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The center beam is not strong enough to span 24 feet. The Steel Tube Institute (see link below) gives the moment of inertia for a 6x6x3/16 section at 22.3 in**4 The section modulus is given as 7.42 in**3 Your center beam supports a 12 foot tributary width of roof. Even in very mild climates it is typical to design roofs to a minimum of 25 pounds per square foot live + dead load (to handle wind loads if nothing else -- and you should be doubly careful of wind loads if you plan to leave any side of this addition open -- 3 sided buildings are notoriously prone to wind failures). So 25 psf loading and 12 foot tributary gives us a distributed load of 25*12 or 300 pounds per foot on the center beam. (25 pounds per inch). The beam is to free span 24 feet, or 288 inches. The formula for center deflection of a simply supported beam with a distributed load w is 5*w*(l**4) / 384*E*I E (modulus of elasticity) is typically 2900000 for steel. Plugging in the numbers 5*25*(288**4) / 384 * 2900000 * 22.3 The mid-point deflection under this loading is 3.5 inches. This considerably exceeds even a very liberal guideline of (span/180) for a roof beam. The bending moment under this loading is w * l**2/8 or 259200 inch pounds The stress at the midpoint is M/S or 259200 / 7.42 or about 35000 psi It is unclear what kind of steel you have; it seems imprudent to assume better than A36 steel with a tensile strength of 36000 psi. Allowable bending stress in A36 steel is 0.66 * 36000 or 24000 psi. You would be overloading the beam by 50 percent. In fact, you are basically at the point of outright failure.
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