Structural analysis is a rather broad field of study, but there are some basic concepts that sum up most of what can be done. The most basic form of structural analysis involves two force members, but structural members with bending loads are also very common in real situations.
Two Force Members
Shear and Bending Moment Diagrams
Tensile Zone Analysis
Full Structural Analysis
A structural member which is only in either tension or compression is known as a "two force member". Calculating static loads on two force members is known as the field of "statics", although the same principles often apply to slowly moving situations also. As long as the two force members are not accelerating so fast as to be significantly loaded by their own acceleration then the same methods of analysis used in statics can be applied. The field of statics is really just applied geometry, with some additional analytic techniques. Most real situations do however also involve bending loads on structural members, so additional types of analysis are required.
The analytic technique known as a "shear and bending moment diagram" is the simplest way to get some idea about how real structural members, such as beams, are loaded. The shear and bending moment diagram is however only one portion of structural analysis. It does not in itself address the actual stresses in the structural member. A shear and bending moment diagram only provides a means of easily approximating the shear load and bending moment magnitude at all points along the structural member. This is very useful in itself, but it is only the first step in predicting the loads that a structural member will be able to sustain. An important point that is often made is that it is the additive combination of shear load and tensile load that is significant, although how this actually relates to shear and bending moment diagrams themselves is a bit murky.
Tensile zone analysis is something that I came up with shortly before dropping out of engineering school and graduating with a German language degree in 2004. Tensile zone analysis is a gross simplification of full structural analysis that actually works. The basic idea is that there is a portion of the cross section of a structural member that carries the bulk of the tensile load when a bending moment is applied. Just where this tensile zone is located and what shape it has depends on the shape of the structural member and the material it is made out of. Generally the tensile zone is at the surface and in somewhat towards the center of the structural member. Just how far the tensile zone extends depends on the shape of the structural member and the material it is made out of.
The simplest structural member is of course an I-beam, as the tensile zone is clearly the square outer rail of the beam. An I-beam may have a tapering web that is much thicker towards the outer rail, but this generally does not contribute much to the tensile zone. The tapering web just allows the center portion of the web to be thinner and lighter while still holding the two rails securely in position. Other types of structural members generally require either lots of sophisticated analysis or some creative educated guessing to determine just where the tensile zone is located.
The actual analytic technique of tensile zone analysis simply involves finding the pivot point of the compression loaded side of the structural member to approximate the lever arm on the tensile zone. The pivot point is somewhere just in from the outer edge of compression loaded side. Just how far in depends on the actual shape of the structural member and what it is made of. Most hard materials don't compress much, so the pivot point tends to be rather close to the edge. Materials which are soft but significantly strong, such as rubber or plastic, will compress a large amount, so the pivot point ends up much farther in towards the center of the structural member. Unless plastic structural members have extreme shapes such as a box section and/or substantial buttressing they generally just bend, stretch and break without doing anything interesting.
Harder materials hold their shape under a load, until finally breaking if the load exceeds structural limits. Again though the shape is significant. A one inch square mild steel bar will just deform and permanently bend out of shape. One inch mild steel tubing on the other hand will actually break, provided that it is not simply smashed flat with sharp point loading. Even three inch mild steel tubing can crush if it is not supported to avoid sharp point loading. If the three inch mild steel tubing is used as a simple point to point beam over a long span then it can break in the middle before deforming at the ends. If on the other hand the three inch mild steel tubing is cantilevered then it may crush before breaking if not otherwise supported at the central contact point. Generally the larger the dimensions of the structural member the more likely it is to break before dramatically deforming. The shape of the structural member and the material it is made out of are however very important. Square mild steel tubing with more wall thickness tends to deform, where square mild steel tubing with a thinner wall thickness tends to break. Provided of course that the comparisons are made for pieces of square tubing with roughly the same external dimensions.
Tensile zone analysis involves a lot of guessing, but there are ways to determine how realistic any one guess might be. Looking at a broken piece of the same material provides a lot of clues. The area around the tensile zone often has a clean break appearance, where the inner parts of the structural member get more deformed and often break more irregularly. There is often a round or dimpled look to the area near the pivot point or just inside from the pivot point. There are also ways to get good information about these guesses without breaking anything. Just how much a structural member bends under a partial load gives a lot of information about the likely location of the pivot point. Anything that is stiff and rigid is going to have a tendency to have a pivot point far out towards the outside surface. If the structural member bends substantially before getting close to breaking then the pivot point is likely somewhat farther in from the surface.
Putting some numbers to this involves comparing the amount of bending before permanent deformation occurs to the amount of deformation before braking. The terminology is "yield strength", "ultimate strength" and "plastic region". Mild steel and some other types of steel get stronger during permanent deformation, this is known as "work hardening". This is why things made out of mild steel usually get bent out of shape somewhat before actually breaking. Once the yield strength is exceeded the material is in the plastic region, meaning that any further movement or loading results in permanent deformation. For materials that are highly "ductile" such as mild steel the plastic region often extends beyond what would be considered a breaking point. Mild steel will undergo work hardening up to it's ultimate strength, but it will continue to stretch and deform somewhat before finally separating and breaking.
