ELSEVIER 0141-0296(95)00153-0

Engineering Structures, Vol. 18, No. 4, pp. 332-337, 1996 Copyright © 1996 Elsevier Science Ltd

Printed in Great Britain. All rights reserved 0141~)296/96 $15.00 + 0.00

On the collapse Alan Jennings

identification mechanisms

of yield-line

Civil Engineering Department, Queen's University, Belfast BT7 INN, Northern Ireland, UK

(.Received December 1994; revised version accepted June 1995)

A theorem is proved which is useful in establishing topologies of yield-line mechanisms likely to cause collapse of an important class of slabs (or plates). It is also shown, through another theorem, that the search amongst concave mechanisms for the most probable one to cause collapse can use a uniqueness property which ensures that the function to be searched has no false maxima.

Keywords: plates, slabs, reinforced concrete, yield-line theory, col- lapse mechanisms

1. Introduction

Yield-line theory for predicting the collapse load of slabs and plates was first proposed by Ingerslev I and developed by Johansen 2, Wood 3 and others. It has been used exten- sively for analysing the ultimate load capacity of under- reinforced concrete slabs 3,4 where the moment-curvature characteristics of the slab are dominated by yielding of the tension steel. For the analysis of steel plates, yield-line the- ory is most applicable when the slabs are thick in relation to the plan dimensions. In that case, small deflection theory assumptions hold sufficiently well up to the collapse con- dition. Thus, yield-line theory may be used for predicting collapse of components of bolted and welded joints s. When used for this purpose, 'collapse' may be interpreted as the situation where gross deformations occur causing distress (although not necessarily complete failure) of the joints. In the ensuing text, the term 'slab' will be used as referring to either a slab or a plate.

The fundamental assumption is made that elastic defor- mations are negligible, which means that deformation only takes place by plastic rotations occurring at 'yield-lines' within the slab. For a slab to collapse, the following must hold:

(a) the kinematic condition that there is a mechanism which involves only plastic rotations at the yield-lines;

(b) the static equilibrium condition that there is a set of internal forces in equilibrium with the applied forces for which the bending moment in the slab nowhere exceeds the yield moment;

(c) the yield condition that the statically admissible set of internal forces in (b) contain plastic moments acting at all the yield lines of (a) in the correct sense for plastic rotations to occur.

In common with other problems involving rigid-perfectly plastic solids, it is possible to use one of the extremum

principles for solution in which either the minimum load to satisfy condition (a) or the maximum load to satisfy con- dition (b) yields the collapse load 6. In this paper, the kine- matical theorem is employed by seeking the yield-line pat- tern which, according to the virtual work principle, has the minimum collapse load. If the search is incomplete, the resulting load will be an upper bound to the collapse load or, in other words, an unsafe solution. (However, if the search encompasses the true minimum, the use of terms 'upperbound' or 'unsafe' would be misleading since the true theoretical solution is obtained.)

The so-called 'equilibrium method' for analysing yield- line patterns was adopted by Ingerslev and others. This is not a statical equilibrium method in the sense of the extremum principles, but is a kinematic formulation which can be derived from the virtual work method by differen- tiation with respect to geometrical parameters 3. Whereas it would seem to allow the user to improve the yield-line pat- tern so as to approach the critical one, it does not appear to have been established in any rigorous way whether the collapse mechanism will be obtained.

In 1987, Munro and Da Fonseca developed a method, suitable for computer implementation, of predicting the col- lapse load of slabs by assuming that yield-lines occur on the boundaries of triangular finite elements 7. Although this constraint on the yield-line geometry impairs the accuracy and reliability of the results, methods are being developed for incorporating variations in the finite element geometry 8 15. However, since the user is more remote from the sol- ution process than is the case with hand solution, particular care needs to be taken to ensure that the yield-line patterns producing collapse are not missed j6.

Uniqueness properties have been identified for some sol- utions involving rigid-plastic continua Iv 21. These relate to situations in which there exist=:

(1) statically admissible stresses that nowhere exceed the yield limit; and

332

Yield-line collapse mechanisms: A. Jennings 333

(2) kinematically admissible velocities associated with strain rates that are compatible with the stresses and differ from zero in at least part of the continuum.

