So the center beam has to carry half of each of the 2x6x20 purlins, plus it's own weight. So that's 10 BF for each of the 11 purlins, plus a 3x10x20 beam at 50 BF fpr 160 BF. If I used the Green WRC value, I get 2.67 lb/bf, for a total load of 427 lbs. From the red book, I chose the following values for WRC:

Modulus of elasticity (E): 1,000,000
Bending Strength (Fb): 975
Horizontal Shear Strength (Fv): 70

Treating this as a distributed load the deflection calc will be:

D = 5PL^3/384EI

where I = bd^3/12 = 3x10^3/12 = 250

D = 5 x 427 x (20*12)^3 / (384 * 1,000,000 * 250) = .3 inches = No problem.

Your bending moment is Fb * bd^2/6. Using 975 for Fb:

M = 975 x 3 x 10^2 / 6 = 48750

The required Moment is given by WL/8:

427 * 20 / 8 = 1067.5, so again, no problem at all.

Finally, we should check horizontal shear.

H = 3 * P / (4 * b * d)

= 3 * 427 /(4 * 3 * 10) = 10.675 psi

The value for Cedar of Fv is 70 psi, so again, you're fine.

Someone should check my math, but my calcs show no problems. CB.

Qs that may help in design and construction details:

How are your posts anchored to the ground or piers - when your friends at the Roman toga party are leaning on the columns what resists their weight and keeps the columns from racking?

Jim's conceptual drawing and your mention of notching lead to the next, and important topic of joints. If you fatten your three main beams, the proportions will look better atop the 8" diam columns, plus you'll have more wood to do some strong functional joinery - won't have to use ugly steel pins or Simpson/Teco plates. More about loads and change of use in a minute Bob.
Right away I'll say go with the clear cedar for longevity and dimensional stability for all the members. They'll be exposed to all weather conditions, right?
How will you keep the 2x6 joists/purlins from leaning/rolling over? Toe nail 22' joists at three points of contact? You'd have to have some mighty fine quality lumber to keep unmoving contact to some fine, straight beams for 5, 10 or 25 years. Use stainless steel fasteners for rain and for cedar. You could fasten a decorative 2x8 band joist at front and back ends of the 11 2x6 joists. Or add the strapping you mentioned running over top of the joists - for better stability run strapping diagonally.

Here's a big "What If?" - your wife (and you) some time in the future decide a better use for the pergola is to have a solid decked roof for shade or protection from rain. See Stuart Brand's excellent book "How Buildings Learn". You may want to recalculate dead load which includes a wood board roof deck with some waterproof membrane, and resize beams and joists to carry same. Daiku has shown you the formulas, easy enough to plug in some new #s to see what size and spacing of beams and joists are required. Now you need to have slight roof pitch to direct drainage and runoff. Do you get snow where you are? snow loads? wind loads and uplift?

Finally, a design Q that is a matter of taste - round Italian columns (painted?) supporting sawmilled rectangular cedar beams? would you consider tapered cedar columns with a rectangular cross section, a Craftsman influenced style and proportion? Or what's vernacular and pleasing to you in your region? You could have fun laying out and cutting them, and doing some beautiful, strong joinery from post tops to the beams.
If your wife is Italian and she insists on round, you better do it her way Bob.
I hope this will be an enjoyable and rewarding project. What are you paying per boardfoot for clear WRC from your local sawyer?
Steve

Can't use edit function so I'll add... Rereading your description Bob, you said 10" diam columns. More reason to consider proportions of beams to columns. Do an accurate scale drawing as you get closer to conceptual final design.
Beauty is in the eyes of the beholder. You have to live with it and like it. And you are paying for it, so you get to say and build what is beautiful to you and your wife.
Steve

Thanks Steve. I have been pondering the "scale" question all along. 8" decorative columns have 12" end caps that the beams appear to sit on. I agree, 3" timbers sitting on those would appear chinsy. 10" columns have 15" end caps.

I was looking for single beam sizing but I think the final project will have pairs of 3x10 beams bolted through and through at each post. On each side of the 6x6 post I'm planning a 1" rabbit to accept the each beam. (Did I describe that well enough?) Along with the bolts, the shoulder will partially support the beam and provide some resistance to racking. Thus each "beam" will be two 3x10s with a 4" gap in between.

Q1: Does changing from "beamns on posts" to bolted beams concern anyone?
Q2: Do I need a deeper shoulder?
Q3: Will that be enough to resist side to side racking?
Q4: Now that beams are "doubled up" can they be 2x10s?

I had planned to knotch the bottom 2" of each 2x6 rafter, and double them up, like the beams, on the perifery and through the centre. Thus, each post will have two 3x10s on the north/south sides, topped with two knotched 2x6s on the east/west sides. In the dead center of the pertgola, a short decorative 4x4 will tie the center pair of rafters to the center pair of beams. A similar 4x4 will tie the front pair of rafters to the front pair of beams.

Q5: Will the knotched rafters be enough to resist front to back racking?

If not, trhe end caps can hide some small angle brackets for further racking resistance. I agree, showng a lot of hardware ruins the look.

As to pricing, I'm in Vancouver, Canada. We have a University research forest where WRC trees get rough sawn and sold at $1.60/bf in sizes below 2x10 and $1.80 above. 50' lengths are not uncommon. I can order my six 22' 3x10s and fourteen 22' 2x6s and seven 9' 6x6 posts, and get them in a week.

Thanks for everyone's assistance. Pease feel free to poke holes in my plans.

Cheers, Bob

We built a non-timber frame cedar pergola a few years ago. The back is attached to the house, the front sits on two posts, and we have two layers of larger beams that criss cross before the final 2x6s that sit on top. Everything is notched and nailed together. The back end is strongly attached to the house using brackets. When we thought we were done, if you leaned on a post the whole thing moved. Certainly gave us the impression that it wasn't permanent and needed some stiffening. Our solution was to bolt the front to the side of the porch that was only inches away. I think you should consider braces in your plans. If you design it so that your last step is to add or bolt on the braces, you can leave them off if it's solid. But if you don't plan for them, you may build it too short so that you don't have enough headroom if you have to add the braces.