Two Main Methods for Yield Line Analysis of Slabs

Transcription

1 Two Main Methods for Yield Line Analysis of Slabs Valentín Quintas 1 Abstract: There is a controversy about whether the classical yield lines analysis methods are in fact different methods or simply different ways to develop basically the same method. In this paper two methods are proposed that, without invalidating previous ones, really correspond to the two different ways of performing yield line analysis and therefore facilitate a better comprehension of the general problem of the failure of slabs. These methods are the nearly abandoned normal moment method and a new skew moment method. In normal moment method only bending moments are supposed to act at yield lines. In skew moment method, twisting moments in addition to bending moments act along yield lines. The normal moment method is general only if yield patterns are correct, that is, they are composed by possible yield lines. If yield lines are incorrect, or not possible, yield line analysis can only be performed, in general, by means of skew moment method. As shown in this paper, many of the classical solutions of yield line analysis correspond to incorrect yield patterns. This work demonstrates that Johansen s nodal force theory or equilibrium method and work method are only partial applications of skew moment method. This generalization of yield line analysis allows defining new equilibrium conditions not included in classical yield lines theory and permits obtaining more accurate solutions. CE Database keywords: Slabs; Analytical techniques. Introduction Ingerslev 1923 performed for the first time a yield analysis for a simply supported rectangular slab by means of which in what follows will be referred to as the normal moment method ; in essence, by simply assuming the equilibrium between loads and only bending moments acting alone at yield lines. This method is the natural approach to yield line analysis because, as Johansen himself recognized Johansen 1962, p.17, at real yield lines only the greatest principal moment acts. However, when Johansen applied the principle of virtual work to the yield mechanism of certain yield patterns the so-called work method he found that results of that method did not agree with that of the normal moment method. He correctly deduced that in those particular cases shears and twisting moments act at yield lines, in addition to bending moments, and therefore, they do not correspond to real yield lines. This type of yield line will be described in what follows as an incorrect yield line. On the contrary, if they correspond to possible real yield lines, they should be described as correct yield lines Quintas By means of his third theorem, Johansen 1962 restricted the application of normal moment method to the particular case in which only yield lines of the same sign meet at a point. Using the terms of this work, he should have stated that correct yield lines are those that accomplish his third theorem. Some paradoes found for certain yield patterns Jones and Wood 1967 show that fulfilling the third theorem is only one of the conditions that a yield line needs to be correct. 1 Dept. de Estructuras de Edificación, E.T.S.A.M. Univ. Politécnica de Madrid, Avda. Juan de Herrera, 4, Madrid, Spain. Note. Associate Editor: Victor N. Kaliakin. Discussion open until July 1, Separate discussions must be submitted for individual papers. To etend the closing date by one month, a written request must be filed with the ASCE Managing Editor. The manuscript for this paper was submitted for review and possible publication on February 28, 2002; approved on July 11, This paper is part of the Journal of Engineering Mechanics, Vol. 129, No. 2, February 1, ASCE, ISSN /2003/ /$ For the rest of the yield patterns Jonansen developed nodal force theory or equilibrium method. In this method resultants of shears and twisting moments are reduced to forces acting at nodes of the yield pattern. As it has been demonstrated Kemp 1965; Morley 1988, at least for isotropic slabs, nodal forces are only pairs of forces equivalent to the resultants of twisting moments acting at each yield line. Since in this equality nodal forces and twisting moments are multiplied by the same length of the yield line, nodal forces are equivalent to twisting moments, but never to shear forces. If nodal forces are substituted by twisting moments acting at yield lines, a more general method of analysis can be performed. This method will be named in this