To determine the lightest weight shape for the inner beams, we need to:

1. Assess the span of the beams, which is given as 20 ft.
2. Refer to the chart that typically provides the section properties of wide flange (W) beams, specifically for A36 steel columns. This chart would usually list options like W8, W10, etc., with details about moment capacity, weight per foot, and other relevant structural characteristics.
3. Calculate or verify the load that each beam must support, considering factors such as distributed loads, point loads, and any additional factors if given.
4. Based on the load capacity and span requirements, identify the beam with sufficient strength that also has the lowest weight per foot, ensuring it meets the structural demands without exceeding weight limits unnecessarily.

Since I can’t view the chart directly, please provide any relevant details from the chart, such as specific weights, section properties, or load capacity for each beam shape. Alternatively, if you have a particular table or values in mind, you could upload that as well, and I could guide you on how to select the appropriate beam.

Would you like details on load calculations, or do you have specific values from the chart?

Here are some follow-up questions that might expand on this information:

1. What are the load values or types of loads acting on these inner beams? (e.g., uniform load, point load)
2. Do you need help calculating the load-carrying capacity of each beam shape?
3. What factors of safety are required for these beams according to your design standards?
4. Do you have any constraints regarding the depth of the beams or any specific design limitations?
5. Are there other beams in the truss that need a similar weight optimization approach?

Tip: Always ensure that the selected beam meets both weight efficiency and structural integrity requirements to avoid unnecessary material use and to optimize cost.