Con 1_SB01_Forces Moments PDF

This document offers a facsimile of a range of sources, which have been reproduced here for academic purposes to serve as required reading material for the course CONSTRUCTION 1 | DESST 1507 pursuant to Part VA...

This document offers a facsimile of a range of sources, which have been reproduced here for academic purposes to serve as required reading material for the course CONSTRUCTION 1 | DESST 1507 pursuant to Part VA of the Copyright Act 1968 of the Commonwealth of Australia. The material has been sourced from the following publications, and the copyright resides with the original authors/publishers. Any further copying or communication may be subject to copyright protection. Cowan, Henry J. and Forrest Wilson. Structural Systems. New York : Van Nostrand Reinhold, 1981. Schodek, Daniel L. Structures. Englewood Cliffs, N.J : Prentice-Hall, 1980. For further information please contact the Course Coordinator: Dr. Amit Srivastava INTRODUCTION Structural systems are necessary for transferring all loads acting on a building safely to the foundation, without interfering with any of the functions of the building. The design of structural systems for buildings, thus, involves estimating all the loads and other actions that the building would be subject to during its life time, and selecting or synthesizing an efficient and economic structural system, whose behaviour is within acceptable limits. Today no reasonable architectural design is impossible because of structural limitations, but some structural solutions may cost a great deal more than others. While structure is no longer a limiting factor, it remains a very important part of the design, and the basic structural decisions should be made at an early stage. Particular attention should be given to the location of the columns and load-bearing walls, since they affect the plan of the building at every level. Columns that transmit the load from the upper stories can be eliminated at a particular level, but only if the loads are transmitted to other structural members, which may greatly increase their size and cost. Fortunately, the need for many interior columns required at the beginning of the 20th century have been eliminated by modern structural technology. Auditoria can today be built without the interior columns that in the past obstructed the view from some seats. Most tall buildings now have floor structures capable of spanning from the service core to the facade, so that there is no longer any need for interior columns which took up valuable space and restrict flexibility. Column layout is not merely a planning problem, it also affects the appearance of the facade, which is often dominated by the spacing and size of the columns. A structure with uniform and symmetrical column spacing is generally more economical and more attractive in appearance than one with arbitrary departures from regularity. UNDERSTANDING STRUCTURAL SYSTEMS At the most basic level we can consider the System to be a combination of Members and Joints. Different configurations of members and joints lead to different assemblies and sub-systems, and a synthesis of these may compose a complete structural system. To develop a workable assembly or system we must understand that kinds of forces these members and joints are subject to and what are the properties of the materials that they are composed of. Thus, an introduction to Forces and Material Properties is an essential starting point to the design of structural systems. Let us first begin with an understanding of forces. FORCES Simply stated a Force is a kind of influence or a capacity to cause change in an object which results from an interaction with another object. As such, the force is not an intrinsic part of the material object but a relational quality between two objects. We are all aware of Gravity as a pulling force that keeps our feet firmly planted on the ground. Gravity also plays an important role in the design of structural systems for buildings, but different members of the system might experience this in different ways. A structural member may be subjected to: Pushing or Compression (a force) Pulling or Tension (a force) Cutting or Shear (a force) MOMENTS A force tends to move a body in the direction in which it acts, and it also tends to rotate the body if it does not pass through its centre of gravity. A Moment is the rotating effect of a force. So in addition to the forces discussed above the member might also be subject to : Bending or Flexure (a moment) Twisting or Torsion (a moment) MANAGING FORCES FOR A STABLE SYSTEM Let’s consider a system with a square made out of 4 members which are joined together with movable joints. Any kind of pushing or pulling force applied to this system will allow for the system to transform and alter in shape. Obviously, this is not an appropriate system for architectural structures as we would want our buildings to be as stable as possible. To make this system more stable we could either add another member or make the joints non-movable. Since a system is a combination of members and joints, we can either opt for a member based solution or a joint based solution. Let us first consider the joint based solution. Movable joints or Pinned Joints allow complete freedom of rotation, while non-movable or Rigid Joints restrict rotation. As a result, a Pinned Joint can transmit forces like Tension and Compression, but it cannot transmit any Bending Moment (see above). For purposes of structural calculation, we can say that the Bending Moment at a Pinned Joint is always zero. In general construction, we may not use actual pin-joints, but we try to ensure that the joint offers little resistance so that the bending moment is negligible. This is because design calculations are greatly simplified by having joints that transmit either no bending moment at all (pin joints) or the entire bending moment (rigid joints). Returning to our previous structural configuration of 4 members connected through pinned joints, let us now consider the member based solution for stability. As long as there are insufficient members in relation to the number of joints, we have a system that will collapse at the slightest push. Such a system can be regarded as a Mechanism. This ability of the mechanism to move is regarded through its Degrees of Freedom. Thus, a mechanism can have one, two, or three degrees of freedom depending on the number of members. We may increase the stability of the system by introducing extra members, and this would reduce its degrees of freedom. When we eliminate the last degree of freedom, i.e. when we have