadrilaterals with distortion. So in the Proposed method, the closed form and Gauss numerical method is used explicitly for interior elements and outer elements respectively. The time efficiency of proposed method is compared with conventional Gauss quadrature that is used for entire domain. It is found that the proposed method is much efficient than Gauss Quadrature.

A closed form solution for the static analysis of continuous skew-curved beams

1980 ◽  
Vol 37 (1-2) ◽  
pp. 53-64 ◽  
Author(s):  
D. E. Panayotounakos ◽  
P. S. Theocaris
Keyword(s):  
Closed Form ◽  
Static Analysis ◽  
Closed Form Solution ◽  
Form Solution ◽  
Curved Beams

scholarly journals Stability Optimization of Trapezoidal Frame with Rigid Members and Flexible Joints

2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
Chao Wu ◽  
Ya-Nan Li ◽  
Lik-Ho Tam ◽  
Li He
Keyword(s):  
Parametric Study ◽  
Moving Load ◽  
Closed Form Solution ◽  
Form Solution ◽  
Frame Structure ◽  
Fe Model ◽  
Flexible Joints ◽  
Analysis And Design ◽  
Portal Frame ◽  
The Stability

Stability has been an important subject in the design of a portal frame structure. Conventional stability analysis of the portal frame is normally conducted assuming that all the joints are rigid. However, the joints of a portal frame in real applications are not always rigid and semirigid connections often exist. AISC design code requires that the effect of the joint flexibility on the behavior and buckling capacity of the portal frame should be taken into account in the analysis and design procedures. To address this issue, a portal frame with flexible joints and rigid members was theoretically analyzed in the literature and closed form solution was derived for its global buckling load. However, when more parameters are involved, e.g., different leg lengths, asymmetric frame shape, and moving load, the solution to the governing equation of the stability of the frame becomes impossible. This paper presents a comprehensive parametric study on the stability of an asymmetric portal frame with flexible joints and rigid members through finite element (FE) analysis. The FE model was first validated by the existing theoretical solution in the literature. Parameters including the position of the moving load, the lengths of the two frame legs, and the span of the frame were analyzed. Design curves were developed based on the parametric study, from which the stable, unstable, and catastrophically unstable states of the portal frame were characterized. This paper contributes benchmark results for the stability optimization in the design of the portable frame of a general shape.

scholarly journals Analytical Solution of Laterally Loaded Free-Head Long Piles in Elasto-Plastic Cohesive Soils

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1961
Author(s):  
Ayman Abd-Elhamed
Keyword(s):  
Closed Form ◽  
Closed Form Solution ◽  
Bending Moment ◽  
Cohesive Soil ◽  
Form Solution ◽  
Cohesive Soils ◽  
Fe Model ◽  
Soil System ◽  
Soil Medium ◽  
Laterally Loaded

This research study presents a closed form solution of responses of laterally loaded long piles embedded on cohesive soils with a constant subgrade modulus. The surrounding soil medium is modelled as elastic-perfectly plastic. The closed form solution is derived by solving the governing differential equation of the pile–soil system. The most popular numerical computation software package MATLAB is utilized for the implementation of solutions. The provided analytical method reliably calculates the pile head deflection and bending moment required for engineering design purposes. Results are discussed and verified with solutions of an equivalent three-dimensional finite element (FE) model developed using ANSYS software. It was concluded that the proposed analytical model could efficiently provide the exact solution of embedded piles in elasto-plastic cohesive soil under lateral loads.

Design of Statically Balanced Spatial Mechanisms With Spring Suspensions

2012 ◽  
Vol 4 (2) ◽  
Author(s):  
Po-Yang Lin
Keyword(s):  
Stiffness Matrix ◽  
Computer Software ◽  
Closed Form Solution ◽  
Form Solution ◽  
Static Equilibrium ◽  
Design Parameters ◽  
Gravitational Forces ◽  
Spatial Mechanisms ◽  
Generalized Coordinate ◽  
Loaded Mechanism

This paper proposes a general approach for designing spatial statically balanced mechanisms with articular joints utilizing ideal zero-free-length springs. The proposed statically balanced mechanism can counterbalance the gravitational forces and provides a perfect static equilibrium at any configuration. The method of the paper is based on the energy approach, and a generalized coordinate system is developed to define the configuration of a spatial mechanism and to be a vector basis for the derivation of potential energy. By incorporating the gravitational forces and the spring forces into the system, the stiffness matrix of a spring-loaded mechanism is proposed. The perfect static balance is observed when the stiffness matrix is a diagonal matrix, from which, the design equations can be readily obtained. The closed-form solution of spring design parameters of a statically balanced, spatial, three-articular arm is obtained as a design example. The simulations of the conceptual design are performed by commercial computer software, and the static equilibrium of a quasi-static continuous motion is verified.

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