For purposes of estimating the pivot point for tensile zone analysis it is the amount of deformation between the yields strength and the ultimate strength that is significant. Some hard materials, like high alloy spring steel, will bend quite significantly before yielding, but this does not necessarily mean that the pivot point then tends to be far in from the surface. Anything that will bend substantially does end up with a somewhat more central pivot point, but it is softer materials that tend towards a significantly more central pivot point. It is both the shape of the structural member and the material it is made out of that determines the location of the pivot point. Even a very soft material such as cross linked polyethylene can have a pivot point far out towards the surface if the structural member is of a shape that provides a large amount of stiffness and rigidity.
The rigidity of a structural member also has a lot to do with the size and shape of the tensile zone. A structural member that bends little before breaking will tend to have a tensile zone that extends farther down towards the central portion. In the case of an I-beam or a piece of square tubing this generally means that the entire outer portion is the tensile zone. A structural member that bends substantially before breaking though tends to have a smaller tensile zone concentrated close to the outer surface. A round piece of plastic will often crack at the surface when subjected to a large bending load. Once the surface begins to crack the structural member then easily breaks all the way through. For both the location of the pivot point and the extent of the tensile zone the rigidity of the structural member is very significant.
What ends up being significant is a comparison between the amount of bending of a structural member before permanent deformation compared to the amount of bending before it breaks or completely fails. If little or no deformation occurs before the structural member breaks then it is likely that the pivot point is rather close to the surface and the tensile zone likely extends deeper into the center portion of the structural member. When the pivot point is rather close to the surface though this does not mean that it is actually at the surface. Even for the hardest materials shaped into extremely rigid structural members the pivot point is going to be somewhat in from the surface. If it is a piece of square tubing that is considered then a very good guess for the pivot point is the middle of the side in compression. If it is a square steel bar then the pivot point is going to be quite far in from the surface. Just how far depends on the size of the bar and the material it is made out of.
Some materials have a very low tensile strength with a substantial compressive strength. Concrete is a good example of this. A little bit of hard structural carbon steel towards the tensile side of a concrete beam dramatically increases it's strength and the pivot point is far out towards the surface. If a larger amount of softer steel is used then the concrete will crush and fail on the compression side.
The basic analytical technique of tensile zone analysis is simply multiplying the area of the tensile zone by the lever arm and dividing this into the bending moment. That's very simple, but getting good results requires making good educated guesses about the location and size of the tensile zone and the location of the pivot point.
Tensile zone analysis is at best only an approximation. Often an easy and very useful approximation, but it is not the full story. A full structural analysis is a methodic approach that involves modeling the movement and loads on all parts of a loaded structural member. This is something that I don't know anything about from formal classes or from reading any books or articles on the subject. I dropped out of engineering school before any of this was introduced. The following description of full structural analysis is just what I now know it must be like.
The first big challenge to full structural analysis is that the material does deform, even hard structural steel in a rigid shape. The result is that the outermost portion of the structural member is more heavily loaded and breaks first. The more centrally located portions of the structural member do contribute directly to the strength of the structural member, but the amount of contribution drops off dramatically as the distance from the outer edge increases. Modeling the loads on each portion of the cross section of a structural member involves predicting exactly how each portion moves relative to the rest of the structural member. This is further complicated by the fact that movement is not limited to compression along a cross section either, although this three dimensional movement can probably be ignored in some types of analysis while still providing fairly good results.
The starting point would be some form of iterative analysis, although once the principles are defined the whole thing can be done by integrating multiple functions across the physical dimensions of the structural member. I should be well prepared to do this mathematically after two full years of university level four unit calculus classes, but that all came before any real engineering classes and most of it was quickly forgotten.
The first calculus class I took was as a freshman in engineering school. I was advised against this as I had not taken any high school calculus class. I had done well in the other high school mathematics classes, even scoring High Honors on the Golden State Exam for geometry in my sophomore year with a score in the 97th percentile of all high school students in the state of California who took the test that year. I just never got around to high school calculus as I wanted to take second year Spanish (only first year foreign language was required to apply to universities), physics (only high school chemistry was required to apply to universities) drivers education (most drivers paid a private California state licensed company instead of taking the high school class), auto shop, welding, machine shop, mechanical drawing (three years) and CAD/CAM. I was lucky that my high school had these shop classes at all (many did not at that time), but I had to juggle things around to get a chance at all of them without ever doing summer school. I did come an hour early for two years to squeeze machine shop in, and I didn't leave early senior year like many students did, but there was no way I was giving up three months of summer vacation for calculus. I hardly ever got a chance to ride my dirt bike during the school year, especially before I got my own car and a motorcycle trailer, so forgoing summer vacation in the mountains was like death. It was all I looked forward to all year every year of high school. The formal University requirements were clear, high school calculus was not a pre-requisite. Only a high score on the mathematics placement exam along with high school geometry, trigonometry and algebra classes was required to sign up for the first calculus class.
