Beam/column

[chigozie]

iceman84 wrote:-
.....At maximum bending moment, the concrete is cracked, and its portion in the tension zone is neglected in the calculations of the moment of inertia....Once cracks have developed, the assumption of uncracked section behavior under small loads does not hold."...Again, ...the reduced flexural rigidity of cracked beams may lead to a bending moment to the columns larger than that calculated on the basis of gross section stiffnesses....To avoid significant moment redistribution, it may be better to base the moment of inertia of the beams on an approximate transformed cracked section (e.g. 0.5Ig) and the moment of inertia of columns on a the gross section Ig." ..

Comment:-

One of the very good things about books and reference is that they tell us or give us information that we have never come across or that we will never come across in their absence. They also remind us of the information that we had come across that are still fresh in our memories and/also those that we have come across but have forgotten in the passage of time. Science and engineering have the added advantage that apart from giving us the information, try in different modes to convince us on the information by demonstrating through means (including mathematic-thus calculations), the validity of the information. Having this in mind, we would like to ask (humbly) on how the figure of 0.5 of the gross moment of inertia for the beam and that of the entire gross moment of inertia for the column were arrived at. Why shouldn’t it be say, 1.2 or 0.6 of the gross moment of inertia for the beam and say 1.0 or 0.8 of the gross moment of inertia for the column or vice versa? It would be most appreciate if we are to be convinced on how these figures were arrived at (demonstrate).
On the other hand, lets enumerate on why the stiffnesses -thus moment of inertia should not be modified but taken as the gross moment of inertia for both the beam and column:-
It is a common knowledge that when we design under the ultimate limit condition, we assume that the concrete in tension has cracked as such its strength in tension is discounted. Since beam is a structural element that sustains loading principally in bending, this situation is more relevant to beam than to column (that supports loading principally in compression). This assumption is not always correct as columns also have to support bending and situations could be reached in which they also have to obey same law in bending as applicable to beam. This situation could be most realized under the ultimate limit condition. Thus under certain conditions (minimum bending)it could be rightly assumed that column’s concrete had not cracked, as such its design moment of inertia is equal to its gross moment of inertia; whilst that of the beam is equal to that of the equivalent cracked section as such less than the gross moment of inertia of the un-cracked beam. But by how much has this cracked section detracted from the un-cracked section? To answer this question, let’s take a look at the bending moment diagram of a simply supported beam or that of beam resting on a column with their joints constructed as a pinned joints.


