Employee Matthew Dutton fell more than four metres from the unprotected edge of a flat roof at Coronation Youth Centre on Green Lane in Shirley.
Solihull Magistrates' Court heard how the 36-year-old and a colleague had been moving a piece of silver ducting up a ladder to the roof on 12 November 2009 when he fell.
He shattered his wrist, where two bones disintegrated, and he will never regain full use of his hand. He also suffered concussion, a black eye and other facial injuries and was off work for four months.
The HSE investigation found that the company had failed to plan a safe system of work, assess the risk, supervise Mr Dutton adequately or provide any safety barriers or other means to prevent a fall.
The company was aware that work would have to be carried out from the roof at height but had not considered the risk of an employee falling from height, or taken any steps to reduce the risk.
Dodd Group (Midlands) Ltd, of Stafford Park 13, Telford, pleaded guilty to breaching Regulation 4(1) of the Work at Height Regulations 2005 and was fined 拢20,000 and ordered to pay 拢3,511 in costs on Wednesday 23 March.
HSE Inspector Paul Thompson said: "This entirely preventable incident is a very sad consequence of Dodd Group Midlands's failure to plan and execute a safe system of work.
"If the company had taken proper precautions, both while the work was in the planning stage and when the work was being undertaken, Mr Dutton's fall and resulting injuries could have been completely avoided.
"Falling from height remains the biggest cause of deaths within the construction industry, and the company's failings could easily have led to this incident being fatal.
"It's vital that any company which is planning to carry out work at height assesses and mitigates the risk of any falls occurring, plans the work effectively and carries it out in a safe manner. HSE provides comprehensive guidance on all aspects of safe work at height, to prevent incidents of this kind from happening."
Got a story? Email news@theconstructionindex.co.uk