It was a mistake, as I was the only person in the class who had not previously taken a calculus class. Most were freshman engineering students like me, but who had taken a year or even two or three years of calculus in high school. Some were second year students who had either failed the year before, or had taken remedial mathematics classes the year before as I had been advised to. The instructor was brutal to everyone. He would not tell me any of the things that the other students already knew from high school calculus, and he was absolutely fiendish about computational ability in grading everyone. More than half the class failed with F or D grades, and those of us who passed got C grades. A D grade was a mandatory repeat in engineering and support classes, so it was considered failing the class. Interestingly though not many D grades were actually awarded. Students who seemed to have no chance of getting a C grade would often just give up and end up with very low final scores. I managed a C+ in that first calculus class, which was actually considered a high grade and something of an accomplishment in itself. There were two B grades and one A- grade in that class of about 40 students. In the next class the professor started out the first lecture on the first day with a five minute summary of previous calculus classes, and I figured out for the first time what differentiation and integration actually are. Nobody had previously presented any clues about this to me. I already understood the concepts from high school physics, but nobody bothered to say that distance is the integral of linear velocity, linear velocity is the integral of linear acceleration, linear acceleration is the derivative of linear velocity and linear velocity is the derivative of linear position. That one phrase "velocity is the derivative of position", I never heard it. Nobody ever said that, or anything that would have given a clue about the connection. I very easily got an A in that second university calculus class, as did a rather large portion of the other students. I got a C in the next calculus class without taking notes, studying or doing the assigned practice problems. That was followed by an F without even going to class. After skipping a large number of the lectures I went into the final exam with almost no idea of what had been covered. When I took that class again it was an easy B despite demanding grading, and it was fairly easy B grades for me from there on out in the calculus classes although I did take notes and do at least some of the homework.
Taking notes was one of the things that seemed to be of great general use in succeeding academically. It was like a game trying to figure out what to write down. If I could keep up with the lecture and get all of the important information scribbled down in some way (and later make sense of it) then I was usually able to get an A no matter how demanding the testing and grading happened to be. Of course note taking itself is a skill that has to be developed. In my freshman year of university I often couldn't write fast enough to keep up, and if I focused too much on writing then I would miss important things that were said. The trick is in quickly figuring out what is going to be important.
In the general education, science support classes and lower division engineering classes I usually did get A grades with a sprinkling of B grades and even a few stray C grades when I stopped attending lectures. I usually got A grades even with unbelievably demanding grading where none of the other students were able to manage an A grade. I even got on the college of engineering "Dean's List" for academic excellence one quarter. Mathematics was easy for me, but I hated the classes. I knew that all these techniques of integration and methods for solving partial differential equations would be useful for something, but the upper division structural engineering classes all came after the calculus classes.
All that mathematics was however very useful in understanding the basics of chemistry, physics and electronics, and I managed to take three four unit chemistry classes and five four unit physics classes. One of the chemistry classes and two of the physics classes not even required for the engineering curriculum.
What I finally figured out years later was that the engineering curriculum had really been oriented towards people who had already been working as design engineers, but who had little or no formal science or engineering education. It started out with heaps of very useful mathematics, but as an 18 and 19 year old student I was clueless about what to apply it to. The science support classes and lower division engineering classes were of little help as they were all "not calculus based". It was usually necessary to follow and at least mostly understand the mathematics intense introductions to concepts and topics to grasp the subject material, but that was a very small part of the classes. I say mostly understand because the notation was nearly always somewhat different than what we were seeing in the calculus classes, and this made it more difficult to follow all of the shortcuts and omitted steps in the manipulation of the integrals. The science and engineering professors themselves often said that we weren't really expected to be able to follow these whirlwind introductions, but the mathematical introductions were often still very useful in understanding the subject material. The functional parts of the curriculum were "not calculus based", so we had nothing to apply what we were learning in all those calculus classes to. It was just a matter of 5 minutes of the professor mathematically proving the validity of a set of equations, and then 40 minutes of applying those equations to sample problems followed by a three hour lab playing with (and being graded on) those equations theoretically and with real live electricity. The equations themselves were so useful in dealing with everything presented in the class that the calculus faded away as unimportant. This was especially true of the physics and core engineering classes, but the chemistry and "fluff" engineering classes were structured in the same way. I think we were actually being actively discouraged from using calculus.
Part of it was that I didn't care much. I didn't try to apply the mathematics to anything outside of the engineering curriculum, I just let it be. This attitude was fostered by professional engineers I knew at the time who said over and over again "I never, ever, use calculus in my work". I kind of thought the entire engineering curriculum was going to be "not calculus based".
