We extend a method of Kharaghani and obtain some new constructions for weighing matrices and orthogonal designs. In particular we show that if there exists an OD(s1,...,sr), where w = ∑si, of order n, then there exists an OD(s1w,s2w,...,8rw) of order n(n+k) for k ≥ 0 an integer. If there is an OD(t,t,t,t) in order n, then there exists an OD(12t,12t,12t,12t) in order 12n. If there exists an OD(s,s,s,s) in order 4s and an OD(t,t,t,t) in order 4t there exists an OD(12s²t,12s²t,12s²t,12s²t) in order 48s²t and an OD(20s²t,20s²t,20s²t20s²) in order 80s²t.