07-12-2010, 12:13 PM

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

.....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