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MECH3300 Energy methods of hand calculation

MECH3300 Energy methods of hand calculation. Statically-indeterminate examples. Expansion loop problem. A pipe contains a loop to allow for thermal expansion as shown below. The material has a linear coefficient of thermal expansion a (strain/degree C).

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MECH3300 Energy methods of hand calculation

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  1. MECH3300 Energy methods of hand calculation Statically-indeterminate examples

  2. Expansion loop problem • A pipe contains a loop to allow for thermal expansion as shown below. The material has a linear coefficient of thermal expansion a (strain/degree C). • Find the deflection D that would occur if expansion were allowed, and the reaction force F that would occur if expansion were prevented. R f D R q A C D B F R F L L Approach: symmetrical, so look at left half. Treat AB, CD as rigid axially (except for thermal expansion). Pinned at ends so assume AB and CD have zero Mb . Find Mb in loop at arbitrary angles q, f to estimate the energy.

  3. F Expansion loop problem - 2 f E R(1+sinf) D R q F F B R(1 - cosq) L L For B to E, Mb = FR(1 - cosq) For E to F, Mb = FR(1 + sinf) Hence D = 3pFR3/(EI) = thermal expansion horizontally = a DT(2L + 4 R) F to prevent expansion can be found for some temperature change DT. Note, in finding the trig. Integrals, I used

  4. 2 beam problem from R.D. Cook A cantilever beam of length 2L rests on one with the same cross-section of length L as shown, and is loaded at the tip. Find the largest bending moment present. P C B A L L The upper beam is curved at B due to the moment PL there. The lower beam however can only be loaded from B to C, and will not curve as much. Hence we expect the two beams to separate, contacting only at B. To model this, place an internal force H, acting at B on both beams. To find H, we argue it does no resultant work, so U/H = 0

  5. P 2 beam problem continued x1 C B A The bending moment only depends on H from B to C. Hence we only need to integrate from B to C. H x2 In top beam, from B to C, Mb1 = Px1 - H(x1 - L) In the lower beam, Mb2 = Hx2 Max. Mb is HL=1.25PL in the lower beam.

  6. A doubly-indeterminate problem • A beam is simply supported, but its motion is also restrained by torsional springs of rotational stiffness kT at each end and a vertical spring of stiffness k in the middle. Find the reaction force or moment in these springs. • This is symmetrical, so the torsional springs transmit equal moments, but there remain 2 unknown reactions to find, force R in the centre spring, and moment MReach end. • Need to write the equations U/R = 0 and U/MR = 0. P kT kT k L L

  7. A doubly-indeterminate problem - 2 MR MR x P kT • Bending moment at x from the left end is Mb = -(P-R)x/2 - MR • The spring stores energy R2/(2k). A torsional spring stores MR2/(2kT). kT k R P/2-R/2 L L Evaluating the integrals gives two equations, which in matrix form are

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