(A kinematically admisslible set of velocities corresponds to the virtual displacement of a mechanism.) To date, uniqueness has only been verified for a limited number of situations, an example being a uniform isotropic slab of arbitrary shape subject to a single concentrated load 23. The difficulty of obtaining statically admissible stresses which are known not to exceed the yield limit make this technique difficulty to extend to more general cases of loading, slab geometry and boundary conditions.

In such circumstances., knowledge about critical yield- line patterns and whether local minima for collapse load may or may not be global minima would be particularly valuable. This would help in the development of computer search strategies and would also help to give confidence in the computer results obtained. The work reported in this paper is aimed at identifying which from a class of mechan- isms is likely to precipitate collapse. This knowledge could be used to enhance the development of computer-based search strategies.

Notation

ci coefficient of (I) i in the energy dissipation equation dA element of slab itrea h normal distance to boundary edge of facet i subscript for positive (sagging) yield-lines j subscript for negative (hogging) yield-lines k subscript for all simply supported boundaries l length of a yield-line (l > 0) lx,ly projected lengths of a yield-line in x- and y-direc-

tions MN negative yield moment for an isotropic slab MA,;MNy negative yield moments for an orthotropic slab AIp positive yield moment for an isotropic slab

Mpx ,Mpy n p(x,y) U V Wi

w(x,y) wl(x,y) x,y O~

0 Ox, Oy

h ~kcrit

positive yield moments for an orthotropic slab number of boundary facets applied load (p->: 0) loss of potential energy of applied forces energy dissipation at yield-lines displacement for case i lateral deflection (w > 0) lateral deflection of facet l orthogonal coordinates of slab inclination of yield-line to x-axis slab rotation at a yield-line rotation of lines parallel to x- or y-axis where they cross a yield-line load factor collapse load factor rotation of facet i about boundary modification to rotation of facet i

2. Use of energy and work concepts

Consider a virtual displacement of a mechanism. If U is the Joss in potential energy of the applied forces arising from this displacement, A U will be the equivalent loss in potential energy if the loads are factored by h. Letting V be the energy dissipation at the yield-lines due to the virtual displacement, the slab will collapse by movement of this mechanism when:

x u = v (1)

A. lthough the minimum value of h is sought, the formula hcrit = min(V/U) is only a valid criterion whilst U > O. A universal criterion which is correct even when U -< 0 is:

hc,t = 1/max (U/V) (2)

The identification of how to determine which yield-line mechanism--from a set of possible ones--will be activated first when loading is increased can be utilised in computer searches to reduce the number of possibilities requiring to be investigated.

3. Theorems

The theorems stated below, which are justified later in the text, are applicable to uniform slabs, which have no internal supports, which are simply supported or built-in on all sides of a convex polygonal boundary. They are also applicable in cases of partial collapse where the undeformed part of the slab may be considered to provide built-in support in such a way that the same conditions hold. In all cases, the load acting on the deformed region must be non-negative.

Mechanisms are considered which have zero deflection round the boundaries and which are of convex polyhedral shape. All deflections will be positive and all internal yield- lines will be positive (sagging). Since elastic deformations are ignored, unyielded parts of the slab deflect as fiat plates. These undeformed regions will be called 'facets'. Since the displacement function must be continuous, two neighbour- ing facets must intersect on a straight yield-line.

Theorem 1 on topology For any n-sided polygonal boundary to the deformed region, the mechanism requiring the lowest load to activate it will contain n facets, each having one side in common with the boundary and no other connection with it.

Theorem 2 on uniqueness In the space defined by the n-1 independent edge rotations for n-faceted mechanisms of the type satisfying Theorem 1, the reciprocal of the activation load is a convex function (thus having one unique maximum and no false maxima).

