The study of BIBDs (balanced incomplete block designs) is one of the main topics in combinatorics. BIBDs also have the applications in experimental designs, coding and information theory,cryptography,and finite geometries. We assume that p is an odd prime number and q=pα. For any number k with gcd (k, p(q-1))=1 and 3 ≦ k s≦ q-4. we construct simple BIBDs with parameters (q, k,k(k-1)). By applying the first result, similar construction is obtained for any number k with gcd (k-1, p(q-l))=1, p t + k, and 4≦k≦q-3.