Mid-term 2.pdf - UBC Mechanical Engineering

MECH 260 Mechanics of Materials
Mid-term Examination 2
Duration: 50 minutes
Important Note: Closed-book/notes; One 8 1 2′′
× 11
′′
(2-sided) student-prepared fact
sheet is allowed; calculator (Only the arithmetic mode, if it is a programmable calculator)
is allowed.
Question 1
A light rod with rectangular X-section (a × b) is vertically supported at its head on a
rigid and fixed platform. The rectangular head has dimensions b × h where b = width and
h = thickness, so that it is flush with the stem of the rod on one side. The stem of the rod
passes through a hole in the platform without touching it, as shown.
Figure: (a) Perspective view (b) Front view.
The rod supports a vertical load P as shown.
The yield strength of the rod in normal stress is σ Y = 400 MPa.
The yield strength of the rod in shear stress is τ Y = 160 MPa.
Determine the maximum value of the dimension ratio a/h so that, when sufficiently
loaded, the rod will fail first in tension in its stem rather than in shear in its head.
Use the same safety factor for the two types of failure.
(55 points)
Hint: First obtain expressions for normal stress (average) in the stem of the rod and the
shear stress (average) on a vertical section of the head, in terms of P, a, b and h. Equate
the normal stress of the stem to the allowable normal stress (i.e., normal yield strength
divided by the safety factor) and determine the corresponding P. For this value of P,
make the shear stress in the head less than the allowable shear stress (i.e., shear yield
strength divided by the safety factor).

Question 2
A schematic diagram of the anchoring cable (steel) of a rigid/fixed sail post at Canada
Place in Vancouver is shown. The cable is firmly attached to the top of the sail post at
one end. The other end of the cable has a rigid end cap, the position of which may be
adjusted by means of a tensioning device. The tensioning device has an identical pair of
steel bolts with nuts, and is rigidly attached to a fixed support base. The bolts pass
through the two holes in the end cap of the cable. The tensioning is done by tightening
the two bolts by equal amounts.
Under unstrained conditions:
L = length of the cable
L/10 = length of each bolt (from the nut position to the head).
Also,
δ = screw pitch of a bolt/nut
E = Young’s modulus of cable = Young’s modulus of bolt.
Assume that the area of X-section of the cable = area of X-section of the bolt.
If both nuts are tightened through n turns each, in the same direction, determine an
expression for the:
(a) Tensile stress in the cable
(30 points)
(b) Tensile stress in each bolt
(15 points)
in terms of E, L, n and δ.
Figure: Sail-post anchoring cable system.
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