Why do I get "Truncated incorrect date value" when using str_to_date()?
you have to pass your datetime column into date() function like this :
create table test (
  id int AUTO_INCREMENT,
  datetime_string datetime,
  date_date date,
  PRIMARY KEY (id)
);

insert into test(datetime_string) values('2014-12-14 22:49:13');
update test set date_date = date(datetime_string);

http://sqlfiddle.com/#!9/cc200b/1