Q1
This question covers structural elements
Consider the Euler{Bernoulli beam AB supported by an elastic rod BC at point B as
shown in Fig. 1. The beam is subjected to a uniform load p(x). The Young’s modulus
for the entire structure (i.e. the beam and the rod) is given by E. The cross-sectional
area of the rod is A and moment of inertia of the beam is I. Model the beam with
a single nite element and the rod with a single nite element. The element sti ness
matrix for the beam is given by

Figure 1: The beam-rod structure.
(a) State the order of the di erential equations governing the response of the beam
and the rod respectively. [2]

(b) With the aid of a sketch de ne C1-continuity.
Why is a C1-continuous approximation of the de ection w required in the nite
element implementation of Euler{Bernoulli beam theory? [5]

(c) The element sti ness matrix in Eq. 1 for Euler{Bernoulli beam theory contains
entries that scale di erently with the length of the element `e
. With reference to
the vector Be
and the de nition of Ke
explain this di erence. [6]

(d) State the essential boundary conditions for the structure shown in Fig. 1. [3]

(e) Solve for the de ection at point B. [10]

(f) The shape functions for Euler{Bernoulli beam theory are given by
1 `e

Sketch the de ection w(x) over the beam.
The sketch should capture all the key features of the de ection. In addition,
indicate the value and the gradient of the de ection at the points A and B. [6]
(g) Use Gauss quadrature to compute the following integral over the length of the
beam:

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