Euler's buckling formula.

Parameters:

• $P$ = Force [N] = Allowable compression force on the column
• $E$ = Young’s Modulus [Pa] or [N/m²] = material properties
• $I$ = Moment of Inertia [m⁴] = shape properties
• $K$ = Constant, depending on boundary conditions
• $L$ = Length [m]

Data:

• Euler has presented around 1759 the basics of the buckling formula for columns.
• In many engineering fields this formula is used to calculate the safety on collapsing by buckling of column type girders. Because it concerns individual items of a structure and not the integrated structure, it is used to calculate each member individually. So it is probably not easy to design a Slide Chart to covers many girders.

Question:

1. Do you know Slide Charts for calculating Euler’s allowable force for specific columns of different shape and materials as steel, concrete, timber, aluminium, etc.
   This kind of formula would typically be implemented by William Cox. Fred Stahl
   This calls for “home made” slide rules using blanks like those from AWF (Seehase) or Dynaslide (Chicago). Karl Kleine
   Perrygraf produced Beam Computers. Eamonn Gormley
   Keuffel & Esser produced a Structural Engineer slide rule. Rod Lovett
   
   Animate this calculation
   Using a generic Rietz slide rule, with A = B CI C = D scales, the following steps are needed:
   • Move B index to $E$ on A.
   • Move cursor to $I$ on B. ($E \cdot I$) is under the cursor on A.
   • Move $K$ on C under the cursor. The C index is now opposite $\frac{E\cdot I}{K^2}$ on A.
   • Move the cursor to $L$ on CI. This places the cursor at $\frac{E\cdot I}{(K\cdot L)^2}$ on A.
   • This can be multiplied by $π^2 = 9.87 $ if desired, by moving the B index to the cursor, then moving the cursor over 9.87 on B and reading the result under the cursor on A. But since $π^2$ is so close to 10, this final step may not be necessary.
   Eamonn Gormley
   Auguste Esnouf has patented the Struttograph “for the computation of safe loads on, and of crippling stresses and factors of safety for, struts, columns, stanchions, and the like” (GB 20424/1913)
   Following Thomas Claxton Fidler, he combined $$ p = \frac{f+R-\sqrt{(f+R)^2-2.4fR}}{1.2} $$ $$ \rm{in \ which \ } \it{p} = \rm{ crippling \ stress} $$ with $$ R = E\pi^2\frac{r^2}{l^2} $$ to get $$ p = \frac{f+\frac{E\pi^2r^2}{L^2} - \sqrt{(f+\frac{E\pi^2r^2}{L^2})^2-2.4\frac{E\pi^2r^2}{L^2}}}{1.2} $$ Now that is a formula!
   See also: Auguste Esnouf, "Some interesting and useful curves", Engineering 19 (1915) pg. 113-115 and pg. 182-183 Andries de Man