STRUCTURAL THEORY PDF

STRUCTURAL THEORY ANALYSIS OF STATICALLY DETERMINATE STRUCTURES IDEALIZED STRUCTURE An exact analysis of structure can never be carried out, since estimates always have to be made of the loadings and the strength of the materials composing the structure. Furthermore, points of application for...

STRUCTURAL THEORY ANALYSIS OF STATICALLY DETERMINATE STRUCTURES IDEALIZED STRUCTURE An exact analysis of structure can never be carried out, since estimates always have to be made of the loadings and the strength of the materials composing the structure. Furthermore, points of application for the loadings must also be estimated. It is important, therefore, that the structural engineer develop the ability to model or idealize a structure so that he or she can perform a practical force analysis of the members. SUPPORT CONNECTIONS Structural members are joined together in various ways depending on the intent of the designer. The three types of joints most often specified are the pin connection, the roller connection, and the fixed joint. A pin-connected and a roller support allow some freedom for slight rotation, whereas a fixe joint allows no relative rotation between the connected members and is consequently more expensive to fabricate. Examples of these joints, fashioned in metal and concrete, are shown in the following figures: For most timber structures, the members are assumed to be pin connected, since bolting or nailing them will not sufficiently restrain them from rotating with respect to each other. Idealized models used in structural analysis that represent pinned and fixed supports and pin-connected and fixed-connected joints are shown in the figure. In reality, however, all connections exhibit some stiffness toward joint rotations, owing to friction and material behaviour. In this case a more appropriate model for a support or joint might be that shown in the figure. If the torsional spring constant k = 0, the joint is a pin, and if k → ∞, the joint is fixed. When selecting a particular model for each support or joint, the engineer must be aware and whether the assumptions are reasonable for the structural design. For example, consider the beam shown which is used to support a concentrated load P. The angle connection A is like that in typical “pin-supported” connection and can therefore be idealized as a typical pin support. Furthermore, the support at B provides an approximate point of smooth contact and so it can be idealized as a roller. The beam’s thickness can be neglected since it is small in comparison to the beam’s length, and therefore the idealized model of the beam is as shown in the next figure. The analysis of the loadings in this beam should give results that closely approximate the loadings in the actual beam. Other types of connections most commonly encountered on coplanar structures are given in Table 2.1. It is important to be able to recognize th symbols for these connections and te kinds of reactions they exert on their attached members. This can easily be done by noting how the connection prevents any degree of freedom or displacement of the member. In particular, the support will develop a force on the member if it prevents translation of the member, and it will develop a moment if it prevents rotation of the member. For example, a member in contact with a smooth surface (3) is prevented from translating only in one direction, which is perpendicular or normal to the surface. Hence, the surface exerts only a normal force F on the member in this direction. The magnitude of this force represets one unknown. Also note that the member is free to rotate on the surface, so that a moment cannot be developed by the surface on the member. As a another example, the fixed support (7) prevents both translation and rotation of a member at the point of connection. Therefore, this type of support exerts two force components and a moment on the member. The “curl” of the moment lies in the plane of the page, since rotation is prevented in that pane. Hence, there are three unknowns at a fixed support. In reality, all supports actually exert distributed surface loads on their contacting members. The concentrated forces and moments shown in Table 2.1 represent the resultants of these load distributions. This representation is, of course, an idealization; however, it is used here since the surface area over which the distributed load acts is considerably smaller than the total surface area of the connecting members. IDEALIZED STRUCTURE Having stated the various ways in which the connections on a structure can be idealized, we are now ready to discuss some of the techniques used to represent various structural systems by idealized models. As a firdt example, consider the jib crane and trolley in the figure. For the structural analysis we can neglect the thickness of the two main members and will assume that the joint at B is fabricated to be rigid. Furthermore, the support connection at A