# 7.2: Pole-vaulting

- Page ID
- 7822

Pole vaulting as an athletic activity dates back to the ancient Greeks. Modern competition started around the turn of the 20 th century, when the Olympic Games were restarted. A sharp increase in the achievable height coincided with the advent of composite (fibreglass) poles, about 50 years ago. These are sufficiently strong and flexible to allow substantial amounts of energy (kinetic energy of the athlete) to be transformed into elastic strain energy stored in the deformed pole, and subsequently transformed again into potential energy (height of the athlete) as the pole recovers elastically. The mechanics of beam bending is clearly integral to this operation.

DoITPoMS standard terms of use

Visual inspection of a bent pole (see photo) is all that's needed to estimate the distribution of axial strains (and hence stresses) within its cross-section. The pole has a diameter of about 50 mm and it can be seen in the photo that it is being bent to a (uniform) radius of curvature, *R *, of the order of 1 m (~ length of the athlete's legs!). Considering a section of unit length (unstrained) in the diagram below, the angle *θ* (~tan *θ* ) ≈ *1/R * after bending (where *R * is the radius of curvature). From the two similar triangles in the diagram, *θ* is also given by the surface strain *ε * divided by *r *, the radius of the pole . The surface strain, *ε*, is thus given by the ratio *r */ *R *, which has a value here of about 2.5 %. This strain is compressive on the "inside" surface of the pole (coloured blue) and tensile on the "outside" surface (coloured red).

The stresses induced by such bending can be high. The axial stress is given by the product of Young's modulus, E, and strain,* ε*.

\[\sigma=E \varepsilon=E r / R\]

For example, assuming the composite to have an axial stiffness of ~ 40 GPa, the axial stresses at the inside and outside surfaces of the pole must be about 2.5 % of this, ie ~±1 GPa. Composites are able to sustain such high stresses, although it's not unknown for vaulting poles to fracture.