Que :- 1. Differentiate Static analysis and Dynamic analysis with proper representing pictures (30 marks) Ans :-When I was doing my first civil engineering design I hardly thought about dynamics. Static analysis was “all there was” for me. And to some degree, it might have been even justified back then. Now,when I understand…
Que :- 1. Differentiate Static analysis and Dynamic analysis with proper representing pictures (30 marks)
Ans :-When I was doing my first civil engineering design I hardly thought about dynamics. Static analysis was “all there was” for me. And to some degree, it might have been even justified back then. Now,when I understand a bit more, I would like to take you on a trip! We will learn about the differences between statics and dynamics! The main difference between static and dynamic analysis is TIME! If the load is applied so slowly, that inertia effects won’t play a role, all you need is static analysis. Dynamic analysis handles impacts and other “fast” happening situations, but also vibrations (which happen in time). But of course, there is implicit and explicit, and all the exciting stuff! So let’s get rolling!
-: Starting Slow :-
This part will be short (and slow!) because it’s about static analysis.
The basic idea is, that the load you have applied to your structure just is there. Furthermore, it could have already been there those precious seconds after the Big Bang! In short, it doesn’t matter how
this load “got there”, but it’s there now and it won’t change later. If you want a slightly more scientific description, this more or less means that the load is applied extremely slowly! So slow in fact,
that the speed of the load application can be omitted!
Static analysis in a nutshell: It doesn’t matter how you apply the load. Solver assumes that this is happening extremely slowly. This means that the way you apply the load has no impact on structural behavior. The load is not changing in time – it is just there… and that’s that! Structural response to the static load CAN differ in time (you know, things like creep, relaxation, etc.). Engineers usually view such analysis as more specialistic, and not “simple static”. Still, structural response changing in time is an option in static design. The fact the load is not changing doesn’t mean that the structural response is linear! All sorts of fun things can happen! If the load is high enough it can cause buckling, yield and all sort of other cool things. This, however, doesn’t change the fact that the analysis is “static” in nature! Without a doubt, the static design is really popular. In fact, in Poland when someone will do calculations of a structure they would say that they are going to do “static design” or simply “statics”. I know the same is true in several other countries as well. This is mostly because static is much easier to calculate than dynamic, and requires less sophisticated software to do so! This also means that people will prefer doing statics. Which in turn leads to something pretty interesting and that is…
-: Static Load Equivalent :-
You see, “back in the day” it was almost impossible to calculate impact, etc. Simply put software was to “weak” to do so. I assume you could do such things at Universities, etc. But in a typical structural office, it was out of reach – at least in civil engineering. But of course, this doesn’t mean that impacts didn’t happen! There were stone crushers, things were thrown out of trucks on structures (in gravel plants and similar facilities) and myriads of other things (including even a car hitting the building you are designing). But how people handled those if they couldn’t perform the dynamic analysis? well, they increased the static load!
Static load equivalent :-In essence if something will impact our structure you may not have to calculate the actual impact. I know it would be super cool to do such things! But often you don’t have the software, and more importantly, the time to do such analysis. This is where static equivalent of dynamic load comes in play. The idea is simple: just increade the load with the “dynamic factor”! Then you can treat it as static load in your analysis. I saw various “dynamic factors” in my career, starting from a humble 1.5 and going up to around 10. If I would have to give the most popular value it would definitely be 4.0. However, 2,0 would be close behind it. Of course, the value of “dynamic factors” depends on the industry and what you are trying to do. Often times, those were estimated and then “passed along” for decades. The origins of many of the values are long forgotten. But it doesn’t mean that this approach doesn’t work! Far from it! I think that most civil engineering structures with impact loads were designed this way! This approach is so popular, that many manufacturers of various technical equipment supply this information in their datasheet! For instance, you get the machine drawing (to know how to connect it to the structure, etc). Usually, this drawing contains machine weight (the real one) and the “static weight” you should use in design. If the machine can cause horizontal loads, the manufacturer should provide those as “static equivalent” as well. Cool huh! You will also get the frequency of the machine, along with all of the above. And this nicely leads us to another part of this article!
Dynamic Or Not Dynamic - Vibrations :-
This is where things start to be a bit more interesting. I think it’s obvious that “vibrations” are a dynamic thing. But, you most likely won’t need all the fancy stuff to analyze vibrations! This is the realm of “linear dynamics”. In essence, you can use modal analysis to predict vibration modes of your structure (as long as the structure itself behaves in a linear way). And what is interesting is, that loads in this analysis don’t change in time. In fact, the solver will “change” the loads you select into the mass of your model. It will simply ignore the rest of the loads! So there is no “load changing in time” component yet! You may think about modal analysis as about Linear Buckling Analysis (LBA) of dynamics! It’s there, it does help, but it’s not the pinnacle of human achievement in the field!
Modal analysis :-Modal analysis allows you to predict natural frequencies of your linear model. This way, you can check if you may have vibration problems. Of course, you don’t want the applied load frequency to be close to the one you’ve got for your structure in modal analysis. If that is the case, it’s better to be careful, since your structure can enter resonance, and this hurts! Resonance is of course dangerous. It’s the situation where the amplitude of vibrations increases A LOT! I don’t want to say that it increases to “infinity” because of the damping. But still, it increases enough to destroy your structure if the source of vibrations isn’t shut off quickly enough!
When I was a student I was in a building during resonance once. And nothing really bad happened, so I live to tell the tale. In Wrocław, the civil engineering department is a 10 story building, and they were making a parking space behind it. Since they needed to compact sand they used those small “hoppers”. You know the small machines that basically “jump” up and down to compact the ground below them. It so happened that the frequency of the hopper “jumps” they used matched almost perfectly to the natural frequency of our building. So naturally, after some time the building started to
shake! I was in the lecture on the 1st floor so it wasn’t so bad. But people from the 10th run down in panic on the stairs (they were afraid to use elevators). Luckily someone realized what was going on! They run to the guys doing the parking space and ask them to take a break… and things stabilized! But I also helped in the design of repairs of the steel structure that entered resonance! This time the source was some technology thing, and before they shut it down, many of the welds and bolts cracked. Luckily the crew was wise enough to run off and shut down the machine by killing the electricity from afar! So yea… you may want to pay attention to natural vibrations, and modal analysis will help you here. Sadly, it’s pretty costly to change the natural frequencies of the structure once it is built! So it’s better to pay attention!
High - End Shaking :-
Of course, modal analysis isn’t all you can do. It’s more or less the beginning I would say. Just as LBA in buckling, the modal analysis doesn’t paint a “full picture”. In fact, you can do a forced response analysis, that would be an equivalent of nonlinear buckling (just to drag the LBA analogy a bit more). This would be a type of more advanced analysis, and I won’t have anything against calling it “dynamics”! In essence, you define the loads/accelerations that are applied to the structure. Those loads/accelerations change in time of course! Many programs have some “historical earthquakes” already implemented there. But you can let your imagination run wild as well (i.e. to fit your load to a certain machine, etc.). Then you run the analysis and see how your structure will react to such loading changing in time.
Forced response advantages:- Of course, forced response analysis is more difficult and time-consuming to perform. So it has to have some advantages because no one would use it otherwise! This is the main one: Forced response allows you to see, how your structure reacts to various frequencies at once. Modal analysis tells you what are the “clean” frequencies your structure will vibrate in (eigenmodes). And what if in reality, you will have more than one frequency present? You also learn how your model reacts to vibrations that have different frequencies than eigenmodes. The fact that frequency of the vibration doesn’t “hit” the eigenmode doesn’t mean that you can simply ignore it… There is a trick here. If you have a load, that changes 20 times in 2 seconds… you could actually calculate 20 different linear static analyses (for each different load). The “animation” between those 20 static analysis outcomes may look as if you really did forced response analysis. I’ve seen people who claimed such a thing… This is of course not what this is about! In “real” forced response analysis your structure will vibrate even after the load ends. Since you usually define just a few seconds of the load, it’s simple to check! Just see what is happening 3 seconds after the load ends. And it’s shouldn’t be “nothing”, unless you have some seriously crazy damping! You are expecting that the structure will still vibrate because of the load you applied before.
Dynamic Or Not Dynamic - Fatigue :-
So we explored vibrations so far, tackling the idea of loads changing in time in force response analysis. Another interesting phenomenon that you can meet in your designs is fatigue! Here, the situation is literally opposite to the one I in modal analysis! The load changes in time, but we will do static analysis to handle it (in most cases at least)! I’m mentioning it briefly here since I don’t think most people would classify fatigue as “dynamic”. Although fatigue is often associated with vibrations. But since I used “load changing in time” as my dynamic definition it’s only fair to mention it here! The idea is, that repeating load cycles can cause cumulative damage to the material . So it’s not only about the stress that is higher than yield. Rather, that 130MPa of stress in S235 steel changing from tension to compression may cause failure over time as well! The thing is, that usually, load cycles are “static in nature” and they don’t happen very fast. This means that there will be no “inertia effects” in your analysis. And in such a case, it’s “murky” to classify fatigue as a dynamic problem. Simply put you will solve static cases to see the maximal and minimal stress in any given place. Afterward, you perform the fatigue checks “outside of FEA” (with scripts or even hand calculations). Those static cases usually will be linear, unless low-cycle fatigue will be considered. In such a case, you have to include yielding you our analysis. Of course, it can happen, that the cycles are “dynamic” in nature with vibrations. That would be a nice mix of forced response and fatigue case in such a situation! Definitely, we are getting closer and closer to the analysis of the actual dynamics. But before we start there is one thing I should mention, and that is “dummy time”. You see some solvers (like Adina used in SOL 601 of NX Nastran) want you to define time even in static analysis! It is seemingly stupid, but this is only a way of introducing the “load factor” into the analysis. While it may look like your load “depends” on time it’s not the case! Time (in such a case “dummy time”) is only used as a “counter” or load multiplier if you prefer. The idea is simple: define how the load changes in the “dummy time”. Usually, you want a linear dependence. You know if dummy time is “0” then the Load is “0”. Of course, you also define maximal dummy time “X” when the load has “full value”. Just be aware that in the case of dummy time “X” means… some measure of dummy time. It is not in seconds (nor any other time unit). Dummy time is just a measure of how much load is applied. In essence when dummy time is equal to “X/2” then 50% of the load is applied.
