Hw2.pdf

ME 202 – Solid Mechanics – Winter 2019 st Homework -2 Due Date: 31 January 2019 Q1. A cantilever beam of rectangular cross section is loaded by a force 𝐹 directed along the diagonal AC of the section as shown in Figure 1. Show that the neutral axis in this case coincides with the other diagonal BD for all values of the dimensions a, b of the rectangle. Q2. The Z-section of Figure 2 has second moments of area 𝐼!! = 560,000 𝑚𝑚 ! , 𝐼!! = 290,000 𝑚𝑚 ! , 𝐼!" = 300,000 𝑚𝑚 ! . It is used for a beam of length 2m, which is simply supported at its ends and loaded by a uniformly distributed vertical load of 1000 𝑁/𝑚. Find the location and magnitude of the maximum tensile stress. Determine the magnitudes of the principal second moments and the inclination of the principal axes to the x and y axes. Identify the stiff and flexible axes as ‘1’, ‘2’, respectively. Q3. Find the location of the centroid and the second moments of area 𝐼! , 𝐼! , 𝐼!" for the thinwalled section of Figure 3. Also compute the principle axes and the orientation of the neutral axis with respect to the stiff axis. Assume some moment M is applied about x and y axes. Q4. Figure 4 shows the cross section of an unequal angle beam for which the principal second moments of area are 𝐼! = 23. 95×10! 𝑚𝑚 ! , 𝐼! = 2. 5×10! 𝑚𝑚 ! . The location of the centroid and the inclination of the principal axes are as shown. The beam is subjected to a bending moment of 10 𝑘𝑁 − 𝑚 about the x -axis. Find the location and magnitude of the maximum tensile and compressive stress. Q5. The beam section shown in Figure 5 has second moments of area 𝐼!! = 317, 600 𝑚𝑚 ! , 𝐼!! = 791, 700 𝑚𝑚 ! , 𝐼!" = − 298, 500 𝑚𝑚 ! . The beam is subjected to a bending moment 𝑀! = 1000 𝑁𝑚 about the x -axis as shown. Find the equation defining the neutral axis and hence the location and magnitude of the maximum tensile stress. Q6. The I-beam shown in Figure 6 is subjected to a bending moment, which is intended to be aligned with the axis 𝑂! , but which is actually inclined at 1! towards 𝑂! because of a manufacturing error. Find 𝐼!! , 𝐼!! , and hence the direction of the neutral axis, (which is also the axis about which bending occurs).

Figure 1





Figure 5

Figure 2





Figure 3



Figure 6

Figure 4