Refinable　function-based　affine　frames　and　affine　bi-frames　have　been　extensively　studied　in　the　literature.　All　these　works　are　based　on　some　restrictions　on　refinable　functions.　This　paper　addresses　what　are　expected　from　two　general　refinable　functions.　We　introduce　the　notion　of　weak　(quasi-)　affine　bi-frame;　present　a　refinable　function-based　construction　of　weak　(quasi-)　affine　bi-frames;　and　obtain　a　fast　algorithm　associated　with　weak　affine　bi-frames.　An　example　is　also　given　to　show　that　our　construction　is　optimal　in　some　sense.