How does the bank robbery furthur the plot as well as the story of "Hard Times"?
The robbery of Brounderby bank occurs in Chapter 7 "Gunpowder," in Book II "The Reaping."
The immediate suspect is Stephen Blackroot, who had been observed hanging around the bank. But Stephen has been set up. In earlier chapters, we learn that Stephen has refused to be strong-armed into joining the Union. Blackroot explains:
"Sir, I canna, wi' my little learning an my common way, thell the genelmen what will better aw this -- though some working men o' this town could, avove my powers -- but I can tell him what I know will never do't. Agreeing fur to mak one side unta'rally awlus and for ever right, and toother side unnat'rally awlus and for ever wrong, will never, never do do't."
The bank robbery gives Slackbridge, and others, a way to not only ostracize Blackroot, but to punish him.
The robbery itself furthers the corruption of Bounderby, Harthouse and all like them: those who are ready to accuse without proof, to exploit the common man, and to line their own pockets. Though it seems counter-intuitive, Dickens himself distrusted Unions and orators intensely. He had an unwavering belief in the power of the individual; corporations like banks, and unions, as in the one depicted here in Coketown, were, for Dickens, pure evil.