Vayigash elav Yehudah - Yehudah approached
The expression hagisha - "approach" is found as having one of three meanings. "Judgment, reconciliation, or war" (based on Bereishit Rabbah 93:4) "and they approach the court and they shall judge them." It occurs as meaning reconciliation in Yehoshua 14:6 "the members of the Tribe of Yehudah approached Yehoshua..." Finally, it occurs as a prelude to war in 2Shmuel 10:13 "Yo'av and the people with him marched into battle." In this instance, Yehudah combined all these three meanings of the word when he "approached" Yosef. When he said bi adoni, this meant that he gave Yosef the chance to choose which of the three kinds of hagisha he wanted to make the basis of this meeting. He spoke like someone who is fully armed, ready for a military confrontation. He meant: "you may find in me a partner for discussion on any one of these three levels. If you want to be conciliatory, fine and good. If you want to debate the rights and wrongs of what has happened also well and good. If, however, you want to use your military power to enforce your decree I am willing to confront you also on that basis.
Alternatively, the word bi here means, "[the guilt is all] mine." He said this seeing that superficially it appeared that the entire guilt was that of Binyamin in whose sack the goblet had been found. He pleaded with Yosef not to treat Binyamin as the sole guilty party but to consider him, Yehudah, the guilty party instead. He asked Yosef to keep him as a slave in lieu of Binyamin and to treat him as if he had been the guilty party. He wanted Yosef to set Binyamin free. Throughout the entire story Yehudah focused on subsitituting for Binyamin. When he said the words yedaber-na avdecha davar he meant that this "word" would be instead of any that Binyamin would have to say. Yehudah had no other purpose than to serve as the surrogate slave instead of his brother Binyamin.
He went on by explaining why it was seemly for Yosef to listen to his plea and to his reasons, seeing ki chamocha kePar'oh "you are of equal stature to Pharaoh," and as such it is incumbent upon you to at least listen to my arguments. Yehudah then proceeded to explain that seeing he had guaranteed the safety of Binyamin it was logical that he should take it upon himself to pay any penalty Binyamin had incurred.