So what to waffle on about tonight [21:46]? In terms of yesterday, I checked purlins in AS4600 steel designer module of Multiframe with the biaxial bending and also in my spreadsheet ignoring the biaxial bending, either way bridging is required.
Beams and Bending Moments
Went looking for a spreadsheet, landscape format for multiple point loads on a beam. Didn’t find one, I’m sure I set one up already, but only one I could find was portrait format and set up specifically for hip rafters: eaves overhang and double span. Initially I set up that spreadsheet using goal seek to solve the equations for compatibility of deflections, then later determined that it was easy to solve the equations and calculate the required answer directly. I left the spreadsheet as is, but modified the cell that was being iterated to calculate the required value directly. Sometimes using goal seek to get an answer is easier and faster than solving the algebra. On the other hand solving the algebra can collapse a series of calculations into a simple one line formula.
Since I couldn’t find the spreadsheet I was actually looking for I threw one together from scratch. I setup it up landscape so that I could get more calculation points in the span: my portrait layouts typically have around 8 points, with landscape I was able to get 16 points. I used the beam formula in my TechLib to calculate the bending moments. This highlighted another issue of numerical error and computers. The moment at the two ends should be zero, and my function specifically checked for the position being zero or equal to the span. However the sum of the increments does not exactly equal the span. I therefore modified the function to allow for a tolerance on the span, except that didn’t entirely fix the problem, as it still doesn’t meet that condition. Therefore added an extra test on the result of the calculation and set it equal to zero if less than an accepted tolerance.
When I plotted the bending moment diagrams, I got some strange shapes, not the triangles expected. When I set the spreadsheet up, I was aware that my points along the beam were not aligned with my loads, and unlikely to be so, but I was still expecting the bending moment diagrams to be triangles. That is I was expecting the moment for the points not calculated to lie on the lines of the triangles plotted from the calculated points. A flawed expectation given I know the values not calculated are the maximum values and located at the apexes of the triangles. The bending moment diagrams were distorted because the apexes had been cut off.
Since these are the maximum values, is it significant? If my memory is correct, the steel designers manual indicates that if have more than 20 point equal loads, then, can replace them with a uniformly distributed load (udl). Most structures are designed with uniformly distributed loads even though the actual loading is a series of point loads. For example portal frames to sheds are designed with UDL’s even though the loads come from the point load reactions from the girts and purlins. For small sheds and canopies however, there is likely only a few point loads, such as from a single central roof purlin. If considering simple beams then it’s a comparison of M=WL/8 versus M=PL/4 where W is the total load (W=wL), for which situation it is clearly preferable to consider the point loads if wish to avoid the structure collapsing.
We can typically add UDL’s together if they are of equal span and in the same location, from the resultant UDL we can they calculated the maximum resultant bending moment. As long as the maximum moment from a load condition is at the midspan of the beam, then we can calculate maximum moments and add them together to get the resultant. For example we cannot add point loads and UDL’s, but we can add the moments induced by these loads. If the point load is not at midspan, then the maximum moment is not at midspan, and since the moment from the UDL or series of point loads are not at the same position then cannot add the maximum moments. Though adding all the maximums does give a quick estimate.
To get better than the quick estimate then need to calculate the moment at various positions along the span of the beam for each different load, and then add the moments at the same position to get the resultant moment at that position. When doing so it is preferable to calculate the moment at the position of each point load, and the position of the maximum moment for other load types.
With a computer program it is relatively easy to calculate moments at uniform increments along the span, and also calculate moments at any special positions and then merge the values together and sort into correct order. Setting up the calculations in a table to present the results and perform the calculations in a spreadsheet is not so easy.
For example I have 16 points available, labeled 0 to 15, with points 0 and 15 being the supports, that leaves 14 points for internal positions. Say need a minimum of the quarter point moments, that is 3 internal positions, then that leaves 11 positions for point loads. It is easy enough to input the possible 11 point loads in sequence from one end of the beam to the other. But relatively cumbersome formula would be required to merge the position of the point loads with the quarter point positions. The points could be manually sorted, but that would hinder automatic calculations expected from a spreadsheet.
Another option is to reduce the increments along the span, by dividing into 20 to 100 increments. My typical portrait format spreadsheet includes about 40 to 50 point division, for 8 to 10 load cases.
This is where the debate between MS Excel and MathCAD becomes nonsense. The focus of MathCAD is presentation of the formula: not an entirely useful function when the primary task is to use numbers to make decisions and take action. The presentation limitations of an A4 sheet of paper should not be controlling how we carry out calculations. The objective of the calculations is to produce a valid specification, either written or drawn. The purpose of the specification is to provide a point of reference for production, from which material schedules and production drawings are to be produced. From production instructions a physical article is manufactured. With factory automation, hydraulic and pneumatic control systems, PLC’s, CNC machine tools, 3D printers, and the industrial internet of things, the results of calculations can bypass the generation of specifications and directly produce the desired artefact.