The best way to think about dummy time is as a load multiplier. In fact, there is only one difference. The load multiplier for the load you applied is 1.0. This means that the load multiplier of 0.5 always means that 50% of the applied load. With dummy time it’s not the case! You can actually say, that maximal load is applied when the dummy time is equal to whatever you like. Of course, it is still wiseto use just to avoid mistakes. But technically you can use a dummy time of 2346 to represent the “full value” of the load you applied. In such a case, at dummy time 2346/2=1173 you will get 50%of the load. Whenever you play like this with dummy time never forget that the solver uses it to iterate the solution. So when you change the dummy time at max load from 1 to let’s say 100 that is not all! It would be wise to adjust “dummy time stepping” as well: Using dummy time Let’s assume we have a load of 1000kN. We want to apply it in 100 equal steps of 10kN each. Firstly the “simple case”. Let’s say that when the full load is applied (1000kN) we have the dummy time = 1.0. In such a case, each step should be 1/100=0.01 units of dummy time to get our 10kN per step. However, we may want to have the max dummy time = 300 units when the load is 1000kN. In that case, each increment of the analysis should be 300/100=3 units of dummy time. In essence, we still get 10kN in each step. It is easy to forget changing the “dummy time stepping” in our analysis! This is why it’s best to stick with dummy time = 1.0 for the full load. Just to avoid weird mistakes! Dummy time has its uses, even though they are not as “grand” as you may think. I consider it a “perk” of certain solvers. Some use it, some don’t but in the end, all work the same. It just a matter of understanding how your solver increments loads. There is only one “benefit” you get from dummy time. You don’t have to toy with “steps” in your analysis when you use it. Let’s imagine you want to do such a multi-step analysis:
Step 1: apply 100% of the load
Step 2: decrease of 50% of the load
Step 3: increase the load to 75%
Step 4: Decrease load to 0…
Normally you will have to set up “steps” in your analysis. Effectively there would be 4 analyses one after another, as described above. But with “dummy time” you can make a single analysis step. All
you have to do is to say that the dependence between time and load is:
Of course, you must say that the analysis should be from Time “0” to Time “1”… and that is it. You don’t have to learn how to make steps or how to restart analysis with other loads. Perhaps this is abit easier this way. That is one of the differences between SOL106 and SOL 601 in NX Nastran by the way. Of course, it’s not all sunshine and rainbows. Dummy time can be “irritating” to understand. Especially if your solver needs it for nonlinear static and you don’t know about it. It took me some time to figure it out for the first time! And even now I forget to set “dummy time” on occasion when I do SOL 601 analysis in my NX Nastran. There is one important thing to remember! The fact that you defined “time dependence” for your loads doesn’t mean automatically that you are making a dynamic analysis! There is a chance this is a static analysis with a “dummy time”. It is always worth checking that in your solvers manual. Sadly, in many cases “dummy time” is described as “time” in your software. It is super easy to get confused! If you are unsure if you are using “dummy” or “real” time test it! Set a time at max load to 0.0001 and do analysis. If you have a static analysis with “dummy time” it will work just fine. If you are really making a dynamic analysis applying load in 0.0001s will cause some funky effects! Most likely you will see impact waves in your model and stuff. It is actually quite possible that your dynamic analysis won’t converge with this setting without some “fighting” for it. Just remember, that if you want to apply a load in 100 steps, each of those steps should be 0.0001/100=0.000001 units of time! It’s easy to forget to change the incrementation settings in the solver!
The True Dynamic Analysis :- Finally, we got to the heart of this. I guess that if you would like to “oppose” static analysis with a dynamic one – this is it! Sure, along the way we have discussed some interesting topics on vibrations, etc. Some of those analyses can easily be called “dynamic”. But the “real” dynamic analysis starts here! The difference between static and dynamic analysis is simple. As I wrote at the beginning, static analysis means, that the load “is just there” and does not change in time (which means it was applied really slowly!). Dynamic analysis is precisely on the opposing side of the scale. Here, we wonder how the load is applied and how fast it happened. We take into account inertia effects and all the jazz. On the left, you see a nonlinear static analysis. Rotation is applied to the handle… and the entire thing just rotates. Nothing fancy really. This is what would happen if you would apply the rotation to the handle very slowly. This is the static domain, the load is applied so slowly, that you can basically ignore inertia effects!
On the right – the same thing… but with a twist! This time I applied the rotation “fast”, which called for dynamic analysis. I actually had to set how fast the 90der rotation will happen (in seconds) during load definition. Notice, that at the beginning handle moves before the tip realizes that there is a movement to be made. Then the tip tries to “catch up” and stuff begins to shake
Que :- 2. Explain the following with relevant force-displacement graph (40 marks)
a.Elastic behavior
b.Inelastic behavior
c.Plastic behavior
d.Non-linear Inelastic behavior
Ans :- A force-displacement graph will have force (in N) on the vertical axis and displacement (in m) on the horizontal axis. The area of the graph is = Fs. This quantity represents the work done on the object. From a force-displacement graph we can: Read forces directly from the graph. Read displacements directly from the graph. Use the area under the graph to find the work done by the force. This is equal to the kinetic or potential energy the object gains due to the application of the force.
Example 1:
The graph below shows the force applied to an object as it was moved a distance of 10m. Answer the following:
a) When was a constant force applied to the object?
b) What force was applied at a displacement of 6m?
c) How much work was done on the object?
a) Reading directly from the graph, a constant force of 10N was applied between a displacement of 0-5m.
b) At a displacement of 6m, a force of 5N was applied to the object
c) To calculate the work done on the object, we must calculate the area under the graph. This is done by dividing the graph into a series of rectangles and triangles and calculating the sum of these
areas;
Area 1
Area 2
Total area
Therefore
a. Elastic Behavior :- Elasticity is the ability of a body to resist any permanent change to it when stress is applied. When stress application ceases, the body regains its original shape and size. Different materials show different Elastic Behavior . The study of the elastic behaviour of a material is of much importance. Almost every engineering design requires knowledge of the elastic behaviour of materials. In the construction of various structures like bridges, columns, pillars, beams, etc. knowledge of the strength of the materials used in the construction is of prime importance. For example: while constructing a bridge, the load of traffic that it can withstand should be adequately measured beforehand. Or while constructing a crane used to lift loads, it is kept in mind that the extension of the rope does not exceed the elastic limit of the rope. To overcome the problem of bending under force the elastic behaviour of the material used must be considered primarily. To study the elastic behaviour of materials let us consider a beam of length l, breadth b and depth d supported at the ends and loaded at the centre by load W.
Using the above equation we can easily say that to reduce the amount of bending for a certain load, Young’s modulus of elasticity of the material used must be large. Also, depth d must be considered since sag is inversely proportional to the cube of depth. But the problem faced with increasing the depth is that bending increases and this is known as buckling. Therefore, a compromise is made between the different cross-sectional shapes.
b. Inelastic behavior :- The Quaternary tectonics of the Japan island arc is characterized by the strong crustal deformation for which the mode and rate differ substantially from those of the late Pliocene. The Niigata–Kobe Tectonic Zone (NKTZ; Fig. 1) has been identified as a high-strain zone in central Japan and shows marked E–W shortening, but there is a discrepancy between the geodesic strain rate (10−7/year; Sagiya et al. 2000) and the geological strain rate (10−8/year; Wesnousky et al. 1982). Comparison of the dynamic response of the NKTZ with the deformation before and after the 11 March 2011 Mw = 9.0 megathrust earthquake (herein the 2011 Tohoku-oki earthquake) based on continuous GPS data has allowed inelastic deformation in the northern NKTZ to be recognized (e.g., Meneses-Gutierrez and Sagiya 2016). Meneses-Gutierrez and Sagiya (2016) showed that the short wavelength patterns of the strain rate on the E–W direction reveal a persistent localized contraction zone with the inelastic deformation around northern NKTZ before and after the 2011 Tohoku-oki earthquake. In contrast to the rapid or instantaneous changes corresponding to the elastic response during a seismic cycle, inter- and post-seismic deformation patterns reflect a cumulative deformation process (Meneses-Gutierrez and Sagiya 2016). Meneses-Gutierrez et al. (2018) explained present crustal deformation in the mid-Niigata region in terms of (1) the contribution from both elastic and inelastic behaviors and (2) the existence of mechanical decoupling between weak sedimentary layers and basement. Those authors suggested that the difference in strain rates before and after the 2011 Tohoku-oki earthquake represents the direct influence of the shallow part of the upper crust in the Niigata region.
Fig. 1
aIndex map of the study area. The area surrounded by the dotted line is the Niigata–Kobe Tectonic Zone (NKTZ; Sagiya et al. 2000). The study area is located in the NKTZ. N: Niigata region.
PAC: Pacific Plate. EUP: Eurasian Plate. σHmax is the maximum horizontal stress (Terakawa and Matsu’ura 2010). b Geological map of the mid-Niigata region, Central Japan (Niigata Prefectural Government 2000). c Geological cross-section along the line X–X’ in b. The Katagai site is located in the eastern limb of the Matto Anticline
Geological Setting :-The Niigata sedimentary basin (Fig. 1), one of the largest rift basins in the eastern margin of the Japan Sea, formed mostly during the early Miocene (Yamaji 1990). Neogene sequences were deposited in rifts that formed during the early Miocene concurrently with the opening of the Japan Sea, and these sequences have been folded and faulted by E–W- to NW–SE-compression since the Pliocene (Sato 1994; Okamura et al. 1995). The sediments in the basin are up to ~ 5000 m thick and have been folded during the last 2 to 3 Myear (e.g., Niigata Prefecture Government 2000). We investigated the mid-Niigata basin (Fig. 1), which contains the following major folds (from west to east): the Sabaishigawa Syncline, the Hachikoku Anticline, the Shibumigawa Syncline, the Yamaya Anticline, the Fudousawa Syncline, the Matto Anticline, and the Ojiya Syncline (Fig. 1). These folds are actively growing in the mid-Niigata region (Otsubo and Miyakawa 2016). The axes of these folds are oriented NNE–SSW or NE–SW, and both longer folds (axial lengths of 10–20 km and wavelengths of 3–5 km) and smaller folds (axial lengths of 2–3 km and wavelengths of 2–3 km) are present. The folds are of kink-fold type with a subhorizontal hinge line and an interlimb angle of ~ 90°. The average horizontal shortening in the mid-Niigata region has been estimated at about 13% (Sato 1989). The studied Katagai site (Figs. 1, 2) is located on the eastern limb of the Matto Anticline. The Katagai site provides a high-quality exposure of a cross-section measuring > 50 m across in the mid-Niigata region and oriented approximately normal to the fold axis of the Matto Anticline (Fig. 1). Sediments deformed by the anticlines range in age from middle Miocene to Pleistocene (Yanagisawa et al. 1985, 1986; Kobayashi et al. 1988, 1991). Miocene sediments are composed mainly of mudstone and turbidites that were deposited in deep- to shallow-marine environments. Pliocene sediments comprise a coarsening-upward sequence that thickens to the west, indicating deposition on a westward-deepening slope along the eastern margin of the rift basin. The Uonuma Formation consists of latest Pliocene–middle Pleistocene fluvial to shallow-marine sediments, and is 2000–2500 m thick. Numerous tuff beds have been identified as key beds in the Niigata region (Kurokawa 1999). Based on the detailed mapping of the key beds and unconformable contacts, Kishi and Miyawaki (1996) showed that the folding activity has migrated progressively eastward since 3 Ma. Growth of anticlines in the study area has continued for < 1 Myear (Kishi and Miyawaki 1996). The east–west horizontal shortening detected with the GPS observation (Sagiya et al. 2000) implies the active shortening in the study area is still ongoing.
