6-36 DESIGN OF LAMINATES 



Y or the slenderness ratio is associated with the ability of a section to resist buckling. 



Since the ^ ratio is based on the geometry and length of the element it can be readily 



determined by the cross-section of the member. Also by substituting various values of stress 



with the appropriate modulus of elasticity, E, in equation 6. 18, corresponding values for 



— can be obtained, 

 r 



When the member is in the second group or does not follow Hooke's Law, the critical 

 buckling load can be determined if the value of E used is the value corresponding to the 

 particular stress. The buckling formula then becomes: 



_ K2 E a I (6.20) 



cr Ll2 



where the value of E a is the tangent modulus and is obtained from a stress -strain curve. 

 Consequently for every P cr there exists a value of E a . 



When the stress exceeds the proportional limit, the buckled member no longer obeys 

 Hooke's Law. The critical stresses are now dependent not only upon the critical load P cr 

 but also upon the bending of the member. The compressive stresses will increase on the 

 concave side and decrease on the convex side. 



Under this condition two values of the modulus of elasticity are involved. A new tangent 

 modulus E a for an increased load on the concave side will be necessary while on the convex 

 side of the material the usual modulus E for the decreased load will be applicable (13). 



The critical load, therefore, will now be a function of a new reduced bending rigidity 

 E r I. It can be shown that the value of E r can be determined by the following expression: 



E, - . UE X Ea (6.21) 



j r 



(y/r* 7v 



where E r = reduced modulus of elasticity 



E = as previously defined 



E a = modulus of elasticity above the proportional 

 limit, or tangent modulus of elasticity 



The critical buckling load is then: 



TC 2-p, T 



P = r (6.22) 



cr o 



P cr , I and L' as previously defined. 



Figs. 6-25, 6-26 and 6-27 indicate ultimate compressive strength curves for mat, woven 

 roving and 10 ounce cloth laminates for various slenderness ratios; — . The curves are for 

 laminates in the wet condition and tested parallel to the warp. Tables 6-1, 6-2 and 6-3 give 

 the slenderness ratios at various compressive stresses and for the appropriate moduli of 

 elasticity obtained from the stress-strain diagrams. Further testing should be conducted to 

 verify these curves or similar curves for other laminates. 



