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# Finite Element Method MCQ – Shear Deformable Plate Model

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

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Answer: b

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.

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

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Answer: b

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.

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

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Answer: c

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.

Which equation correctly describes Hamilton’s principle used in FEM?

a) 0=∫t2t1[δK-(δU+δV)]dt

b) 0=∫t2t1t[δK-(δU+δV)]dt

c) 0=∫t2t1[δK+(δU+δV)]dt

d) 0=∫t2t1t[-δK+(δU+δV)]dt

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Answer: a

Explanation: Governing equations of displacement-based plate theories are derived using the principle of virtual displacements. The principle of virtual displacements or Hamilton’s principle requires that 0=∫t2t1[δK-(δU+δV)]dt where δU, δV and δK denote the virtual strain energy, virtual work done by externally applied forces, and virtual kinetic energy, respectively. These quantities are expressed in terms of actual stresses and virtual strains, which depend on the assumed displacement functions and their variations.

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

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Answer: d

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.

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

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Answer: c

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.

Which option specifies an assumption made in Shear Deformation Theory for a plate lying in the plane XY?

a) εzz≠0

b) εxz=0

c) εyz≠0

d) εxy=0

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Answer: c

Explanation: 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, but In the SDT, we relax the normality assumption of CPT, i.e., transverse normal may rotate without remaining normal to the mid-plane. Thus, for a plate in the XY plane, the assumptions are equivalent to specifying εzz=0 only whereas εyz and εxz are non-zero.

In SDT, what are the boundary conditions for a plate that is clamped if ф represents the rotation of the transverse normal about an in-plane axis and w is the transverse deflection?

a) w=0,∂w∂n=0

b) w=0,ф=0

c) w=0,∂w∂n≠0

d) w=0, Mnn=0

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Answer: b

Explanation: Geometrically, plate problems are similar to the plane stress problems except that plates are also subjected to transverse loads that cause bending about axes in the plane of the plate. In SDT, the boundary condition for a clamped plate is the absence of deflection and rotation of the transverse normal about any in-plane axis, i.e., w=ф=0. Because a simply supported end does not restrict rotation, the reactive moment is zero, i.e., w=Mnn=0. For a free end, both, the reactive moment and the shear force are absent, i.e., Mnn=Qn=0.

In SDT, what are the boundary conditions for a plate that is simply supported if ф represents the rotation of the transverse normal about an in-plane axis and w is the transverse deflection?

a) w=0,∂w∂n=0

b) w=0,ф=0

c) w=0,∂w∂n≠0

d) w=0, Mnn=0

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Answer: d

Explanation: Plate problems are geometrically similar to the plane stress problems except that plates are also subjected to transverse loads. In SDT, a clamped plate has no deflection and rotation of the transverse normal about any in-plane axis, i.e., w=ф=0. In a simply supported end, rotation is not restricted; thusthe reactive moment is zero, i.e., w= Mnn=0. A free end does not have reactive moment and the shear force, i.e., Mnn=Qn=0.

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

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Answer: c

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.

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

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Answer: b

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.

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

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Answer: d

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.