can be modelled as a fixed support and the details of the trolley excluded. Thus, the members of the idealized structure are represented by two connected lines, and the load on the hook is represented by a single concentrated force F. Ths idealized structure shown here as a line drawing can now be used for applying the principles of structural analysis,which will eventually lead to the design of its two main members. Beams and girders are often used to support building floors. In particular, a girder is the main load-carrying element of the floor, whereas the smaller elements having a shorter span and connected to the girders are called beams. Often the loads that are applied to a beam or girder are transmitted to it by the floor that is supported by the beam or girder. Again, it is important to be able to appropriately idealize the system as a series of models, which can be used to determine, to a close approximation, the forces acting in the members. Consider, for example, the framing used to support a typical floor slab in a building, shown in the figure. Here the slab is supported by floor joists located at even intervals, and these in turn are supported by the two side girders AB and CD. For analysis it is reasonable to assume that the joints are pin and/or roller connecterd to the columns. The top view of the structural framing plan for this system was also being shown in the figure. In this “graphic” scheme, notice that the “lines” representing the joists don not touch the girders and the lines for the girders do not touch the columns. This symbolizes pin- and/or roller-supported connections. On the other hand, if the framing plan is intended to represent fixed-connected members, such as  those that ar welded instead of simple bolted connections, then the lines for the beams or girders would touch the columns as shown in left. Similarly, a fixed-connected overhanging beam would be represented in top view as shown in right. If reinforced concrete construction is used, the beams and girders are represented by double lines. These systems are generally all fixed connected and therefore the members are drawn to the supports. For example, the structural graphic for the cast-in-place reinforced concrete system in the first figure is shown in top view in the next figure. The lines for the beams are dashed because they are below the sla. Structural graphics and idealizations or timber structures are similar to those made of metal. For example, the structural system shown in the next figure represents beam-wall construction, whereby the roof deck is supported by wood joists, which deliver the load to a masonry wall. The joists can be assumed to be simply supportedon the wall, so that the idealized framing plan would be like that shown in idealized framing plan. TRIBUTARY LOADINGS When flat surfaces such as walls, floors, or roofs are supported by a structural frame, it is necessary to determine how the load on these surfaces is transmitted to the various structural elements used for their support. There are generally two ways in which this can be done. The choice depends on the geometry of the structural system, the material from which it is made, and the method of its construction. ONE-WAY SYSTEM A slab or deck that is supported such that it delivers its load to the supporting members by one-way action, is often referred to as a one- way slab. To illustrate the method of load transmission, consider the framing system shown where the beams AB, CD, and EF rest on the girders AE and BF. If a uniform load of 100 lb/ft^2 is placed on the slab, then the center beam CD is assumed to support the load acting on the tributary area shown dark shaded on the structural framing plan in the figure. Member CD is therefore subjected to a linear distribution of load of (100 lb/ft^2)(5ft) = 500 lb/ft, shown on the idealized beam. The reactions of this beam (2500 lb) would then be applied to the center of the girders AE (and BF), shown in idealized girder. Using this same concept, do you see how the remaining portion of the slab loading is transmitted to the ends of the girder as 1250 lb? For some floor systems the beams and girders are connected to the columns at the same elevation, as shown in the figure. If this is the case, the slab can in some cases also be considered a one-way slab. For example, if the slab is reinforced concrete with reinforcement in only one direction, or the concrete is poured on a corrugated metal deck, as in the next photo, then one-way action of load transmission can be assumed. On the other hand, if the slab is reinforced in two directions, then consideration must be given to the possibility of the load being transmitted t the supporting members from eithr one or two directions. For example, consider the slab and framing plan in the figue. According to the code, if 𝐿2 > 𝐿1 and if the span ration (𝐿2 /𝐿1 ) > 2, the slab will behave as a one-way slab, since 𝐿1 becomes smaller, the beams AB, CD, and EF provide