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There are other factors, like impact loading, uniform distribution loading, multiple span loading, dynamic (moving) loading, etc.

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What will answer my question: "I did a project exactly like that where I placed 12 boards side by side and it passed code by an inspector instead of using a beam of double height"

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Beam strength goes directly as the width (twice as wide is twice as strong) and as the cube of the depth (you call height), so twice as deep is 8 times as strong.

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Answers that in the form of "I think..." or "I believe" or "Probably what would work" or "you could do it this way or that way with this material or that material. No. Useless. If you do not know the answer then no answer of any type is excellent. Answer that try to tell or educate me about modulus of elasticity or bending force. I know about all that and do not require a class on it. I want a simple practical example of a proven job or the proven mathematical relationship.

I understand you will not be able to answer the question, so if you can't, don't. I want an expert, a contractor or engineer (not a civil engineer) intelligent summary of the results and nothing more.

So, to answer your question, beams with short spans tend to fail in horizontal shear, beams with extra long spans tend to fail in deflection and middle span beams tend to fail in Extreme Fiber in Bending.

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Optical fiber is tiny and made of silica (glass), chosen for its ability to reflect light in a way that allows the light to transmit data efficiently.

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The mathematically proven relationship is, for wood, 1/2 the height means you have to increase to 4.323 times the width to get the exact same load capacity.

Bending Stress is based on the stress of a given quality of material (species and grade) AND the geometric shape at the cross section. This stress is also known as the “Extreme Fiber in Bending” because you are checking the fiber of the wood at its extreme outer edges (top in compression and bottom in tension). A common grade for SPF is 1500 psi under normal conditions of loading as established by West Coast Lumbermen’s Bureau.

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I am trying to find an accurate comparison of two beams of equal strength but different dimensions. The answer will be a PROVEN real world example that passed code by an inspector, OR a PROVEN mathematical relationship. I have scoured the internet but no one seems to have written anything about it that I can locate.

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Deflection is the stress that tends to make a beam bend. If a beam is overloaded it can bend so much that it’s noticeable and unsightly.

Shear Stress is also called “Horizontal Stress” because you are checking the fiber stress in the horizontal direction. To illustrate this stress: stack 3 - 2x4’s flat up on 2 chairs about 10’ apart. Then stand in the middle and you’ll see that the boards tend to slide past each other. A common grade for horizontal stress for SPF is 120 psi.

The mathematics of nominal lumber are annoying, (2X12 and 2X6 are not any of those dimensions) but your ballpark is going to be 4 2x12 is 8 times as strong as 4 2x6, so you'd need 32 2x6 for the equivalent strength.

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Determining the maximum load (or ultimate load) or as you called it “maximum strength” is determined by calculating three issues: 1) bending stress, 2) shear stress, AND 3) deflection.

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Example: I KNOW that a four 2x12's fastened together will easily carry some amount of deck load. What I want to know is HOW MANY 2x6's fastened together will carry the same load.

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