This おり is basically the same as いて (the te-form of いる) except that it sounds more formal and/or bookish.
You probably know the basic idea of how 中止法 works. It refers to the use of masu-stem to join clauses. However, 中止法 does not work very well with short ichidan verbs such as 着る, 見る, 得る and いる. The masu-stem of those verbs consists of only one kana.
- 学校に行き、授業を受け、家に帰った。: OK
- ぐっすり寝、服を着、学校に行った。: Questionable/uncommon
In particular, い is unacceptable as 中止法 even though the masu-stem of いる is い. And that's where おり comes into play. Instead of い, you can use おり and preserve the formal/bookish mood of 中止法:
- ❌ 目視による検査を行ってい、強度に問題はないという。: Wrong
- ✅ 目視による検査を行っていて、強度に問題はないという。: Correct but less formal
- ✅ 目視による検査を行っており、強度に問題はないという。: OK (formal)
Although おる is basically a humble version of いる, おり used in this way does not carry any humble sense. It's just a replacement of い you must use in formal documents. You have to be able to use this type of おり properly even when you don't need to pay respect to anyone (for example, when describing a chemical reaction).
On the other hand, おる at the end of a sentence sounds very different:
- 目視による検査を行っております。: Humble expression (keigo)
- 目視による検査を行っておる。: Sounds like a pompous aristocracy in fiction
Note that (-て)いる/おる after 行う is necessary because 検査を行う is an habitual action that explains why the bridge is safe.