the greater stiffness to carry the load. TWO-WAY SYSTEM According to the code, if 𝐿2 > 𝐿1 and the support ratio (𝐿2 /𝐿1 ) ≤ 2, then the load is assumed to be transferred to the supporting beams and girders in two directions. When this is the case the slab is referred to as a two-way slab. To show how to treat this case, consider the square reinforced concrete slab in the figure, which is supported by four 10-ft-long edge beams, AB, BD, DC, and CA. Here 𝐿2 /𝐿1 = 1. As the load on the slab intensifies, numerous experiments have shown that 45° cracks form at the corners of the slab. As a result, the tributary area is constructed using 45° lines as shown in the next figure. This produces the dark shaded tributsry area for beam AB. Hence if a uniform load of 100 lb/ft^2 is applied to the slab, a peak intensity of (100 lb/ft^2)(5 ft) = 500 lb/ft will be applied to the center of beam AB, resulting in the triangular load distribution shown. For other geometries that cause two- way action , a similar for example, if 𝐿2 /𝐿1 = 1.5 it is then necessary to construct 45 ° lines that intersect as shown in the next figure. This produce the dark shaded tributary area for beam AB. A 100- lb/ft^2 loading placed on the slab will then produce trapezoidal and triangular distributed loads on members AB and AC, shown in the two figures respectively. LOAD PATH The various elements that make up a structure should be designed in such a way that they transmit he primary load acting on the structure to its foundation in the most efficient way possible. Hencee, as a first step in any design or analysis of a structure, it is very important to understand how the loads are transmitted through it, if damage or collapse of the structure is to be avoided. This description is called a load path, and by visualizing how the loads are transmitted the engineer can sometimes eliminate unnecessary members, strengthen others, or identify where there may be potential problems. Like a chain, which is ”as strong as its weakest link”, so a structure is only as strong as the weakest part along its load path. To show how to construct a load path, let us consider a few examples. In the figure, the loading acting on the floor of the buildin is transmitted from the slab to the floor joists then to the spandrel and interior girder, and finally to the columns and foundation footings. The loading on the deck of the suspension bridge in the next figure is transmitted to the hangers or suspenders, then the cables, and finally the towers and piers. PRINCIPLES OF SUPERPOSITION The principle of superposition forms the basis for much of the theory of structural analysis. It may be stated as follows: The total displacement or internal loadings (stress) at a point in a structure subjected to several external loadings can be determined by adding together the displacements or internal loading (stress) caused by each of the external loads acting separately. For this statement to be valid it is necessary that a linear relationship exist among theloads, stresses, and displacements. Two requirements must be imposed for the principle of superposition to apply: 1. The material must behave in a linear-elastic manner, so that Hooke’s law is valid, and therefore the load will be proportional to displacement. 2. The geometry of the structure must not undergo significant change when the loads are applied, i.e., small displacement theory applies. Large displacement will significantly change the position and orientation of the loads. An exampl would be a cantilevered thin rod subjected to a force at its end, causing it to bend. EQUATIONS OF EQUILIBRIUM It may be recalled from statics that a structure or one of its members is in equilibrium when it maintatins a balance of force and moment. In general this requires that the force and moment equations of equilibrium be satisfied along three independent axes, namely, ∑ 𝐹𝑥 = 0 ∑ 𝐹𝑦 = 0 ∑ 𝐹𝑧 = 0 ∑ 𝑀𝑥 = 0 ∑ 𝑀𝑦 = 0 ∑ 𝑀𝑧 = 0 The principal load-carrying portions of most structures, however, lie in a single plane, and since the loads are also coplanar, the above requirements for equilibrium reduce to ∑ 𝐹𝑥 = 0 ∑ 𝐹𝑦 = 0 ∑ 𝑀𝑜 = 0 Here ∑ 𝐹𝑥 = 0 and ∑ 𝐹𝑦 = 0 represent, respectively, the algebraic sums of the x and y components of all the forces acting on the structure or one of its members, and ∑ 𝑀𝑜 = 0 represents the algebraic sum of the moments of these force components about an axis perpendicular to the x-y plane (the z axis) passing through point O. Whenever these equations are applied, it is first necessary to draw a free-body diagram of the structure or its members. If a member is selected, it must be isolated from its supports surroundings and its outlined shape drawn. All the forces and couple moments must be shown that act on the member. In this regard, the types of reactions at the supports can be determined in Table 2.1. Also, recall that forces common to two members act with equal magnitudes but opposite directionson the respective free-body