For cross sections consisting of stringers and 'plates, we make the following assump- 

 tions in order to calculate the tension stresses: 



1. All of the area has been concentrated into points which shall be called nodes. This 

 is done by assigning the areas of the plates and stringers to the nearby nodes. By this 

 means, the integrals on page 46 can be replaced by sums. 



2. For effective members, the strain is a linear function of position K^ + K2y + KgZ. 

 Some members may end near the cross section to be analyzed and, hence, their stress would 

 be less than a completely effective member. For the nodes, an "effectiveness" is assigned 

 which is 1.0 for effective members* and less for others. Thus the assumed form for stresses 

 is (see page 156 of Reference 8 or page 209 of Reference 9): 



'^xx = Ki(kE) + K2(kEy) + K3(kEz) 



where K,, K,, K, are unknown constants to be determined as shown above, 



k is effectiveness, 



E is Young's modulus, and 



y, z are coordinates. 



Except for the addition of effectiveness and the possibility of having different moduli at 

 the nodes, this is exactly the same as ordinary beam theory and would give the usual equa- 

 tions (this means that "ordinary beam theory" is based on a form of distribution of the 



E 

 tensile stress such as ct = K, + K^y + K,z). Another factor k'= k (E = reference 



xxl2''3' P° 



value of modulus) is defined so that the tension at node i is given by: 

 exx)i = E, [Ki(k:) + K2(k:y.) + K3(k'.z^)] 



(This expression for a^^ is the same as that previously used where F, = 1, Fj = y, F, = z 

 except that a term k, providing for the effectiveness of the section, is included). The 

 values of y and z are given by (A. is the node area and y., z. are the node coordinates): 



S k 'y .A. 2 k ' z.A. 

 I'' 1 1 .111 

 1 1 



y = ; z = 



Sk'A. 2k:A: 



*Since the effect of cutouts (such as doors and hatches) and regions near the ends of members is to reduce 

 the stress in that region, we introduce a tension effectiveness factor k, — k — 1. k = 1 if there are no 

 cutouts or ends nearby in the axial direction. References 10 and 11 give rules for determining effectiveri^ess. 



52 