Already as a Freshman at university I had realized that there was something strange going on with the curriculum. It just didn't seem to be adding up to anything meaningful or even particularly useful. When I saw a flyer advertising a California State study abroad program for a year in Germany I was intrigued. I had wanted to study German in high school, but only Spanish and French had been offered. With the spark of inspiration from the flyer I signed up for my first university level foreign language class. My friends said I was nuts, why are you switching from Spanish to German? Some of us had been to Mexico one long weekend, and I had amazingly been able to pull off some pretty big feats with just two years of high school Spanish classes, like talking our way out of being arrested without a cash bribe when my friend had made an illegal U-turn. A girl with us who had taken three years of high school Spanish wouldn't say more the "hola". None of us had ever been to Mexico before, but I had spent time in western Canada listening to ridiculous attempts on the part of westerners at speaking French with visiting easterners.
Again the vast majority of students in the first German class had already taken a year or more of high school German, but there were others like me who were starting from zero. For us newbies the vocabulary was tough. It was just stacks and stacks of flash cards that had to be memorized right away to have any chance at the grammar lessons and discussions in class which very quickly were nearly entirely in German.
Friends of mine kept saying, "Oh, it must be easy for you since your family is German". Yea, right! All four of my father's grandparents might have all been immigrants from Germany, but they were all long gone with 30 years between the generations. Exactly three phrases came down through two generations of non-German speakers, and as I discovered they were botched pretty bad. Aside from the standard German words that everyone knows like "Kindergarten and Gesundheit" the most common word used in my family was in relation to a giant eight inch wide and four foot long carved wooden spoon hanging on the wall. It was called the "Suffluffel", pronounced exactly as that spelling would be in English. Stuff as in stuffing a turkey and luff as in sails luffing into the wind. The other two phrases were longer, but mostly just as nonsensical and they were not introduced to me until I had already started studying Spanish in high school. One was "Ist dass ist ein?", which was said to mean "Is that?...Yes it is..." I could see right away that this supposed translation was bogus, the four words with one repetition obviously could not convey that meaning. The third one was longer still and seemed more likely to be valid. "Sauerkraut und Speck macht die aldi wider fett.", and this was said to mean "Sauerkraut and ham makes the old woman fat". I later figured out what all three of these phrases had originally come from, but at first I just had to ignore them as they obviously were severely garbled. I also had to consciously ignore Spanish at first. Right at first in the first few weeks I was constantly mixing up Spanish and German vocabulary. Not only was this interfering with learning German, but I couldn't reliably say anything in Spanish anymore either without stumbling and starting to say German words. The first foreign language is difficult, but the second one requires a whole different approach. I had to just consciously push Spanish out of my mind to make room for German. Years later I was again able to use Spanish a bit, but at first I had to just replace it with German.
It turned out that "Stuffluffel", was Stoffloeffel, which is a reference to force feeding Geese and has severely negative connotations. "Ist dass ist ein?" is probably "Ist dass ein...Ja, dass ist ein..." which itself is mostly nonsense. "Sauerkraut und Speck macht die aldi wider fett." is most surely "Sauerkraut und Speck macht die alte Witwe fet", which I have heard pronounced as "Sauerkraut und Speck macht die alte Witfer fett."
What had come down through my family correctly is a description of there being multiple versions of what they called "low German" in addition to the standardized language. This was also emphasized in the first university German classes I took, but only as an academic study. Our instructor was from Northern Germany, and she had left as a young woman in her 20's. She wasn't an academic either, but she had gained an interest in foreign language accusation when she was faced with the challenge of actually using English all the time. She had started working as a German language instructor, and had gotten so good at it that the University eventually hired her to teach the lower level classes. The program was run by a German academic who did have a doctoral degree in language learning from a German university, and she was the head of the entire foreign language program at that California university including Spanish, French, Portuguese, Italian, Russian, Japanese and others. There was no Chinese program though, I guess she didn't know much Chinese.
The requirement of the California State study abroad program was one year of university level language classes with B grades or higher and an overall high grade point average as well as Junior level standing. I met all of these requirements just barely, except the high grade point average which I met with flying colors even though I was in the College of Engineering which overall tended to have very low grade point averages what with professors refusing to give anyone an A or a B if he thought none of them were up to some arbitrary level of understanding.
Again I was advised that the basic one year language requirement was not realistic, and that most applicants actually had two full years of university level German classes in addition to two, three or even four years of high school German. By this time this argument was getting old. The problem was that the curriculum wasn't going anywhere, not that I wasn't able to perform academically. By spring of that first year of German language classes I had decided that I was going to Germany. I had gotten straight A grades in the German classes, and conversationally I seemed to be able to keep up with the second year students other than a continuing lack of vocabulary. The entire grammar of the German language had already been covered by the end of that first year, and I was able to use it fairly reliably both conversationally and in written papers and exams. I knew that I lacked vocabulary, but that had always been my perspective on foreign language. I had always severely lacked vocabulary in Spanish also.
What appealed to me about the California State study abroad program was that it was advertised as working with any form of financial aid and that credit would apply towards non-language majors. I was the only Engineering or Science student accepted to the program, and no one in the administration of the program had any engineering or science background either. I knew that I was not going to be able to keep up with real engineering classes in German, but I figured I could get a few of the general education and support classes out of the way during the year long program. It seemed there was no way that the German universities could do a worse job at the science support classes than what I had been seeing so far.
