Question

A 2024-T3 aluminum bar has rectangular cross section dimensions of 2" x 0.5". The length of the column is 9.0 inches.

A) Calculate the transition slenderness ratio (Sr)_D for this column. [Answer should be 69.9]

B) Calculate the slenderness ratio for your assigned column length. [Answer should be 62]

C) Calculate the maximum unit load (P_cr/A) for the bar. Indicate whether the column is a Johnson or Euler column. [Numerical answer should be 25296 psi but I don't know Johnson or Euler]

Please show steps for all parts.

Answer 1

Here I get an answer are different because you not specified column condition for different condition K will change so all answer is changed. You can modified calculation easily with a different value of K and get your result easily. I attached all the information I used, maybe its help you lots.

A.)

The transition slenderness ratio can be calculated for a column which indicates the cutoff between long and intermediate columns. The transition slenderness ratio is the value of the slenderness ratio at which the Euler critical stress, σcrσcr, is equal to half the material yield strength (Sy).

( Note. The relevant yield strength (Sy) for column buckling problems is the compressive yield strength. For ductile materials, compressive strength is approximately equal to tensile strength. For brittle materials, compressive strength is higher than tensile strength. If compressive strength is not known, it can be conservatively assumed that compressive strength is equal to tensile strength.)

The Euler formula is valid for predicting buckling failures for long columns under a centrally applied load. However, for shorter ("intermediate") columns the Euler formula will predict very high values of critical force that do not reflect the failure load seen in practice. To account for this, a correction curve is used for intermediate columns. The Johnson formula (or "Johnson parabola") has been shown to correlate well with actual column buckling failures and is given by the equation below which already used in a calculation.

σcr=Pcr/A=Sy−((Sy*KL)/(2pir))² X (1/E)

The plot below shows the Euler curve for a pinned-pinned column with a 1 inch diameter circular cross section and a material of 6061-T6 Aluminum:-

Note the Euler curve shown in red in the plot above. As column length approaches zero, the Euler critical stress approaches infinity. The Johnson formula is shown as the blue curve, and it corrects for the unrealistically high critical stresses predicted by the Euler curve for shorter columns.