150 



STRENGTH OF MATERIALS 



The bending stress on any cross section can easily be found geo- 

 metrically from formula (98) by carrying out the following directions. 



1. Plot the given cross section and transform its boundary into a 

 new section, as explained at the beginning of this article. 



2. Determine the position of the center of gravity and calculate 

 the moment of inertia of this transformed section by the graphical 



^| method given in Article 47, Chapter 111. 



3. Substitute the values of </ ami /' 

 i 



found by this method, and also the 

 values of M, p, and // in formula (98). 



It is to be noted that formula (98) 

 gives the unit stress due to pure bending 

 strain only. In the case of a hook or 

 Y I / / W]os L link there is also a direct stress to be 



added to the above, of amount wlu-iv 



P is the total axial load and /' is 

 the area of the cross section (compare 

 Article 123). 



Problem 148. The wrought -iron crane hook, 

 shown in V\^. 1O4. is designed to support a load 

 of ten tons. Find the maximum stress in the 

 hook under this load, ami thence determine tin- 

 factor of safety. 



Solution. Let a cross section OCT of the ho,, k 



be taken at the position of maximum moment, as shown in the shaded pn.j. 

 in Fig. 104. 



In Fig. 105 let the curve numbered 1 represent this projection. The -ra\ity 

 axisDjPof this section, perpendicular to the ;ixi< of symmetry OCY. is IIIM deter- 

 mined, which may be done by the graphical method explained in Article 47, or other- 

 wise.* Curve 1 is then transformed into curve 2 by the method explained at the 

 beginning of Article 125, the light construction lines on the left of OY fho\\iiiu r 

 how this is accomplished. 



The moment of inertia I' of curve 2 is then found graphically by the method 

 explained in Article 47. This method consists in first transforming curve 2 into 

 curves 3 and 4, as there explained, then measuring the areas between <)Y and 

 curves 3 and 4 by means of a planimeter, and finally substituting the areas so 

 found in the formulas for the moment of inertia /' of curve 2 and the distance c 

 of its center of gravity from AB, given in Article 47. 



* A simple method of determining a gravity axis which is sufficiently accurate for all 

 ordinary purposes consists in cutting the section out of cardboard and balancing it on 



a knife edge. 



FIG. 104 