4. Change of slope across yield-lines and boundaries

Consider a yield-line of length 1 (where l is always taken to be positive) which is situated at angle oL to the x-axis as shown in Figure 1. The projected lengths of the yield-line onto the x- and y-axes are therefore

/,

I - -1 Figure 1 Projections for a yield-line (plan view)

334 Yield-line collapse mechanisms: A. Jennings

lx =/cos a , ly : / s i n a. (3)

If the yield-line rotates through a small angle 0, lines drawn on the slab parallel to the x- and y-axes (see Figure 1) will change their slopes by 0~ and 0y, respectively, at the yield line, where:

0~ = 0sina, 0y = 0cosot. (4)

It follows from Pythagoras's theorem that

ll0[ = IZyll0~l + II~ll0y[, (5)

This same equation applies also to boundary lines for the slab provided that the area exterior to the slab is considered to have zero slope.

5. Evaluation of total energy dissipation

Consider an infinitesimal strip of width dy parallel to the x-axis which traverses the entire slab. By including the rotations across all the intersecting yield-lines and also the intersecting simply supported boundaries, the fact that the slope at both ends of the strip must be zero gives:

~] 0x+~] 0x+~] 0x=0 (6) i j k

where the summations i, j and k are for all positive (sagging) yield-lines, all negative (hogging) yield-lines and all simply supported boundaries, respectively, which the strip crosses (see Figure 2). Furthermore, because the boundary lines must all have negative (hogging) angles of rotation, equation (6) yields:

ZI0xl = ZI0xl + ZI0~l. (7) i j k

Multiplying this equation by dy, and integrating for all such strips which intersect with the slab, gives:

i j k

Negative yield -~ines Simpty \ '~ '~ \ ' ~ \ \ \ \ k \ \ \ \ \ \ \ - . \ \ supported..~,~ < o \ " ~ / , /Bu i t t - in boundary ,~ ~.~ 1 _~_ - / N ~ ¢~ boundary

. . . . . . . . - / - - f

, - t - - , " ! " V , . . . . T-- i i-.4,.. " '

: ,\~yl IL¢~. ~.'>-12 - ~ ~ l "----Positive

• \ - , \ \ \ , ~ \ \ \ - , , \ \ \ \ \ \ ~ , - ~ \ \ \ \ \ "

(a ) Deflection contours

(b)

Figure2 An arbitrary mechanism. (a) Plan; (b) deflection of strip AA with signs of rotations

the summations being for all yield-lines and boundary lines. The similar equation obtained by considering strips in the y-direction is:

~lexll0yl : ~lZxllOyl + ~ll~ll0~l. (9) i j k

Where the slab is orthotropic with positive yield moments in the x- and y-directions of M e and M e , respectively, and corresponding negative yield m~)ments o'f M~ x and Msy, the total energy dissipation is:

v = Mpx ~lZyll0J + M~,, ~llxll0yl i i

+ M~ E[lyll0x[ + M~ ~[/Jl0yl. (10) J J

Substituting equations (8) and (9) into (10) gives:

w = (Me~ ÷ M~) ~lZyll0xl + (Mey + M~,) ~lz, ll0yl J J

+ Mex ~]llyll0xl + Mp ~]l/~.ll0yl. ( l l ) k k

Thus energy dissipation can be determined by examining the rotations of only the simply supported boundaries and the negative (hogging) yield-lines which for convex mech- anisms will only occur at built-in boundaries.

Note that--if the slab is isotropic---equation (5) may be used to reduce equations (8) and (9) to:

~1101 : ~/10l + ~lt01 i j k

and equations (10) and ( 11 ) simplify to:

(12)

and

V=Me ~el01 ÷ MN ~/101 (13) i j

v = (Mr + MN) ~¢101 ÷ ge ~ll01 (14) j k

6. Facet types in a concave mechanism All intersection points for yield-lines and/or boundary lines will be called nodes. Because the whole slab area under investigation is assumed to deflect, boundary nodes mark- ing vertices of this boundary polygon must impinge on at least two facets (see Figure 3(a)). Furthermore, nodes can-

\ , \ \ \ ~ 3

(a) (b)

Figure3 Yield-l ines and deflection contours adjacent to boundaries. (a) Feasible situation for a boundary vertex node; (b) node on a straight boundary violating concave requirement

Yield-line collapse mechanisms: A. Jennings 335

not occur part-way along a straight boundary without violating the condition that there be no negative internal yield-lines (see Figure 3( b ) ).

