The compatibility equation (1.88) for the nonsymmetric deformation of shallow caps were given in Chapter I. This equation reveals the connection between the membrane stress function F(r, ϕ) and the changes of the Gaussian curvature in the initial and deformed state Γ − Γ ¯ .

How do you find the airy stress function?

Obtain the displacement field for the plane strain solution of the axisymmetric stress distribution from that for the plane stress solution obtained in Section 5.28. Let the Airy stress function be φ = f(r) sin nθ; find the differential equation for f(r).

What is Prandtl’s stress function?

Prandtl stress function (Ludwig. Prandtl 1875-1953) • A trick to reduce three unknown stresses to a single. unknown stress function. • Solves exactly differential equations of equilibrium.

What is theory of elasticity?

The theory of elasticity treats the relationship between forces applied to an object and the resulting deformations. In practice, the analysis of the elastic behaviour of a material is reduced to the study of simple deformations and the determination of the corresponding elastic constants.

How do you do compatibility equations?

How many compatibility equations are there?

The compatibility conditions in linear elasticity are obtained by observing that there are six strain-displacement relations that are functions of only three unknown displacements. This suggests that the three displacements may be removed from the system of equations without loss of information.

Which of the following is Laplace equation?

Laplace’s equation, second-order partial differential equation widely useful in physics because its solutions R (known as harmonic functions) occur in problems of electrical, magnetic, and gravitational potentials, of steady-state temperatures, and of hydrodynamics.

What do you understand by an airy stress function in two dimension?

The Airy stress function ( φ {\displaystyle \varphi } ): Scalar potential function that can be used to find the stress. Satisfies equilibrium in the absence of body forces. Only for two-dimensional problems (plane stress/plane strain).

What is plane stress and strain?

In continuum mechanics, a material is said to be under plane stress if the stress vector is zero across a particular plane. … A related notion, plane strain, is often applicable to very thick members. Plane stress typically occurs in thin flat plates that are acted upon only by load forces that are parallel to them.

What is prandtl membrane analogy?

The elastic membrane analogy, also known as the soap-film analogy, was first published by pioneering aerodynamicist Ludwig Prandtl in 1903. It describes the stress distribution on a long bar in torsion. The cross section of the bar is constant along its length, and need not be circular.

What is torsion torsion concept?

Torsion refers to the twisting of a structural member that is loaded by couples (torque) that produce rotation about the member’s longitudinal axis. In other words, the member is loaded in such a way that the stress resultant is a couple about the longitudinal axis and the response is a twisting motion about that axis.

What is elastic theory in RCC?

Fundamental assumptions of elastic theory of bending The concrete and steel reinforcement are perfectly bonded.It means that the tensile strain in steel reinforcement is equal to the tensile strain in concrete surrounding the steel. All tensile stresses are taken up by steel and none by concrete.

Which of the following are known as governing equation of theory of elasticity Mcq?

Explanation: For plane elasticity problems, the equations of motion are one of the governing equations. The vector form of equations of motion is D*σ+f=ρü, where f denotes body force vector, σ is the stress vector, u is the displacement vector, D is a matrix of differential operator and ρ is the density.

What is the ratio of bulk modulus?

Bulk Modulus is a measure of the resistance of a fluid to compression. It is defined as the ratio of pressure stress to volumetric strain. The value of bulk modulus equals the pressure change x 100 required to cause a one percent change in volume.

Why do we need compatibility equation?

This compatibility condition is an equation which must be satisfied by the strains at all material particles. When all material particles in a component deform, translate and rotate, they need to meet up again very much like the pieces of a jigsaw puzzle must fit together.

What is being compatible?

adjective. capable of existing or living together in harmony: the most compatible married couple I know. able to exist together with something else: Prejudice is not compatible with true religion.

What is stress equilibrium equation?

The correct stress distribution for a problem in solid mechanics must satisfy these equations of equilibrium. … The axial stress is given by s x = P/A ; all other stresses are zero, and you see by substitution into the above equations that equilibrium is satisfied.

What is the compatibility equation for above mentioned condition?

Explanation: First b refers to the point where load is specified, second B refers to the point where load is applied. 6. What is the compatibility equation for the above mentioned condition? a) Δ/ bb – Δb = 0.

What is the other name of flexibility method?

In structural engineering, the flexibility method, also called the method of consistent deformations, is the traditional method for computing member forces and displacements in structural systems.

What is compatibility theory?

The idea that a solution’s character does not change for a particular perturbation. In linear programming the character could be an optimal basis, and the theory is concerned with whether a particular basis remains optimal when the data is changed in a prescribed direction.

Is Laplace equation linear?

Because Laplace’s equation is linear, the superposition of any two solutions is also a solution.

How do you do Laplace equations?

What is Laplace and Poisson equation?

Laplace’s equation follows from Poisson’s equation in the region where there is no charge density ρ = 0. The solutions of Laplace’s equation are called harmonic functions and have no local maxima or minima. … But Poisson’s equation ∇2V = −ρ/ǫ0 < 0 gives negative sign indicating maximum of V .

What is stress tensor in engineering?

The Stress Tensor Stress is defined as force per unit area. If we take a cube of material and subject it to an arbitrary load we can measure the stress on it in various directions (figure 4). These measurements will form a second rank tensor; the stress tensor.

What is the relationship between strain and stress?

Stress is the force applied to a material, divided by the material’s cross-sectional area. Strain is the deformation or displacement of material that results from an applied stress.

How do you calculate plane stress?

What is the plane strain?

Plane strain refers to the physical deformation of a body that is characterized by the displacement of material in a direction that is parallel to a given plane. The occurrence of plane strain acts as a source of stress corrosion in metals.

What is meant by plane stress analysis Mcq?

This set of Materials Science Multiple Choice Questions & Answers (MCQs) focuses on “Stress Transformation”. … Explanation: Plane stress is a two-dimensional stress in which stress components on any one direction is zero.