## Potential Energy Derivation

47 From section ??, if U0 is a potential function, we take its differential

Q-"CTjj

Offij

48 However, from Eq. 13.4

thus,

ij deij

^ij deij

49 We now define the variation of the strain energy density at a point1

Applying the principle of virtual work, Eq. 13.37, it can be shown that

1 Note that the variation of strain energy density is, ¿Uo = &£ij, and the variation of the strain energy itself is &U = fn ¿U0dQ.

Figure 13.4: Single DOF Example for Potential Energy

sn = o |
(13.66) |

n d=f u - we |
(13.67) |

= f U0dQ - |
I u-bdQ+ I u-tdT + u-PJ (13.68) |

Jq |
Wo J r / |

51 We have thus derived the principle of stationary value of the potential energy:

Of all kinematically admissible deformations (displacements satisfying the essential boundary conditions), the actual deformations (those which correspond to stresses which satisfy equilibrium) are the ones for which the total potential energy assumes a stationary value.

52 For problems involving multiple degrees of freedom, it results from calculus that

53 It can be shown that the minimum potential energy yields a lower bound prediction of displacements.

54 As an illustrative example (adapted from Willam, 1987), let us consider the single dof system shown in Fig. 13.4. The strain energy U and potential of the external work W are given by

We = mgu = 100u Thus the total potential energy is given by and will be stationary for n = 250u2 — 100u

/ | ||||||

- Total Potent ---Strain Energ ----External Wo |
al Energy | |||||

y rk |
y y |
y y y | ||||

Displacement [in] Figure 13.5: Graphical Representation of the Potential Energy Displacement [in] Figure 13.5: Graphical Representation of the Potential Energy Substituting, this would yield Fig. 13.5 illustrates the two components of the potential energy. |

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