Again it was a huge mistake. I couldn't understand anything that the science professors said, I didn't have any of the vocabulary in German. The language part of the program was excellent. It started off with several months out in the countryside at a small rural language institute where people from all over Europe and the world came to pay handsomely for professional instruction in the German language. Most of them had already been speaking German for many years, so the level of instruction was sort of "expert level". I still felt like I severely lacked the vocabulary required, but the formal instruction was really very good.
Shortly after the program moved to the university where it was based I began being called a "Germanistic" student, which is an antiquated German term for the scientific study of the German language. I thought this was silly, as I wasn't studying the language like that at all. I was just trying to figure out how to talk so that I could study Engineering, but it was a complement.
The University level language program was excellent also, but it was only for us Californians. One of the professors was a full time German professor at the University who had become interested in foreign language acquisition, and the other was an instructor who had worked in California for many years before returning to her native Germany. All that was great, we became experts in German language, literature and related fields, but I was lost at a University that had only a very limited Engineering program. Instead there was a large Theology department and a world renowned Physics department. I signed up for what I thought was the next Physics class. It's pre-requisite was a year of experimental physics classes, which I had already done in California. It was beyond what was required for an Engineering degree in California, so I didn't have to worry about getting the credit to transfer over.
As it turned out it was just impossible to do that kind of stuff in a foreign language. I had no idea what the professor was saying in the lectures, and I had a hard time following the mathematics he wrote on the black board for a variety of reasons. One was that I didn't know what he was saying most of the time, again the notation was somewhat different than what I had seen before, and it was actually an upper division theoretical physics class for physics majors. When I spoke with the professor he offered to help me in his office hours in both German and English, and he also offered to recommend English language books on the subject. The only part of his advise that I took was his recommendation to not buy all the expensive German language books that the other students in the class had. As it turned out that was the part that was the biggest mistake. I didn't want extra help in English just because it didn't matter how well I did. I was doing it just to see what I could do in German. Not having the same books that the other students had though was a huge disadvantage. I really should have bought those same books that everyone else was using. I had plenty of money, as the California based estimates of how much it would cost to live and work in Germany were gross exaggerations. No pun intended. The German students were for the most part living on small federal government stipends, and everything was very cheap.
A good (usually, but not always) meal at the cafeteria without a drink was under one dollar, and bottled beer at the supermarket was about 15 cents for a half liter if you returned the glass bottles and the plastic case they came in. A student buss pass for the entire city was practically free, and even a regular bus pass like everyone else used to get to work was very inexpensive by U.S. standards. Even though it was a modest sized University town busses came every five or ten minutes on the main routes, and sometimes during the rush hour periods busses on the same route followed each other a minute apart. On weekends, at night and on peripheral routs though it was necessary to have a schedule and plan ahead, as it could be hours before the next bus. With a large portion of the population using the bus system as their main means of transportation it was cheap and worked well. Because most people only used the bus to get to work on weekdays though that was when it worked well. On weekends ridership was low, the schedule was dramatically abbreviated and getting around at best required careful planning.
Fresh food at grocery stores, meat markets, bakeries and Saturday farmers markets was also very cheap and good, but the packaged foods were extremely expensive and mostly inedible. Restaurants were also rather inexpensive, but still somewhat out of reach for students as a regular means of sustenance. Hence the federally subsidized university cafeteria. A really nice Italian restaurant run by an immigrant from Italy would cost a small group about $10 a person plus drinks. It was cheap as a form of entertainment, but not what students could afford day in and day out. A half liter beer at a bar or restaurant was about $2 to $2.50 and up to $3 in some places, and a 0.30 liter Coca-Cola was about the same price. Since I didn't drink caffeine this left just beer or even more expensive bottled "Mineralwasser". I did develop a habit of sometimes drinking Coffee socially while in Germany, but mostly I drank only the city water which was bad but not as bad as Coffee, Coca-Cola or beer. I also bought large five liter non-returnable plastic bottles of plain water at the grocery stores sometimes, but they were expensive, difficult to move around, and an insulting waste of plastic.
The beer was mostly pretty good. The cheap stuff from the grocery store was sort of like what you got out of a soda machine, only much cheaper and somewhat better. Yes, if you found a soda machine in Germany in the 1990's it nearly always had beer as one or two of the choices. Sometimes three choices of beer in half litter cans, three choices of very strong tasting mineral water in smaller plastic bottles and Coca-Cola in small aluminum cans. I hadn't previously been much of a soda drinker, so the lack of choices other than Coca-Cola wasn't as alarming to me as it was to other Californians. I was however a bit confused about how everyone denied drinking anything on a regular basis. Older Germans I knew personally and spent time with in their homes tended to drink a lot of the expensive bottled "Mineralwasser mit wenige Kohlensaeure" mixed with fruit juce, red wine, coffee or even beer to cut the strong taste and rather high acidity.
