Both companies face charges under the Health and Safety at Work Act 1974.
Yarm Road, formerly known as Kvaerner Cleveland Bridge, was charged with breaching section 3(1) of the act. The firm pleaded guilty to failing to ensure the safety of its employees and persons not in their employment.
Costain was charged with failing to ensure the health and safety of its employees.
The firms admitted breaching the law when the case was heard in July.
The workmen, who died after falling 80 ft, were Paul Stewart, 23, Ronnie Hill, 38, Jeff Williams, 42, and Andy Rogers, 40.
n Contractor Atlas Building and Civil Engineering will be sentenced next year after admitting liability for the death of a nine-year-old boy who died on one of its sites.
Derby-based Atlas pleaded guilty at Wakefield magistrates' court last week to a charge that people other than its employees were exposed to risks to their health and safety.
The firm is due to be sentenced at Leeds Crown Court on 4 January next year.
Simon Golding died when a sewer pipe at a building site near his home in Normanton, Derby, shattered on 8 April this year.
The inquest into his death heard that the impact of the concrete was so great that it split Golding's heart into two halves.