I work on some approximate method for analysis of elastic Euler-Bernoulli cantilever beam. There is a concentrated mass on beam tip. Method works exelent for m/M5. I assume that the higher values of concentrated mass violates the assumption of small deformations of beams which I introduced.
First of all I seek a clarification. The approximate solution is said to become erroneous when the concentrated mass increases. In that case the condition should be M/m > 5 and not m/M >5.
According to Aleksandar his approximate dynamic analysis of an Euler-Bernoulli beam becomes erroneous when M/m > 5 (if my correction is accepted) where m = beam mass and M = tip mass. The exact solution to this problem is known and it is based on a linear elastic beam response. In that case the error in his approximate method is not due to violation of the linear elastic assumption when M/m > 5. All the same it will be interesting to answer Aleksandar's question on the limiting deflection of a cantilever beam beyond which the beam responds inelastically (violates small deflection assumption).
The tip deflection of a cantilever beam of self weight w loaded by a tip weight W is given by Δ = (W/3+w/8)*L^3/EI--- (1) The slope at the tip is given by . θ = (W/2+w/6)*L^2/EI ----- (2) The maximum bending moment occurs at the base of the cantilever and is given by M_B = L* (W+.5w) ----(3) It is at the base that a uniform beam will begin to behave inelastically and subsequently plastically. Let the limiting linear elastic stress that can be sustained by the beam be σ_y and the section modulus of the beam be Z, then at the limit of linear elasticity L* (W+.5w) = Z*σ_y ----(4) Then the ratio of the tip mass to beam mass at the limit of small deformation is given by M/m = W/w = Z*σ_y/(wL) - 0.5. By substituting the expression for W into Eqs (1) and (2) the limiting deflection and slope can be computed. A hypothetical example shall now be considered. What is the M/m ratio for a 2 metre long 100 by 100 mm concrete beam to violate small deformation if σ_y = (2/3)*20 N/mm^2?. Solution: Assuming unit weight of concrete to be 24 KN/m^3 then w = 24*.1*.1*2 = 0.48KN, Z = bh^2/6 = 10^6/6 mm^3. Using consistent units M/m can be computed to be M/m = 1.81. As M/m increases beyond 1.81 the beam response deviates more and more from the linear elastic beam theory. The calculated limiting M/m ratio would have been much lower for a longer beam. However, I am suggesting that the discrepancy observed by Aleksandar is not due to nonlinear effects because the exact solution for the problem is based on linear elasticity. I do not know of an exact closed-form formula for beam vibration based on nonlinearity. In conclusion, the violation of small deformation restriction is not likely to be the problem. Aleksandar should please check carefully the theory behind his approximate method to discover some hidden errors. In addition, possible numerical instability in the solution scheme should be investigated.
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