Layer - Parallel Slip During Folding :- We identified 36 layer-parallel slip surfaces at the Katagai site on the eastern limb of the northern Matto Anticline (Fig. 2). The outcrop consists of steeply dipping (50°–70°) beds of the Uonuma Formation. The Uonuma Formation is overlain unconformably by terrace gravel deposits, on which topsoil has formed. The Uonuma Formation, the terrace deposits, and the topsoil have estimated at the ages of 850, 150–130, and 8 ka, respectively (Kishi et al. 1996; Hayatsu and Arai 1982; Suzuki et al. 2008). The recognized bedding-plane slip surfaces show a reverse sense of movement at the Katagai outcrop (Fig. 2). Bedding planes that have accommodated layer-parallel slip can be recognized as the planar surfaces where slight roughness has disappeared. Most observations of surface slip were made at the interfaces between siltstone and mudstone layers. Slickenside striations are present on all faults and indicate approximately dip-slip movement (Fig. 2b). Fault movement directions are approximately perpendicular to the hinge of the Matto Anticline. Asymmetric microstructures on fault surfaces (Petit 1987) were used to determine the sense of slip, including the “comet patterns” made by micro-breccia on these surfaces (e.g., Mino et al. 2001). The identified slip surfaces have bedding-plane-normal spacings of ~ 3.6 ± 0.7 m (Standard error of 36 layer-parallel slip surfaces, Fig. 3).
Fig. 3
Shear Experiments To Determine The Friction Coefficient :- We measured the friction coefficient of gouge generated by bedding-plane slip and that of mud around non-slip surfaces (Fig. 4). Double-direct shear tests of four disaggregated samples (KAT01, 02, 03, and 04) on two sliding planes were performed at a constant velocity of 3.0 µm/s under dry and room temperature conditions (see Kawai et al.2015 for experimental details). In all experiments, the applied normal stress was 12, 23, 35, and 46 MPa during sliding, and samples were sheared up at least four times at each normal stress to achieve a steady-state friction. These normal stresses correspond to shallow crustal depths of 0.5, 1.0, 1.5, and 2.0 km, respectively (density of the Uonuma Formation = 2360 kg/m3; Hoshino et al. 2001)
Fig. 4
Friction coefficients of fault gouges (KAT01, 03) generated by bedding-plane slip and of mudstones (KAT02, 04). Double-direct shear tests of four samples (KAT01, 02, 03, and 04) on two sliding planes were performed as in previous studies (Kawai et al. 2015). D: Depth
Figure 4 shows the results of the frictional experiments for the four samples. The friction coefficient (µ) was calculated by dividing the shear stress by the normal stress assuming zero cohesion. The estimated friction coefficients at steady state were 0.74–0.79, 0.74–0.75, 0.70–0.76, and 0.67–0.73 for normal stresses for samples KAT01, 02, 03, and 04, respectively (Fig. 4).
Estimation of effective elastic thickness We estimated the strength of shallow sedimentary units under contraction in the mid-Niigata region using a two-dimensional buckling model. In general, the bending moment, M, of the elastic plate is proportional to the curvature, K, of plate so that M1 = - DKM = - DK
where D is the flexural rigidity of a plate and is the constant of proportionality. Following Pollard and Johnson (1973) and Jackson and Pollard (1988), we employed D as follows
D2 = Yt3e12 ( 1 - V2 )D = Yte312 ( 1 - V2 )
where Y is Young’s modulus, ν Poisson’s ratio, and te the elastic thickness. The value of D was estimated for each plate to characterize the resistance to bending. The quantity te is the thickness assuming that the sedimentary units act as an elastic plate. Here, we consider a plate composed of two thick elastic layers; i.e., a composite plate. In this model, no friction is assumed on the boundary between layers. From Eq. (2), the flexural rigidity of a composite plate is D = Y12 (1-V2)[(te2)3+(te2)3] = Y12(1-V2)2(te2)3 = Y12(1-V2)(te4-under root 3 )3.D = Y12(1-V2)[(te2)3+(te2)3]= Y12(1-V2)2(te2)3= Y12(1-V2)(te43)3. As 1/4 - under root3 = 0.631/43 = 0.63 , a plate composed of two equally thick elastic layers has an effective elastic thickness of about 63% of the thickness of a plate. It is obvious from the above argument that when a composite plate comprises many layers, its effective elastic thickness, te, satisfies
t3e = t31+t32+....+t3n , te3 = t13+t23+....+tn3
where ti is the thickness of the ith layer. The effective elastic thickness decreases as te = n - 2/3 (nt)te = n-2/3(nt) when the plate is divided into n layers with the same thickness t and the same elastic constant (Fig. 5). Therefore, the flexural-shear folding occurs readily in sedimentary piles (Jackson and Pollard 1988). The thickness of the deformed layer in the mid-Niigata basin is ~ 5000 m (Niigata Prefecture Government 2000), and this layer is regarded here as the effective elastic thickness te. Y and ν were as constant during folding. The average spacing of bedding-plane slip surfaces is ~ 4 m at the Katagai site, meaning that the sedimentary rock mass can be divided into n = 1250 elastic layers, whereby the effective elastic thickness decreases to < 0.01% of the elastic thickness of ~ 5000 m using n = 1250 at the Katagai site (Fig. 5).
Fig. 5
Relationship between the effective elastic thickness and the spacing of bedding-plane slip surfaces. In the case of the maximum elastic thickness, the sedimentary mass contains no bedding-plane slip surfaces. In the sedimentary units, the numerous bedding-plane slip surfaces act to reduce the mechanical strength (effective elastic thickness) of sedimentary rocks under contractional deformation. T: Thickness of a bed as an elastic layer
c. Plastic behavior :- A property of various metals that describes their ability to undergo permanent stains is called plasticity. The uniaxial tension test is a convenient method toindicate plastic behaviours of a material. A curve for typical material - soft steel in the system nominal stress-Cauchy strain presented in Fig. 6.1 indicates thefollowing characteristic
points: 1: proportional limit; 2: elastic limit; 3: yield point; 3-4: platform of ideal plasticity; 4-5: plastic hardening; 5: necking point;6: rupture point. The typical curve shown does not describe the character of stressstrain curves for other materials. Fig. 6.1. The stress-strain curve for soft steel Many materials do not give evidence of the existence of a linear segment (annealed aluminium for example). The stress-strain relations for chosen materials are presented in Fig. 6.2. In order to describe the behaviour of material in the best possible way the modeling process is necessary by introducing thermodynamical notions and relations connecting these notions. The behaviour of an elementary system is said to be elasto-plastic, if in the present state and in the space of stress tensors , a domain exists that entirely contains a loading path to which the elastic evolutions are associated. In general, all materials and their mechanical properties are sensitive to temperature and strain rate. Fig. 6.3 illustrates the influence of temperature on flow stress for aluminium. Fig. 6.2. Stress-strain curves of different materials at room temperature [Krause, 1962] Fig. 6.3. Stress-strain curves of aluminium at different temperatures [Stüwe, 1965] In this chapter we assume that the variation of strain depends on the loading history and is independent of the loading rate. The plastic models with the viscosity effects are not considered here and are analyzed in the next section. The plasticity modeling is carried out in the framework of physical linearization and infinitesimal transformations. Since permanent strain is involved, the plastic evolution of an elementary system is irreversible. This evolution can be viewed as the superposition of thermodynamic equilibrium states.
d.Non-linear Inelastic behavior :- Every year, many people die because of earthquakes around the world. Lateral stability has been one of the important problems of steel structures specifically in the regions with high seismic hazard. The Kobe earthquake in Japan and the Northridge earthquake that happened in the USA were two obvious examples where there was lack of lateral stability in steel structures. This issue has been one of the important subjects for researchers during the last three decades. Finally they came up with suggesting concentric, such as X, Diagonal and chevron, eccentric and knee bracing systems and these were used in real life projects by civil engineers for several decades.One of the principal factors affecting the selection of bracing systems is inelastic performance. The bracing system which has a more plastic deformation capacity prior to collapse, has the ability to absorb more energy while it is under seismic excitation. Eccentrically braced frames (EBF’s) are a lateral load-resisting system for steel building that can be considered a hybrid between conventional moment-resisting frames (MRF’s) and concentrically braced frames (CBF’s). EBF sare in effect an attempt to combine individual advantages of MRFs and CBFs, while minimizing their respective disadvantages. Figure 1 illustrates several common EBF arrangements.
The distinguishing characteristic of an EBF is that at least one end of brace is connected to that the brace force is transmitted either to another brace or column through shear and bending in a beam segment called a link. The link length in Fig. 1 are identified by the letter ‘e’. The excellent performance of EBFs under severe earthquake loading was demonstrated on one-third-scale model frames at the University of California in 1977. Soon after thisstudy, several major buildings were constructed incorporating EBFs as part of their lateral seismic resisting systems. The most attractive feature of EBFs for seismic-resistant design is their high stiffness combined with excellent ductility and energy-dissipation capacity. The bracing members in EBFs provide the high elastic stiffness characteristic of CBFs, permitting code drift requirements to be met economically. Yet under very severe earthquake loading, properly designed and detailed EBFs provide ductility and energy dissipation characteristics of MRFs. During the recent decades, nonlinear response of bracing systems has been studied and consequently parameters such as, seismic behavior factor, R, over strength factor, W, and displacement amplification factor, Cd, were introduced to loading codes of practice like UBC (Uniform Building Code) and IBC (International Building Code). These design codes are widely used in the USA and also throughout the world In the process of the earthquake load calculation of a structure, seismic behavior factor is the parameter illustrating the impact of nonlinear performance of the bracing system that is fundamentally affected by the system ductility. The efficiency of bracing systems is influenced by these key parameters because they directly affect the reduction of the earthquake loads in the structure. In accordance with the loading codes, specific R, W and Cd factors were introduced for various structural systems (illustrating the distinction of their nonlinear behavior), such as concrete moment frame and steel moment frame with high, medium and low ductility, steel frames with concrete shear walls and steel braced frames.Producing a frame that can remain substantially elastic outside a well-defined linkage is the most important factor that EBF designing is done on the basis of it. While being undergone huge loading, it is foreseen that the link will be distorted inelastically with great ductility and dissipation of energy. The provisions of codes are provided so that they guarantee the beams, braces, columns and their connections to stay and remain in elastic phase and also the links remain stable. A. Three Important Variables in the Designing of EBF Bracing System,
1) Bracing configuration 2) The link length 3) The link section properties
When these elements are taken into consideration, then the rest of the designing process of the frame can be executed with minimal effect on the link size, configuration or link length. Designating a systematic procedure to assess the effect of the prominent variables is crucial to EBF design. If attention is not paid identify their effect, then the designer may have to iterate through a myriad of probable combinations. The strategy suggested by (Roy Becker & Michael Ishler, 1996) in their guide is as follows:
1) Establish the design criteria.
2) Identify a bracing configuration.
3) Select a link length.
4) Choose an appropriate link section.
5) Design braces columns and other components of the frame.
B. Bracing configuration: UBC 2211.10.2 the selection of a bracing system configuration is related to various elements. These factors encompass the size and position of required open areas in the framing elevation and the height to width proportions of the bay elevation. These constraints may substitute structural optimization as designing criteria. UBC 2211.10.2 requires at least one end of every brace to frame into a link. There are many frame configurations which meet this criterion.
C. Frame proportions: Michael D.Engelhardt, and Egorp.popov, p. 504 (1989) in designing EBF systems, the proportions of frames are typically opted to increase the application of the high shear forces in the link. Frame properties of typical eccentric braces are shown in figure 2 below. Shear yielding is very ductile and its capacity for inelastic behavior is very high. This characteristic, as well as the benefits of frames with high stiffness, generally make short lengths desirable.
The desirable angel of the brace as shown in the above picture should be kept between 35° and 60°. If the angle is beyond or below this range, then it will result in awkward details at the brace- to- beam and brace-to-column connections. Meanwhile small angles are also apt to result in a huge axial force member in the link beams (Michael D.Engelhardt, and Egorp.popov, p. 504) (1989).
