I'll be honest, I find this thread "inconsistent"..
You titled the thread "DH shouted at my daughters".
Then changed it to "raised his voice" because you say he speaks so quietly as a default.
Now it's back to a "shouting episode".
You said your daughter claimed he pushed her forcefully by the shoulder but that you didn't see it.
Then changed it to he definitely didn't push her forcefully, because the other daughter said he didn't.
Isn't that something you'd establish before posting on a forum with potentially 100s of replies? Wouldn't it be a rather significant thing to establish - given it takes this scenario into violence against a female child - to call a spade a spade.
That's presuming you established that after you started the thread.
If you established it before you started the thread why would you write that your DD claimed he pushed her forcefully but that you didn't see it ...and not mention anything about her sister saying he didn't (thus she must be lying/it didn't happen).
It's very hard to give advice when the main facts are being flip flopped on.
All I know is it sounds like an adult man shouted at your daughters and pushed one by the shoulder because he was stressed out by his child's tantrum and because she'd scraped him.
They were laughing at the situation...well, kids will do things like that. Maybe they should have been talked to about it before. Even if they had been talked to before now, and didn't stop doing it ..... An adult man shouting at them and pushing one is hardly a constructive, healthy way of dealing with it. Kids giggling and being silly doesn't mean they deserve abuse.
Let's face it, he did it cause he lost his temper. They shouldn't be being treated by their step Dad like that in their home. Not exactly a good male role model for their lives.