In any mechanism defiarmation, it is possible to define the plane of each facet in space by three variables. How- ever, if a facet impinges on the boundary, the condition that boundary nodes have zero deflection can be used to reduce the number of independent variables required to define its displacement. Furthermore, a facet cannot be per- mitted to impinge on two nonconsecutive boundary nodes otherwise internal parts of the slab would be constrained to have zero deflection (see Figure 4). Three types of facet can occur as follows:

(a)

(b) and (c)

Boundary facets which impinge on two adjac- ent boundary nodes and therefore have an edge in common with the boundary. Only one inde- pendent displacement variable is needed to define the position of each such facet and this can be the rotation about the boundary line. Partly and fully internal facets which impinge on one or no boundary node, respectively. The number of independent position variables will be either 2 or 3.

Figure 5(a) shows a mechanism which includes different types of facet for a slab, with the number of variables required to define the position of each facet indicated. In this case, the total number of variables required to specify the deflected shape is 14. To find the positions of the yield- lines and nodes from these variables, it is necessary to determine where the neighbouring facets intersect. Hence when facet variables are altered, the node positions change. If, for instance, the variables for facet ABCDEF are adjusted so that the displacements over the facet are all increased, a point is reached in which the facet has crossed all the intersection points for the planes of the neighbouring facets and hence disappears. Then the resulting yield-line pattern has the form shown in Figure 5(b) which may be defined by only 11 variables.

7. Justification of theorem 1 on topology

Consider the energies U and V associated with a mechanism obeying the above characteristics such as that shown in Fig- ure 5(a). If ~ , - . . ,On are the rotations of the boundary facets in a positive sense, equation (12) shows that the energy dissipation will have the form:

V= c l~l +-..+ c.~. (15)

~ sible cef

/ - -~~2- -p '~ " ~n~°?enSlreoi°n~de' (e c f i° n

Figure 4 An infeasible facet impinging on nonadjacent bound- ary nodes

(a)

\

(b)

(c)

Figure 5 A concave y ie ld- l ine pattern and modi f icat ions show- ing the number of independent facet variables. (a) Initial mech- anism; (b) mechanism wi th facet ABCDEF removed; (c) mech- anism wi th all internal facets removed

where coefficients c~,...,c, must all be positive and will depend on the yield moments, the boundary conditions and the lengths of each boundary. The most important feature of this formula is that energy dissipation is independent of the variables defining the positions of the internal facets.

On the other hand, the loss in potential energy of the applied forces is given by:

U=fpwdA (16)

where integration is over the whole area of the slab. How- ever, because of the concave displacement profile:

w(x,y) = min wt (x,y) (17)

where wt(x,y) is the displacement of the plane associated with any facet l at coordinate (x,y). Of all the mechanisms with boundary rotations dp~,...,~,, the one which is most easily activated must be the one which has all the internal

336 Yield-line collapse mechanisms: A. dennings

facets removed. Thus U is maximised, whilst V is unchanged. The effect on the configuration of Figure 5(a) would therefore be to convert it to that shown in Figure 5(c) which is entirely defined by the six boundary rotations and contains only boundary facets (of type (a) in the above list). A mechanism with internal facets cannot be activated at a lower load, and will only be activated at the same load if the internal facets do not carry any of the external load. Thus, Theorem 1 is established.

point A). If this point also lies in facet i for cases 1 or 3, then

o r

Wl = hdp~ (e.g. for points A and B in Figure 6)

w3 = h(dPi + 2~bi)

(e.g. for points A and C in Figure 6)