By far the cheapest bottled beverage in Germany was beer by the case from grocery stores. This was great for having a dinner party on the cheap, but not so good for staying hydrated on a daily basis. For a Saturday, Friday or even Thursday dinner party a case of beer, a kilogram of dried pasta (about the only packaged food that seemed at all edible), a can of tomato sauce (about the only canned food that seemed at all edible), "drihundert gramm" pork sausage and variety of fresh local vegetables such as onions, tomatoes (in season), green bell peppers (in season), squash and/or whatever else happened to be available and seemed to go with tomatoes made for a very cheap way for a group of four to six to eat, drink and be merry. The gallons of water to drink in the middle of the night and in the morning to mitigate the hang over from drinking too much beer though was harder to come by and more expensive.
I continued to drink several liters of water every day as I always had, but it was sometimes hard to get. I carried a "stolen" returnable plastic one liter soda bottle that I filled at home from the tap, but if I was out walking around and ran out of water when the stores were closed (which seemed to be always) I would have to beg at a restaurant for them to fill my bottle for me. A very degrading experience.
The local breweries made much better beer, but it was usually only available from bars and restaurants at the much higher prices. Nearly everyone seemed to go to the bars to have a beer. For most people it was about a once a week outing, and there were constantly jokes about which table would have who sitting at it which day of the week.
Living in Germany was interesting, challenging, entertaining and generally a positive experience, but I still didn't find any meaningful engineering curriculum. I knew a few engineers in Germany, an older guy that drove a substantial distance several times a week to the Volkswagen plant in Stuttgart, and a young guy who had just graduated with a mechanical engineering degree, but who had already been working in his father's small buissness for many years before he went to the university. I met with both of these guys quite a number of times, but we didn't have much to talk about other than bier, Sunday adventures to restaurants etc. I didn't know any of the technical vocabulary in German so casual conversations strayed to other topics. These German Engineers seemed to want to talk to me in English, but I avoided this for two reasons. One was that I was in Germany to learn German, so I avoided speaking English in all situations. And the second was that my technical knowledge, even in English, was more of an industrial floor, shop based practical understanding which I wanted to transcend and build upon to become a professional engineer.
I did talk to both of these German engineers a bit in English about physics and "not calculus based" engineering, and it seemed to just make them uncomfortable. I guess it was the fact that I did have a pretty good understanding of engineering in English and could speak German rather well but didn't know any of the technical vocabulary in German. In the German language I probably seemed like a partially retarded liberal arts major who was well spoken and knowledgeable but slow to understand spoken language and somewhat prone to confusion. Where in English I was a successful academic with substantial engineering knowledge.
Both of these engineers had learned English through the field of engineering and because of the field of engineering. They knew lots of vocabulary, especially technical vocabulary, but weren't all that good at actually talking in English. Mostly sticking to short sentences and stings of phrases to avoid making the grammar mistakes that they were aware they were prone to. I on the other hand had learned German only through the German language education program, and I was pretty good at it, so I could speak in complete sentences with correct grammar and I could also understand the nuances of speech and written German. I just didn't know any technical vocabulary and sometimes I got totally lost and had no idea what people were talking about on any subject. I could read anything in German, but the more words there were that I didn't know the slower it got because I had to not only simply look the word up in a word list, but I also had to figure out how it was actually used. Sometimes this was very slow, requiring several books. I couldn't do that when casually talking to people, so if there were too many words that I didn't know I would lose the trend of thought and just be totally lost. To be honest it was painful to talk to the German engineers in English, it took forever for them to get ideas across, and then when I spoke I was never sure if they understood the grammar or not. And these were people who used English on a regular basis in their work. They said that large portions of corporate engineering work in Germany was done in English, especially when it involved business partnerships with companies anywhere outside of the German speaking world. Speaking to these two German engineers in German worked fine, we just didn't have much to talk about.
Back in California the reason I finally just graduated with a German language degree was that some of the classes for my concentration in energy systems were going to be unavailable for some unknown length of time when one key professor moved to Turkey. Instead of the high paid and prestigious staff engineering position at a large corporation I had been expecting I just bailed with a German language degree that I had nearly finished anyway as a second major. I also got out with only a few thousand dollars in student loans, as I had stopped taking the interest deferred loans years earlier and instead drove nearly free junker cars that I nursed along and I lived in about the smallest and cheapest apartment in town cooking nearly all of my meals myself. For entertainment and recreation I rode my street legal 1991 Husqvarna WMX 610, which was cheap aside from tires because no replacement parts were available anyway.
As a point of clarification I need to add that I did cook all of my own meals as a University student before I went to Germany also. That wasn't something I picked up in Germany, although many of the other Californian students did apparently find that they were forced to learn to cook for the first time when they moved to Germany.
So that should add up to an ability to integrate across the cross sectional area of a structural member and put the functions together to model both the deformation of the material and the tensile load on each portion, right? Instead though I just came up with my simplified tensile zone analysis technique and used that every time I needed to predict the breaking strength of a bending loaded structural member. Not in school mind you, I was never expected to do anything that meaningful as an engineering student. It was later when I started building big difficult things that I first applied my own analytical techniques. Partly because I could not remember much of what had been taught in engineering school, and also because I had developed a distaste for everything that was mathematics intense but "not calculus based".