work skew moment method, and it envisages, in addition to nodal force theory and work method, new equilibrium conditions. It will be concluded that yield line analysis can be approached more successfully using two basic ways: normal moment method and the skew moment method. The scope of these two methods is outlined by the use of two basic principles that define whether a yield line is correct or incorrect. Principle of the Yield Mechanism. Curved Yield Lines and Simulated Annealing Method The first basic principle of yield line analysis or principle of the yield mechanism is that: yield lines must divide the slab in such a way that it is transformed into a mechanism. In order to simplify yield analysis Johansen, by means of his first two Theorems, restricted this basic principle to straight yield lines that, consequently, divide the slab into plane regions. However, as it can be seen in real load tests, real yield lines, and consequently regions bounded by them, are very frequently curved. This curvature can be produced by elastic deformations or by partial cracks, very visible in real tests. The eistence of curved yield lines for certain boundaries is very important for this work because, as we shall see later, in those cases correct and real yield lines must be necessarily curved. All this was confirmed using simulated annealing method Vazquez The simulated annealing method is an optimiza- JOURNAL OF ENGINEERING MECHANICS / FEBRUARY 2003 / 223

2 Fig. 1. Annealing method results tion technique based on the selection of random sequences of design emulating the reduction of temperature in a bar that goes from a high to a low temperature Kirkpatrick et al Although this technique has been applied to the optimal design of structures Tzan and Pantalides 1996, the application to the failure of concrete slabs has been done for the first time by Vazquez In this application the random designs selected are the yield patterns of a slab. The results obtained by Vazquez agreed very closely with real tests and confirmed the curvature of yield lines for certain edges. They also showed that this curvature was produced mainly by partial cracks that bound partial plane regions Fig. 1. Consequently it was found that many classical solutions of yield line analysis as that of Fig. 2a obtained by Johansen 1962, pp. 77 and 78 are only approimations, while the best yield pattern obtained is very frequently curved, rather different and corresponds to a yield bending moment 26% greater Fig. 2c Vazquez Notation, Representation, and Assumptions Used As we must use etensively Mohr s circle, the calculation of bending and twisting moments acting at any direction become much easier if bending moments are represented as vectors normal to those lines and twisting moments as vectors with the same direction of lines along which they act Fig. 3. In other words, we will represent bending moments and twists as vectors with the same direction of the stresses produced by these moments. The two bending moments acting at a point of a slab will be designated as M a and M b. Twisting moments will be designated as M ab and M ba, or simply as M ab, since M ab M ba. The two principal bending moments will be designated as M and M. Fig. 3, and the shear force acting at a yield line as T a. Finally, a yield positive line should be represented as one crooked line, a yield negative line as two crooked lines, a free edge as a line, a simply supported edge as two lines, a clamped edge as a family of parallel lines, and a column as a circle. In what follows it will be assumed that the slab yields at any point and in any direction with a positive yield bending moment, designated M p, and a negative yield bending moment, designated M p. This corresponds to a uniformly reinforced concrete slab with different reinforcements for positive and negative bending moments, that is, to an isotropic slab. The use of isotropic slabs clarifies the following eposition and, in any case, it can be easily generalized to orthotropic reinforcement. Principle of Normality of Yield Moments. Correct and Incorrect Yield Lines A correct yield line occurs in the direction perpendicular to the largest principal bending moment. Consequently the yield line is Fig. 2. Johansen s solution compared with annealing method solutions only submitted to a bending moment normal to its direction, and shear forces in some cases. This statement can be easily shown simply by looking at Lame s ellipse of moments Fig. 4. The maimum of all the internal moments that act at a point has to be the greatest principal bending moment M and, therefore, yield lines must appear in the direction normal to that bending moment when M reaches the value of M p. This has also been stated by Sobotka 1989 and Johansen This general property implies that along real or correct yield lines twisting moments have to be always zero. 224 / JOURNAL OF ENGINEERING MECHANICS / FEBRUARY 2003