D. Link length: The inelastic performance of a link is a great deal affected by its length. If the link length becomes shorter, then the inelastic behavior will become greater as a result the influence of shear forces. Shear yielding has a tendency to occur uniformly alongside the link. Shear yielding has a high ductility and also considerable inelastic performance capacity which is more than that predicted by the web shear area, if the web is braced enough against buckling. (Michael D.Engelhardt, and Egorp.popov, p. 499, 1989). Often the behavior of the links are like short beams which are exposed to equal shear loads applied in opposite directions at the ends of the link. According to this style of loading, the moments produced at both ends are identical and also in the same direction. The shape of the link deformation is like the letter (S), which is distinct at mid span by a point of counter flexure. The measure of moment is equal to 1/2 the shear times the length of the link.
E. Link lengths generally behave as follows: If E< 1.3 Ms/Vs Guarantees shear performance, and are recommended as upper limit for shear links (Egor p. popov, Kasai, and Michael, p. 46) (1978) If E< 1.6Ms/Vs Link post - elastic deformation is controlled by shear yielding. UBC2211.10.4 rotation transition. (“Recommended lateral force requirements and commentary”, p. 331, C709.4) (1996) If E=2Ms/Vs theoretically, the behavior of Link is balanced between shear and flexural yielding. If E<2Ms/Vs Link behavior considered to be controlled by shear for UBC 2211.10.3 ((“recommended lateral force requirements and commentary”, p. 330, C709.3) (1996) If E>3 Ms/Vs By flexural yielding, Link post-elastic deformation is controlled. UBC2211.10.4 rotation transition. (“Recommended lateral force requirements and commentary”, p. 331, C709.4) (1996)
F. Evaluation of nonlinear static procedures: Nonlinear static procedures are recommended by FEMA 273 document in assessing the seismic performance of buildings for a given earthquake hazard representation. Three nonlinear static procedures specified in FEMA 273 are evaluated for their ability to predict deformation demands in terms of inter-story drifts and potential failuremechanisms. Two steel and two reinforced concrete buildings were procedures. Strong-motion records during the Northridge earthquake are available for these buildings. The study has shown that nonlinear static procedures are not effective in predicting inter-story drift demands compared to nonlinear dynamic procedures. Nonlinear static procedures were not able to capture yielding of columns in the upper levels of a building. This inability can be a significant source of concern in identifying local upper story failure mechanisms. The American Society of Civil Engineers (ASCE) is in the process of producing an U.S. standard for seismic rehabilitation existing buildings. It is based on Guidelines for Seismic Rehabilitation of Buildings (FEMA 273) which was published in 1997 by the U.S. Federal Emergency Management Agency. FEMA 273 consists of three basic parts:(a) Definition of performance objectives; (b) demand prediction using four alternative analysis procedures; and (c) acceptance criteria using force and/or deformation limits which are meant to satisfy the desired performance objective. FEMA-273 suggests four different analytical methods to
estimate seismic demands:
I. Linear Static Procedure (LSP) II. Linear Dynamic Procedure (LDP) III. Nonlinear Static Procedure (NSP) International Journal of Latest Trends in Engineering and Technology (IJLTET) Vol. 4 Issue 1 May 2014 275 ISSN: 2278-621X
IV. Nonlinear Dynamic Procedure (NDP)
Given the limitations of linear methods and the complexity of nonlinear time-history analyses, engineers favor NSP as the preferred method of analysis. Following the analysis of a building, the safety and integrity of the structural system is assessed using acceptance criteria. For linear procedures acceptance criteria are based on demand-to-capacity ratios and for nonlinear procedures, they are based on deformation demands.
Que :- 3. Explain Mass, Stiffness & Damping components in the equation of motion (30 marks)
Ans :- Damping of Ship Hull Vertical Vibrations Numerical (deformation) methods compute the vertical vibrations of the Ship Hull using mass, damping, and stiffness matrices. The damping matrix is generally expressed as follows: (5.32) Dij=∫Ld(x)·qi(x)·qj(x)dx d(x) is the damping force per length, divided by the vertical velocity of the cross-section. qi and qj are the shape functions for the vertical deflections of the ship cross- sections. If we use the vibration natural modes for qi, the mass matrix and the stiffness matrix are simple diagonal matrix. The damping of vertical ship hull vibrations is weak. Therefore, the off-diagonal elements in the damping matrix can be neglected and the individual vibration modes can be considered separately. We can estimate the hydrodynamic damping as follows: (5.33) Dii=∫L(qi2+U2ω2(qi′)2)Ndx+ρ∫LcwB|w|qi2dx+U2ω2qiTqiT′NT+UqiT2mT″+ρ4nDPP2qiP2 The first term is due to the waves radiated from the vibrating hull and can be computed in a strip method as for the rigid-body motions. U is the ship speed and qi′ the derivative of the shape function (natural mode) with respect to x. N(x) is the damping constant (per length) for a given strip. N depends on section shape and frequency. For high-frequency radiated waves (with wave lengths much shorter than the section width), we can approximate: (5.34)N(x)≈ρgB(x)tanα2ω B(x) is the local section width and α the flare angle in the waterline measured against the vertical (α = 0°, i.e. N = 0, for wall-sided sections). The second term is due to the pressure resistance of the vertically moving cross-sections. The contribution of the vibrations by themselves is negligibly small, but the interaction with rigid-body motions is considered here by the average vertical velocity |w|. cw is the vertical motion resistance coefficient of the section. Lacking better data, one employs here steady flow resistance values, typically 0.5 < cw < 1 for the midbody sections. At the ship ends, where the sections are well-rounded, cw is negligibly small. The third and fourth terms consider the effect of the transom stern (thus index T, e.g. qiT = qi(xT) is the value of the shape function at the transom). NT is the damping constant, m″T the added mass, both for high-frequency and the transom shape. These terms assume a detaching flow at the transom, similar to that in the strip method. Note that the wetted transom stern in operation (with ship wave system and motions in seaways) differs from that at rest. The third term containing NT is typically much smaller than the fourth term and can usually be neglected. The last term is due to the propeller. qiP is the value of the shape function at the propeller, DP the propeller diameter, P the propeller pitch, and n the propeller revolution (in 1/s). The terms depend on ship speed, motions in seaways and propeller action. Therefore vibration damping in port will be different from actual operation conditions. Besides hydrodynamic damping, material damping and component damping (due to floor and deck coverings) play a role in damping. In the literature, widely different values are stated for damping characteristics and the uncertainty increases for higher frequencies. For simple practical estimates, Asmussen et al. (1998) give:
for ship in loaded condition: ϑ = min (8; 7·f/20 + 1)%
for ship in ballast condition: ϑ= min (6; 5.5·f/20 + 0.5)%
The frequency f is taken in Hz. The degree of damping ϑ is coupled to the logarithmic decrement Λ: (5.35) A=2π·ϑ/1−ϑ2 The logarithmic decrement describes the ratio of two successive maxima: eΛ=A1/A2. Undamped vibration: The type of vibration is undamped if there is no loss or dissipation of energy due to friction or other resistance during vibration of a system. Damped Vibration : The system is considered damped if energy dissipition occurs in the presence of damping components during vibration. Although finding dynamic characteristics is easier and simpler if neglecting damping, a consideration of damping becomes extremely important if the system operates near resonance.
Free Vibration : If a system oscillates only due to an initial disturbance, the system is said to undergo free vibration. Here, no external force is applied after the initial disturbance (or after time zero). Forced Vibration : The system is said to be under forced vibration if the system vibrates due to the application of an external force . Linear vibration: If all the basic components (mass, spring, and damper) of a vibrating system act linearly, the resulting vibration is called linear vibration. The governing equation of motion in linear vibration must be a linear differential equation. Nonlinear Vibration : If any of the basic components of a vibrating system behave nonlinearly, the nonlinear vibration occurs. The equation governing nonlinear vibration will be a nonlinear differential equation. Self-excited vibration: This is a periodic and deterministic oscillation. There are systems in which the exciting force is a function of the motion variables (displacement, velocity, or acceleration) and thus it varies with the motion it produces. A few illustrations of self-excited vibration are friction-induced vibration (in vehicle clutches and brakes, vehicle-bridge interaction), flow-induced vibration (circular wood saws, CDs, DVDs, in machining, fluid-conveying pipelines), and undesired oscillation of an airplane wing. The study on vibration of beams and plates is an extremely important area owing to the wide variety of applications involved. Since these structural members form integral parts of structures, prior knowledge of their vibration behavior is essential for an engineer before finalizing the design of a given structure. In particular, beams and plates with different shapes subjected to boundary conditions at the edges are often encountered in several engineering applications such as Aueoronatic Engineering, automobile and telephone industries, machine design, nuclear reactor technology, naval structures, and earthquake-resistant structures. Accordingly, the vibration charatracstics may be evaluated with the help of analytical and different computational techniques In general, the laws of the universe are written in the language of mathematics. Algebra is sufficient to solve many static problems, but the most important natural phenomena involve change and are described by equations that relate changing quantities. These are generally referred to as differential equations (Penny, 2004). In a similar fashion, vibration problems associated with structural members are governed by higher-order PDEs. However, it is not always possible to find the analytical solutions for such problems. Accordingly, numerical methods may be applied to solve such differential equations. Although various numerical methods exist to handle these PDEs, these are sometimes problem dependent and may not handle all sets of boundary conditions, along with complicating effects, with ease. A brief idea on theoretical formulation associated with mechanics of different structural members may be found in the available resources (Timoshenko and Woinowsky-Krieger, 1959; Wang et al., 2000; Reddy, 2000; Rao, 2004; Bhavikatti, 2005; Chakraverty, 2009) and the literatures provided therein. In earlier decades, the isotropic structural members along with composites have played crucial role in different engineering applications, strutural design , and architectures. On the contrary, the thermal withstanding behavior of FG structural members have become significant in these applications. It can easily be noted in Abrate (2008) that FG plates behave like Homogenous plates that require no special tools to analyze their mechanics. It is therefore a challenging task to develop numerical methods that may be general to undertake the investigation. As such, the present study aims to develop mathematical models based on different computational procedures for vibration of FG structural beams and plates to obtain the corresponding generalized 1.3.1 Beam and Plate Theories Generally in vibration problems, the displacement fields of deformed beam (or plate) may be decided by shear deformation beam (or plate) theories. Instead of classical beam (or plate) theory, different forms of deformation thories may also be observed in open literature. It may occur due to the fact that classical beam (or plate) theory neglects transverse shear deformation effects. Based on certain assumptions given in Reddy (1984) and Aydogdu (2009), one may easily provide alternate forms ofshear deformation theories. Accordingly, a list of deformation beam and plate theories related to isotropic members is available in Wang et al. (2000). Further, Reddy (1984) has given a refined nonlinear thory accounting for the von Karman strains to find exact solutions for simply supported plates. A new higher-order shear deformable laminated composite plate theory is proposed by Aydogdu (2009) using 3D elasticity bending solutions in an inverse method. Infinitesimal deformations of a homogeneous thick elastic plate have been investigated by Xiao et al. (2007), using a meshless Petrov-Galerkin (MLPG) method and a higher-order shear and normal deformable plate theory. Moreover, Reddy (2011) has developed microstructure-dependent nonlinear Euler-Bernoulli and Timoshenko beam theories using the principle of virtual displacements. A general assessment of inverse trigonometric shear deformation theory is developed by Grover et al. (2013) to analyze structural responses of laminated-composite and sandwich plates. A general higher-order shear deformation theory is proposed by Qu et al. (2013) for free and transient vibration analyses of composite laminated beams with arbitrary combinations of classical and non-classical boundary conditions. Thai et al. (2014) have recently introduced an inverse tangent shear deformation theory for the static, free vibration and buckling analysis of laminated composite and sandwich plates. In a similar fashion, a major topic in deriving shear deformation\ theories can also be found in Sina et al. (2009), Xiang et al. (2009), Şimşek (2010a), Thai and Vo (2012), Şimşek and Reddy (2013), Qu et al. (2013), Vo et al. (2013) and the literatures mentioned therein. 1.3.2 Boundary Conditions Owing to the solution of governing PDEs of continuous systems, we may need the value of the dependent variable and probably their derivatives at more than one point or on the boundary. These problems are said to be boundary value problems (BVPs). In particular, this book considers three classical boundry condition, viz. clamped, simply supported, and free, and the corresponding conditions are mentioned below:
Clamped: This case considers both displacement and slope (or rotation) to be zero
Simply supported: The displacement and bending moment must be zero in this boundary condition.