just the sufficient number of members in relation to the number of joints to prevent the system from collapsing at the slightest push, we obtain a Statically Determinate Structure. So we can say that a statically determinate structure has zero degrees of freedom. If we add further structural members, we obtain a Statically Indeterminate Structure with one, two, three or more Degrees of Redundancy. For any given configuration of joints, which indicate the general shape of the structure, the statically determinate solution is therefore the borderline between a whole range of mechanisms (with more and more degrees of freedom) and a whole range of statically indeterminate structures (with more and more redundancies). This relationship of Freedom and Redundancies is also applicable to joint based solutions discussed above. To develop an efficient and economic structural system we should aim to design a statically determinate structure, but we may consider a certain degree of redundancy for safety in regard to other unconsidered forces. THINKING THROUGH TRIANGLES We have already discussed that increasing the number of members could help a mechanism become stable. In the example introduced above, a mechanism of 4 members connected through pinned joints can be stabilised through the introduction of a fifth member that connects the two diagonally opposing joints. Since we have just introduced a diagonal member to brace the collapse of the system, this solution is simply called Diagonal Bracing. But why does this work? If we consider this solution properly, we will realise that we have just converted our quadrilateral (4 sided) configuration into a set of triangles (3 sided). We are already aware that triangles are a very stable geometric configuration, so a simple way of ensuring the stability of a structural system is to think through triangles. This process of developing a structural system as a series of triangles is called Triangulation. Based on this we can define the principle of triangulation as follows: Three members connecting three pin joints make a statically determinate frame, and for every additional joint we require two additional members if the system is to remain statically determinate. This principle is commonly used in construction and results in a Truss. A truss is an assemblage of individual linear elements arranged in a triangle or combination of triangles to form a rigid framework that cannot be deformed by the application of external forces without deformation of one or more of its members. The individual elements are typically assumed to be joined at their intersections with pinned connections. A pitched truss is characterized by its triangular shape and can be seen in the design of roofs in small residential buildings. The parallel chord truss results from a configuration of triangles which are contained between two straight lines forming the top and bottom chords. As a pin-jointed system these trusses do not transmit Bending Moment and so the members in these trusses are only subject to Tension or Compression. It is extremely important that trusses be loaded only with concentrated loads that act at joints for truss members to develop only tensile or compressive members. If loads are applied directly onto truss members themselves, bending stresses will also develop in the loaded members in addition to the basic tensile or compressive stresses already present, with the consequence that member design is greatly complicated, and the overall efficiency of the truss is reduced. INTERNAL FORCES IN MEMBERS Continuing with our discussion of Truss systems, where members are joined together in a triangulated configuration with pinned joints, we know that the members of such a system can be subject to either Tension or Compression. One way of determining the sense of the force in a truss member is to visualize the probable deformed shape of the structure as it would develop if the member considered was imagined to be removed. The nature of the force in the member can then be predicted on the basis of an analysis of its role in preventing the deformation visualized. Consider the diagonals shown in truss A in Figure 4-3(a). If the diagonals were imagined removed, the assembly would dramatically deform, as illustrated in Figure 4-3(b), since it is a non- triangulated configuration. In order for the diagonals to keep the type of deformation shown from occurring, it is evident that the left and right diagonals must prevent points B-F and points B-D, respectively, from drawing apart. Consequently, diagonals placed between these points would be pulled upon. Hence, tension forces would develop in the diagonal members. The diagonals shown in truss B in Figure 4-3 must be in a state of compression since their function is to keep points A-E and C-E from drawing closer together. With respect to member BE in both trusses, it is fairly easy to imagine what would happen to points B and E if member BE were removed. In truss A, points B and E would have a tendency to draw together, hence compressive forces develop in any member placed between these two points. In truss B, however, removal of member BE leads to no change in the gross shape of the structure (since it remains a stable triangulated configuration), hence the member serves no direct role for this loading. It is a zero-force member. Note that members AF, FE, ED, and DC in truss B could also be removed without altering the basic stability of the remainder of the configuration, and hence are also zero-force members. This is obviously not true for the same members in truss A. Final forces in both truss A and truss B are illustrated in Figure 4- 3(c). Another completely different way of visualizing the forces developed in a truss is to use an arch and cable analogy. It is evident, for example, that truss A can be conceived of as a cable with supplementary members [see Figure 4-3(d)]. Truss B can be conceived of as a simple linear arch with supplementary members. Diagonals in truss A are consequently in tension while diagonals in truss B must be in compression. The forces in other members can be determined by analysing their respective roles in relation to the ‘arch’ or ‘cable’. (Also, see section on FUNICULAR STRUCTURES) EFFECTS OF BENDING (Internal Tension and Compression) When a member carries external loads that are applied transversely to the long axis of the member (rather than along the axis of the member), the action of the external forces is to tend to cause the member to bow. A member that carries transverse forces, undergoes a bowing because of them, and consequently develops internal tensile and