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I am building a lightweight single axle trailer that needs to haul 600lbs cargo loaded evenly over the length of the trailer. Dimensions are 24.5 ft long including the tongue, by 3 ft 6 inches wide with an axle width of 5 ft 6 inches. Axle placement will be at roughly 18 feet from the tongue.

I plan on having a single center beam with 1 inch steel angle welded perpendicular at 3 ft increments to support the load. The steel angle will be welded to eachother with 1 inch angle in an "X" shape between them. No decking will be used.

The center beam I came up with is 2x2 inch steel box, 0.125 inch wall thickness, with a flange welded vertically across the bottom.

The flange is 0.200 inch thick flat steel and will be 1.5 inches tall at either end of the box tube, and 4 inches tall in the center. The steel box will be bowed to form a long shallow arch.

Is this a death trap, or will it be safe to use?

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Short answer, impossible to know with the given info.

Long answer:

Some assumptions have to be made, and in the absence of detailed information, conservative design factors can cover the gaps.

Let's start with a design factor of 5. Total load we design for is 5 x 600 lbs = 3000 lbs.

The load is assumed to be picked up by the "1 inch steel angles". This callout doesn't give enough information to understand the shape you're using. So, we can't analyze whether they will be overstressed.

The total length is 24.5' and there is an angle every 3.5'. Therefore, there are 24.5 / 3 = 7 angles on each side along the length.

loaded evenly over the length of the trailer

is an assumption I would question, but for simplicity's sake, we'll use it. 6000 lbs / 14 angles in total = 215 lbs / angle. (If we knew properties of the angles, we could check if they were overstressed.)

Not sure how you're going to attach your angles to the center beam, so no analysis of the weld or connection is possible. But I'd be concerned with the webs of the angles landing against the edge of the center beam.

Assume the evenly distributed load is picked up by the center beam alone. Model this as a simply supported beam with an overhang, and a uniformly distributed load along the length. Beam diagram

I imagine bending will control over shear. Max moment for this arrangement is 45,002 lbs-in. Section modulus of the center beam you described (using the 4" depth) is 1.48 in³.

Max bending stress = 45,002 lbs-in / 1.48 in³ = 30.4 ksi.

Assuming a yield stress of your steel at 32 ksi, center beam is OK. I want to stress again that there is not enough info to confidently say this is acceptable. These are quick, back of the napkin calculations, and shouldn't be mistaken at all as an approval.