diagrams of the members. If the internal loadings at a specified point in a member are to be determined, the method of sections must be used. This requires that a “cut” or section be made perpendicular to the axis of the member at the point where the internal loading is to be determined. A free-body diagram of either segment of the “cut” member is isolated and the internal loads are then determined from the equations of equilibrium applied to the segment. In general, the internal loadings acting at the section will consist of a normal force N, shear force V, and bending moment M, as shown in the figure. DETERMINACY AND STABILITY Before starting the force analysis of a structure, it is necessary to establish the determinacy and stability of the structure. DETERMINACY The equilibrium equations provide both the necessary and sufficient conditions for equilibrium. When all the forces in structure can be determined strictly from these equations, the structure is referred to as statically determinate. Structures having more unknown forces than availble equilibrium equations are called statically indeterminate. As a general rule, a stable structure can be identified as being either statically determinate or statically indeterminate by drawing free-body diagrams of all its members, or selective parts of its members, and then comparing the total number of avaible equilibrium equations. For a coplanar structure there are at most three equilibrium equations for each part, so that if there is a total on n parts and r force and moment reaction components, we have r = 3n, statically determinate r > 3n, statically indeterminate In particular, if a structure is statically indeterminate, the additional equations needed to solve for the unknown reactions are obtained by relating the applied loads and reactions to the displacement or slope at different points on the structure. These equations, which are referred to as compatibility equations, must be equal in number to the degree of indeterminacy of the stucture. Compatibility equations involve the geometric and physical properties of the structure and will be discussed further in the succeeding discussions. STABILITY To ensure the equilibrium of a structure or its members, it is not only necessary to satisfy the equations of equilibrium, but the members must also be properly held or constrained by their supports regardless of how the structure is loaded. Two situations may occur where the conditions for proper constraint have not been met. PARTIAL CONSTRAINTS Instability can occur if a structure or one of its members has fewer reactive forces than equations of equiibrium that must be satisfied. The structure then becomes only partially constrained. For example, consider the member shown in the figure with its corresponding free-body diagram. Here the equation ∑ 𝐹𝑥 = 0 will not be satisfied for the loading conditions, and therefore the member will be unstable. IMPROPER CONSTRAINTS In some cases there may be as many unknown forces as there are equations of equilibrium; however, instability or movement of a structure or its members can develop because of improper constraining by the supports. This can occur if all the support reactions are concurrent at a point. An example of this is shown in the figure. From the free-body diagram of the beam it is seen that the summation of moments about point O will not be equal to zero (Pd ≠ 0); thus rotation about point O will take place. Another way in which improper constraining leads to instability occurs when the reactive forces are all parallel. An example of this case is shown in the next figure. Here when an inclined force P is applied, the summation of forces in the horizontal direction will not equal zero. In general, then, a structure will be geometrically unstable – that is, it will move slightly or collapse – if there are fewer reactive forces than equations of equilibriu; or if there are enough reactions, instability will occur if the lines of action of the reactive forces intersect at a common point or are parallel to one another. If the structure consists of several members or components, local instability of one or several of these members can generally be determined by inspection. If the members form a collapsible mechanism, the structure will be unstable. We will now formalze these statements for a coplanar structure having n members or components with r unknown reactions. Since three equilibrium equations are available for each member or component, we have r < 3n, unstable r ≥ 3n, unstable if member reactions are concurrent or parallel or some of the components form a collapsible mechanism If the structure is unstable, it does not matter if it is statically determinate or indeterminate. In all cases such types of structures must be avoided in practice. APPLICATION OF THE EQUATIONS OF EQUILIBRIUM Occasionally, the members of a structure are connected together in such a way that the joints can be assumed as pins. Building frames