Yes strain does mean to ‘endeavour’ or ‘try’ to do something. But it also suggests that it is very difficult to do. My car was straining to get up the hill this morning. I was straining to look at the bright lights.

Dictionary definition of the verb 'strain' As a transitive verb: 1. To pull, draw, or stretch tight: strained the sheets over the bed. 2. To exert or tax to the utmost: straining our ears to hear. 3. To injure or impair by overuse or overexertion; wrench: strain a muscle. 4. To stretch or force beyond the proper or legitimate limit: strain a point. 5. Physics To alter (the relations between the parts of a structure or shape) by applying an external force; deform. 6. a. To pass (gravy, for example) through a filtering agent such as a strainer. b. To draw off or remove by filtration: strained the pulp from the juice. 7. To embrace or clasp tightly; hug. As an intransitive verb 1. To make violent or steady efforts; strive hard: straining to reach the finish line. 2. To be or become wrenched or twisted. 3. To be subjected to great stress. 4. To pull forcibly or violently: The dog strained at its leash. 5. To stretch or exert one's muscles or nerves to the utmost. 6. To filter, trickle, or ooze. 7. To be extremely hesitant; balk: a mule that strained at the lead.