3 Fig. 3. Representation of bending moments If M p is a mathematical local maimum, shear forces have to be zero along the yield line, as T a M a /a M ab /b, M ab 0 and M a /a 0. Fig. 3. It must be noted that M a can reach the value of M p simply by being the largest value of all bending moments in that region without being a local maimum. In this case T a has a definite value along yield lines, though M ab continues to be zero. T a must be used to establish the equilibrium at the yielded plate, but in no case T a should be the cause of yielding. On the contrary, a yield line that needs the aid of twisting moments to equilibrate loads, or fulfill the boundary conditions, will be a virtual or incorrect yield line, since there always eists at any point of that line a direction in which there is a bending moment greater than the assumed yield bending moment M p ; that is, the principal bending moment M. Using the above general property, several laws can be applied simply by using Mohr s circle in order to distinguish between correct and incorrect yield lines and to find the static laws that must relate forces acting at incorrect yield lines. 1. Intersections of Yield Lines At the inner part of the slab it can be established that: 1a. The intersection of yield lines having the same sign is always possible, and therefore correct. These intersections represent the case where M a M M b M M p and M 0; that is, when Mohr s circle degenerates into a point, all the directions are principal directions and the value of the twisting moment is zero Fig. 5a. This apparently agrees with Johansen s third theorem. Nevertheless, it can be only stated that these intersections are correct in some cases where the yield pattern has aial or double symmetry, as those shown in Fig. 5b. In other cases, as that of Fig. 5c, only for a specific slope the yield lines Fig. 5. Correct intersections of yield lines having the same sign are free of twisting moments, for simple reasons of equilibrium. Finally, as shown below, cases can be found in which twists are necessary to fulfill boundary conditions. 1b. Intersection of yield lines having different signs is only possible when they are only two and they cross at a right angle. This corresponds to the case in which M a M M p, M b M M p, and M ab 0 see Mohr s circle in Fig. 6 with the angle a90. Following this, the corner lever pattern of Fig. 7a is incorrect, while the correct solutions are the fans shown in Fig. 7b, in which positive yield lines intersect at right angles the negative curved yield lines. Two possible equilibrium states for those incorrect yield lines can be postulated Fig. 8. In the first state Fig. 8a it is assumed that one of the yield lines is the correct one and therefore subjected to the yield bending moment M p whereas the other yield lines are incorrect ones at which, in addition to bending moments M a, twists M ab act. The relationship of M a and M ab to M p and M p is that of moments in any direction with principal bending moments, and can also be deduced from Mohr s circle in Fig. 8a M a M p tan am ab M ab M p M p sin 2a (1) 2 M a M p cos 2 am p sin 2 a Fig. 4. Lamè s ellipse of bending moments Fig. 6. Correct intersections of yield lines having different signs JOURNAL OF ENGINEERING MECHANICS / FEBRUARY 2003 / 225

4 Fig. 7. Yield lines at corners of simply supported slabs a angle at which the incorrect yield line intersects one of the principal directions or the correct yield line. In the second state Fig. 8b it is assumed that all the yield lines are incorrect and therefore, all are submitted to twisting moments M ab1,m ab2,... in addition to bending moments M a1, M a2,... with a smaller value than that of the principal bending moment M p. The relationship of those moments to M p continues to be Eq. 1, making M a1 M a, M ab1 M ab, and M a2 M a, M ab2 M ab. A relationship between the internal forces at the two incorrect yield lines can also be obtained by means of Mohr s circle of Fig. 8b, or eliminating M p and M p from the relations between M a1, M a2, M ab1, M ab2 and M p, M p M ab1 M ab2 M a2 M a1 