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Chapter 14

Q. 14.10.2

Using Eqn 14.10-26, obtain an upper bound on the shakedown load factor for the uniform two-span beam in Fig. 14.10-2, if the loads P and Q may each vary at random within the range 0 to \lambda M_{P}L.

\qquad \lambda\left[\sum\mathscr{M}^{max}_{i}\theta^{+}_{i} – \sum{\mathscr{M}^{min}_{i}\theta^{-}_{i}} \right] =\sum{(M_{P})_i\left|\theta_{i} \right|}                        (14.10-26)

fig14.10.2

Step-by-Step

Verified Solution

Consider the mechanism in Fig. 14.10-5. The elastic moments in Table 14.10-1 are
relevant. Substituting appropriate values into Eqns 14.10-26,

\begin{matrix} \lambda \left[(0.188 M_{p})(\theta )- (-0.203 M_{P})(2\theta )\right] &=M_{P}(\theta +2\theta )\\ whence \hspace{17em}  \lambda &= 5.05\end{matrix}

Ans. λ = 5.05 is an upper bound on the shakedown load factor.

Section Due to P Due to Q Combined Loading
\mathscr{M}^{max}_{i} \mathscr{M}^{min}_{i} \mathscr{M}^{max}_{i} \mathscr{M}^{min}_{i} \mathscr{M}^{max}_{i} \mathscr{M}^{min}_{i}
B 0.094 M_P 0 0.094 M_P 0 0.188 M_P 0
D 0 -0.203 M_P 0.047 M_P 0 0.047 M_P -0.203 M_P
E 0.047 M_P 0 0 -0.203 M_P 0.047 M_P -0.203 M_P
Table. 14-10-1

COMMENT For any arbitrary mechanism, Eqn 14.10-26 only gives an upper bound on the shakedown load factor. However, in this particular case it is seen that the mechanism in Fig. 14.10-5 is the correct incremental collapse mechanism; hence λ =5.05 is the correct shakedown load factor.

fig14.10-5