Advanced Structural Analysis/Part I - Theory/General Properties of Materials/Measures/Elasticity/Broad Implications

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Broad Implications[edit]

As is the case with many elementary physical laws, Hooke's law, stating the linear correlation between load and deformation, spans the small and simple, and the big and complex. For instance, the theory is a good approximation of simple test specimens subjected to axial loading. Naturally, it is also an accurate physical model for linear springs. And it is even valid for very complex structures, so long as they respond linearly. These are all trivial and perhaps self-evident statements, yet they are powerful and important.

Load and Stress[edit]

To expand on this, let's consider an initially stress free quasi-static linear-elastic mechanical system that is subjected to one load F_0 which has some fixed direction and point of application. The global equilibrium of the system is sustained by a pattern of internal stresses in the structure. The specific shape of the stress pattern is required to achieve equilibrium at each point in the structure. That is, if the stress magnitude \sigma_i at some point P differed from the equilibrium state \sigma_0 by a factor k, the system would automatically strive to reach the unique pattern of equilibrium. One could however obtain a new equilibrium state that abides the new condition at P by altering the stress levels in the material surrounding P by the same factor k. A state of equilibrium is thus reached if

\begin{equation} \sigma_i = k \sigma_{i0} \end{equation}

If the refactoring of the stress magnitudes is propagated throughout the system, including the point at which the load is applied, global equilibrium will be achieved. Consequently, also the magnitude of the load itself would have to be altered by the factor k in this scenario. Hence we can write

 F = k F_0 (2)

Equation () and () yields

 \sigma_i = \frac{\sigma_{i0} F}{F_0} = k_i F (3)

Where k_i is a constant.

Equation () is only true if the normalized stress pattern is unaffected by the magnitude of k_i in the considered interval. If instead, the system is nonlinear, i.e. responds differently depending on the magnitude of k_i, equation () is not be true/reliable.

Load and Strain[edit]

Moreover, Hookes law and equation () gives us

\epsilon_{xi} = \frac{\sigma_{xi} - \nu (\sigma_{yi} + \sigma_{zi})}{E}
= F \frac{k_{xi} - \nu (k_{yi} + k_{zi})}{E} = F k'_{xi}

\epsilon_{yi} = \frac{\sigma_{yi} - \nu (\sigma_{xi} + \sigma_{zi})}{E}
= F \frac{k_{yi} - \nu (k_{xi} + k_{zi})}{E} = F k'_{yi}

\epsilon_{zi} = \frac{\sigma_{zi} - \nu (\sigma_{xi} + \sigma_{yi})}{E}
= F \frac{k_{yi} - \nu (k_{xi} + k_{yi})}{E} = F k'_{zi}

Load and Displacement[edit]

Let's consider a path \Delta S_{12} that runs through a continuous linear elastic body. \Delta S_{12} has two endpoints P_1 and P_2.

The relative change \Delta S_{12} in distance in some direction between the two points can be formulated

\Delta S_{12} = F \int_{P_1}^{P_2} \! k(s) \, ds = F k_{12}

Displacement and Strain[edit]

From equation () and () we get

\Delta S_{12} = F k_{12} = \frac{k_{12}}{k'_i} \epsilon_i = k'_{i12} \epsilon_i

Displacement and Stress[edit]

From equation () and () we get

\Delta S_{12} = F k_{12} = \frac{k_{12}}{k_i} \sigma_i = k_{i12} \sigma_i

Generalization[edit]

Moving on to a more general point of view, we state the following definition

\psi = C \psi' (3)

Where C is a constant and \psi and \psi' are some linearly dependent structural measures at P and P', respectively.

From equation (3) and the superposition principle we conclude that

\psi = c_0 + c_1 \psi'_1 + c_2 \psi'_2 +...+ c_{n-1} \psi'_{n-1} + c_n \psi'_n     (4) 

The independent parameters of (4) can be obtained by performing >=n tests/calculations. It follows that it is easy to, for instance, scale linear test results according to different safety factors.

It is important to know whether or not the structure is linear before drawing any critical conclusions from the theory of elasticity. Common sources of non-linear behavior in a structure are material, geometric, contact and dynamic effects.