cot b where bangle of intersection between yield line 1 and yield line 2. This equation is identical to the one obtained by Johansen 1962 for nodal forces in isotropic slabs, if M ab1 M ab2 is replaced by the sum of nodal forces Q A, and the difference between M ab1 and M ab2 by that of yield bending moments that are supposed to act at the incorrect yield lines. It can be shown that: a nodal forces correspond only to the twisting moments of that particular state of equilibrium, and b that the first state is not included in nodal force theory. Fig. 8. Equilirium at incorrect intersections of yield lines Fig. 9. Yield lines at free edges 2. Edges The following conditions at the different boundaries of a plate can be established. 2a. Yield lines have to be normal to free edges. As known, if the boundary condition is M a 0, M ab 0, principal directions must be parallel and perpendicular to edges, and the principal bending moment normal to the edge must be zero. In these conditions Mohr s circle is that of Fig. 9a with an angle a90. This law implies in many cases that correct lines must be curved ones Figs. 9b, c, and d. Therefore a yield line that reaches a free edge at an angle a different from 90 is incorrect. It can be deduced from Mohr s circle of Fig. 9a that these lines must be submitted to bending and twisting moments M a and M ab related to the principal bending moment M by M a M p cos 2 a, M ab M p sin a cos a (2) and related between them by M ab M a tan a (3) This is eactly the epression of nodal forces obtained by Johansen, provided we make Q 1 M ab, 90a, M p 0, and M p M a. As always happens on incorrect yield lines, a principal moment M, larger than M a, eists acting at another direction. In this particular case M has a defined value that can be deduced from Eq. 2: M M a /cos 2 a M a (1tan 2 a). 226 / JOURNAL OF ENGINEERING MECHANICS / FEBRUARY 2003

5 Fig. 10. Yield lines at simply supported slabs Fig. 11. Yield lines at corners of clamped edges Therefore the value M p of the real yield bending moment can be bounded if we consider M a an upper-bound solution and M a lower-bound solution M a 1tan 2 am p M a (4) 2b. Yield line can reach a simply supported edge at any angle. Since, assuming Kirchhoff s hypothesis, the boundary condition for a simply supported edge is M a 0, M ab 0, an infinite number of Mohr s circles can be found fulfilling this condition at point A of Fig. 10a, and therefore also infinite angles a, with the only condition that the two principal bending moments have to be of opposite signs. However, the twisting moment M ab acting at the edge must be taken into account. From Mohr s circle of Fig. 10a the relation between this moment and the principal bending moments M M p and M M p can readily be obtained: M ab M p cot am p tan a, and the condition tan 2 a M p M 1 (5) j p where jratio of M p to M p. This condition was also obtained by Johansen 1962, p. 22 for real yield lines, and studied by Nielsen 1984, but it was not included in equilibrium method. The above law means that a yield line is never incorrect if it arrives alone at a simply supported edge. However, if two yield lines intersect at a simply supported edge the only correct way of doing it is being of opposite signs and acting at normal directions. The patterns of Fig 10c are therefore incorrect and the value of twists M ab acting at those lines must be obtained as in the case of yield lines of different signs Eqs. 1 of law 1b. 2c. Yield lines can reach a clamped edge at any angle. This is obvious, as the boundary condition is M a 0, M ab 0. Nevertheless, if a straight clamped edge were to be a yield line, and therefore subject to the principal bending moment M p M, following law 1b, positive yield lines would be normal to the edges Fig. 11a. It is impossible to fulfill this condition at corners and, therefore, fans must always develop at clamped corners Figs. 11a and b. The results of the simulated annealing method Vázquez, 1994 confirm this conclusion. Incorrect yield lines are subject then to the efforts of intersections between yield lines of different signs described in law 1b by Eq. 1. Equilibrium Conditions. Normal Moment Method and Skew Moment Method In the usual process of designing a slab the value of the yield load p of the slab is known, and therefore the aim of yield analysis is to obtain the values of yield bending moments M p and M p that appear at the failure state of the slab. This can be done by following two equilibrium conditions: 1. At each region of the yield mechanism of a slab, internal forces acting at yield lines must balance loads and reactions. 