Introductory Solid Mechanics

Axial Loading

Saint-Venant's Principle: Slender beam case

Stress analysis very near to the point of application of load $P$. Saint-Venant's principle: "the stress and strain produced at points in a body sufficiently removed* from the region of external load application will be the same as the stress and strain produced by any other applied external loading that has the same statically equivalent resultant and is applied to the body within the same region".

*farther than the widest dimension of the cross section

Force-Deformation Relation

\[\delta = \frac{PL}{EA}\]

Derivation: $P = \sigma A$; $\sigma = E\varepsilon$; $\varepsilon = \delta / L => P = E\frac{\delta}{L}A$

Axial Flexibility: $\delta = fP => f = \frac{L}{EA}$

Axial Stiffness: $P = k\delta => k = \frac{EA}{L}$

Axially Varying Properties

For non-uniform load, material property and cross-section area:

\[\delta = \int_0^L\frac{F(x)}{E(x)A(x)}dx\]

Assume variations with $x$ are "mild" (on length scale longer than cross-sectional length scales)

Principle of Superposition

Nomenclature convention: internal forces and resulting displacements are positive when they cause/represent elongation, negative for compression. Displacement of a point (e.g. $Z$) with respect to a fixed point: $\delta_z$. Relative displacement of one point (e.g. $A$) with respect to another (e.g. $D$).

\[\delta_{AD} \equiv \delta_{DA} \equiv \delta_{A/D} \equiv \delta_{D/A}\]

Superposition: If the displacements are (1) small and (2) linearly related to the force components acting, the displacements caused by the components can be added up:

\[\delta = \sum_i \delta_i = \sum_i \frac{F_i L_i}{E_i A_i}\]

General Procedure:

  1. Draw a FBD
  2. Equilibrium equations: force balance and moment balance
  3. Constitutive equations: stress-strain or force-displacement relations
  4. Compatibility equations: geometric constraints

Statically Indeterminate Problems

Equilibrium does not determine all internal forces.

Thermal Effects: Temperature changes

Rod rests freely on a smooth horizontal surface. Temperature of the rod is raised by $\Delta T$. Rod elongates by an amount

\[\delta_{T} = \alpha \Delta T L\]

Linear coefficient of thermal expansion $\alpha$, $[\alpha] = \frac{1}{K},\frac{1}{°C},...$. This deformation is associated with an average thermal strain:

\[\epsilon_{T} = \frac{\delta_T}{L} = \alpha T\]

$\delta_T$, $\varepsilon_T$ present in addition to elastic $\delta_E$, $\varepsilon_E$ (from internal forces). Superposition (small strains):

\[\varepsilon_{tot} = \varepsilon_{E} + \varepsilon_{T}\] \[\delta_{tot} = \delta_{E} + \delta_{T}\]

Initially, rod of length $L$ is placed between two supports at a distance $L$ from each other. With no interal forces, there is no stress or strain.

Equilibrium:

\[R_{A} = R_{B} = 0\] \[R_{A} = F\]

After raising the temperature, the total elongation of the rod is still zero. The total elongation is given by:

\[\delta = \frac{FL}{EA} + \alpha L \Delta T = 0\] \[F = -\alpha E \Delta T\] rod is under compression, if \[\Delta T = 0\]

The stress in the rod due to change in temperature is given by:

\[\sigma = -\alpha E \Delta T\]

Misfit Problems

A misfit problem is one in which there is difference between a design distance and the manufactured length of a material. Some misfits are created intentionally to pre-stain a member. (e.g. spokes in a bicycle wheel or strings in a tennis racket). This type of problem neither modifies the equilibrium equations (1) nor the force-extension relations, (2) but the compatibility equations, (3) need to be modified.

Stress Concentration

Highest at lowest cross-sectional area.

Stress concentration factor:

\[K = \frac{\sigma_{max}}{\sigma_{avg}}\]

  • Found experimentally
  • Solely based on geometry