Free: Here, the bending moment and shear force must be zero.
In addition, the solution also depends on certain assumptions and on the appropriate choice of a shape or trial function, which is dependent on the physical characteristics of the system. The selected shape functions should satisfy certain boundary conditions of the problem. Thus we may find two classes of boundary condition: essential (or geometric) and natural (or dynamic) boundary conditions. These conditions may be defined as follows:
Essential: These boundary conditions are demanded by the displacements or slopes on the boundary of a physical body. Essential boundary conditions are also referred to as dirichlet boundary conditions.
Natural: These conditions are demanded by the condition of bending moment and shearing force balance. As such, natural boundry condition are also known as neumanaa boundry condition. Based on essential and natural boundary conditions, the BVPs may also be divided into three categories: Dirichlet, Neumann, and mixed. Problems in which all the boundary conditions are of the essential type are called Dirichlet problems, and those in which the boundary conditions are of the natural type are said to be Neumann problems. On the other hand, the problem known as mixed type assumes both essential and natural boundary conditions at the edges. There are rubber joints at both ends of the hydraulic buffers of locomotives and vehicles. When the equivalent damping of shock absorbar is large, the rubber nodes and the elasticity of the liquid itself cannot be ignored. Domestic and foreign shock absorber manufacturers and scholars have conducted long-term research on the dynamic model of the hydraulic buffer, and have established many types of dynamic simulation models. Hydraulic buffers require different parameters based on the different degrees of complexity of various hydraulic buffer models. Based on the DIN prEN 13802, the simplest maxwell model (with dampers and springs in series) is suitable for application as a standardized hydraulic buffer model for vehicle dynamics simulations. A diagrammatic representation is shown in Figure 4.5. Both the damping component C and the equivalent stiffness K of the model can adopt nonlinear properties.
Figure 4.5. Maxwell Damper Series Model.
Hydraulic buffers can be described using static and dynamic properties. Only the dynamic properties of hydraulic buffers can accurately reflect the essential characteristics of buffers. Assuming that the coefficient of the buffer’s equivalent stiffness in series is k; Damping coefficient is c; displacement of the shock absorber piston is x0; end of the buffer subjected to vibration amplitude is A; and the sinusoidal displacement input is x with a frequency of w, i.e., x=Asin(wt).Then the differential equation of vibration of the shock absorber system can be expressed as: (4.1)k(x0−x)+cx˙0=0 Figure 4.6 shows the relationship between the displacement x˙, velocity x˙, Piston Displacement x0, piston velocity x0, relative displacement of the spring in series xsti=x- x0, and damping Force F.
Figure 4.6. Relationship between damping force-velocity-displacement.
Figure 4.6 shows that when we consider the series stiffness of the shock absorber, the damping force F is no longer in phase with the excitation speed x˙=Awcos(wt), and a lag in phase π/2−φ occurs. The damping force amplitude then reduces commensurately, becoming a multiple, k/k2+c2w2, of the ideal damping force. According to the requirements of the standardized BS EN 13802:2004 Railway applications suspension component hydraulic dampers, the measuring dynamic damping coefficient in the experiment is: (4.2) c=kwtanφ where w is the excition frequency, φ is the phase angle between damping forces, and k is dyanamic stiffness. Figure 4.6 also shows that the dynamic stiffness of the buffer is the ratio of the amplitude of the axial force Fmax of the shock absorber to the deformation xsti,max of the spring deformation. (4.3)k=Fmaxxsti,max=Fmaxxmaxcosφ=Fmaxxmax1+tan2φ The dynamic stiffness is not to test the ratio between the measured axial force Fmax and the displacement amplitude xmax, but to test the influence of dynamic effects. The dynamic stiffness and damping of the hydraulic buffers can be processed based on the test results described above. The beam is undamped and all the SMD sets have no damping components in them; namely, [D] = 0. Under the circumstances, Eq. (14.10) becomes (14.14)ω2[M]{u}=[K]{u} where λ2 = −ω2 has been used. The corresponding characteristic equation is (14.15)det(−ω2[M]+[K])=0 which has n nonnegative roots 0≤ω12≤ω22≤⋯≤ωn2. The parameters ω1, ω2,…, ωn are called the natural frequencies of the system. The associate eigenuvetor {u1}, {u2},…, {un} are real, and enjoy the orthogonolity relations (after proper scaling) (14.16){uj}T[M]{uk}=δjk, {uj}T[K]{uk}=δjkωk2
An SLS (Simulated LoudSpeaker) load is a network of passive, reactive and damping components (capacitors, inductor and resistors) that more-or-less behaves like a given loudspeker, at least under the kind of test conditions it was designed for. Like a resistive test load, an SLS can be designed to withstand any duration and power level of continuous test signal. But with even modestly high powers, above 50 to 100 watts, specially made, physically bulky inductors are needed, as even ‘high power’ conventional ferrite cored types begin on the slope to saturation, causing the load to be unacceptably level dependent. This could mean a lower powered amplifier seeing an easier load than a higher powered one. For fair measurements, the power handling Vs. impedance curve of an SLS should be measured before first use, so the maximum voltage and power levels for consistent loading are identified, and then marked up on the load. A simple SLS to simulate a single drive-unit (Figure 7.9) still needs a conventional, high power-rated resisitive load. This defines the DC resistance, and remains the sole part dissipating significant power (as heat). The capacitor may be electrolytic but must have a high ripple current rating, and to achieve this and also be adequately rated for handling high swing amplifiers, also a high voltage rating, up to 550v DC. The inductor must practically be an air- or steel-cored type for consistent, saturation-free inductance over the range of current (0 to 75A, sometimes more) and frequency (at least 1000:1)
The above two examples, as well as those introduced in Section 5.8.3, demonstrate that the RAM/V controller can play a significant role in balance stability and control, especially when the angular momentum damping component from the RNS is used. The formulation is simple and yet quite efficient—there is no need to modify the contact model to account for the transitions between the stable and unstable states. When the contacts are stable, the RNS-based control input (5.100) should be switched off to avoid unnecessary arm movements. When the contacts are unstable (the feet rotate), the control input can be used to ensure the recovery of the stability via the induced arm motion. Thereby, the system is forced to behave as a CRB, for any desired base-link state. The recovery is swift, with a subtle arm motion, for relatively small deviation of the feet from their equilibrium states. For larger deviations, the recovery can be prolonged; this results in a “windmilling” arm motion pattern for stabilization that is also sometimes used by humans to stabilize their posture in a critical situation. The performance of the RAM/V controller can be further improved by a reformulation in terms of acceleration (to be presented in Section 5.11.2). With such a reformulation it becomes possible to control the angular momentum damping injected into the system. With such damping, the convergence to a stable contact state can be ensured, e.g. without the direct involvement of the foot rotation error, as in the rolling feet example. The second-order mechanical equation of a synchronous machine can be decomposed into two first-order differential equations: one for the mechanical angular velocity of the rotor ωr and the other for the mechanical rotor angle δ. The constant kD is usually incorporated into the angular velocity equation to add a damping component that is proportional to the difference of angular velocities ωr and ωB:
(4-15)dωrdt=ωB2HTm−Te−kDωr−ωB.
(4-16)dδdt=ωr−ωB, where
(4-17)H=12·JωB2VAbase.
In the above equations ωB is the rated angular velocity of the rotor, kD is the damping factor (pu torque/pu angular velocity), H is the unit inertia constant (watt·s/VA at rated angular velocity), J is the axial combined (rotor and prime mover) moment of inertia (kg m2), and (VA)base is theapparent powerbase Sbase. Static and dynamic component stiffness are measured in different ways. static stiffness c (Fig. 14.11(a)) is measured with slow sawtooth-like signals betweenforce limits representing normal usage of the full vehicle (or also maximum durability loads). The time history of applied force and resulting mount deflection is recorded and displayed in a four-quadrant graph. The htsteristes of the curves is a measure of the component damping. A single value for the stiffness can bemobtained from the slope of the hysteristic curve at a working point (with or without preload).
is measured with sinusoidal stepped sweeps of constant amplitude. The stiffness at a specific frequency is the quotient of the maximum force and maximum deflection, the phase angle between both quantities being the loss angle δ. Stiffness and damping values are plotted against frequency in the figure. For data compatibility, it is important to define the speed of measurement as well as the amplitude of both the static and the dynamic measurement. Common values for the measurement speed are 10–30 mm/min for the static data with application-specific maximum force values. The amplitude of the sinusoidal dynamic measurement can vary, but for ‘pure’ Road NVH load cases it may go down to e.g. ± 0.025 mm, while spanning frequencies up to 500 Hz, establishing high requirements to the measurement rig hardware. Note that a mount will be measured softer with increasing amplitudes, which is valid for static as well as for dynamic load cases. For the amplitude-related reversible and non-reversible effects on the stiffness (Payne effect, Mullins effect), see Walter (2009). From Eq. (8.22), Section 8.3.1, we know that the natural frequency of a single spring–mass system is expressed as ω=k/m. At crictical damping, the damping constant has a value of cc2=4mk. From these two relationships, we write the critical damping constant as a function of m and ω as (8.84)cc=4mk=4m×mω2=2mω. We can declare that the damping constant can be represented by linear combination of m and k, that is (8.85)c=αm+βk, where α is the mass damping component and β is the structural- or stiffness-damping component. We declare this relationship because the magnitudes for m and k are usually larger than the magnitude of c. Also, direct modeling of damping is very complex and is considered more of an art than a science. From Eq. (8.85), the damping ratio is calculated as (8.86)ζ=ccc=α2ω+βω2. Eqs. (8.85) and (8.86) are related to a single mass–spring–damper system. We can analogously proclaim that the same relationship holds true for a large system of masses, springs, and dampers as (8.87)[C]=α[M]+β[K] Eq. (8.87) is called the equation of Rayleigh damping, named after lord relight as described in Section 8.3.3. Although the rheological or physical association of this equation is not clear, acquiring the damping matrix through this method and using it to solve transient responses was found to be quite successful. If we consider only the mass damping (i.e., neglect stiffness damping), the damping ratio can be calculated from α and ω using Eq. (8.86) (see results shown in Fig. 8.9). We can see from this figure that the damping ratio ζ is inversely proportional to the frequency. Additionally, as the frequency becomes higher, the effect of mass damping diminishes. Thus, mass damping is most suitable for damping out oscillations due to low frequency and high amplitude. Because only a single α value is allowed as an input parameter to an FE model, the specific frequency that dominates the impact responses should be the one used to calculate the α value to damp out unwanted vibrations of the system.
