# Finite Element Method Questions and Answers – The Euler-Bernoulli Beam Element

This set of Finite Element Method Multiple Choice Questions & Answers (MCQs) focuses on “The Euler-Bernoulli Beam Element”.

1. Normal stress parallel to the cross-sectional plane is equal to zero. Keeping Euler-Bernoulli Beam Theory in mind, state whether the statement is true or false.
a) False
b) True

Explanation: The Euler-Bernoulli Beam Theory has a total of 3 assumptions. This happens to be the 2nd kinematic assumption from the above mentioned theory. The other important assumption is – plane sections remain plane and perpendicular to the reference line of the deformed beam.

2. What is the main difference between the Timoshenko Beam Theory and Euler-Bernoulli Beam Theory?
a) Both are the same
b) Shear deformations in Timoshenko Beam Theory are zero values
c) Shear deformations in Euler-Bernoulli Beam Theory are nonzero values
d) Shear deformations in Timoshenko Beam Theory are nonzero values

Explanation: The two beam theories under consideration are given utmost importance while solving any beam related problems. In Euler-Bernoulli Beam Theory, the shear deformations are zero values; whereas in the Timoshenko Beam Theory, shear deformations are nonzero values.

3. Which of the following defines slender beams?
a) Cross sectional depth is lesser than the length
b) Cross sectional depth is more than the length
c) Cross sectional depth is negligible
d) Length is negligible

Explanation: In the case of slender beams, the cross sectional depth is much smaller than the length of the beam. It is considered to be 3 to 5 times smaller when compared. Euler Bernoulli beam theory is applicable only in the case of slender beams.

4. Which necessary condition is required for the existence of non zero shear stress values corresponding to zero strain values?
a) Shear modulus of the material tends to zero
b) Young’s modulus of the material tends to infinity
c) Shear modulus of the material tends to infinity
d) Young’s modulus of the material tends to zero

Explanation: It is assumed that the shear stress values are zero corresponding to zero strain values. This is not the case in practice; non zero shear stress values do exist in practical situations. In order to enable this, the material’s shear modulus is assumed to tend to infinity.

5. The Euler Bernoulli theory states that the deformed beam angles are small.
a) True
b) False

Explanation: The given statement is true. Deformed beam angles correspond to the slope of the deformed beams. For x location on the beam, if Δ corresponds to the displacement; the angle θ = dΔ/dx. As the slope is minimal θ2 = (dΔ/dx)2 corresponds to zero value.
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6. Which of the following is the mathematical expression for Euler Bernoulli beam equation?
a) d2/dx2(EP(d2w/dx2)) = I
b) d2/dx2(EI(d2w/dx2)) = p
c) d2/dx2(d2w/dx2) = p
d) d2/dx2(E(d2w/dx2)) = p

Explanation: The mathematical expression for Euler Bernoulli beam equation is given by –
d2/dx2(EI(d2w/dx2)) = p
where,
E = Elastic Modulus of the material
I = Mass moment of Inertia
w = beam deflection
p = load acting on the beam

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