At the column/beam interface (center of column), moment is zero as such, there is no crack recorded on the beam (remember that structural cracks are results of bending actions, at least in the case that we are considering). In this situation, the column and beam are operating at the same level as such their design moment of inertia is equal to their respective gross moment of inertia. (Hope we do not disagree on that). At mid span, where the maximum moment and deflection are recorded, the bottom portion of the beam is in tension as such the concrete is assumed to have cracked. But at this point, the upper side that is in compression-thus not cracked (as such the concrete definitely contributes to moment of inertia), has its flange (the supported slab) to count on. At that point, the breadth of the beam is no longer the width of the beam measuring from the underside of the slab but the flange width. So the moment of inertia is calculated as that of a T, L, Z or whatever section that results at that point due to the structural arrangement. This more than off-sets the effect of the crack registered at the bottom part of the beam at that section. Let’s leave this line of reasoning for a while and assume that the flange of the bean does not contribute to the moment of inertia of the beam (which is not the case as exposed above) and also let’s assume a cracked section for the beam and un-cracked section for the column. The crack that we are talking of only penetrates but only up to the link level. So if we have a 400mm x 1200mm beam and 400mm x 1200mm column and we calculate the moment of inertia about the principal axis, we have bh^3/12 = 400x1200^3/12 = 5.76 x10^10mm^4 for both column and beam if we assumed that the beam is not subjected to bending as such has not cracked as is the case at the bean/column interface for pinned support as discussed above. At the point of maximum bending (mid-span), neglecting the flange effect of the supported slab, then the effective depth of the beam = 1200-35 = 1165mm (35mm is the concrete cover which is the depth to which the crack penetrates). So the moment of inertia = 400x1165^3/12 =5.3x10^10mm^4. Ratio of gross moment of inertia to the equivalent moment of inertia of the cracked section = 5.76/5.3=1.087 (practically=1) which does not make any structural engineering difference. The implication is that the proposed reduction in gross moment of inertia for beam from 1 to 0.5!!!, 1 to 0.7!!!, 1 to 0.2!!! or whatever has no engineering justification. It would be noted that:-
· The proposal quoted (……To avoid significant moment redistribution, it may be better to base the moment of inertia of the beams on an approximate transformed cracked section (e.g. 0.5Ig) and the moment of inertia of columns on a the gross section Ig." ….), definitely, was not in reference to structural design at the ultimate limit state. The writer might have made the statement in reference to a situation quite different to the one that we are discussing as such his proposal must have been quoted out of context. But if we assume that the quote was relevant to our context, then that might turn out to be a personal opinion of the writer, to which he, like any other person has the absolute right to. If this idea is a convincing one, the codes would have given it the desired consideration and at least included it as an alternative to the gross moment of inertia option (as implemented to date). The analysis as above demonstrated to the opposite !!!!.
· The codes are the engineers’ bible. The engineers have to follow it as much as is possible. So designing outside the codes (unless when absolutely necessary-as is not in this case), should be avoided, at all costs. The codes are designed by very competent personals that have at their disposal resources and wealth of knowledge that could not be surpassed by mere speculation.
· The beam enjoys lots of advantages over the column. One of these advantages stems from the fact that it is restrained along its axis by the members that it supports that form component with it (these components stiffen the beam). But in the calculation of the moment of inertia for the beam, these advantages are usually disregarded because they err on the positive side-thus conservative and for the fact that consideration of these advantages will make little or no structural difference. Again, life is already difficult for the engineer, given the simplifications adopted in design but will become almost impossible if we should cross all the “tees” and dot all the “I’s”, that would at the terminal point produce very close if not same result if we should continue to adopt the simplifications as presently.
Regards
Teddy

[iceman84]

1.How did you assume that the crack penetrates up to the link level? Cracks can't occur between links?
2. Have you ever calculated a moment-curvature relation (by hand or with a software) for a RC element? If you have I suggest to look at the stiffness in behavior stage II (cracked section) and compare it to the uncracked section stiffness? You will probably obtain a result of 0.3-0.5. That is way it is taken 0.5 (it is an average value) for beams.
3. The writer you are talking about is Thomas Paulay. I'm afraid I can never agree with your opinion on this issue. The writer's personal opinions are sustained by TESTS. Your opinion is sustained by a calculation, which in my opinion is not correct!
4. I have quoted 2 books, as requested. Unless I will be given quotes from different books I (humbly) refuse to answer in this thread.
Regards

[chigozie]

