# Finite Element Method Questions and Answers – Three Dimensional Problems – Finite Element Formulation

This set of Finite Element Method Multiple Choice Questions & Answers (MCQs) focuses on “Three Dimensional Problems – Finite Element Formulation”.

1. Which are the forms of three dimensional elasticity?
a) Weak & Strong form
b) Long & short form
c) Thick & thin form
d) Axial and flexural form

Explanation: There are two forms of three dimensional elasticity. They are Strong form of three dimensional elasticity and Weak form of three dimensional elasticity. The weak form is not required to satisfy all conditions; hence is applied over integrals. Whereas, the strong form is required to satisfy all the pre requisite conditions.

2. Unlike two dimensional elasticity, it is preferred to separate the equivalent nodal force element of each element into 3 parts.
a) True
b) False

Explanation: The given statement is false. For both two dimensional, as well as three dimensional elasticity, separation of the equivalent nodal force element takes place. It is preferred to separate the element into 2 parts for simplification of calculations.

3. When different shape functions are used to specify relation between global and local co ordinate systems, such an element is referred to as iso-parametric element.
a) False
b) True

Explanation: The given statement is false. When same shape functions can be used to specify the relation between global and local co ordinate systems, such an element is referred to as iso-parametric element. An eight node hexahedral element is one such element.

4. Which of the following is not a step involved in the formulation of a finite element model?
a) Boundary conditions definition
b) Construction of geometric model
c) Conversion to finite element model
d) User data input

Explanation: User data input is not a necessary step for the formulation of any finite element model. Boundary conditions, geometric model, finite element model are important steps required to formulate a finite element model. Some other steps required are result interpretation, load application etc.

5. What is the generalized form of a strain vector for a three dimensional element?
a) {E} = {ex ey ez vxy vyz vzx}
b) {E} = {ex ey vxy vyz}
c) {E} = {ex ey ez vxy vyz}
d) {E} = {ex ey vzx}

Explanation: The generalized expression for a strain vector of a three dimensional element is given by – {E} = {ex ey ez vxy vyz vzx}; where all expressions refer to the strain tensors of the 3D element under discussion.
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6. Which of the following is one of the three invariants of a (3*3) stress tensor?
a) I = ux + uy + uz
b) I = ux – uy + uz
c) I = ux + uy – uz
d) I = ux * uy + uz

Explanation: I = ux + uy + uz is the first invariant of a (3*3) stress tensor. This is used to calculate the principal stresses of the matrix. The problem usually has three unknown variables that can be obtained by solving the three invariant equations.

7. What is the generalized expression for the displacement of a three dimensional element?
a) U = [u, v]T
b) U = [u]T
c) U = [u, v, w]T
d) U = [u, v, w]

Explanation: The generalized expression for the displacement of a three dimensional element is given by U = [u, v, w]T; where U = displacement, u, v, and w are the displacement components along the x, y and z axes respectively.

8. What is the relation between the stress and strain components of a three dimensional element?
a) Strain = Integration of displacement component
b) Strain = Multiplication of displacement component
c) Strain = Division of displacement component
d) Strain = Differentiation of displacement component

Explanation: The strain component is equal to the differentiation of that particular displacement component along that axis. So, if u is the displacement along the x direction, then the strain component = du/dx.

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