MECH2040
2017 SPRING TERM FINAL EXAM
Part I: Basic Concepts (40 points)
1. For problems (a), (b) shown in Fig.1, write down the boundary conditions
at point O and A for problem (a) and (b), and the continuity conditions at
point B for problem (b). (10 points)
Fig.1
2. (1) For problems (a) and (b) in Fig.2, determine the minimum critical force
Pcr of in-plane buckling and draw a schematic buckling curve at the minimum
critical force Pcr. The in-plane bending stiffness is EI and the effective length
of column for various end conditions is shown in Fig.3. (10 points)
Fig.2
Fig.3 Effective length of column for various end conditions
(2) For the three kinds of possible buckling modes under critical compressive
forces (PA, PB, PC) shown in Fig.4, which of the following is correct? ①PA> PB>
PC, ② PB> PC> PA, ③ PC> PB> PA, ④ PA> PC> PB, (Tip: The Buckling force for the
buckling curve y(x) = A sin
𝑛𝜋𝑥
𝐿
is P𝑐𝑟 =
𝑛2 𝜋2 𝐸𝐼
𝐿2
)
(5 points)
Fig.4
3. (1) A cubic element is in plane stress state and the stress components on
surface A is marked on its Mohr’s Circle as shown in Fig.5, ① Please draw the
stress state of this element; ② The maximum tensile stress of this element
=
and is oriented at
degree with the positive x axis. (5 points)
Fig.5
(2) A cubic element is in plane stress state and the stress components on
surface A is marked on the Mohr’s Circle as shown in Fig.6, ① Please draw
the stress state of this element; ② The maximum shear stress of this element
=
and is oriented at
degree with the positive x axis. (5 points)
Fig.6
(3) A cubic element is in plane stress state and the Mohr’s Circle of this
element shrinks to a point (σ = −2𝜎0 ) as shown in Fig.7, ① Please draw the
stress state of this element; ② The maximum shear stress =
. (5 points)
Fig.7
Part II: Calculation (60 points)
1. For the beam and loading shown below,
(a) find the state of stress at point A in the Cartesian coordinate
system indicated in the figure.
(b) use Mohr’s circle to determine (i) the principal stress and principal
plane; (ii) the normal and shearing stresses acting on a plane
oriented at 450 indicated in the figure.
The parameters are: 𝐿 = 1𝑚, ℎ = 20𝑚𝑚, 𝑏 = 10𝑚𝑚, 𝑀 = 1𝑘𝑁 ∙
𝑚, 𝑃 = 150𝑘𝑁.
Figure 1
2. A mass 𝑚 at ℎ = 1𝑚 falls onto a beam at point A. Given the length
of the beam 𝐿 = 1𝑚, 𝑚 = 2𝑘𝑔, 𝐸𝐼 = 100𝑀𝑁 ∙ 𝑚2 , 𝑥 = 0.3𝑚, and
neglect the gravity, calculate
(a) the maximum deflection of point A;
(b) the maximum deflection of the entire beam.
Figure 2
3. The overhanging beam ABCD supports a concentrated force P and a
uniform load with intensity q as shown below. For what ratio P/(qa)
will the deflection at point D be zero? Use the method of
superposition to solve this problem and you are allowed to use any
formula given.
Figure 3