Yield Line Theory of Slabs
Reinforced concrete design methods under the present ACI Code are based on the results of an elastic analysis of the structure as a whole , when subjected to the acting of factored loads such as 1.4D + 1.7L where D and L refer to service dead and live loads . Actually the behavior of a statically indeterminate strcture is such that after the ultimate moment capacities at one or more points have been reached , discontimuities develop in the elastic curve at those points and the results of an elastic analysis are no longer valid . If there is sufficient ductility , redistribution of bending moments will occur until a sufficient number of sections of discontinuity , commolly called “plastic hinges” , form to change the structure into a mechanism , at which time the structure collapses or fails . The term “ultimate load analysis” , as opposed to “ elastic analysis “ , relates to the use of the bending moment diagram at the verge of collapse as the basis for design . Other than the provisions for redistribution of moments at the supports of continuous flexural members the present ACI Code has as yet made no allowance for ultimate load analysis .
The chief concern of this chapter is to develop the yield line theory for two-way slabs . Although not yet adopted by the ACI Code , slab analysis by yield line theory may be useful in providing the needed information for understanding the behavior of irregular or single-panel slabs with various boundary conditions .
Although the study of flexural behavior of plates up to the ultimate load may date back to the 1920`s , the fundamental concept of the yield line theory for the ultimate load design of slabs has been expanded considerably by K . W . Johansen . In this theory the strength of a slab is assumed to be governed by flexure alone : other effects such as shear and deflection are to be separately considered . The reinforcing steel is assumed to be fully yielded along at collapse and the bending and twisting moments are assumed to be uniformly districuted along the yield lines .
Yield line theory for one-way slabs is not much different from the limit analysis of continuous beams . On a continuous beam the achievement of flexural strength at one location , say in the negative-moment region over a support , does not necessarily constitute reaching the ultimate load on the beam . If the section having reached its flexural strength can continue to provide a constant resistance while undergoing further rotation , then the flexural strength may be reached at additional locations . Complete failure theoretically can not occur until yielding has occurred at several locations ( or along several lines in case of one-way slabs ) so that a mechanism forms giving a condition of unstable equilibrium .
Consider for example , the one-way slab of finite width shown if Fig . 11-1 . A uniform loading on the slab will cause uniform maximum negative bending moment along AB and EF and uniform positive bending moment along CD , which is parallel to the supports . When the uniform load is increased until the moments along AB , CD , and EF reach their respective ultimate moment capacities , rotation of the slab segments will occur with the yield lines cating as axes of rotation . Once the ultimate moment capacity is achieved , angle change can occur without additional resisting moment being developed . Thus , under the limiting condition with the slab segments able to rotate with no change in resisting moment , the slab system is geometrically unstable . This condition is known as a “collapse mechanism . ”
Yield line theory for two-way slabs requries a different treatment from limit analysis of continuous beams , because in this case the yield lines will not in general be parallel to each other but instead form a yield line pattern . The entire slab area will be divided into several segments which can rotate along the yield lines as rigid bodies at the condition of collapse or unstable equilibriun .
The slab of Fig . 11-2a has nonparallel supports . At the collapse condition this slab will break into segments ; one segment will have an edge rotating about I and the other will have an edge rotating about II . The exact position of yield line III will depend on the reinforcement amount and direction , both in the positive and negative moment regions .
For the case of Fig . 11-2b where a rectangular panel is either simply supported or montinuous over four linear supports , the collapse mechanism consists of four slab segments . The exact locations of points a and b will depend on the moment capacitise at the supports and the positive moment reinforcement in each direction .
The slab in Fig . 11-2c is supported along two edges and in addition is supported by two isolated columns . The rotational axes for the slab segments at collapes must occur along the supports , and additional rotational axes must pass through the isolated columns . The critical position of the positive moment yield lines a , b , c , d , and e is a function of the reinforcement amount and direction : in the meantime compatibility of deflection along the yield lines must be maintained during the rigid body rotations of the slab segments .
Prestressed Concrete
Modern structural engineering tends to progress toward more economic structures through gradually improved methods of design and the use of higher strength matericals . This results in a reduction of cross-sectional dimensions and consequent weight savings . Such developments are particularly important in the field of reinforced concrete , where the dead load represents a substantial part of the total design load . Also , in multistory buildings , any saving in depth of members , multiplied by the number of stories , can represent a substantail saving in total height , load on foundations , length of heating and electrical ducts , plumbing risers , and wall and partition surfaces .
