Pergola Advice

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That beam was doomed from the start!

beam.png

And that's under its own weight
 
Been tasked with replacing this 3.6m 47 x 150 joist. The sagging has occurred over 20 years ( not my work!) and I was considering attaching a metal bar to the inside face to add some strength along it’s length.
Questions
How long should that bar be on 3.6m span ?
Is screwing it to the face enough or should it be routed in?
Easy enough to get 1m bars from Screwfix or similar or should I source just 1 bar of the recommended length if the view is that the bar needs to be longer than 1m?
Thanks
You could make a slightly upward curved wood laminated arc, the way the romans used to do. Roman stone arcs are still standing after a couple of millennia, there must be a good reason.... In this case the posts might be tensioned with a steel cable under the arc, to avoid them from collapsing outward because of the weight and outward momentum... I think that if you copy the geometry of timber roofs you'd get a good stable structure...
 
Unless the metal bar is deep 100mm l+ and preferably a L section it's not going to help sagging.

Tbh for the price of £20 for some timber I'd be replacing it!
You are incorrect as a flat strip works perfectly. A structural engineer advised this 20 years ago and it has been solid ever since on a floor support in a listed building.
 
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