2. Internal forces must be in equilibrium at each side of a yield line. These two conditions can be performed using directly equilibrium equations or, alternatively, applying the principle of virtual work to the whole mechanism of the slab supposing that internal forces that act at each side of each yield line are equal and, therefore, fulfilling simultaneously the equilibrium conditions 1 and 2. The results must be identical as principle of virtual work is only a way of using equilibrium equations. In this way we can always obtain a relation between internal forces and the geometrical parameters of the yield pattern of the slab. The following step is to find the values of those parameters that approach best the real yield pattern. This can be done by two basic methods: the normal moment method and the skew moment method. In the normal moment method, it is assumed from the beginning that only a constant bending moment M a acts at yield lines, plus shear forces if M a is not a local maimum. In skew moment method a constant moment, M a, and a constant twisting moment, M ab, whose resultant is a skew moment are assumed to act at yield lines, plus shear forces if applicable. As their application is very different for correct and incorrect yield patterns, the two methods will be separately considered depending on the nature of the yield pattern concerned. Correct Yield Patterns If the yield pattern is correct, in normal moment method we first obtain the relationship between the bending moment M a acting alone at yield lines of each region and a shear T a if JOURNAL OF ENGINEERING MECHANICS / FEBRUARY 2003 / 227

6 Fig. 13. Virtual displacements used to find separated bending moments at each region For region B: M B a l y l /2 2p 1 l 2 2 tan a l 2 3 Fig. 12. Regions of a simply supported rectangular slab applicable and the loads. Since the unknown M ab has been eliminated, the geometry of the correct yield pattern can be obtained directly by equating moments at each side of each yield line. The bending moment corresponding to that correct yield pattern is then supposed to be the yield bending moment M p. In skew moment method, we must fulfill the two equilibrium conditions 1 and 2 at the yield mechanism, in order to obtain a relation between the internal forces and the geometrical parameters of the yield pattern. The geometrical parameters that define the correct yield pattern are obtained making zero all twisting moments at every internal yield line. Alternatively, this can be performed using maimum principle, as we shall see later. A simple eample may clarify all this. Eample 1 Consider the very well known yield pattern of a simply supported slab of Fig. 5c. This yield pattern can be correct and M a corresponds to a local maimum; therefore both methods can be applied assuming T a 0. If we use the normal moment method and equilibrium equations we can obtain separated moments M a A and M a B in regions A and B, taking moments about the edges Fig. 12 M A a p l 2 24 tan2 a, M B a p l 2 8 p l 2 12 tan a with l y l (6) We can perform the same calculations using work equations, assuming regions A and B are isolated, and the virtual displacement of Fig. 13 For region A: M A a l l /2tan a p 1 2 l l 2 tan a 3, pl y l tan a l 2 And we obtain the same epressions of M a A and M a B. The correct yield pattern is the one for which M p M a A M a B. This equality results in the equation: tan 2 a 2/tan 2 a 30, that gives the value of the angle a of the correct yield pattern: tan a(1/) 2 3 1/. In this way Ingerslev 1923 solved this problem for the first time. It must be noted that law 2b and condition 5 give the value of the yield negative bending moment that must support the slab: M p M p /tan 2 a. If we use the skew moment method, we must assume that both constant bending moments M a and constant twisting moments M ab act at yield lines, with the eception of the central yield line, at which, for obvious reasons of symmetry, M ab 0, and the edges, at which M a 0. If we use equilibrium equations, and we take moments about edges 1-1 and 2-2, then equilibrium in each region gives For region A: For region B: M A a M A ab tan a p l 2 24 tan2 a M B a M B ab p l p l 12 tan a where l y /l. Equilibrium at yield lines implies that: M a M A a M B a and M ab M A ab M B ab, and the two equations become a system of two equations with two unknowns whose solution is M ab p l 2 24 M a p l 2 8 tan a/3 1/tan a tan 2 a 2/tan a3 1/ tan a The same epressions of M a and M ab can be obtained using work equations and virtual displacements that make zero the virtual work of the other unknown: In order to obtain M a, the well-known virtual displacement of Fig. 14a can be assumed. Making, as in equilibrium equations, M a M a A M a B we obtain the same epression of M a Eq. 7.To obtain M ab, we can assume the virtual displacement of Fig. 14b, in which regions B moves upwards a vertical value, and region 2 (7) (8) 228 / JOURNAL OF ENGINEERING MECHANICS / FEBRUARY 2003