8. Justification of Theorem 2 on uniqueness

Consider case 1 to be a mechanism comprising just n boundary facets, the deflection profile, of which can be determined entirely from the (positive) boundary rotations qb~,...,dpn. Because it is the relative values of the boundary rotations that are significant, there will be only n-1 inde- pendent boundary rotation variables. The other dependent variable may be used to ensure that the energy dissipation V is equal to a prespecified value. Let case 2 be a mechan- ism in which the boundary variables are changed to dP~+~bl,...,~n+~b, such that the energy dissipation V remains unaltered. On account of the linear relationship (15) between V and the boundary rotations, it follows that:

n

i=1

(18)

Hence, if a case 3 is designated with boundary rotations dP~+2+~,...,qb,+2qb, it follows that:

V~ =V2=V3 (19)

(where the subscripts denote the case number). The values of ~i and (~i may be taken to involve any feasible move- ment in the n-1 dimensional space of the independent boundary variables provided that they remain positive, that is

(I) i > 0 and qb i + 2+~ > 0 (20)

for all facets. Since the boundary variables for the three cases are not

proportional to each other, the positions of at least one of the yield-lines must change. If a point (x,y) is located in facet i for case 2, its deflection is given by:

W 2 : h (qbi+~bi) (21)

where h is the perpendicular distance of the point from the boundary associated with this facet (shown in Figure 6 for

x \ \ x x \ \ X N N \ \ \ \ \ \ \ \ ~ \

Facet i

"" Case 3

"" (lose 2

Case I

\

Figure 6 Three linearly related concave mechanisms involving only boundary facets

However, where the point lies in a different facet

Wl < hqbi (e.g. for points C and D in Figure 6)

o r

w3 < h(Cbi +2~bi) (e.g. for points B and D in Figure 6)

Hence

w2 -> ½ (wl+w3) (22)

If U~, /-]2 and U3 are the losses in potential energy of the applied forces for the three cases, it follows from equation (16) and (22) that

(-]2 -> ½ (U,+U3) (23)

Here the inequality sign will apply unless no lateral loading occurs over the regions of the slab swept by the yield-lines as they move between cases 1, 2 and 3. The inequality will always apply to cases where the whole slab is loaded, even when it is not loaded uniformly.

Since case 2 can describe any feasible mechanism within the solution space being investigated, and cases 1 and 3 represent any feasible direction of movement in that space, the function U must be convex and have a unique maximum with no false (local) maxima. Furthermore, because V has been held constant, equation (2) shows that the lowest acti- vating load parameter can be determined from this maximum value of U.

9. The equilibrium method

The equilibrium method employs facets of type (a) which are adjusted until the load parameter has a stationary point in the n-1 dimensional space defined by the edge rotations. The theorems presented in this paper show that, if the resulting mechanism is convex, the collapse load will be obtained except for the possibility that a mechanism having negative internal yield-lines may be activated at a lower load.

10. Situations of partial collapse

Cases exist in which the collapse mechanism does not encompass the whole area of the slab. Examples of this are when comer levers or fans are present or when a single concentrated load gives rise to a circular fan mechanism. The theorems cannot then be applied to the total slab since these mechanisms contain negative yield-lines. However, they do apply if the boundary is adjusted to exclude all the areas of slab which do not deform (i.e. ADEC instead of

Yield-line collapse mechanisms: A. Jennings

Simply supported ~ / boundary

N

~ \ P ~ N N \ N \ N N N \

E E Undeflecfed region

(a)

Figure 7 Boundary modif ication to

~ ./built-in boundary

Undeflected region

(b)

take account of corner leaves and fans. (a) A corner lever; (b) a fan

ABC for the comers shown in Figure 7). Whereas the possibility of partial collapse needs to be considered in any computational procedure for yield-line analysis, the the- orems developed in this paper are still of relevance when applied to slabs with modified boundaries.

11. Conclusions

Theorems have been developed which relate to the rigid- plastic behaviour of plates and slabs under lateral loading. Mechanisms have been considered which are of convex polyhedral shape, with zero deflections at the slab bound- aries. It has been shown that the least load to activate any such mechanism can be obtained unambiguously by search- ing amongst the set of feasible mechanisms using standard optimisation procedures. The theorems are also applicable in cases of partial collapse, the possibility of which will also need to be allowed for in computational procedures.

