Tropical Cyclone Sidr, packing sustained winds of up to 240 kilometers (150 miles) per hour, was expected to create storm surges as high as 6 meters (20 feet) that could inundate widespread low-lying areas in 15 coastal districts, Bangladesh's Meteorological Department said.
Coastal areas within a 250-kilometer (155-mile) radius of the storm's eye were buffeted by gale-force winds, driving rain and high waves, forest official Mozharul Islam said in Khulna, 135 kilometers (85 miles) southwest of the capital, Dhaka.
No casualties were immediately reported, but rescue teams were on standby, Islam said. Communications with remote forest areas and offshore islands were cut, he said.
In the coastal districts of Bagerhat, Barisal and Bhola, residents said the storm flattened thousands of flimsy straw and mud huts and uprooted trees and electric poles. Road, rail and river transport were also hit, they added.
"We are sitting out the storm by candlelight," said Bishnu Prashad, a resident of Bagerhat.
The coastal area borders eastern India and is famous for the mangrove forests of the Sundarbans, a world heritage site that is the home of rare Royal Bengal Tigers.
The storm was also likely to trigger flooding along coastal areas of West Bengal and Orissa states in eastern India, the Indian Meteorological Department said.
The cyclone, which was moving in a northeasterly direction, caused rainstorms and high winds across much of southern and central Bangladesh, the weather office said.
Thousands of villagers moved to cyclone shelters - concrete buildings on raised pilings, or sought refuge inside "mud forts" - mud walls built along the coast to resist tidal surges. Schools, mosques and other public buildings were also turned into makeshift shelters.
At least 650,000 people had so far moved into official shelters, where they were being given emergency rations, Ali Imam Majumder, a senior government official, told reporters in Dhaka. Some 3.2 million people were expected to be evacuated by late Thursday, he added.
Authorities dispatched dry foods, medicines, tents and blankets to the affected areas, he said.
"We have taken all precautions," Majumder said.
Authorities suspended operations at the country's two main seaports - Chittagong and Mongla - while ferry services and flights were halted across the coastal region, authorities said.
The sea resort of Cox's Bazar was deserted after warnings of the storm, but dozens of tourists were stranded in the offshore coral atoll of St. Martins by rough seas, an AP reporter in the area said.
Bangladesh, a low-lying delta nation, is prone to seasonal cyclones and floods that cause huge losses of life and property.
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