iceman84 wrote:-
1.How did you assume that the crack penetrates up to the link level? Cracks can't occur between links?
Comment: - You had already answered the question by yourself “Cracks can't occur between links” or better said “Structural cracks can't occur between steel reinforcements”. Will cracks develop if we have complete steel section (steel beam)? The answer is no. Why do we have cracks in the first place? Cracks develop in reinforced concrete structural members due to the fact that at certain conditions, they have to be subjected to bending which induces tensile stresses on some parts of the member. If that situation develops, the ability of the structural member to resist tensile stresses is called into action. We know that concrete is a pseudo plastic material as such performs poorly in withstanding tensile stresses. Its strength in tension is about 1/10 its strength in compression. So if we have grade 25 concrete, the design strength in compression = 0.45x25 = 11.25n/mm^2. Then its strength in tension = 0.1x 11.25 = 1.1. If we have 400mmx1200mm concrete beam with section modulus Z = bd^2/6 = 400x1200^2/6 = 9.6x10^7mm^3, then this section can resist in tension only a moment = stress x section modulus Z = 1.1x9.6x10^7 = 1.05x10^8Nmm = 105KNm. The weight of reinforced concrete =25KN/m^2 which implies that for the 400mmx1200mm concrete beam, weight/unit length (w) = 25x0.4x1.2 = 12KN/M or the design weight W = 1.4x12 = 16.8. For simply supported beam, moment M =WL^2/8. This implies that L = (8M/W)^0.5 = (105x8/16.8)^0.5 = 7m ; so we will find out that at a length greater than 7m, the section had already cracked under its own weight (without imposed load). But for steel section, this is not the case as steel is very good in resisting either compressive or tensile stresses equally. So if you impose steel on the way of the cracks, the steel stops the crack from developing or progressing. That was why it said that the crack only penetrates up to the link level and stops since at that level, you have the longitudinal reinforcement (like your good self also confirmed that “Cracks can't occur between links”). This is because the steel absorbs the tensile stresses so that the crack stops at that level.
2. Have you ever calculated a moment-curvature relation (by hand or with a software) for a RC element? If you have I suggest to look at the stiffness in behavior stage II (cracked section) and compare it to the uncracked section stiffness? You will probably obtain a result of 0.3-0.5. That is way it is taken 0.5 (it is an average value) for beams.
Comment:- moment-curvature relation has nothing to do with the present discuss. It may comes in when you are discussing deflection and accessing the capacity of such structures like water tanks that strength does not govern design but are designed based on the serviceability limit condition. This is not same thing as when you are designing under the ultimate limit condition.
3. The writer you are talking about is Thomas Paulay. I'm afraid I can never agree with your opinion on this issue. The writer's personal opinions are sustained by TESTS. Your opinion is sustained by a calculation, which in my opinion is not correct!
Comment:- I am neither challenging the writer’s integrity nor his opinion. What I am saying is that you must have quoted him out of context. Situations could arise in which you have to discount part of the concrete strength in flexion or whatever (like while assessing the strength of leaking concrete tank-serviceability condition). I had already hinted on that in my last post. But that situation does not arise in the present discuss. Again you said …Your opinion is sustained by a calculation, which in my opinion is not correct!...”.
Comment:- Please proof me wrong. Demonstrate (we are engineers as such we have to support our discuss with facts and figures).
4. I have quoted 2 books, as requested. Unless I will be given quotes from different books I (humbly) refuse to answer in this thread.
Comment:- I do not need to quote any books. The codes have already confirmed that that is the right thing to do. Is what I said in line with the codes of practice? Check out this example (on analysis of reinforced concrete frame) in Mosley and Bungey “Reinforced concrete design” pages 37-42. You will see that your proposed modification of moment of inertia from “I” to “0.5I-0.3I for the beam” was not implemented. This is because it lacked bases. Again check out page 264 of “Reinforced concrete designer’s handbook edited by Charles Reynolds et allies”. These books that I quoted are “authorities” as long as civil engineering is concerned. The Reinforced concrete designer’s handbook, on that page calculated the moment of inertia of un-cracked concrete beam section and that for column and found their ratio, It went on to calculated the moment of inertia of transformed cracked concrete beam section and that for column and found their ratio section and compared the differences between the two approaches and arrived at same ratio as such concluded that there is no gain in using the more fatigues approach as it will at the end give same result as for the untransformed section as such will amount to a waste of precious time that will achieve nothing.
One of the gains of belonging to or having access to an association like ours(CIVLEA) is the ability to share and to learn. We all learn every day of our lives. We would be most fortunate if we could have avenues to learn particularly the positive things like we get from this community. It is said that two heads are better than one. One of the reasons why is because, I may be having an idea that I thought of to absolutely correct but which actually was not very correct. If you are existing (solely) on your own, you will have nobody to point it out for you. But if you have the opportunity to share with others, you might get the benefit of the others helping you to correct that idea. When that becomes the fact, like happens to all of us in life, you do not have to be sad over it, but be grateful and thankful to your stars that you got that opportunity. So I see no reason why you should be annoyed. No one is here to humiliate the other. We are all learners and like I said before now, we learn all the days of our lives.
Regards
Teddy