Acknowledgments The author would like to thank A. Thavalingam, J. J. McKeown and T. D. Sloan for many discussions on this topic, the EPSRC for funding work on computational methods for yield-line analysis and design, and a reviewer for valuable comments on the original script.

References 1 Ingerslev, A. 'The strength of rectangular slabs', J. Inst. Struct.

Engrs. January 1923, 1, 3--14

337

2 Johansen, K. W. 'Yield line theory', translation by C & CA, Lon- don, 1962

3 Wood, R. H. 'Plastik and elastic design of slabs andplates', Thames and Hudson, 1961

4 Jones, L. L. 'Ultimate load analysis of reinforced and prestressed concrete structures', Chatto and Windus, London, 1962

5 Dowling, P. J., Knowles, P. R. and Owens, G. W. 'Structural steel design', Steel Construction Institute, 1988

6 Prager, W. and Hodge Jr, P. G. 'Theory of perfectly plastic solids', Wiley, New York, 1951

7 Munro, J. and Da Fonseca, A. M. A. 'A yield-line method for finite elements and linear programming', The Struct. Engnr. June 1978, 56B, 37-44

8 Shoemaker, W. L. 'Computerised yield-line analysis of rectangular slabs', Concrete Int. August 1982, pp. 62-65

9 Bauer, D. and Redwood, R. G. 'Numerical yield-line analysis', Comp. Struct. 1987, 26, 587-596

10 Dickens, J. G. and Jones, L. L. 'A general computer program for the yield-line solution of edge supported slabs', Comp. Struct. 1988, 30, 465-476

11 Jennings, A. Thavalingam, A., McKeown, J. J. and Sloan, D. 'On the optimisation of yield-line patterns', in "Developments in compu- tational engineering mechanics', Topping, B. H. V. (ed.), Civil- Comp Press, Edinburgh, 1993, pp. 209-214

12 McKeown, J. J., Jennings, A., Thavalingam, A. and Sloan, D. 'Optimisation techniques for generating yield-line patterns' in 'Advances in structural optimisation', Topping, B. H. V. and Papad- rakakis, M. (eds), Civil-Comp Press, Edinburgh, 1994, pp. 161-169

13 Damkilde, L. and Krenk, S. 'Limits - - a system for limit state calcu- lation of collapse load or optimal material layout', in 'Advances in structural optimisation', Topping, B. H. V. and Papadrakakis, M. (eds), Civil-Comp Press, Edinburgh, 1994, pp. 171-178

14 Johnson, D. 'Mechanism determination by automated yield-line analysis', The Struct. Engnr 1994, 72, No. 19/4, 323-327

15 Thavalingam, A., Jennings, A., McKeown, J. J. and Sloan, D. 'The use of optimisation techniques in yield line analysis', in Proc. Int. Conf. Computational Methods in Structural and Geotechnical Engin- eering, Hong Kong, 1994

16 Hilleborg, A. 'Yield-line analysis', Concrete Int. May 1991, pp. 9-10 17 Lee, E. H. 'On the significance of limit load theorems for an elastic-

plastic body', PhiL Mag. 1952, 43, 549-560 18 Hill, R. 'On the problem of uniqueness in the theory of a rigid-plastic

solid', J. Mech. Phys. Solids 1956, 4, 247-255 19 Bishop, J. F. W., Green, A. P. and Hill, R. 'A note on the deformable

region in a rigid-plastic solid', J. Mech. Phys. Solids 1956, 4, 256- 258

20 Baldaci, R., Cerodini, G. and Giangreco, E. 'Plasticity', Vol. IIA, Halsider, Genova, 1971

21 Save, M. A. and Massonnet, C. E. 'Plastic analysis and design of plates, shells and disks', North Holland, 1972

22 Prager, W. 'An introduction to plasticity', Addison-Wesley, Reading, MA, 1959

23 Haythornthwaite, R. M. and Shield, R. T. 'A note on the deformable region in a rigid-plastic structure', J. Mech. Phys. Solids 1958, 6, 127-131