# Finite Element Method Mcqs

Q:

In the Shear Deformation plate theory, what characteristic contributes to shear locking?

 A) Transverse shear strains in thick plates present computational difficulties B) Transverse shear strains in thin plates present computational efficiency C) For thick plates, the element stiffness matrix yields erroneous results for the generalized displacements D) For thin plates, the element stiffness matrix becomes stiff and yields erroneous results

Answer & Explanation Answer: D) For thin plates, the element stiffness matrix becomes stiff and yields erroneous results

Explanation: The transverse shear strains in the element equations of Shear Deformation Theory cause computational difficulties when the side-to-thickness ratio of the plate is large. Shear locking is observed when the transverse shear strains in thin plates are negligible, and consequently, the element stiffness matrix becomes stiff and yields erroneous results for the generalized displacements.

60
Q:

In the Shear Deformation plate theory, when does the transverse shear strains in the element equations present computational difficulties?

 A) If the plate is thick B) If the side to thickness ratio of the plate is large C) If the side to thickness ratio of the plate is small D) If higher-order finite elements are used

Answer & Explanation Answer: B) If the side to thickness ratio of the plate is large

Explanation: The transverse shear strains in the element equations of Shear Deformation Theory present computational difficulties when the side-to-thickness ratio of the plateis large (say 50, i.e., when the plate becomes thin). For thin plates, the transverse shear strains are negligible, and consequently, the element stiffness matrix becomes stiff and yields erroneous results for the generalized displacements. This phenomenon is known as shear locking.

67
Q:

In FEM, which option is correct for a linear plate theory based on infinitesimal strains and orthotropic material properties?

 A) The plane elasticity equations govern the transverse deflections B) The transverse deflections are coupled with in-plane displacements C) The in-plane displacements are zero in the absence of in-plane forces D) The transverse deflections are zero in the absence of in-plane forces

Answer & Explanation Answer: C) The in-plane displacements are zero in the absence of in-plane forces

Explanation: For a linear theory based on infinitesimal strains and orthotropic material properties, the in-plane displacements (ux, uy) are uncoupled from the transverse deflection uz=w. The plane elasticity equations govern the in-plane displacements (ux, uy). The in-plane displacements are zero if there are no in-plane forces and hence, we discuss only the equations governing the bending deformation and the associated finite element models.

87
Q:

In displacement-based plate theories, which assumption of Classical Plate Theory is relaxed in Shear Deformation Theory?

 A) A straight-line perpendicular to the plane of the plate is inextensible B) A straight line perpendicular to the plane of the plate remains straight C) A straight line perpendicular to the plane of the plate rotates such that it remains perpendicular to the tangent to the deformed surface D) A straight line perpendicular to the plane of the plate rotates

Answer & Explanation Answer: C) A straight line perpendicular to the plane of the plate rotates such that it remains perpendicular to the tangent to the deformed surface

Explanation: In the SDT, we relax the normality assumption of CPT, i.e., transverse normal may rotate without remaining perpendicular to the mid-plane. The Classical Plate Theory is based on the assumption that a straight line perpendicular to the plane of the plate is (1) inextensible, (2) remains straight, and (3) rotates such that it remains perpendicular to the tangent to the deformed surface.

80
Q:

In displacement-based plate theories, which option is correct about Shear Deformation Theory (SDT)?

 A) It is also called Kirchhoff plate theory B) It is an extension of Euler-Bernoulli beam theory from one dimension to two dimensions C) It does not involve Timoshenko beam theory D) It is often known as Hencky-Mindlin plate theory

Answer & Explanation Answer: D) It is often known as Hencky-Mindlin plate theory

Explanation: The two most commonly used displacement-based plate theories are the Classical Plate Theory (CPT) and first-order Shear Deformation Theory (SDT). CPT is an extension of the Euler-Bernoulli beam theory from one dimension to two dimensions and is also known as the Kirchhoff plate theory. Shear Deformation Theory is an extension of the Timoshenko beam theory and it is often called the Hencky-Mindlin plate theory.

83
Q:

In FEM, what are the primary variables in the Shear Deformation Theory of plate deformation (w)?

 A) The transverse deflection w only B) The transverse deflection w and the normal derivative of w C) The transverse deflection w and the angles of rotation of the transverse normal about in-plane axes D) The angles of rotation of the transverse normal about in-plane axes only

Answer & Explanation Answer: C) The transverse deflection w and the angles of rotation of the transverse normal about in-plane axes

Explanation: An examination of the boundary terms in the weak form of Shear Deformation Theory suggests that the essential boundary conditions involve specifying the transverse deflection w and the angles of rotation of the transverse normal about in-plane axes (φx, φy), which constitute the primary variables of the problem (like in the Timoshenko beam model). Hence, the finite element interpolation of w must be such that w, (φx and φy are continuous across the inter-element boundaries in SDT elements.

65
Q:

In displacement-based plate theories, if a linear theory based on infinitesimal strains and orthotropic material properties is used, then the in-plane displacements are coupled with the transverse deflection.

 A) True B) False

Explanation: For a linear theory based on infinitesimal strains and orthotropic material properties, the in-plane displacements(ux, uy) are uncoupled from the transverse deflection uz=w. The plane elasticity equations govern the in-plane displacements (ux, uy). The in-plane displacements are zeroin the absence of in-plane forces, andhence, we discuss only the equations governing the bending deformation.

70
Q:

In FEM, which theory is an extension of the Timoshenko beam theory?

 A) Classical Plate Theory B) Hencky-Mindlin plate theory C) Kirchhoff plate theory D) Shell theory

Explanation: The two most commonly used displacement-based plate theories are the Classical Plate Theory (CPT) and first-order Shear Deformation Theory (SDT). CPT is an extension of the Euler-Bernoulli beam theory from one dimension to two dimensions and is also known as the Kirchhoff plate theory. Shear Deformation Theory is an extension of the Timoshenko beam theory and it is often called the Hencky-Mindlin plate theory.