6. Displacements in Elastic Structures

6.1 Assumptions for Analysis

  1. Internal and extrnal forces are in equilibrium.

  2. Material is linearly elastic.

  3. Internal and external deformations are compatible.

    • no gaps, overlaps, or discontinuities form in the structure as it deforms.
    • for every given set of external deformations, there is exactly one set of internal deformations (distortions, strains) that is compatible.

  4. Principle of super-position

Figure

An effect is:

  • a stress at a point, or
  • a strain at a point, or
  • a shear force, at a point, or
  • a bending moment, at a point, or
  • a reaction, or
  • a displacement, at a point.
  • etc.

When does super-position not hold?

  1. Non-linear stress strain.

    Figure

    Most engineering materials have non-linear stress-strian behaviour, where the slope of the stress-strain curve decreases as stress increases. For stresses in the significantly non-linear portion, the principle of super-position does not hold. (It should be noted that at normal working stresses for most materials, linearity is not a bad assumption).

  2. Proportional limit exceeded.

    Figure

    Similar to the above, when proportional limits are exceed, the stress-strain behaviour is no longer linear, and the principle of super-position is not valid. The above figure shows a hypothetical case for structural steel which has a well-defined yield point (except for the very high-strength steels).

  3. Geometric changes.

    Figure

    The above truss may have non-linear load-deformation behaviour, even if the material stays in the linear stress-strain region. As $P$ and $\Delta$ increase, if $\Delta$ becomes of significant size relative to $H$ and $W$, the tension in the two members will decrease due to their more favourable angle to provide vertical equilibrium of the joint.

    When this happens, they elongate less, proportionally, then they do under lower loads (and higher bar tensions).

    Figure

    This kind of structure “stiffens” as displacements increase, so that a load addition at high loads results in less displacement than the same addition at low loads. This is illustrated in the above load-displacement curve.

    Non-linear load-deformation behaviour can also result in “softening” of the structure. Imagine the above truss inverted so that the load, $P$, produces compression in the bars rather than tension. There increased load will “worsen” the geometry and lead to increased displacements.