# truss' moment of inertia

my handbook of formulas has a myriad of entries for the moment of inertia of cross sections of various shapes. but there's nada for
trusses. i've got a canti-levered beam which is too heavy for the motor i'm using to rotate it. i'm trying to avoid buying a new motor by replacing the hollow cylindrical beam with a triangular warren truss. but i need equations to tell me how to dimension it to retain the same stiffness (young's modulus of elasticity). everything i've found on the net about trusses is all about how to analyse a custom truss for compression, tension, and displacement, but there are no tables which parameterize the standard truss designs and give you the formulas for things like moment of inertia. anyone know where such things can be found?
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Look up Moment of Inertia of built up sections. One source is "Design of Welded Structures" from The James F. Lincoln Arc Welding Foundation, P.O. Box 3035, Cleveland, Ohio; Section 2.2 Properties of Sections. This same text has the method of designing and calculating the properties of an expanded beam which may suffice for your application.
One thing that bothers me is the original support is a cylinder. A cylinder offers the stiffness, and all properties, 360 degrees about the longitudinal axis which may be a requirement. Have you considered another cylinder - larger OD, thicker wall - to get better properties? Commercial steel pipe can be obtained with 42 inch OD, 1.50 inch wall , 39181 in^4 Moment of Inertia, 1865.7 in^3 Section Modulus and weigh 649 lb/ft. You may not need something that big, but look up another size of pipe to do the job.
Hope that helps, Jim Y
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Why don't you just analyze/design it as a regular truss? you can find the procedure in a text book. If the truss was a flat truss you could estimate the inertia of the truss and stiffness. You could do this by using the center of the truss as the CG and calculate the inertia of the top and bottom chords then multiply the chord areas times the distance to the CG squared then add these together. The web members act as shear components which consist of tension and compression forces (strut and tie method). For a triangular truss you can't do this because the truss then comes a beam with a variable moment of inertia and is more complicated and not worth the effort.
CID...
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but it's not. it's need's to be triangular or square so that it's rigid in all three dimensions.

as you say, it's "complicated", which is why i was hoping to just find a table of formulas. do they really not exist? i've made a fairly extensive search of the web and can't find anything. all i want is a formula for the young's modulus of elasticity for a triangular warren truss parameterized with variables for the relevant spacings and diameters of the members. it seems to me a table of such equations for the various truss designs would be very handy for structural engineers and i'm finding it difficult to understand why i can't find it anywhere. such tables exist for beams of various cross sections, why not trusses of various designs?
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Structural beams are made to a standard, or actually one of several standards, and thus there are a limited number sizes. There are many, many truss designs, that can be made out of a wide variety of different sized elements, with different truss angles and spacing. An individual truss can be loaded in a large variety of different ways. The number of variables is just too large for a table to be a solution, so to speak.
Note that Young's modulus of elasticity is a material property, not a shape property. This throws another set of variables into the list.
Space trusses are evaluated with the same tools as 2d trusses. There are a number of relatively simple procedures to analyze them, but the number of members and joints can quickly cause a big bookkeeping job. In practice, most engineers rely on software of one sort or another beyond very simple cases.
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Charly Coughran
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In a previous post Charly Coughran wrote...

It sort of depends on why you need the information. For a quick deflection check, Chuck's method of treating the truss as a deep beam with top and bottom flanges only (the top and bottom chords) should be conservative. Formula for Moment of Inertia (I) = A*(d^2), where A is the area of the chord and d is the distance between the centroids of the top and bottom chord. Of course this assumes that the top and chords are the same size.
Anything else will require an FEM model with member sizes in order to compute more exact deflections and member forces.
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Bob Morrison, PE, SE
R L Morrison Engineering Co
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thanks for all the suggestions guys. i took roleic's (e-mailed privately) and chuck's idea of computing the inertia of the cross section of the longitudinal beams and did some strength-to-weight calculation's comparing trusses to thin-walled tubing (which i'm currently using and which jim y suggested), and surprisingly, at least initially to me, the two are roughly equivalent. (keep in mind i'm a neurobiologist trained as a computer scientist.) so for example, unless i'm doing something wrong, a 0.5" cylinder with 0.01" walls is about as strong and weighs about as much as a truss with three 0.08" diameter longitudinal members forming a triangle circumscribed by a 0.5" circle. and this neglects the diagonal supporting members, so the truss actually has a *worse* strength-to-weight ratio for similar outer dimensions. same is true when everything is scaled up 100x. i guess looking at the equations this all makes sense b/c the dominant term is the outer dimension. but now i'm confused b/c i had always thought that trusses had superior strength-to-weight ratios compared to beams. so am i doing something wrong? the only advantage i can see to trusses is windage, which in some apps, mine being one of them, is certainly a consideration. just seems to me i'm missing something here. thanks again for the help.
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In a previous post bjarthur wrote...

I'm actually not surprised by this. The tube section has about 50% more material located away from the neutral axis than the two much smaller tubes in the truss chords. A simple drawing (cadd helps) will confirm this, although you could calculate the difference.
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Bob Morrison, PE, SE
R L Morrison Engineering Co
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bjarthur wrote:

When comparing the strength-to-mass ratio (aka specific strength) of different structures you have to consider all failure modes limiting their strengths not just global bending. E.g. Don't forget buckling (global instability) under compressive, shear and bending loads.
Beams made of very thin walled cross sections are limited by cylinder buckling under bending or shear load. Trusses can of course also be limited by instabilities. Their instability type is beam buckling under compressive load of the truss members where the buckling length of the members is the free truss length btw. the truss nodes.
Buckling strength is a bit more complicated to calculate than bending strength. Where as the formulas for compressive beam buckling are still rather simple and easy to find in lit., for cylinder buckling under compression or shear or bending load you need special charts from reference literature for standard geometries like cylinders, cones, cyl. shell panels (e.g. Bruhn,1973, Analysis and Design of Flight Vehicle Structures or ESDU-sheets) or a finite element model for non-standard geometries.
So don't jump to conclusions before you have considered all relevant physics:-)
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This isn't generally true. A truss is statically determinate which, under some loading conditions, means you can use the material more efficiently than a statically indetermiante structure (which most beam structures are). However, there are many types of loading where a beam or shell is more efficient.

This is can be an advantage, but there's also ease of pass-through (same roots as low windage, but different design considerations), ease of manufacture, ease of transport (trusses are generally easy to break up), and ease of analysis (static determinacy). Ease of manufacture is a huge reason for seeing trusses in large structures...a 10' deep solid beam would be a nightmare to build, a 10' deep truss is trivial.
Tom.
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bjarthur wrote:

Don't believe trusses can always be treated as beams. They may not have diagonal (shear) members to make it bend like a beam.
Assuming your truss behaves like a beam, you can treat it like a sheet-stiffener design. The axial members react bending moments and axial loads and the diagonal members react shear loads. This should at least get you an initial sizing. More complicated analysis or modeling will help in final sizing.