With the use of computers everyone can be empowered to carry out calculation dependent tasks without need to view such calculations. If I can collapse 40 hours of hand written calculations down to an hours worth of spreadsheet calculations, then the next generation can collapse that down to 5 minutes. The obstacle to faster processing times is understanding what the spreadsheet or other software does and its limitations. For example it may only take 2 minutes to enter values into a spreadsheet but otherwise take 2 hours to read the resultant spreadsheet report. At which point it may become apparent that the spreadsheet is calculating exactly what is wanted, and then an extra 5 hours is spent modifying the spreadsheet to get the desired answers.
Therein lies another problem. With pencil and paper only concerned about getting an answer, not so much with presentation, typically stick to a vertical format, and consume as much paper as necessary to reach a solution. With spreadsheet calculations, I can ignore the presentation on an A4 sheet, and just use the full space of the worksheet as is convenient to do the required calculations. For example I once checked some standard carport and verandah designs, by first creating a single spreadsheet row of calculations for the first design, then copying the row about five times, modifying the spans to get the results for each of the standard designs. The report I was reviewing was about 40mm thick, and contained multiple print outs from MicroStran and several pages of handwritten calculations. Initially my view was the calculation process was not correct, but then I was advised my task was to check if I agreed with the results not the process. That is: did my own calculations refute the specifications or confirm them? So dismissing the several hours of reading the original calculations, it took me a couple of hours to create the first row of calculations, and a few seconds to copy. The original report probably took at least a week to produce, I collapsed the time to a few hours, however the original report was more informative. So we have an issue of getting answers versus presenting answers.
However, my contention is that we do not need to present the calculations, as a true independent technical check does not involve looking at the designer’s calculations but rather checking the validity of the specification. Previously I have suggested that design involves multiple iterations and that checking only involves one iteration. This is not entirely correct. Checking involves a single pass through detailed assessment calculations, to either accept or reject a proposal, it does not require iteration until find a valid solution. However, both design and checking involve iteration through incrementally more detailed assessment methods.
For example I can make a first pass of checking a beam by calculating M=wL^2/8, with w=1 kN/m, that is a unit load, since we largely carry out linear elastic design, I can multiply the result by any other number to consider smaller or large loads. If I am working with steel then there are published design capacity tables and I can look up the size of a suitable steel beam. If I am working with other materials then I can produce my own design capacity tables, or safe load tables, which is what I did for cold-formed c-sections when I started using AS1538 and then revised for AS4600. Similarly produced capacity tables for different grades of timber to AS1720.
Span tables have their uses but not very flexible. A builder once came into the office looking for a quick check on spans for a carport, so I looked in the carport and verandah tables. The tables however did not cover the configuration he wanted, he kept looking at one set of tables similar to what he wanted, but not similar enough to be valid. I kept looking at other tables, ignoring the configuration and just trying to find a table which was close enough to represent a span table for the steel sections. My net conclusion was none of the tables were any use, and I would have to do the calculations of the loads and use the c-section load capacity tables.
The timber industry has its timber framing code, with span tables for various structural elements, but there are no load capacity tables, and no design capacity tables for structural sections: thus limiting the use of timber. The steel industry has design capacity tables, but lacks spans tables. The cold-formed steel industry has load capacity tables, but lacks design capacity tables (DCT’s) and lacks span tables.
Yet all industries should produce the tables, even the concrete industry. Start with section property tables, then produce design capacity tables, from there produce load capacity tables, and then following on from there span tables for specific structural elements and purposes.
Tables and curves are more useful and valuable than pocket calculators and computers. Just because I know the formula for a circle and can calculate the area of any circle with a calculator doesn’t make the calculator more useful than tables of circle areas. The reason is because none of us need to calculate the area of any circle, we typically need to calculate the area of a specific circles associated with specific pieces of technology. With tables we can get answers relatively quickly and can compare a specific circle with other smaller and larger circles. If such areas are important to what you do, then with common use, you will start to memorise the values.
The same goes with the use of other tables. Keep using span tables to design timber house framing, chances are you always pick the same structural sections project after project. This may be because you are locked into using a specific structural section and haven’t yet met its limitations, or it maybe because there is a trend in building design and the results keep coming out the same. From such span tables will also know the limitations of timber for the current structural form, and know when spans start to trend near the maximum that need to start generating some new span tables for an alternative structural form which can get greater spans from the available material.
Rather than needing more computer software and smartphone apps, for point-value calculations, we rather need more prescriptive solutions, design curves and design tables.
Sure some people may have difficulty using such tables, and therefore some people write computer programs to help look up the tables. As to whether such lookup tables are faster than calculations, depends on how complex and time consuming the calculations are compared to the speed of the lookup capabilities of a program. Such apps however take us back to looking at point-values rather than looking across the domain and the limiting suitability of the technology.
When I refer to point-value, I am referring to a single design-solution out of many possible solutions. To looking at a series of design curves through a black piece of paper with a small window cut in it. Whilst can move around the space occupied by the curves, you can never see the whole picture.
Civil engineering focuses on designing and constructing buildings one at a time. Manufacturing engineering is concerned with designing all buildings in one hit, or in more practical terms designing a large range of buildings, and then automating the production. Point-value calculations are not much use for the latter exercise.
- [19/02/2017]: Original