If we consider only the structural damping (i.e., neglect mass damping), the damping ratios can be calculated from β and ω in Eq. (8.86), and results are shown in Fig. 8.10. It can be seen from this figure that the damping ratio ζ is directly proportional to the frequency. Additionally, as the frequency becomes higher, theeffect of damping increases linearly. Thus, stiffness damping is good for damping out oscillations due to high frequency and low amplitude. Again, to damp out unwanted vibrations, responses due to the most dominant frequency should be used to determine a single β value as the input parameter to an FE model.
In real-world problems, both mass and stiffness damping are desired to damp out vibrations from both low and high frequencies. Assuming that we want to have a constant damping ratio of 0.5 for frequencies ranging from ω1 to ω2 rad/s. From Eq. (8.86), we can write (8.88)α2ω1+βω12=0.5α2ω2+βω22=0.5. We further assume that ω1 = 1 and ω2 = 10. Eq. (8.88) represents two equations for solving two unknowns. We find that α = 0.909 and β = 0.0909. Inserting these values into Eq. (8.86) gives us the calculated damping ratios are shown in Fig. 8.11 as functions of frequency. It can be seen that mass damping dominates the effect in the low- frequency segment, while stiffness damping dictates the high-frequency portion of the response.
In an undamaped system, the peak magnitude of each oscillation decreases logarithmically over time. The damping ratio of the system can be determined from experimental data. Assuming the period of a damped vibration is T=2πω for one cycle within which there are two consecutive peaks with peak values of x1 and x2. The logarithmic decrement Δ between these two consecutive peaks is calculated as (8.89)Δ=lnx1x2. The damping ratio can be calculated as (8.90)ζ=Δ(2π)2+Δ2.
Que :- 4. Provide relationship between Natural period (Tn) & Natural frequency (f) and provide their definitions (25 marks)
Ans :- The Frequency ( f ) Of A Wave Is The Number Of Full Wave Forms generated Per Second. This Is The Same As The Number Of Repetitions Per Second Or The Number Of Oscillations Per Second. Time Period ( T ) Is The Number of Second Per Waveform , Or The Number Of Second Per Oscillation It Is Clear That Frequency And Time Period Are Reciprocals. That Is , T = 1/f
For the preliminary design stages, some empirical relationships for building period Ta (Ta = 1∕n1) are available in the earthquake-related chapters of every building code and design standard. However, these expressions are based on recommendations for earthquake design with inherent bias toward higher estimates of fundamental frequencies, because shorter periods result in higher seismic loading and are therefore conservative. For wind design applications and drift calculations, these values may be unconservative because an estimated frequency higher than the actual frequency would yield lower values of the gust-effect factor and thus a lower design wind pressure. However, the estimated period based on the analytical model is limited by the upper bound (Cu*Ta) to avoid overestimation of the period and prevents the use of an unusually low base shear for design of a structure that is, analytically, overly flexible because of mass and stiffness inaccuracies in the analytical model. So, in general, if the fundamental period, T, from a properly substantiated analysis (per Section12.8.2) is less than upper bound of the period, then T should be used for the design of the structure. For wind design (n < 1), the general recommendation is to use the lower bound approximate natural frequency based on the equations mentioned in the code for wind design or (in a rare case) higher frequency based on the stiffer analytical model with realistic lateral stiffness. As an alternative to using the approximate period formulae, we can use structural software to perform a dynamic analysis to determine the building periods as the software calculates only the stiffness of the bare lateral frame, without any stiffness contribution from the gravity framing, interior partitions, infill walls and cladding, etc. Also note that higher mass and lower stiffness assigned in the model, will increase the estimated fundamental period. So, these calculated fundamental periods are generally too long (lower frequency), that would yield higher values of the gust factor, and considered conservative. This is especially true for short-term wind loading and elastic-level forces such as those considered for serviceability. Most engineers are now aware of these, since this information moved into the body of ASCE 7-10, rather than the code commentary in prior codes. In general, buildings with a mean roof height < 60’ (18 m) can be considered rigid, but for those that are flexible the following text from ASCE 7-16 should be reviewed carefully.
ASCE 7-16Section 12.8.2.1 Approximate Fundamental Period.The approximate fundamental period (Ta), in seconds, shall be determined from the following equation:
Historically, the exponent, x, in Eq. (12.8-7) has been taken as 0.75 and was based on the assumption of a linearly varying mode shape while using Rayleigh’smethod. The exponents provided in the standard, however, are based on actual response data from building structures, thus more accurately reflecting theinfluence of mode shape on the exponent. Because the empirical expression is based on the lower bound of the data, it produces a lower bound estimate of theperiod for a building structure of a given height.
Section 26.11.2.1 Limitations for Approximate Natural Frequency.As an alternative to performing an analysis to determine n1, the approximate building natural frequency, na, shall be permitted to be calculated in accordancewith Section 26.11.3 for structural steel, concrete, or masonry buildings meeting the following requirements:1. The building height is less than or equal to 300ft (91m), and2. The building height is less than 4 times its effective length, Leff, based on Eq. (26.11-1)
Section 26.11.3 Approximate Natural Frequency.The approximate lower bound natural frequency (na), in hertz, of concrete or structural steel buildings meeting the conditions of Section 26.11.2.1 is permittedto be determined from one of the following equations:
Explicit calculation of the gust-effect factor per section 26.11
These equations for gust factor and natural frequency are based on studies for regular rectangular building models, with side ratio (D∕B, where D is the depth of the building section along the oncoming wind direction) from 1∕3 to 3. So wind tunnel studies should be considered for slender and irregular tall buildings or dynamically sensitive structures.Commentary:BUILDING OR OTHER STRUCTURE, FLEXIBLE: A building or other structure is considered “flexible” if it contains a significant dynamic resonant response. Thegust effects and the natural frequency of buildings or other structures greater than 60 ft (18.3 m) in height is determined in accordance with Sections 26.11.When buildings or other structures have a height exceeding 4 times the least horizontal dimension or when there is reason to believe that the natural frequencyis less than 1 Hz (natural period greater than 1 s), the natural frequency of the structure should be investigated. Approximate equations for natural frequency orperiod for various building and structure types in addition to those given in Section 26.11.2 for buildings are contained in Commentary Section C26.11. Gust energy and the resonant response of most buildings and structures with lowest natural frequency above 1 Hz ("rigid structure") are sufficiently small that resonant response can often be ignored. A general guidance is that most rigid buildings and structures have height to-minimum-width less than 4. Approximate fundamental frequency:Lower bound estimates of frequency that are more suited for use in wind applications and are now given in Section 26.11.2; graphs of these expressions areshown in Fig. C26.11-1.
Because these expressions are based on regular buildings, limitations based on height and slenderness are required. The effective length, Leff, based on Eq.(26.11-1), uses a height-weighted average of the along-wind length of the building for slenderness evaluation. The top portion of the building is most important;hence, the height-weighted average is appropriate. This method is an appropriate first-order equation for addressing buildings with setbacks. Observations fromwind tunnel testing of buildings where frequency is calculated using analysis software show that the following expression for frequency can be used for steeland concrete buildings less than about 400 ft (122 m) in height:n1 = 100∕H(ft) average value (C26.11-6)na = 75∕H(ft) lower bound value (C26.11-7)Based on full-scale measurements of buildings under the action of wind, the following expression has been proposed for wind applications:f n1 = 150∕h (ft) (C26.11-8)This frequency expression is based on older buildings and overestimates the frequency common in U.S. construction for smaller buildings less than 400 ft (122m) in height, but it becomes more accurate for tall buildings greater than 400 ft (122 m) in height. The Australian and New Zealand Standard AS/NZS 1170.2,Eurocode ENV1991-2-4, Hong Kong Code of Practice on Wind Effects (2004), and others have adopted Eq. C26.11-8 for all building types and all heights.Studies in Japan involving a suite of buildings under low amplitude excitations have led to the following expressions for natural frequencies of buildings:n1 = 220∕h (ft) (concrete buildings) (C26.11-9)n1 = 164∕h (ft) (steel buildings) (C26.11-10)These expressions result in higher frequency estimates than those obtained from the general expression given in Eqs. (C26.11-6) through (C26.11-8),particularly since the Japanese data set has limited observations for the more flexible buildings sensitive to wind effects, and Japanese construction tends to bestiffer. the construction in India is similar to Japan and buildings tend to be stiffer, so the lower bound 75∕H(ft) may be too conservative. Structural Damping:Because the level of structural response in the strength and serviceability limit states is different, the damping values associated with these states may differ.In wind applications, damping ratios of 1% and 2% are typically used in the United States for steel and concrete buildings at serviceability levels, respectively,while ISO (1997) suggests 1% and 1.5% for steel and concrete, respectively. Damping ratios for buildings under ultimate strength design conditions may besignificantly higher, and 2.5% to 3% is commonly assumed.Damping values for steel support structures for signs, chimneys, and towers may be much lower than buildings and may fall in the range of 0.15–0.5%.Damping values of special structures like steel stacks can be as low as 0.2–0.6% and 0.3–1.0% for unlined and lined steel chimneys, respectively. These valuesmay provide some guidance for design.Damping levels used in wind load applications are smaller than the 5% damping ratios common in seismic applications because buildings subjected to windloads respond essentially elastically, whereas buildings subjected to design-level earthquakes respond inelastically at higher damping levels.
Que :- 5. Explain in detail about Response spectrum & its Graph (25 marks)
Ans :- A response spectrum is a function of frequency or period, showing the peak response of a simple harmonic oscillator that is subjected to a transient event. The respons spectrum is a function of the natural frequency of the oscillator and of its damping. Thus, it is not a direct representation of the frequency content of the excitation (as in a Fourier transform), but rather of the effect that the signal has on a postulated system with a single degree of freedom (SDOF). Analysis of an SDOF System Consider a mass-spring-damper system attached to a moving base. The foundation has a given movement, .
The equation of motion for the mass can, if there are no external forces, be written as
Dividing by the mass, and using customary notation,
Here, the undamped natural (angular) frequency is
and the damping ratio is
It can be seen that the support movement acts as a forcing term and that the solution depends only on the two parameters and , but not on the individual values
of m, c, and k. Instead of using the absolute displacement as the degree of freedom, it is possible to choose the relative displacement between the mass and the base, . This is actually a frame transformation where the oscillator is studied in a coordinate system attached to the base. As in any accelerating frame, there will be inertial forces. The equation of motion can be stated as
(1)
Thus, the support acceleration appears as a gravity-like load. There are two advantages with this representation: The internal forces in the system — that is, elastic and damping forces — depend on relative displacements and velocities. These forces are not affected by a rigid . body motion. . Often, measured data is available in terms of an accelerogram, so that the foundation displacements are not directly available. For given values of , , and , this equation can be solved for a sufficiently long time. The displacement, velocity, and acceleration response spectra are defined as the maximum values caused by the acceleration history .
