Orthogonal Frequency Division Multiple Access (OFDMA) as well as other orthogonal multiple access techniques fail to achieve the system capacity limit in the uplink due to the exclusivity in resource allocation. This issue is more prominent when fairness among the users is considered in the system. Current Non-Orthogonal Multiple Access (NOMA) techniques introduce redundancy by coding/spreading to facilitate the users' signals separation at the receiver, which degrade the system spectral efficiency. Hence, in order to achieve higher capacity, more efficient NOMA schemes need to be developed. In this paper, we propose a NOMA scheme for uplink that removes the resource allocation exclusivity and allows more than one user to share the same subcarrier without any coding/spreading redundancy. Joint processing is implemented at the receiver to detect the users' signals. However, to control the receiver complexity, an upper limit on the number of users per subcarrier needs to be imposed. In addition, a novel subcarrier and power allocation algorithm is proposed for the new NOMA scheme that maximizes the users' sum-rate. The link-level performance evaluation has shown that the proposed scheme achieves bit error rate close to the single-user case. Numerical results show that the proposed NOMA scheme can significantly improve the system performance in terms of spectral efficiency and fairness comparing to OFDMA.