Class Exercise #25 1.050 Solid Mechanics Fall 2004 Here are 6 problems of the same type, drawn from past years’ final exams. i) What do they have in common? What’s different? ii) What must you know in order to do the common part of the problem? iii) Which is the most difficult problem? Why? Which is the easiest? iv) Formualte a new problem - a variation on the common theme - that I might assign on this year’s final. 98 - Problem 2. Truss structure carries a load P at node #1. P Find the force in member 5-9. Which member is liable to buckle if the load P is increased without limit. 1 2 3 4 5 6 7 8 9 L LL all member lengths = L 99 - Problem 1 Truss structure carries a load P at node a. Find the force in members fh, fi, and gi in terms of P. Find the reactions at nodes h and i. Which member is liable to buckle if the load P is increased without limit. Note: all members have the same length. P a b d e f g h i lengths all membe c00 - Problem 1 For each of the three problems 1a, 1b, and 1c, state whether the problem posed is statically determinate or statically indeterminate. In this, assume all information regarding the geometry of the structure is given as well as values for the applied loads. P2 a d b e c P11a) Determine the force in member ab. 1b) Determine the force in member bd 1c) Determine the reactions at the wall. 01- Problem 3 A rigid ring of Radius Ro is fixed to ground as indicated by the shading. An inner, rigid ring of radius Ri is free to rotate clockwise or counterclockwise about an axis through its center and per-pendicular to the plane of the paper when subject to a torque Mt. The inner ring is connected to the outer ring by means of “n” steel spokes. (Only four of the n spokes are shown but the n are symmetrically disposed about the axis of rotation. Think of a bicycle wheel). The inner ring, when subject to a torque Mt experiences a small rotation, φ, which is much less than 1.0 (radian). The spoke is made of steel with Elastic modulus θL φMt, Ri Ro E = 200 E09 Pascals and shows a yield stress σy = 400 E06 Pascals. i) Express the extension of a spoke, call it δ, in terms of Ri, φ, and θ where θ is the angle between the extension of the radius Ri and the spoke. ii) Derive an expression for the rotational stiffness of the system, i.e., KT in the equa-tion MT=KT φ, in terms of n, the number of spokes, k = AE/L, the uni-axial stiffness of the spoke, and the geometric parameters Ri and θ.iii) Express the ratio of the torque required for buckling of the spoke to the torque required for yielding of the spoke in terms of the ratio (r/L) where r is the radius of the spoke and L its length. iv) If L = 0.5m, will the spoke buckle or yield? (Again, think bicycle wheel). 02 - Problem 2. A planar truss carries a concentrated load aat one end as shown. All members have length a, as indicated. A B aa a Which member experiences the greatest C tensile force and what is its magnitude in W terms of W? Which member is likely to buckle first? 03-Problem 2. A W 45o45o a a a a a a/2 C DB A planar truss carries a concentrated load at one end as shown. All horizontal members have length a and the diagonal members are all inclined at 45 degrees, as indicated. Find the force in the cable, BD, as a function of W. Find the forces carried by the two members joined at node A, again in terms of
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