I am currently designing a cable-stayed bridge and my design gave me a 45x50cm column. But now that I see my column's unsupported height is 7m. I checked for slenderness and it is of 93. Now, reviewing other designs of cable-stayed bridges, some similar to mine, having similar heights and dimentions, I checked them and all of them columns are slender. Checking for the minnimum height needed for my column based only on slenderness gave me a height of 1.17 m which is HUGE. Thinking about it, since the columns in cable stayed bridges are basically on equilibrium (or under small actions) I think slenderness can be, not neglected, but considered less important. My point is, is it okay to keep a slender column in a cables stayed bridge?
You have to follow your local code, but just to get an idea of why the slenderness ratio and Euler's critical load are especially important for the design of columns in bridges compared to columns in buildings the following would help.
Bridges are usually uniquely located in the canyons that are the natural paths of high-velocity wind.
The deck of a bridge is vulnerable to vibration in certain winds and will start to resonate in unbounded growing magnitude which will create hard-to-predict loads on the structure and possibly explosive destruction.
Accidents can create traffic jams on the most critical part of the bridge and subject the columns to large push and pull moments, forcing the column to behave like a column constrained by a spring on top exposed to a forcing function (refer to the diagram below) traffic can randomly create the critical natural frequency of the bridge.
The following paper is a research on prismatic uniformly reinforced bridge columns. column slenderness design