Going a few steps further I can say what sort of iterative analysis might be used to model a structural member. First of all an estimate of the deformation across the cross sectional area would be needed. As a first approximation the deformation could be assumed to be linear. The outside portion stretches twice as far as the middle. This approximation of deformation would then be used to develop a stress distribution. The top part would have twice as much tensile load than the middle. The next obvious conclusion is that the more heavily loaded outer portion carries a disproportionately larger amount of the bending load than the lightly loaded central portion. Not only is the outer portion deformed more to have a higher tensile load, but the lever arm for the top portion is also longer. The contribution of each bit of material to resisting the bending load is it's tensile load multiplied by it's lever arm.
Then there is also compressive loading. Every part of the structural member has some compressive loading, but the compressive loading of the outer portion on the tensile side is of course much lower than the compressive loading of the compression side. The farther in from the outer surface on the tensile side the higher the compressive loading. The much higher compressive loading of the middle portions tends to mean that they are not able to support as much tensile load. This results in a structural member breaking more "clean through" than might be expected just considering the tensile loading alone. The area just below the tensile zone also breaks because it is under much higher compressive loading that the outer portion of the tensile zone. Generally though compressive loading can probably be mostly ignored for the tensile zone of steel beams.
Where compressive loading is significant though is around the pivot point. The compressive loading means that the material deforms somewhat due to compression. Steel generally stretches out more than it will compress down, but some compression deformation does occur. This compression deformation tends to actually reduce the tensile loading on the more central portions of the structural member that would be expected to carry a portion of the bending load.
The conclusion I have always come to is that this all adds up to a rather small and isolated portion of the structural member carrying most of the tensile load. More precise analysis is possible of course, but it has never seemed worthwhile. Just getting within about 20% in an estimate of the load carrying capability of a structural member has seemed extremely useful in many situations. One further refinement in the tensile zone analysis would be to integrate a uniform load and lever arm distribution over just the tensile zone itself. It would still be assumed that the bulk of the tensile load is carried by an isolated tensile zone, but within that tensile zone the tensile loading would be larger towards the outside and smaller towards the central part of the structural member. Likewise the lever arm would be longer towards the outside and the lever arm would be shorter towards the central portion of the structural member. That is actually a very simple and easy integration task. If a symmetric six inch I-beam with one inch by three inch upper and lower rails is considered then the approximate breaking strength is very easy to approximate. Using my simple tensile zone analysis method the lever arm is five inches and the tensile zone is three square inches. For 100,000psi (rather soft) structural carbon steel the bending moment capability is 125,000 foot pounds. Integrating over the tensile zone for tensile load variation with the same arbitrary 5 inch lever arm reduces the bending moment capability estimate to 114,000 foot pounds if the maximum tensile load at the outer portion of the tensile zone is assumed to be the same conservative 100,000psi. This is however not realistic to integrate over the tensile zone only for tensile loading. It is necessary to integrate over the tensile zone for both tensile loading and lever arm, which yields a bending moment capability of 121,000 foot pounds if the maximum tensile load at the outer portion of the tensile zone is assumed to be the same conservative 100,000psi. The interesting thing here is that integrating over the tensile zone for both the tensile load and the lever arm yields a bending load capability remarkably similar to using just my simple tensile zone analysis which can rather easily be done with no paper and no calculator. It is actually not all that hard to do the integration over the tensile zone for both tensile load and lever arm with no paper also, as long as you already know what the integrals are going to look like. Being a bit shaky about how to do the calculus, as often seems to be the case for me after not thinking about it for a few years, it is a rather lot of paper that is required.
Lay two 40 foot lengths of this six inch I-beam across a 39 foot gap and you would be pretty safe driving a loaded 10,000 pound GVWR light truck across. I mean safe in terms of the dynamic loading of the soft steel beams, not necessarily safe in terms of staying on the three inch wide beams for the 40 foot distance.
Bolting or welding cross pieces on the ends of the 40 foot beams to hold them parallel to each other increases the reliability of this "bridge". Adding cross pieces to build up a road surface makes the bridge much more practical to actually use, but the added weight reduces the safe load carrying capability of the bridge. The cross pieces close to the ends don't contribute much to reducing the safe load carrying capability of the bridge, but the full weight of the central cross pieces does contribute to a reduction in the safe load carrying capability of the bridge. This bridge of two six inch mild steel I-beams would actually have at least a 25,000 pound static load carrying capability if the load were evenly divided between both beams. A road surface of ten 4x8 sheets of 1-1/8 Douglas fir sub-flooring would only weigh 1,000 pounds when dry and could easily support well centered loads. Each beam weighs 1,200 pounds also, but these weights are remaining rather low compared to the 25,000 pound static load carrying capability. Adding quarter inch thick by eight inch wide plates on top of the beams to distribute the load more evenly on the wood sub-flooring is only another 500 pounds. Add an inch of asphalt over the entire eight foot wide roadbed to protect the wood from sustained heavy use and it is 2,700 pounds more weight. Again though all of these weights are for the entire 40 foot bridge, so the total load carrying capacity still remains substantially high. Driving a 10,000 pound truck across is still a pretty safe bet, and more like 20,000 pounds would be likely to make it across without failure of the beams.