7 Fig. 14. Virtual displacements used to find yield bending moments and twists A moves downward (l y /l tg a), in order to maintain compatibility of horizontal displacements at the corners. The work equation is then 4M A ab l y tan a4m B l ab p l l 2 tan a l y tan a4p 1 l l tan a 3 l p l 2 l yl tan a 2 Making M A ab M B ab M ab we obtain the same epression 8 of M ab. We must now find the geometrical values of the correct yield pattern, provided that for this pattern M ab must be zero. Making zero the epression 8 of M ab, we obtain: tan 2 a (2/)tan 2 a 30, which is eactly the same equation obtained by the normal moment method. The value of tan a introduced in the epression 7 of M a gives the same value of M p M a obtained by the normal moment method, though the epression of M a is very different. It can be observed that calculations performed by normal moment method are, by far, the simplest. In fact normal moment method is the adequate method if correct yield patterns are analyzed. This situation can be very different for incorrect yield patterns as will be shown in what follows. Incorrect Yield Patterns Fig. 15. Yield pattern for a square simply supported slab In many cases, laws defining correct yield patterns of The Principle of Normality of Yield Moments can only be accomplished by means of curved yield lines. Although it is possible to apply normal moment method to that type of yield pattern, calculations are usually simpler if yield lines are supposed to be straight, even if the yield pattern becomes incorrect. If a yield pattern is incorrect, equilibrium is impossible without the aid of twists acting at yield lines. The yield pattern that best approaches the real yield pattern is then the one accomplishing the laws of Principle of Normality of Yield Moments. Correct and Incorrect Yield Lines for incorrect yield lines; that is, the one which is at least in equilibrium: the balanced yield pattern. Therefore, for incorrect yield lines, the laws of The Principle of Normality of Yield Moments substitute the condition of zero twisting moments. As shown later the balanced yield pattern can be also obtained using maimum principle. Anyway, in some cases normal moment method can be used: If the yield lines selected to enter into equilibrium equations are supposed to be correct, twists could not appear in those equations. The balanced yield pattern can thus be obtained as in the correct yield patterns, simply equating bending moments. The following eample may clarify this. Eample 2 Consider the yield pattern of Fig. 15, of a square slab subject to a uniform load p. It is incorrect, as yield lines 1 and 2 cross at an angle a different from 90. We can suppose that yield lines 2 are correct, and therefore subject only to the yield bending moment M p, and yield line 1 is the incorrect one, and therefore subject to a bending moment M a, which is not the yield negative bending moment M p, a shear force T a, and a twisting moment M ab. Since T a and M ab do not enter into the equilibrium equations, we can use the normal moment method, applying equilibrium equations separately to each region. Taking moments about the edges in region B, and about the negative yield line in region A, we obtain For region B: For region A: M p B p l d 3 1tan a 2 12d (9) M p A M a A p l2 12 d2 tan 2 a (10) JOURNAL OF ENGINEERING MECHANICS / FEBRUARY 2003 / 229

