COLUMNS AND STRUTS 



93 



If the elastic curve had been assumed to be a sine curve instead 

 )f a parabola, the result would have been the well-known equation 



7T*EI 9.87 El 



(118) 



J/o 



which is Euler's formula for long columns in its standard form. 



Under the load P given by this formula the column is in neutral 

 equilibrium ; that is to say, the load P is just sufficient to cause it 

 to retain any lateral deflection which may be given to it. For this 

 reason P is called the critical load. If the load is less than this 

 critical value, the column is in stable equilibrium, and any lateral 

 deflection will disappear when its cause is removed. If the load 

 exceeds this critical value, the column is in unstable 

 equilibrium, and the slightest lateral deflection will 

 rapidly increase until rupture occurs. 



53. Effect of end support. The above deduction of 

 Euler's formula is based on the assumption that the 

 ends of the column are free to turn, and therefore 

 formula (118) applies only to long columns with round 

 or pivoted ends. 



If the ends of a column are rigidly fixed against 

 turning, the elastic curve has two points of inflection, 

 say B and D (Fig. 81). From symmetry, the tangent 

 to the elastic curve at the center C must be parallel to 

 the original position of the axis of the column AE, and 

 therefore the portion AB of the elastic curve must be symmetrical 

 with BC, and CD with DE. Consequently, the points of inflection, 

 B and D, occur at one fourth the length of the column from either 

 end. The critical load for a column with fixed ends is therefore 

 the same as for a column with free ends of half the length ; whence, 

 for fixed ends, Euler's formula becomes 



E 



FIG. 81 



(119) 



r = 



Columns with flat ends, fixed against lateral movement, are 

 usually regarded as coming under formula (119), the terms fixed 

 ends and flat ends being used interchangeably. 



