Question 3.3.2: A Y-shaped frame is in the yz plane and is rigidly attached ...
A Y-shaped frame is in the yz plane and is rigidly attached to the central point of a thin square plate in the xy plane (Fig. 3.3-2(a)). The underside of the plate at A is hinged to three non-coplanar bars which are anchored to the foundation at points 1, 2, and 3. The plate is further attached to the foundation by bar B4 in the z-direction, bar B5 in the x-direction, and bar C6 in the z-direction. The frame is acted on by a twisting moment of 2 kN m at F and a horizontal force of 1 kN in the x-direction at H. Determine the forces in bars B4, B5, and C6.
Figure 3.3-2(b) shows that the 2 kN m twisting moment at F has a component m_{y}(F) about the y- axis and a component m_{z}(F) about the z-axis:
m_{y}(F)=(2 kN m) cos 45°=1.41 kN m
m_{z}(F)=(2 kN m) cos 45°=1.41 kN m
For the purpose of calculating support reactions, the 1 kN force at H is equivalent to the following force and moments acting at A :
a force P_{x} (H) of 1 kN in the x-direction, plus a moment m_{y} (H) of 1 kN × 5 m = 5 kN m, about the y-axis, plus
a moment m_{z}(H) of 1 kN ×1 m = 1 kN m, about the z-axis.
In other words, the square plate is acted on by the following force and moments at point A :
(a) A force P_{x} of 1 kN, which is entirely resisted by the three bars at A.
(b) A moment about the y-axis of magnitude
\sum{M_{y}}= m_{y}(F)+m_{y}(H)=1.41+5=6.41 kN m
By inspection, this moment is resisted entirely by a couple constituted by a compressive force in C6 and a tensile force in B4.
Compressive force in C6 = tensile force in B4
=\sum{M_{y}}/(length BC)
=6.41/2 = 3.2 kN
(c) A moment about the z-axis of magnitude
\sum{M_{z}}= m_{z}(F)+m_{z}(H)=1.41+1=2.41 kN m
By inspection, this moment produces a compressive force in bar B5, such that : compressive force in B5 × 1 m = 2.41 kN m
Hence compressive force in B5 = \underline{2.41 kN}