Significant savings can be achieved by the use of high strength concrete and steel in conjunction with present-day design methods , which permit an accurate appraisal of nember strength . However , there are limitations to this development , due mainly to the interrelated problems of cracking and deflection at service loads . The efficient use of high strength steel is limited by the fact that the amount of cracking is undesirable in that it exposes the reinforcement to corrosion , it may be visually offensive , and it may trigger a premature failure by diagonal tension . The use of high strength materials is further limited by deflection considerations , particularly when refined analysis is used . The slender members which result may permit deflections which reduces the flexural stiffness of members
These undesirable characteristics of ordinary reinforced concrete have been largely overcome by the development of prestressed concrete . A prestressed concrete member can be defined as one in which there have been introduced internal stresses of such magnitude and distribution that the stresses resulting from the given external loading are counteracted to a desired degree . Concrete is basically a precompression to the member which reduces or eliminates undesirable value . Prestressing applies a precompresion to the member which reduces or eliminates undesirable tensile stresses that would otherwise be present .Cracking under service loads can be minimized or even avoided entirely . Deflections may be limited to an acceptable value ; in fact , members can be designed to have zero deflection under the combined effects of service load and prestress force . Deflection and crack control , avhieved through prestressing , permit the engineer to make use of dfficient and economical high strength steels in the form of strands , wires , or bars , in conjunction with concretes of much higher strength than nromal . Thus prestressing results in overall improvement in performance of structual concrete used for ordinary loads and spans , and extends the range of application far beyond old limits , leading not only to much longer spans than previously thought possible , but permitting innovative new structural forms to be employed .
The first suggestion for prestressing seem to have been made between 1886 and 1908 by the Americans P . H Jackson and G . R . Steiner , the Austrian J . Mandl , and German J . Koenen . The use of high-strength steel was first suggested by the Austrian F . von Emperger in 1923 , while at about the same time , the American R . H . Dill proposed “full prestressing” to eliminate cracks completely . These proposals remained mainly on paper ; the practical development of prestressed-reinforced-concrete structures is chiefly due to E . Freyssinet and Y . Fuyon ( France ) , E . Hoyer ( Germany ) , and G . Magnel ( Belgium ) . Circular prestressing of cylindrical tanks and pipes , originated by W . H . Hewiti in 1923 , was the first important application of the principles of prestressing in the United States . Important contributions have been made by T . Y . Lin in the design of many types of prestressed concrete structures in the United
State since 1950 .
It is interesting to contrast the development of prestressed concrete in Europe , where a high ratio of material to labor cost prevails , with that in the United States , where the ratio is reversed . In Europe , many sophisticated designs have been executed which minimize the materical used but require a great amount of highly qualified labor to build and to prestress . Long-span bridges , two-dimensional floor systems , shells , and even space trusses have accounted for the bulk of prestressed concrete construction in the United States over the past 20 years . This difference is less apparet , now than previously . European industry now produces standardized precast building componets , utility poles , railroad ties , and other items using techniques periected in the United
State , while engineers here have made use of European experience in the design of special structures of major span .
Fundamentals of Composite Action and Shear Connection
The evolution of satisfactory design methods for composite beams has been a slow process , requiring much theoretical and experimental work in order to provide economic an , at the same time , safe design criteria . The purpose of this Chapter is to describe in some detail the more important fundamentals which have to be taken into account in the design of composite structures .
Historically the first analysis of a composite section was based on the conventional assumptions of the elastic theory which limit the stresses in the case of steel , crushing in the case of concrete . The assumptions inherent in the elastic method are similar to those for ordinaty reinforced concrete . In recent years the concepts of the ultimate load design philosophy nave been applied to composite action and a body of experimental evidence jas shown it to be a safe , economical basis on which to proportion composite sections . Although at the present time ultimate load design methods are directly applicable only to bulidings and not to bridges there seems no teason to doubt that in time the restriction will disappeat .
Before dealing in detail with the two design approaches elastic and ultimate load certain basic points requrie consideration
A clear understanding of the way in which the component materials , steel concrete and shear connection ,react to applied load is an essential preliminary to full analysis of the composite section . Of primary importance are the strss-strain telationships , which must of mecessity be the product of carefully controlled experiment . These experimental tesults are not generally suited to direct application and so simplifications and idealisations adopted in practice . The use of computers has made it possible to reduce the amount of idealisation required with the result that computer ‘experiments’ can now be performed using material stress-strain relationships of considerable complexity .
Composite action between steel and concrete implies some interconnection between the two matericals which will transfer sheat between them . In teinforced concrete members the natural bond of concrete to steel is often sufficient to do this , although cases do arise in which additional anchorage is required . The fully encased filler joist also has a large embedded area ehich is adequate for full shear transfer . However , the situation is quite different with the common type of composite beam in which the concrete slab rests on , or at best enclose , the top flange of the steel beam . It is true that there will initially be shear transfer by bond and friction at the beam-slab contact surface . There is , however , a tendency for the slab to separate vertically from the beam and , should this occur , horizontal shear transfer will cease . A single overload or the fatigue effect of pulsating loading may destroy the natural bond , which once destroyed cannot be reconstituted . The imponderable nature of such shear connection is clearly undesirable ; some form of deliberate connection between beam and slab is required with the two objects of transferring lorizontal shear and preventing vertical separation . A natural bond will exist in the presence of shear connection but it is neither desirable to count on its existence nor possible in all cases to calculate its value . Thus shear connection must be provided to transfer all the horizontal shear force is provided it may in fact not come into operation because the natural bond takes all the shear force , and so ‘if sufficient shear connectors are provided then they are unnecessary .
