WASHINGTON—The House is expected Wednesday to pass modified legislation banning abortions after 20 weeks of pregnancy, months after concerns from centrist and female Republicans derailed a vote on the bill.

The legislation stands little chance of becoming law, as President Barack Obama is unlikely to sign it even if it clears Congress. Still, the bill highlights the challenges Republicans face tackling an issue important to conservatives without alienating some women voters ahead of the 2016 elections.

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