Is there any danger of a whole sentence? :-))

Julie4, why don't you show us the sentence into which you would like to add the word, and I am highly confident that we can offer a direct answer or a way around the problem then.

Please also specify BE or AE. "Glad of" sounds strange to me; I'd use "glad for" or "glad about" instead.

Sorry, the sentence is:

"What these two people have in common is their gladness (about?) the successful surgery of their friend."

It does not matter if it is BE or AE, just has to be correct ;-)

What these two people have in common is that they are both glad their friend's surgery was successful.

For example.
I think using "gladness" is rather unusual and would recommend a work-around.

If forced to pick a preposition for your sentence, I would tend towards "over" or "about".

We'll see what the others say :o)

#5: I think using "gladness" is rather unusual and would recommend a work-around.

Agree.

#3: "Glad of" sounds strange to me; I'd use "glad for" or "glad about" instead.

"Glad of" is fine in BE in e.g. "I'd be glad of your help". (Indeed, I wouldn't use "for" or "about" in that example.)

Muret-Sanders bietet an: at, of, about...

Agree with 5 that the sentence structure seems unusual, and with the recast suggested there.

Does the sentence absolutely have to start with "what these two people have in common" (in counterpoint to their differences, which you have presumably enumerated before), or are you actually trying to simply state "these two people share one thing"?

If you must: gladness about...

gladness regarding ...

I'd say, but I concur re the strangeness. -)

I agree with Lara that 'gladness' is awkward in the sentence. This is another case where German style can use an abstract noun, but English style prefers a clause with a verb. In fact, as you can see, Lara's sentence uses two more clauses:

that they are both glad
(that) their friend's surgery was successful

In each case, 'be' + adjective is more idiomatic.

'Glad of' is certainly possible, but it's more formal or elevated diction, and it has a different meaning. If you're glad of something, you're relieved or grateful to have it; you welcome it.

Travelers in the rainy season will be glad of a sturdy umbrella.
We are glad of this opportunity to express our appreciation.

That may be more dated now than I had thought, perhaps more so in AE than BE. NOAD, for instance, doesn't even mention it, and actually lists a different preposition under that sense:

NOAD:
glad¹ - ... • grateful: he was glad for the excuse to put it off.

That sounds almost wrong to me, or at least, much more informal/colloquial; I wouldn't advise learners to imitate it.

I (BE) would never say 'glad for'.

Hecuba, care to tell us what you would say instead?

Another possibility is:
"What these two people have in common is their happiness about their friend's successful surgery."

#13: I'd use 'about' -- 'I'm glad about that'.

and also 'glad of' in the special sense referred to in #6 -- 'I'd be glad of your help' and #11, 'glad of this opportunity', etc. i.e. glad to have your help/this opportunity.

It did occur to me after I posted my #12 that I might in fact use 'glad for', but only in the sense of 'on someone's behalf', when something good has happened to them: I'm really glad for you -- though I would more probably say 'I'm really pleased for you'.

Sorry, had to leave yesterday. Thank you so much for all your suggestions. I think I'll paraphrase the sentence as most of you did.