The evolution of shear connection devices has been slow and has necessitated a large nolume of experimental work on the static and fatigue properties of a wide range of mainly mechanical connectors
It soon appeared clear to early research workers that some form of connetor fixed to the top flange of the beam and anchored into the slab was necessary . Caughey and Scott in 1929 proposed using amongst other things , projecting bolt ends . Since then a wide variety of types of mechanical connector has been used in experiment and practice . To some extent the proliferation of type has been result of steel fabricators using sections which came easily to hand , since initially a purpose-made shear connector was not available
In any mechanical connection system it is possible to identify parts which transfer horizontal shear and parts which tie the slab down to the beam . Generally horizontal shear resestance is the ruling criterion of shear connector action and with this in mind mechanical connectors may be classified into three main groups-rigid , flexible and bond
Soil Mechanics
Soil mechanics is a branch of engineering which deals with soils under stress . It did not develop into a science until Terzaghi in the 1920`s laid down the principles which still form the basis for most calculations . His principle of effective stress states that the stress normal to a section of the soil is equal to the sum of the intergranular , or effective , stress transmitted form grain to grain and the neutral , or pore , stress transmitted through the water contained in the soil .
Another important idea of Terzaghi concerns the shearing resistance of soils against retaining walls , bulkheads , and braced cuts . This lateral pressure increase linearly with depth in teraining walls and parabolically in braced cuts . His results solved many of the disagreements between practice and older theories .
Grain size is the basis of soil mechanics , since it is this which decides whether a soil is frictional or cohesive , a sand or a clay . Starting with the largest sizes , boulders are larger than 10 cm , cobbles are from 5 to 10 cm , gravel or ballast is from about 5 cm to 5 mm , grit is from about 5 mm to 2 mm , sand is from 2 mm to 0.06 mm . All these soil are frictional , being coarse and thus non-cohesive . Their stability depends on their internal friction . For the cohesive or non-frictional soils the two main internationally accepted size limits are silt from 0.02 mm to 0.002 mm , and clay for all finer material . There are , of course , many silty clays and clayey silts .
Every large civil engineering job starts with a soil mechanics survey in its early stages . The first visit on foot will show whether the site might be suitable , in other words , wheter money should be spent on sending soil-sampling equipment out to it . The soil samples and the laboratory results obtained from the triaxial tests , shear tests and so on will show at what depth the soil is likely to be strong enough to take the required load . For a masonry or steel structure , this is where the soil mechanics survey will end , having rarely cost more than 2 per cent of the structure cost .
Generally , the strength of a soil increases with depth . But it can happen that it becomes weaker with depth . Therefore , in choosing the foundation pressure and level for this sort of soil , a knowledge of soil mechanics is essential , since this will give an idea of the likely settlements .
There are , however , several other causes of settlement apart form consolidation to load . These causes are incalculable and must be carefully guarded against . They include frost action , chemical change in the soil , undergroud erosion by flowing water , reduction of the ground water level , nearby constrution of tunnels or vibrating machinery such as vehicles .
Static load can cause elastic or plastic settlement , consolidation settlement being permanent . However , when plastic flow is mentioned in English , it generally means the failure or a soil by overload in shear . Consolidation settlement occurs mainly in clays or silts .
From dynamic load alone the commonest settements are found in sands or gravels , caused by traffic or other vibration , pile driving or other earth shocks . A drop in the ground water level will often cause the soil to shrink and a rise may cause expansion of the soil . Ground water is lowered by the drainage which can be caused by any deep excavation . The shrinkage which can occur with drying is well shown by the clay underlying Mexico City , a volcanic ash . After seven weeks drying this clay shrinks to 6.4 per cent of its initial volume . It is an unusual clay with the very high voids ratio of 93.6/6.4 = 14.6
Underground erosion is the removal of solids , usually fines , from the soil by the flow of underground water . The solids can be removed as solids or in solution , though only a few rocks are soluble rock . Potassium salts also are soluble .
The permeability of a soil in important for calulations of underground flow , for example , of oil or water to a well , or of water into a trench dug for a foundation , or of water through an earth dam . Of the loose soils which can be dug with a spade , clays are the lesat permeable , silts slightly more , sands yet more , and gravels even more . In other words , the permeability is in direct proportion to the grain size of the soil .
When a well is being pumped , the water flows towards it from every direction and the ground water surface around it sinks . As the distance from the well increases , the water table is lowered rather less , so that around the well it become shaped like a funnle , though it is usually called a cone of depression .
