Category: Finite Element Formulation

  • Will you help me with the assumed natural strain method for my beam?

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    “If I understand it right, the ‘assumed natural strain’ method for beams is to assume that there is no compression, and then calculate the compression and tension stresses from the displacement and deflection. Here is the general procedure: 1. Calculate the displacement, 2. Calculate the deformation, 3. Calculate the stresses, 4. Use the values of stresses to calculate the stresses at the end points. You will see that the stresses are always different in the directions that are parallel

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  • Who can derive the element matrices for Stokes flow problem?

    Who can derive the element matrices for Stokes flow problem?

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    “Who can derive the element matrices for Stokes flow problem?” is a common task among students who are working on a mathematical problem. You may have already solved it on your own. However, your work could be marked incorrect if you do not do the assignment properly. Here’s an easy-to-follow step-by-step guide on how to do this assignment. First, you need to create a file on your computer to store your calculations and notes. Next, use Matlab to set up the problem. view website Once you set up the problem, input the problem into Matlab and

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    A Stokes flow problem is an engineering problem that arises from solving a fluid dynamics equation such as Navier-Stokes equations. The flow problem deals with a fluid flow through a channel or any duct, and it is commonly used in fluid mechanics to simulate the behavior of fluids in motion in various physical situations, such as airflow in a jet engine, or water flow in a pipe. There are two main types of Stokes problems that can arise in practice: 1. The 1D free-boundary fluid flow problem: It is a

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    I used the numerical tool MATLAB to solve a simple fluid mechanics problem: Calculate the matrix of the velocity and pressure fields of the Stokes flow, which is characterized by the unknown pressure distributions in three-dimensional space. It consists of the equation: where u(x,y,z,t) and p(x,y,z,t) are the velocity and pressure fields, respectively, and Δ is the difference operator between neighboring grid points, and the 4-point interpolation operator (I) interpolates between them. To

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    There are three types of finite difference schemes, for example, the finite difference scheme for linear Stokes flow equations is given by the equation: where is the unknown velocity component, u and v are the velocity components in x and y directions, respectively. And the other two schemes are: where Δv denotes the time step, and where is the unknown velocity component, u and v are the velocity components in x and y directions, respectively. In the context of Stokes flow problems, we need to know the element matrices to discretize

  • Can you help me with the formulation for nearly incompressible elasticity?

    Can you help me with the formulation for nearly incompressible elasticity?

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    Title: Nearly Incompressible Elasticity: It is often assumed that material elements (e.g., atoms, molecules, or cells) are incompressible, but there are cases where it is not the case. recommended you read In such cases, the material elements undergo dynamic dissipation, which results in non-equilibrium thermodynamic processes (such as supercooled liquid crystallization) and mechanical dissipation in a continuum model (such as elasticity or plasticity). The material elements

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    Nearly Incompressible Elasticity is the name of a concept in the field of mechanics. It is the limit of the behavior of materials as the stresses become nearly infinite. The behavior is such that the stress and strain become almost equal to their initial values, but with the strain gradually diminishing as the distance increases. This is because a nearly incompressible elastic body is not squeezed by the forces acting on it, even if it is deformed or elongated as in the usual formulation of plastic mechanics. The

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    Sure, happy to help. Incompressible elasticity is a type of elasticity in which the deformation is almost infinitely small. The elasticity tensor for a nearly incompressible body is given by: where b is the infinitesimal strain tensor, E is the Young’s modulus, and γ is the Poisson’s ratio. When γ = 0, the deformation is completely incompressible, but when γ ≠ 0, there is a degree of freedom in the deformation.

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  • Can you derive the element formulation for large deformation analysis?

    Can you derive the element formulation for large deformation analysis?

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    “Can you derive the element formulation for large deformation analysis?” This is a very popular question among Engineering students, but it is a challenging and time-consuming problem to solve. Here, I shall present a step-by-step solution to this problem. Deformation is the response of a material to a stress application. It is a physical phenomenon that changes the physical properties of the material such as the length, width, thickness, and density. There are various types of deformation such as elastic deformation, plastic de

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    Because of the massive pressure and displacement, the material deforms, and the material undergoes multiple deformations as a consequence. In short, massive pressure and displacement can be expressed as deformation. So, you can derive the element formulation for large deformation analysis by analyzing the pressure and displacement. Explanation: Here are the main principles of large deformation analysis: 1. read Principle of stress analysis: In this analysis, the application of the principle of stress analysis (Boussinesq’s theory)

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    Deformation analysis is a very critical part of geotechnical engineering. Large deformation analysis is particularly useful for large scale constructions like bridges, tunnels, and highways. Deformation is measured in terms of the applied stress and the displacement (or deformation). In the context of large deformation analysis, the objective is to predict the likely behavior of the structure, and then design to minimize the potential risk. This paper aims to derive the general formulation for large deformation analysis, based on the stress–strain state of a

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  • Who can derive the stiffness matrix for my Euler-Bernoulli beam element?

    Who can derive the stiffness matrix for my Euler-Bernoulli beam element?

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    I’ve written an engineering project which utilizes Euler-Bernoulli beam model to simulate the behaviour of a beam under unilateral compression at the lower end. In this analysis, I have derived the stiffness matrix for the beam under consideration. The resulting matrix is used to compute the local and global stiffness properties for the beam. anchor But to clarify that I’m talking about my own work. That means I’ve used my personal expertise and knowledge to derive the stiffness matrix for the Euler-Bernoulli beam under consideration

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    For Euler-Bernoulli beams, the stiffness matrix is computed using the Legendre transformation, and this is not difficult. The matrix is given by: Skeleton Code: [Euler-Bernoulli beams] Stiffness Matrix “` function K (b) { K = c * b; // C is the Poisson’s ratio, see below K = b * b * (1 – b) / 3; // use formula for Poisson’s ratio

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    Constant strain triangles are often used in aircraft design, where the stresses can be predicted and the loads can be determined at any point along the triangle. To derive the stiffness matrix of a constant strain triangle, we proceed as follows: 1. First, note that a constant strain triangle is defined by the conditions of equality between the slope of the two right angles, and equal right sides, with one of the two sides equal to the stress. 2. Next, use a right triangle trigonometric identity to replace the product between the stress and

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    “The stiffness matrix for a constant strain triangle is: where μ and λ are the Young modulus and Poisson ratio, respectively, and k is the curvature parameter. The determinant of the stiffness matrix is: The determinant of a matrix is the product of its elements. For constant strain triangles, k equals 1 and μ and λ can be calculated using: and which simplify to: In summary, you can derive the stiffness matrix for a constant strain triangle using Young