How much distance do you have between beds? And do you get equal amounts of morning and evening sun? Or is one side shaded already? Also, is it possible to run your trellis across the north end of all the beds? I'm assuming it's not what you want, but I thought I'd ask anyway.

I would instal it such that casts least shade.
Your bed running NS, I would suggest to put it up on the
NORTH edge. But then 4 feet might not be enough. Then continus
on the EAST edge.
Patial Shading the adjacent beds may be problem. So then, your trellis
should be as short as possible. Also, do not plant the peas
and cukes too close, so they will not totatlly block the sun.
Conside planting veggies in those partial shade area that do not mind it, or they also grow tall, like tomatoes.

There is about 2-3 feet in between the beds. I was planning to have some cole vegetables there in the bed next to it and later beans or something in the fall. I was thinking earlier I had some extra lumber because I was going to build more beds until I was able to get a good size community garden plot so I might build a 2x8 bed and run it lengthwise east-west in the back of the garden and grow just cucumbers on it so they have more freedom to do they want height wise. I will have to research spacing on cukes, I was thinking they might be able to crowd themselves since they go vertical - maybe a 6 inch spacing? How many are expected per plant?

Normally cucumbers are planted in raised rows or individual hills.
Since you want to build a 2' by 8', bed, then what you can do is to
plant the cukes at the center line and slightly hill the center.

Normally, they plant 2 or 3 seeds per hill, spacing them by 2 to 3 inches(within the same hill)
Then make the next hill similarly, space at 16"(center -to-center).

The above way is when you don't have trellis. But with trellis, the consideration for spacing is solely based on the root system. So you may plant them more packed.

Trellises are the sole reason for having beds E-W, and the trellis goes at the N end of the northernmost bed. Anything else, you are shading something. It is not a big deal, though. Put the trellis on the W side, and then plant something that likes summer shade, such as cabbage, collard, kale, lettuce, chard, something leafy. It will like the shade and the nitrogen from the peas.

It's hard to know what's best without seeing your garden. If you had something blocking the sun on a particular side, you might want to do it one way. Nothing blocking the sun, you might want to do it differently. You said you may put in a third bed running east-west, on the north side. If you're going to do that, your problem's solved. If not, and you want to avoid any shading, you could still do something similar. You could put up your trellis on the north end, and fill in the last 2-4 feet of your path with more raised bed space. So you'd have a U shaped garden. Or you could just set some containers there if you didn't want to build additional bed space.

white apple,

i have been using trellis gardening for years to get more out of the space i have.

in a open garden and in the green house i run my trellis N and S.

this shading issue is a mystery to me. in the green house we use 50% shade glazing and we get better production then in the field. try ti figure that out.

also i plant my cucs about 2" apart in a row and trellis them. what i do is to feed them well water them well. not over water but enough to keep them from being bitter.

in the fields i try to plant short crops in between the tall crops. my rows go N to S and are 34" apart. i do this so i can cultivate with my tiller and to allow more sunshine ot get to the plants.

i think that the cultivating helps keep N in the soil and keeping it aerated helps too.

i plant all my trellised crops closer together. i also sucker prune the tomatoes and cucs.

i don't know if this helps but it works for me.

also i read some where that tomatoes are a heat and air plant and don't need as much photosynthesis as some other plants.

of course in the summer we get a lot of sunshine there in this high mountain valley.

Dean