8 where d /l. We must suppose that in region A a principal negative bending moment eists: The yield negative bending moment Mp A. Using 1 and introducing M p A with its sign, M a A M p A cos 2 am p A sin 2 a If we call jm p A /M p A, we can write M a A M p A (1 j tan 2 a)/(1j tan 2 a and the value of M p A is, after introducing that of M a A in Eq. 10 M p A p l2 24 d 2 tan 2 a1tan 2 a 1 1 j/2tan 2 a (11) Making M A p M B p, and therefore equating Eq. 9 to Eq. 11, an equation with d as the unknown is obtained after simplifications 1tan a2 2 1 d 3 tan 2 a d 2 1tan a j 2 tan2 a tan 2 a 1 1j/2tan2 a 0 1tan 2 a It might look as if two unknowns continue to eist: d and tan a, but tan a is defined by the boundary conditions of the slab. In this case, by law 2b for simply supported edges. Therefore Eq. 5 must be accomplished M ab M p cot M p tan, tan 2 M p M 1 j p In our eample Fig. 15: 135a and therefore tan 2 (135 a) 1/j that can be transformed into: tan a (1j)/(1 j). Depending on the values of j, those of M p for this slab can be obtained. The maimum is for j0.40, that gives a yield bending moment: M p p l 2 /20.64, larger than the one obtained by Jones and Wood 1967 by means of vertical equilibrium of nodal forces. For j1 we obtain M p p l 2 /24 and the yield pattern of Fig. 5b. For j 1/(12) 2, M A a 0 and the contour formed by yield negative line and edges can be regarded as a simply supported corner whose edges cross at an angle of 135. The positive line bisects that corner, and we also obtain M p p l 2 /24. In many other cases it becomes impossible to find yield lines that can be assumed as correct, or incorrect yield lines entering into equilibrium equations. The only possible method available is then skew moment method. This case can be best studied by means of the following eample. Eample 3 Consider a square slab, simply supported at two adjacent edges, with the other two edges free, and submitted to a uniform load. The correct yield pattern is the correct solution of Fig. 9d, which implies curved yield lines. The solution obtained assuming a yield line that connects the corner at the intersection of the two supported edges to the diagonally opposite corner is at the same time incorrect and unbalanced, as M ab must be zero for reasons of symmetry. The only possible yield pattern involving one straight yield line is that of Fig. 16, proposed by Johansen 1972 and studied by Nielsen 1984, that is only incorrect. It cannot be analyzed by means of the normal moment method, as twisting Fig. 16. Yield pattern for a square slab with two free adjacent edges moments have to be taken always into account. If we use the skew moment method, we obtain, taking moments about the edges, the equilibrium equations For region A: M A a M A ab tan a p l2 6 tan2 a For region B: M B a tan am B ab p l2 p l2 tan a 2 3 since T a 0, if M a corresponds to a local maimum. A At each side of the yield line, it must be accomplished: M a M B a M a and M A ab M B ab M ab. Introducing these conditions into the above equations, they become a system of two equations with two unknowns, M a and M ab, whose solution is M a p l2 6 M ab p l2 6 3 tan atan 2 a 1tan 2 a tan 3 a2 tan a3 1tan 2 a (12) The same can be obtained using work equations. The condition of equilibrium for this incorrect yield line is that of law 2a and Eq. 3: M ab M a tan a Introducing in this equation the values of M a and M ab obtained above, the free boundary condition becomes the equation tan 2 a(2/3)tan a10, whose solution is tan a (101)/3. Introducing this value of tan a into Eq. 12, we obtain M a p l and M ab p l The principal bending moment is given by Eq. 4: M M a 1tan 2 a p l and therefore the real yield bending moment M p is bounded between p l M p p l This great difference between the two limits should lead us to suppose that this solution is a rough approimation to the real yield bending moment (pl 2 /5.1, as obtained by the annealing method. The negative yield bending moment that appears at each region is given by Eq. 5. In any case it has no influence on the yielding of this pattern, as it appears only at the simply supported corner. Maimum Principle and Work Method The maimum principle corresponds to Johansen s fifth theorem Johansen 1962, and can be stated as Johansen did: The real 230 / JOURNAL OF ENGINEERING MECHANICS / FEBRUARY 2003

9 yield pattern corresponds to the maimum absolute value of the ultimate bending moment M p. It is well known that this principle is only a corollary of the upper-bound theorem of limit analysis when applied to the yield mechanism of an isolated slab. It has two applications: as a criterion to be able to tell which is the real yield pattern or the yield pattern that approimates best the real yield pattern among a family of yield patterns for the same slab, and as a method of analysis of yield bending moments, the so-called work method. Since the balanced yield pattern is always in equilibrium, if we use the skew moment method and obtain the epression of normal bending moment M a, the yield pattern that approimates best the real yield pattern