page_title: Johnson's parabolic formula
section_title: Application of the Johnson Parabola
hierarchical_section_title: Johnson's parabolic formula / Application of the Johnson Parabola
caption_attribution_description: English: Uses Johnson and Euler formulas to illustrate transition between higher and lower slenderness ratios
context_page_description: The Johnson formula is an empirically based formula relating the slenderness ratio to the stress illustrating the critical load required to buckle a column. The formula is based on empirical results by J. B. Johnson from around 1900 as an alternative to Euler's critical load formula under low slenderness ratio conditions.
  Buckling refers to a mode of failure in which the structure loses stability. It is caused by a lack of structural stiffness. Placing a load on a long slender bar will cause a buckling failure before the specimen can fail by compression.
  One way to calculate buckling is to utilize Euler's formula, which produces a critical stress vs. slenderness curve such as the one illustrated to the right.
  However, depending on the geometry of the structure under stress, this equation is not always applicable, and the Johnson parabola should be used.
context_section_description: One common material in aerospace applications is Al 2024. Certain material properties of Al 2024 have been determined experimentally, such as the tensile yield strength (324 MPa) and the modulus of elasticity (73.1 GPa).  The Euler formula could be used to plot a failure curve, but it would not be accurate below a certain  value, the critical slenderness ratio.
  Therefore, the Euler equation is applicable for values of   greater than 66.7.
  Euler:  for 
  (units in Pascals)
  Johnson's parabola takes care of the smaller  values.
  Johnson:  for 
  (units in Pascals)
caption: Plot of both Euler and Johnson formulas for Al 2024. The transition point occurs at the critical slenderness ratio.
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