There is a succession of designs, with varying degrees of structure, linking BIBDs to pergolas (see here for definition of NBIBDs). The concurrence parameter λ will adopt different values according to the design under consideration. Suppose that there exists an OBIBD(v,v(v-1)/k,v-1,k,k-1;2) which is, moreover, a pergola. This can be regarded as a BIBD(v,v(v-1)/k,2(v-1),2k,2(2k-1)), which is the other end of the chain. Next up from this BIBD is an NBIBD(v,b1,b2,r,k1,k2,λ1,λ2) of Preece type, with outer blocks of size 2k and sub-blocks of size k, NBIBD(v,v(v-1)/k,2v(v-1)/k,2(v-1),2k,k,2(2k-1),2(k-1)). (This is twice the size usually studied.) Next is an NBIBD of the more general type just described, where there are two sub-BIBDs in blocks of size k: NBIBD(v,v(v-1)/k,v(v-1)/k,2(v-1),2k,k,2k-1,k-1). Next come the correlated block design in which the blocks of an OBIBD are matched, but not the treatments, and finally there comes the original OBIBD. Beyond that there exist Perfect Mendelsohn Designs: from a PMD(v,v(v-1)/k,v-1,k,1) can be constructed an OBIBD(v,v(v-1)/k,v-1,k,k-1;2), but not necessarily vice versa. Note that there are two OBIBD etc blocks for each PMD block, so results don't exactly correspond.
The same argument applies where there are more than 2 sets of treatments: these correspond to NBIBDs with more than 2 sub-blocks. The argument does not exactly apply to OBIBDs or even pergolas of more general form: for instance, not to pergolas with λ < (k-1).
There is also a more direct relationship between PMDs and NBIBDs: if there is a PMD with block size a compound number, then there will be a NBIBD with sub-block size equal to each of the factors, and of the same size as the original design. Thus, if k=m.n, then a (m,k)NBIBD(v) can be obtained from a PMD(v,k,1) by rearranging the blocks (a1,a2,a3,...,amn) as (a1 a(n+1)...a((m-1)n+1) | a2 a(n+2)...a((m-1)n+2) | ... | an a2n ... amn).
For cyclic designs, a visual representation of the difference patterns of these designs is given here.