is the one corresponding to the maimum of M a and, at the same time, the one which is in equilibrium Nielsen 1984, p This allows obtaining the value of M a without making M ab 0 in correct yield patterns, or without defining the value of twisting moments and shear forces at the yield lines of incorrect yield patterns. As we have seen above, M a can be obtained by equilibrium or work equations alternatively, so it makes no real sense to call this method the work method and it should be best described as the maimum principle applied to the skew moment method. The application of this method to the eamples that we have studied previously may clarify its use. In the rectangular slab of Eample 1 we have obtained by the skew moment method using work or equilibrium equations the epression of M a M a p l 2 8 tan a/3 1/tan a The value of tan a that makes M a a maimum can be obtained by making: M a / tan a 0, and this leads to the equation: tan 2 a (2/)tan a30, which is the same one obtained by the skew moment method, making M ab 0. In Eample 2, the maimum principle has no application, as we have used the normal moment method. In Eample 3, we have obtained by means of the skew moment method the value of M a M a p l2 6 3 tan atan 2 a 1tan 2 a The value of tan a that makes M a a maimum corresponds to: M a / tan a 0 and we deduce the equation: tan 2 a(2/3)tan a 10, which is the same one obtained introducing boundary conditions. Conclusions 1. Yield line patterns can be classified into two types: correct yield patterns that correspond to possible yield lines and incorrect yield patterns that correspond to not possible or virtual yield lines. 2. There are only two main methods in the yield analysis of slabs: the normal moment method, in which only bending moments and shear forces if applicable are assumed to act at yield lines; and the skew moment method, in which bending moments together with twisting moments and shear forces in some cases are supposed to act at yield lines. The problem can be solved with work or equilibrium equations alternatively in all cases. 3. The normal moment method is the adequate method for correct yield patterns composed by curved or straight yield lines 4. The skew moment method can be applied to correct yield patterns using the condition of zero twisting moments. 5. The skew moment method can be applied to incorrect yield patterns if internal forces are added on incorrect yield lines in order to accomplish equilibrium conditions in the slab. Johansen s nodal forces are a particular case of these internal forces. 6. The work method is only the application of the maimum principle to skew moment method. Notation The following symbols are used in this paper: a angle between a yield line and a principal direction; j ratio of negative to positive yield moments; k nodal force; l length of slab; M a, M b bending moments at any point of the slab; M ab twisting moment at any point of the slab; M, M principal bending moments; M p yield positive bending moment; M p yield negative bending moment; p uniform load acting at a surface; Q sum of nodal forces at a point; T a shear force at any point of the slab; angle between a yield line and any other direction; angle that forms a corner of a slab; and virtual displacement. References Ingerslev, A The strength of rectangular plates. J. Inst. Estruct. Eng. December. Johansen, K. W Yield-line theory, Cement and Concrete Association, London. Johansen, K. W Yield-line formulae for slabs, Cement and Concrete Association, London. Jones, L. L., and Wood, R. H Yield-line analysis of slabs, Elsevier, New York. Kemp, K. O Recent developments in yield line theory. The evaluation of nodal and edge forces in the yield-line theory, Cement and concrete Association, London. Kirkpatrick, S., Gelatt, C. D., Jr., and Vecchi, M. P Optimization by simulated annealing. Science , Morley, C. T Nodal forces in slabs and the equilibrium method, Butterworths, London, Nielsen, M. P Limit analysis and concrete plasticity, Prentice Hall, New York. Quintas, V Sobre el método estático en el cálculo de placas de hormigón armado. Hormigón y acero., 190 in Spanish. Sobotka, Z Theory of plasticity and limit design of plates, Elsevier, New York. Tzan, S. R., and Pantelides, C. P Annealing strategy for optimal structural design. J. Struct. Eng. 127, Vázquez, M Recocido simulado: un nuevo algoritmo para la optimación de estructuras. PhD thesis, Universidad Politécnica de Madrid, Spain, Chap. 4. JOURNAL OF ENGINEERING MECHANICS / FEBRUARY 2003 / 231