The next question of course is how much this flimsy bridge will sag with a 20,000 pound truck in the middle. The answer is a lot. Perhaps as much as three and a half feet, and this three and a half foot sag is going to pull the ends in to where the 40 foot bridge is barely 39 feet long. Obviously this is not practical, but the rather soft 6" carbon steel I-beams are strong enough to do it.
Modeling the sag of a structural member is actually a very complex process. The above estimate is just based on a visual approximation. First I assumed that the steel has a modulus of elasticity of 30,000,000psi, which is usually assumed to be pretty typical of most structural steel. This would mean that with a full 100,000psi load on the entire 40 foot beam the upper rail would be 40 feet plus 0.13 inch long. The lower rail might compress by as much as 0.13 inches also, although it is normally a smaller amount of compression than elongation. Of course the full 100,000psi load does not extend anywhere near the entire length, so the actual total difference in rail length is likely to be less than 0.18 inches. That is the easy part. The hard part is modeling what shape the beams will take. A uniform beam with a single point load in the middle takes a uniform parabolic shape. This is not however the shape that real beams take. The weight of the beam itself is distributed over the entire length, and other loads normally are also somewhat distributed as opposed to being concentrated in the middle. The distributed loads result in the beam taking a more circular shape.
In the case of the flimsy I-beam bridge the bulk of the load is just the truck, but even that is not a single point load but is distributed between the front and rear axles. If it is a 20,000 pound truck it likely has at least three axles separated by a distance of more than 20 feet. Of course much of the load may be concentrated on the rear two axles with considerably less on the front axle, but the result is still something considerably more distributed than just a point load. The nearly 7,000 pound weight of the beams, wood, 1/4" plates and asphalt is entirely distributed, so the flimsy bridge certainly does take a significantly more circular shape than would be the case for pure single point loading. The more circular shape actually results in less sag and less shortening of the bridge for the same 0.18" difference in upper and lower rail length for the 40 foot bridge. Three and a half feet of pure parabolic sag would result in 10 inches of shortening of the 40 foot bridge, so a more circular shape clearly would result in somewhat less sag and considerably less shortening of the bridge. Perhaps as little as three feet of sag and seven inches of shortening of the bridge with a 20,000 pound truck in the middle, but that is just a guess.
Once some sort of an estimate of tensile loading is attained a shear and bending moment diagram is a very useful way to check how shear loads may relate to a particular structural member. The shear load is usually assumed to be uniformly distributed across the entire cross section of the structural member, and this is a useful assumption for using shear and bending moment diagrams. It is not actually true that the shear load is perfectly evenly distributed across the cross section of the structural member, but it is pretty close to this. Close enough to be a very good estimate. When the shear load is divided by the entire cross sectional area of the structural member the shear loading of the material usually ends up being much smaller than the tensile load in the tensile zone. In some cases this small shear load can be significant though, so it is very useful to be able to figure out just what the shear load actually is. Somewhat ironically it is in shorter and fatter structural members that shear loading tends to be more significant, where in long slender structural members shear loading is often essentially insignificant. Where shear loading tends to be most significant is in multiple span beams where the beam is supported at intermediate points between the two ends. Where shear loading is least significant is in a long slender structural member supported on only one end.
It is also usually assumed that the total load any bit of material can take is the product of the shear load added to the tensile load. This may not be entirely true either, but it is a useful approximation that is usually considered to have good predictive power. Just how accurate this approximation of adding shear and tensile loads together happens to be depends on the material.
Because the shear load is distributed over the entire cross sectional area and the tensile load is concentrated in the tensile zone the shear stress on the material is usually quite low. This means that shear loading ends up being mostly insignificant in the vast majority of all situations. Since shear loading is usually essentially insignificant there is rarely any problem with assuming that shear and tensile loads are additive, even if reality may be somewhat different than this.
There are however situations where shear loading is significant, so it can't just be entirely ignored. It is necessary to verify in some meaningful way that shear loads are nearly insignificant for a particular situation. This may be as simple as saying "long steel beams have very light shear loading of the material", end of story. If on the other hand a multiple span steel beam is heavily loaded near the supported points and lightly loaded in the middle of the spans then shear loading can become more significant.
Modeling unusual materials can also result in unusual shear loading realities. Dirt, clay and concrete are examples of materials that behave much differently under shear loads than steel. Concrete is very heavy, so it puts large shear loads on beams. Concrete also happens to have a large shear load carrying capability compared to it's very modest tensile strength. Add steel re-enforcements to concrete though and it's tensile strength increases dramatically. The last word is that shear loading can be significant for concrete structures, and that means that the high shear load carrying capability of concrete can't be ignored.