These are all relative spectra. It is possible to do a similar definition of the absolute spectra, by instead using the absolute displacement . Sometimes, a distinction is made between the positive and negative spectra, so that
and similarly for velocity and acceleration spectra. The velocity and acceleration response spectra are often approximated by
Such spectra are called pseudovelocity and pseudoacceleration spectra. For a system without damping, the pseudoacceleration spectrum based on the relative displacement is actually equal to the absolute acceleration spectrum. This can be seen from the undamped equation of motion,
Thus,
The maximum absolute value of the relative displacement must thus occur at the same time as the maximum absolute value of the absolute acceleration. The scale factor between the two is . For systems with low damping, this relation will still be approximately true. Since most mechanical systems have a low damping (often 2% to 5%), it is customary to assume that the spectra for the absolute acceleration and the pseudoacceleration are the same. Another common way of describing the damping in this context is by the Q factor (quality factor). The relation to the damping ratio is given by
How to Create a Response Spectrum
For a certain given time history, , a response spectrum is created in the following way:
Select a frequency range for which the spectrum should be generated
Select a frequency step that determines how many points on the response spectrum should be computed
Select a certain damping ratio,
For each of the selected frequencies a. Solve Equation (1) with for a sufficiently long time b. Keep track of the maximum value of and store it
The equation can be solved by a pure numerical time stepping, but there may be better ways of it. If is given as a number of points in an accelerogram, then itis natural to assume that the acceleration has a linear variation in time between those points. So, for each interval between two measurements, say from to , the equation of motion for the oscillator is
This equation, where the right-hand side is a linear function of time, can be solved analytically for each time interval. The initial conditions are obtained from the final state of the previous interval. The maximum values can actually occur after the end of the driving event. This will happen for low values of (long periods). The time-stepping must thus be continued at least until a full period if the oscillator has elapsed. With this formulation, it is possible to track the extreme values of the:
Relative displacement
Relative velocity
Relative and absolute acceleration
The absolute displacement and velocity are not available, since only is known, but not and . It is, of course, possible to recover the foundation velocity and
by time integration of the acceleration. In practice, this integration will, however, cause a drift, so that the final velocity and displacement turn out to be nonzero. Since nonzero final displacements and velocities are unphysical (at least for many types of events), some numerical filtering has to be applied. For shocks, there is also a question of which initial values should be chosen for the displacement and velocity of the foundation. Example 1: A Half Sine Shock A common description of impact, used in several standards, is a half sine acceleration pulse. The acceleration amplitude, as well as the duration of the shock, can have different values. The acceleration may, for example, be 20 g, 50 g, or 100 g (g = 9.8 m/s2), and the time of duration can be 6 ms or 11 ms. In this example, a 50-g half sine pulse with a duration of 11 ms is used as the acceleration .
The computed response spectra for 5% damping (Q = 10) is shown below.
The absolute acceleration has a maximum at frequencies similar to the frequency content of the input signal. In this case, the peak is at 74 Hz (T = 13.5 ms). For high frequencies, the acceleration spectrum tends toward 50 g. This is a general observation for any signal: At high frequencies, the oscillator will behave as a rigid body, so the mass just follows the base motion. As an effect, the asymptotic value of the absolute acceleration spectrum always equals the peak base acceleration during the event. For low frequencies, the acceleration tends toward zero with a rate that is inversely proportional to the frequency. With a very soft oscillator, the base movement will just compress the spring without significant movement of the mass. It can also be seen that the pseudoacceleration spectrum in this case is almost indistinguishable from the actual absolute acceleration spectrum, even though the damping is 5%. In the figures below, the time response for the oscillator is shown for three different choices of its natural frequency, corresponding to the markers in the response spectrum above.
. At 15 Hz, the load pulse just gives a small initial push, and then the oscillator experiences free vibration at its natural frequency. The peak acceleration occurs a long
. time after the end of the excitation.
. At 75 Hz, there is maximum dynamic amplification. The load pulse is essentially in phase with the relative velocity, and it provides a maximal energy input to the system.
. At 500 Hz, the oscillator to a large extent acts as a rigid body, closely following the base acceleration. The peak acceleration is almost the same as that of the base acceleration.
Absolute acceleration at a natural frequency of 15 Hz and damping of 5%.
Absolute acceleration at a natural frequency of 75 Hz and damping of 5%.
Absolute acceleration at a natural frequency of 500 Hz and damping of 5%.
The relative displacement spectrum is shown below. This is essentially the same as the acceleration spectrum above, but scaled with a factor
The relative displacement response spectrum for 5% damping.
Next, the relative velocity spectrum and the pseudovelocity spectrum are compared. As can be seen, they are quite different. The pseudovelocity and pseudoacceleration
spectra do not represent the true relative spectra. This is a general observation, and the pseudo spectra should be viewed as different representations of the displacement
spectrum.
The relative velocity response spectrum and the pseudovelocity spectrum for 5% damping.
A convenient way to represent a response spectrum is in a tripartite, or four-axis plot. In such plot, the relative displacement, pseudovelocity, and pseudoacceleration are
shown simultaneously. This is possible, since they are related by a factor of frequency and frequency squared, respectively, which in a logarithmic plot just gives lines
with different slopes. The tripartite plot is essentially a pseudovelocity plot but with two extra sets of skewed grid lines that represent the displacement and acceleration,
respectively.
Tripartite plot of the response spectrum for the half sine pulse. The 50-g acceleration level and 20-mm displacement level are highlighted in red and green, respectively. The diagonal displacement (dotted) and acceleration (dashed) levels have the same '1-2-5-10' spacing as used for the velocity axis.
The response spectrum for the half sine pulse is actually somewhat atypical. The reason is that this pulse only has positive acceleration. If it is integrated with respect to time, such pulse corresponds to a resulting nonzero velocity and an ever-increasing displacement. Most events, like earthquakes, have the property that both displacement and velocity are zero both before and after the event. If a complete sine pulse is used instead of a half sine pulse, the characteristic low-frequency decay is also obtained.
Response spectrum for a full sine pulse.
Example 2: The El Centro Earthquake One of the most studied earthquake recordings is that of the "El Centro" earthquake on May 18, 1940. Recorded signals (with some filtering) are shown below.
Acceleration in the N–S direction (left) and E–W direction (right) for the El Centro earthquake. Strong-motion data accessed through the Center for Engineering Strong
Motion Data (CESMD). The networks or agencies providing this data are the (CSMIP) and the (NSMP).* These curves are typical for an event for which response spectrum analysis is relevant. A visual examination of the signal suggests that the main frequency content is in the range of 1–3 Hz, while the duration of the major part of the event is about 30 s. Thus, the conditions are not even close to being considered as a steady state. On then other hand, there is a significant number of cycles (of the order of 100), which can excite a structure having resonances in the 0.5–30-Hz range. The computed response spectra for 2% and 5% damping are shown below.
Pseudoacceleration spectra for the N–S direction for the El Centro earthquake.
Pseudoacceleration spectra for the E–W direction for the El Centro earthquake.
The response spectra exhibit some interesting general properties. Higher damping will give lower response values and a smoother spectrum. Both these properties are related to the fact that the frequency response of an oscillator will have lower but wider peaks at higher damping. Furthermore, there is a significant difference in the amplitudes in the N–S and E–W directions. However, there is a general resemblance between the shapes of the spectra in the two directions. Design Response Spectra
The response spectrum of a single time signal is seldom of interest for an analysis, since it would be better to perform a direct time domain analysis of the structure with the original signal as input. As seen in the El Centro example above, a certain earthquake may give a response spectrum with significant peaks at certain frequencies. The peaks for another similar earthquake may, however, be located at other frequencies. In order to be able to use a response spectrum for analysis of an event that has not yet happened, a design response spectrum is created. The design response spectrum can be seen as an envelope over all known and anticipated earthquakes in a certain geographical region. Such spectra are, for example, provided in building codes like
ASCE 7-16 and Eurocode 8 (). The a(Ref 2,3)cceleration levels in a design response spectrum will typically depend on the geographical location and the type of soil. The design response spectrum is the actual input to the response spectrum analysis. Design response spectra are often provided in terms of the period, rather than the frequency. Since one is the inverse of the other, the two graphs are just mirrored when plotting on a logarithmic scale.
Example of a design response spectrum.
Floor Response Spectrum A typical design response spectrum for earthquakes gives information about the effect of the ground motion on a primary structure like a building. However, if we are interested in analyzing a secondary component or system that is mounted inside the building, the original response spectrum may not provide a suitable description. The secondary system can, for example, be a piping system or a pressure vessel. The secondary system will be subjected to a base acceleration at its location inside the primary structure. This acceleration is, in general, not the same as the acceleration of the ground. A floor response spectrum is a type of design response spectrum developed for a certain location in a primary structure. The primary structure will, through its natural frequencies, act as a bandpass filter for the original signal. Thus, the floor response spectrum will typically have significant peaks related to the natural frequencies of the primary structure. The term floor response spectrum is derived from the fact that this local response spectrum will typically be different between different floors of a building. A large system, like a piping system, may not have the same floor response spectrum at all of its support points. This causes significant complications to the analysis.
Analysis Based on Response Spectra
The Multiple DOF System Assume that a mathematical model of a structure is discretized by FEM so that the equations of motion on matrix form are
The structure is at a number of points connected to a common "ground" that has the base motion . This vector has the same size as (the total number of DOFs), but it contains only three different values: in all x-translation DOF, in all y-translation DOF, and in all z-translation DOF. The relative displacement is now . With no external load, the equation of motion is
or
Here, the fact that a rigid body motion does not introduce any elastic or viscous forces in the system has been used, so that . By solving the undamped eigenvalue problem with the support nodes being fixed, a set of N eigenmodes
can be computed. These eigenmodes can represent the relative displacements (but not the absolute displacements), since all eigenmodes will have zero displacements at the support points
in an analysis By standard operations for mode superposition the decoupled modal equations are
It has been assumed that the mass matrix normalization of the eigenmodes is used and that the damping matrix can be diagonalized by the eigenmodes. The mass matrix normalization is not essential, but it will simplify certain expressions. Here, is the modal coefficient for mode j, so that the relative displacement can be written as a linear combination of eigenmodes, weighted by the modal coordinates:
The support motion can be decomposed along three orthogonal directions as
The vector has the value "1" in all X-translation DOFs and the value "0" in all other. The modal equation of motion is then
The multipliers are the modal participation factors;
Thus, the maximum amplitude of mode j, when loaded by a base motion in direction k described by a response spectrum, is
or, using the pseudoacceleration spectrum
To summarize, the peak amplitude for a certain eigenmode is the product of the response spectrum value at the corresponding natural frequency (which is independent of the structure) and the participation factor (which is a property of the structure but independent of the loading). Summation Over Modes In practice, several modes will have natural frequencies in the frequency range covered by the design response spectrum. This means that some combination of their responses is needed. There are several rules for how this combination can be arranged, as will be described in detail below. These summation rules are nonlinear. For all combination types, all result quantities are strictly positive. As an effect, any quantity must be summed based on its own modal response. For example, stress components must be computed using the modal stresses and cannot be recovered from the summed strains, and strains cannot bem recovered from summed displacements. This has many consequences for the interpretation of results from a response spectrum analysis. Some examples are:
. It is not meaningful to plot displacement shapes, since the individual displacement components do not match.
