yoko1 #6,
‘Max demonstrated’ usually represents the maximum crosswind that the manufacturer has evaluated based on accurate and recorded wind reports, and judged not to be limiting. If higher values are used in service then you are accepting the risk that it could be limiting, particularly that the accuracy of reported wind speeds might be unknown, and in conditions where there could be little margin between just OK and ‘off the side’.
If the manufacturer had tested higher crosswinds and concluded that a particular value or conditions to be unreasonable, i.e. it is limiting, then this value should be published in the aircraft AFM limitations section.
An interesting article relating to crosswind - engine handling, during takeoff
https://safetyfirst.airbus.com/app/t...y_first_27.pdf page 7-