Department of Aerospace Engineering, IIT Kharagpur
AEROSPACE STRUCTURAL ANALYSIS (AE31009)
Tutorial Sheet No.2
BENDING
Instructor: Prof. B.N.Singh
Email ID: bnsingh@aero.iitkgp.ernet.in
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1. It is desired to make a beam of Z-section, as shown in figure 1, of such proportions that the Zand Y-axes will be principal axes for the section. If all the elements of the section have same
thickness t=1 cm. and other dimensions are as shown in figure, what is the proper dimension a
to realize the required condition?
2.
3.
4.
5.
6.
Figure 1
Figure 2
Figure 3
Figure 4
A prismatic rod of hexagonal section is made up of six equilateral triangular segments as
shown in figure 2. The segments are alternately Redwood (E1=84x103 kg/cm2) and soft White
Pine (E2=70x103 kg/cm2) glued together to form a solid section. The dimension a=2.5cm. if
the bar is subjected to pure bending in its longitudinal plane of symmetry, calculate maximum
bending stress, assuming that M=1000 kg-cm.
Referring to the circular cross-section of two materials as shown in figure 3 and denoting the
ratio of moduli of elasticity E1/E2 by α, calculate the distance yc from the horizontal diameter
to the centroid of the transformed section.
The cross-section of a small beam cut from a sheet of 7-ply ply-wood has the dimensions
shown in figure 4. Alternate layers of the plywood have grain parallel to the length of the
beam. The beam is 1.2m long, simply supported, and loaded at the middle. The modulus of
elasticity parallel to the grain is E1=105 kg/cm2, that perpendicular to the grain is
E2=2.5x104kg/cm2. The corresponding working stresses are σ1=84 kg/cm2 and σ2=21 kg/cm2.
Calculate the safe value of the load P.
A uniform cantilever of arbitrary cross-section and length l has section properties, Ixx, Iyy and
Ixy with respect to the centroidal axes as shown in figure 5. It is loaded in the vertical (yz)
plane with a uniformly distributed load of intensity w/unit length. The tip of the beam is
hinged to a horizontal link which constrains it to move in the vertical direction only (provided
that the actual deflections are small). Assuming that the link is rigid, and that there are no
twisting effects, calculate: (a) the force in the link; (b) the deflection of the tip of the beam.
A uniform thin walled beam has the open cross-section shown in figure 6. The wall thickness t
is constant. Calculate the position of the neutral axis and the maximum direct stress for a
bending moment Mx=3.5Nm applied about the horizontal axis Cx. Take r=5mm, t=0.64mm.
Figure 5
Figure 6
Figure 7
7. Part of the wing section is in the form of the two-cell box shown in the figure 7in which the
vertical spars are connected to the wing skin through the angle sections all having a crosssectional area of 300mm2. Idealize the section into an arrangement of direct stress carrying
booms and shear stress carrying panels suitable for resisting bending moments in a vertical
plane. Position the booms at the spar/skin junctions. If the bending moment of 100kNm is
applied in the vertical plane of symmetry, what will be the magnitude and position of the
maximum direct stress in the section?
8. A thin walled cantilever with walls of constant thickness t has the cross-section as shown in
figure 8. It is loaded by a vertical force W at the tip and a horizontal force 2W at the midsection, both forces acting through the shear centre. Determine and sketch the distribution of
direct stress according to the basic theory of bending, along the length of the beam for the
points 1 and 2 of the cross-section. The wall thickness t can be taken as very small in
comparison with d in calculating the sectional properties Ixx, Ixy etc.
Figure 8
Figure 9
9. Construct a lumped sheet-stringer model for the thin-walled cross-section shown in figure 9
Assume that the stringers in the straight sections are spaced at 50 mm and at 450 in the curved
portions. Assume the only loading is a moment, M2. All dimensions are in millimeters. What
will be the maximum bending stress?
10. A uniform beam of the cross-section shown in figure 10.a is supported and loaded as shown in
figure 10.b. if the direct and shear stresses are given by the basic theory of the bending, the
direct stresses being carried by the booms and the shear stresses by the walls, calculate the
vertical deflection at the ends of the beam when the loads act through the shear centre of the
end cross-sections.
11. Taking  = ax+by as the general equation of bending stress due to Unsymmetrical bending.
Deduce the value of constants ‘a’ and ‘b’.
12. Due to load misalignment, the bending moment acting on the channel sections is inclined at
an angle of 3◦ with respect to the y axis. If the allowable flexural stress for this beam is
σal=16ksi what is the maximum moment, Mmax, that may be applied (fig.12).
13. Find the maximum compressive and tensile flexural stresses for the cantilever beam shown in
fig.13.
Figure 10.a
Figure 12
Figure 13
Figure 10.b