and trusses are typical examples that are often constructed in this manner. Provided a pin-connected coplanar structure is properly constrained and contains no more supports or members than are necessary to prevent collapse, the forces acting at thejoints and supports can be determined by applying the three equations of equilibrium (∑ 𝐹𝑥 = 0, ∑ 𝐹𝑦 = 0, ∑ 𝑀𝑜 = 0) to each member. Understandably, once the forces at the joints are obtained, the size of the members, connections, and supports can then be determined on the basis of design code specifications. To illustrate the method of force analysis, consider the three- member frame shown in the figure, which is subjected to loads 𝑷1 and 𝑷2. The free-body diagrams of each member are shown in the next figure. In total there are nine unknowns; however, nine equations of equilibrium can be written, three for each member, so the problem is statically determinate. For the actual soluton it is also possible, and sometimes convenient, to consider a portion of the frame or its entirety when applying some of these nine equations. For example, a free-body diagram of the entire frame is also shown in the figure at the next page. One could determine the three reactions 𝑨𝑥 , 𝑨𝑦 , 𝑪𝑥 on this “rigid” pin-connected system, then analyze any two of its members, (see figure 2), to obtain the other six unknowns.  Furthermore, the answers can be checked in part by applying the three equations of equilibrium to the remaining “third” member. To summarize, this problem can be solved by writing at most nine equilibrium equations using free-body diagrams of any members and/or combinations of connected members. Any more than nine equations written would not be unique from the original nine and would only serve to check the results. Consider now the two-member frame shown in the figure (a). Here the free-body diagrams of the members reveal six unknowns (b); however, six equilibrium equations, three for each member, can be written so again the problem is statically dterminate. As in the previous case, a free-body diagram of the entire frame can also be used for part of the analysis, (c). Alhough, as shown, the frame has atendency to collapse withou its supports, by rotating about the pin at B, this will not happen since the force system acting on it must still hold it in equilibrium. Hence, if so desired, all six unknowns can be determined by applying the three equilibrium equations to the entire frame, (c), and also to either one of its members. The above two examples illustrate that if a structure is properly supported and contains no more supports or members than are necessary to prevent collapse, the frame becomes statically determinate, and so the unknown forces at the supports and connections can be determined from the equations of equilibrium applied to each member. Also, if the structure remains rigid (noncollapsible) when the supports are removed, like that shown in the top figure, all three support reactions can be determined by applying the three equilibrium equations to the entire structure. However, if the structure appears to be nonrigid (collapsible) after removing the supports (c), it must be dismembered and equilibrium of the individual members must be considered in order to obtain enough equations to determine all the support reactions. HOW IMPORTANT IS THE FREE-BODY DIAGRAM? For any structual analysis it is very important! Not only does it greatly reduce the chance for errors, by accounting for all these forces and the geometry used in the equilibrium equations, but it is also a source of communication to othe rengineers, who may check or use your calculations. It seems needless to say, but engineers are responsible for their design and will be held accountable if anything goes wrong. So be neat and accurate in your wor, and draw your free-body diagrams! Geometric instability in a structure occurs when the reactive forces acting on the structure are insufficient or when their lines of action intersect at a single point or are parallel to each other. If there are fewer reactive forces than equations of equilibrium, it means that the structure is underdetermined and cannot be in complete equilibrium. In this case, the structure will undergo slight movements or even collapse due to the imbalance of forces. On the other hand, if there are enough reactive forces but their lines of action intersect at a common point or are parallel, the structure will experience instability. In such cases, the forces acting on the structure are not distributed evenly, leading to potential failures or collapses. The concentration of forces at a single point or their parallel alignment can create conditions of unbalanced loading, causing the structure to become unstable. It is important to note that stability analysis is crucial in the design and construction of structures to ensure their safety and performance. Engineers and architects must consider these factors to create stable and reliable structures.