. An equivalent stress, like von Mises, cannot be computed from individual stress components.
. If there are interaction rules, like the combination of normal force and moment in a beam, the way of doing the summation is sensitive. A conservative interpretation separately determines the two quantities and then works with the sums and differences.
. Particular care must be taken with respect to the signs if the results of the response spectrum analysis are to be added to results from a static load case, like a dead load. Often, the excitation is given in three orthogonal directions. The general approach is to consider the excitation in the three directions separately. First, all modal responses are summed for each direction, and then the results for the three directions are summed. An exception is the CQC3 summation rule, described below, in which the spatial and modal summation is done at the same time. Periodic and Rigid Modes It is often useful to divide the eigenmodes into periodic modes and rigid modes. The distinction is related to the frequency content of the excitation relative to the eigenfrequency of the mode.
In a high-frequency mode, the mass of the oscillator will mainly be translated in phase with the support. Such modes constitute the rigid modes. Their responses are synchronous with each other (and with the base motion). This means that for rigid modes, a pure summation (including signs) should be used. Modes with a significant dynamic response constitute the periodic modes. The maximum values for such modes will be more or less randomly distributed in time, since their periods differ. For this reason, the periodic part of the response requires more sophisticated summation techniques. A plain summation of the maximum values will, in general, significantly overestimate the true response. Modes that are in a transition region will partially contribute to the periodic modes and partially to the rigid ones. In addition, it is sometimes necessary to add some static load cases containing a missing mass correction. Not all analyses require a separation into periodic and rigid modes. If not, all modes are treated as periodic. In the following, denotes any result quantity caused by excitation in direction . can be, for example, displacement, velocity, acceleration, a strain component, a stress component, effective stress, or a beam section force. The periodic part of is denoted , and the rigid part is denoted . Similarly, and denote the results from an individual eigenmode j. Partitioning into Periodic and Rigid Modes There are two different methods in use by which partitioning can be done. In either case, for mode j,
so that
The difference between the two methods lies in how the coefficient is determined. For low frequencies, it should approach the value 0, and for high frequencies, the value
1. In the Gupta method, is a linear function of the logarithm of the natural frequency.
Here, and are two key frequencies. Thus, for eigenfrequencies below , the modes are considered as purely periodic, and above , purely rigid. In the original Gupta method, the lower key frequency is given by
Here, and are the maximum values of the acceleration and velocity spectra, respectively. In the idealized spectrum above, this occurs at the point D.
The second key frequency should be chosen so that the modes above this frequency behave as rigid modes. The frequency can be taken as the one where the response
spectra for different damping ratios converge to each other. In the Lindley-Yow method, the coefficient depends directly on the response spectrum values, not only on the frequency. As a consequence, it is possible that a certain
mode can be considered as having a different degree of rigidness for different excitation directions.
The so-called zero period acceleration (ZPA) is the maximum ground acceleration during the event,
This is also the high-frequency asymptotic value of the absolute acceleration (or pseudoacceleration) in the response spectrum. It corresponds to the F-G part of the idealized spectrum.
Thus,
The value of must, for physical reasons, be in the range of 0 to 1 and increase with frequency. For this reason, NRC RG 1.92 (Ref. 1) requires that be set to zero for
any eigenmodes below point
C. Summing the Periodic and Rigid Modes Once the periodic and rigid responses for all modes have been summed up separately, they are combined as
Summing the Periodic Modes The most conservative method is to sum the maximum response for all N modes, thus assuming that all modes reach their maximum at the same time. In many cases, this approach leads to a design that is significantly overconservative.
In the worst case scenario, the predicted result using N, not closely spaced modes can be a factor larger than what would be obtained using the other methods
below. The most popular method for superposition of the periodic modes is the complete quadratic combination (CQC) method:
The interaction between the modes is determined by the mode interaction coefficient (). Since is symmetric and when , it is more
efficient to use the equivalent expression
This expression is actually valid for several evaluation rules. The only difference is how is computed. Several such expressions are given below. When a method is referred to as CQC, it is usually implied that the Der Kiureghian correlation coefficient is used. Der Kiureghian Correlation Coefficient The mode interaction coefficient is defined as
Here, and are the natural frequencies of the two modes, and and are the corresponding modal damping ratios.
For the common case of uniform damping, the expression can be simplified to
It is possible that the response from two different modes, and , have different signs, so that a cross term can give a negative contribution to the sum. This is intentional, but it is a common misconception that the absolute values of and should be used. However, the underlying analysis contains an assumption about the response being a linear function of the mode shape. If this it not the case, using absolute values is a safer approach. The most common nonlinear result quantities, like effective stresses, are always positive, in which case all terms in the sum will give a positive contribution anyway. The strength of the correlation between two modes depends on the frequency ratio for the modes, but it also strongly depends on the damping.
The Der Kiureghian mode correlation factor for different damping ratios as a function of the ratio between the pair of eigenfrequencies
The double sum method uses a mode interaction coefficient , which is called the Rosenblueth correlation coefficient. It is conceptually similar to the Der Kiureghian correlation coefficient. The double sum method exists in two variants:
The older version of this method is actually erroneous, but the results are more conservative than those of the newer variant, so it can be used without risk. In either case,
where
and
Here, is a separate input. The duration of the dynamic event and are the modal damping ratios. For large values of , the modal correlation factor in the double sum method is rather similar to that of the Der Kiureghian model. For smaller values of , a much stronger correlation is predicted by the double sum method The SRSS method does not include any interaction between the modes; that is,
This method should only be used when the modes are not closely spaced; that is, when no two eigenfrequencies are close to each other. All other methods take possible interaction between the modes into account in various ways.Grouping Method
The modes are grouped according to the following rule:
Start a new group by adding the lowest (in frequency), not yet grouped eigenmode k
Step up through the eigenfrequencies from k
As long as , add mode i to the group
Go back to Step 1
After having exhausted the list of eigenmodes, there is a number of groups, where some may contain just a single eigenmode. The rule for the correlation factor between two modes is
The sign() operator used here is a way of stating that the absolute value of the product of the modal responses is added to the sum, since
Ten Percent Method The ten percent method is similar to the grouping method in the sense that eigenmodes with a natural frequency difference of less than 10% get a special interaction treatment. The modal correlation coefficient is
It can be noted that both the grouping method and the ten percent method are equivalent to the SRSS method if no pair of eigenfrequencies are within 10% from each other. When using the CQC method, however, the modes are also considered as significantly coupled at a larger spacing, unless the damping is very low. Summing the Rigid Modes There are two possible combination methods for summing the rigid modes. Combination Method A The rigid modes are summed as
Here, is the result of the solution to the static load case when solving for missing mass, as described below. Combination Method B This method can only be used when the Lindley-Yow method is used together with the static ZPA method. Then, the rigid mode response is simply
Missing Mass Correction Missing Mass Method Since a mode superposition uses a limited number of modes, some mass that is attributed to the nonused modes will, in general, be missing from the analysis. With the assumption that the higher-order modes do not have any dynamic amplification, it is possible to devise a correction by solving some extra static load cases containing the peak acceleration acting on the "lost" mass. The effect of applying a static correction is usually most prominent when evaluating support forces. So-called static correction can actually be used for mode superposition in general. However, for the case of response spectrum analysis, the expressions will be simplified. In general, the static correction load vector can be computed as:
Here, is the original load vector and are the modal loads; that, is the projection of the physical load on each eigenmode,
In the base excitation context, the load vector related to excitation in direction I is
The modal load is then
Thus,
For the rigid body modes, the peak acceleration is equal to the zero period acceleration (ZPA). This is the maximum ground acceleration during the event,
which also corresponds to the high-frequency asymptote of the acceleration spectrum.
The static load is thus
The extra displacement correcting for the missing mass is now given by solving the standard static problem
where is the stiffness matrix.
The expression
can be viewed as a type of auxiliary mode. In this context, it is a vector with the length of the number of DOFs. In this method, there is no need to deduce the missing mass. This method can only be used together with the Lindley-Yow method for separating periodic and rigid modes. According to the method, all rggjjgigid modes have the acceleration . This acceleration is given to the whole structure. The static load cases are thus just pure gravity loads, but scaled by instead of the acceleration of gravity.
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.
and 
, but not on the individual values
. This is actually a frame transformation where the oscillator is studied in a coordinate system attached to the base. As in any accelerating frame, there will be inertial forces. The equation of motion can be stated as
(1)
. Sometimes, a distinction is made between the positive and negative spectra, so that
. For systems with low damping, this relation will still be approximately true. Since most mechanical systems have a low damping (often 2% to 5%), it is customary to assume that the spectra for the absolute acceleration and the pseudoacceleration are the same. Another common way of describing the damping in this context is by the Q factor (quality factor). The relation to the damping ratio is given by
for a sufficiently long time
and store it
is given as a number of points in an accelerogram, then itis natural to assume that the acceleration has a linear variation in time between those points. So, for each interval between two measurements, say from 
to 
, the equation of motion for the oscillator is
. It is, of course, possible to recover the foundation velocity and
. This vector has the same size as 
(the total number of DOFs), but it contains only three different values: 
in all x-translation DOF, 
in all y-translation DOF, and 
in all z-translation DOF. The relative displacement is now 
. With no external load, the equation of motion is
. By solving the undamped eigenvalue problem 
with the support nodes being fixed, a set of N eigenmodes
is the modal coefficient for mode j, so that the relative displacement can be written as a linear combination of eigenmodes, weighted by the modal coordinates:
has the value "1" in all X-translation DOFs and the value "0" in all other. The modal equation of motion is then
are the modal participation factors; 
separately. First, all modal responses are summed for each direction, and then the results for the three directions are summed. An exception is the CQC3 summation rule, described below, in which the spatial and modal summation is done at the same time. Periodic and Rigid Modes It is often useful to divide the eigenmodes into periodic modes and rigid modes. The distinction is related to the frequency content of the excitation relative to the eigenfrequency of the mode.
denotes any result quantity caused by excitation in direction 
. 
, and the rigid part is denoted 
. Similarly, 
and 
denote the results from an individual eigenmode j. Partitioning into Periodic and Rigid Modes There are two different methods in use by which partitioning can be done. In either case, for mode j,
is determined. For low frequencies, it should approach the value 0, and for high frequencies, the value 
and 
are two key frequencies. Thus, for eigenfrequencies below 
and 
are the maximum values of the acceleration and velocity spectra, respectively. In the idealized spectrum above, this occurs at the point D.
larger than what would be obtained using the other methods
(
). Since 
when 
, it is more
and 
are the natural frequencies of the two modes, and 
and 
are the corresponding modal damping ratios.
and 
, which is called the Rosenblueth correlation coefficient. It is conceptually similar to the Der Kiureghian correlation coefficient. The double sum method exists in two variants:
is a separate input. The duration of the dynamic event and 
, add mode i to the group
is the result of the solution to the static load case when solving for missing mass, as described below. Combination Method B This method can only be used when the Lindley-Yow method is used together with the static ZPA method. Then, the rigid mode response is simply
is the original load vector and 
are the modal loads; that, is the projection of the physical load on each eigenmode,
is the stiffness matrix.
is given to the whole structure. The static load cases are thus just pure gravity loads, but scaled by 