compressive forces at the same cross section is said to be in Bending. The common beam is typically subject to bending. As the member is bowed under the action of the loads, it is evident that some fibres in the member must stretch, while others must compress. It is also evident that both stretching and compressing of member fibres occur at the same cross sections. Associated with this phenomenon are internal tensile and compressive forces. If we bend a beam, it becomes shorter on top and longer at the bottom. Somewhere near the middle of the beam (exactly at the middle if the section is symmetrical) there is a surface that gets neither longer nor shorter, and this surface is called the Neutral Axis. The beam is in compression above the neutral axis and in tension below it. The greatest tension occurs at the very bottom of the beam, the greatest compression at the very top. EFFECTS OF BENDING (Bending Moment) We already know that if a force does not pass through the centre of gravity of an object it tends to cause a rotating effect and that this rotating effect is known as Moment. Considering the transverse loading of a member as discussed in the discussion on bending above, the member may be subject to a Bending Moment. Since the moment is the rotating effect of the force which is acting away from the centre of gravity, we can observe that it simply a product of the force and the distance. The effect of this can be easily observed in the behaviour of a simple mechanical scale or a see- saw. If the loads on either side of a scale or a see-saw are the same and they are at the same distance from the fulcrum they will remain balanced (or in equilibrium). If the distance from the fulcrum is reduced for either side a proportionately larger load will need to be applied to keep the balance. So, in other words, the further the perpendicular distance of the force from the point under consideration, the greater the moment of that force. Since Force (P) is measured in Newtons (N) and Distance (L) in Metres (m), the unit for Moment is Newton-metres (Nm). We have already discussed that a Simply Supported Beam, which has non rigid supports on two ends, is subject to bending and that the beam is in compression above the neutral axis and in tension below it. But it also important to understand the effect of bending in a Cantilever, which is a horizontal member with support on only one end. Since a cantilever tries to rotate downwards at the far end it experiences bending in the opposite direction to a simply supported beam. Therefore, a cantilever is in tension above the neutral axis and in compression below it. Based on our discussion of Bending Moment and the condition that the further the perpendicular distance of the force from the point under consideration, the greater the moment of that force, we can recognise that the limits of the cantilever can be determined by calculating the maximum bending moment it can withstand. (For calculations of Bending Moments see separate resource provided.) STABILITY AND EQULLIBRIUM It is obvious that for a system to be stable we need all forces to be in equilibrium. Therefore, first, all the horizontal forces must balance one another; that is, the sum of all horizontal forces must be zero. And second, all the vertical forces must balance one another; that is, the sum of all the vertical forces must be zero. This ensures that the structure cannot be moved either vertically or horizontally, but it still leaves the possibility that it can be rotated. So evidently, we need a third condition of equilibrium, namely that the sum total of all the moments of all the forces about any convenient point be zero. LOADING TYPES Until now we have only been discussing the effects of single lines of forces. However, in order to understand the various conditions of bending it is important to recognise that multiple forces may be acting along the entire length of a structural member. For now let’s expand this to include one other condition where a given load is uniformly spread across the entire length of the member. This allows for 2 different loading conditions that we can take into consideration: Point Load or Concentrated Load – Where there is a single line of force in action. Uniformly Distributed Load or Continuous (UDL) – Where the load is evenly spread across length. SHEAR AND BENDING MOMENT DIAGRAMS That the shears and moments found by considering alternate parts of the structure should be numerically equal but opposite in sense is reasonable in view of the fact that we need to maintain equilibrium. But this does bring up a difficulty in terms of sign conventions. With respect to moments, note that the tendency of either the external force system acting on the left part or the external force system acting on the right part is to cause the structure to deform or bend, as illustrated to the left in Figure 2-22(a). When the external force system causes this type of concave curvature to develop at a section in the member, the moment present at the section is said to be positive. An equivalent convention is to say that upward-acting external forces tend to cause positive moments to develop at a section. If the external force system tends to produce a convex curvature (concave downward) at a section, the moment present is said to be negative. Equivalently, downward-acting forces tend to produce negative moments at a section. Relating moment to curvature is also a way of looking at structures that is useful in developing a feeling for the physical behaviour of structures. For shear, the convention typically used is that when the external forces acting on the left part of a structure have a net resultant that acts vertically upward, the shear is said to be positive. Positive shear is associated with the tendency of the external forces to produce relative movements of a section of the type illustrated to the left in Figure 2-22(b). As an aid in visualizing the distribution of these shears and moments, the values thus found can be plotted graphically to produce Shear Force Diagrams (SFD) and Bending Moment Diagrams (BMD). (See SFD & BMD Chart provided separately) FUNICULAR STRUCTURES (Cables and Arches) Many whole structures can be characterized as being primarily in a state of pure tension or compression. These are interesting structures deserving special treatment. Structures wherein only a state of tension or compression is induced by the loading are referred to as Funicular Structures. Consider a simple flexible cable spanning between two points and carrying a load. This structure must be exclusively in a state of tension, since a flexible cable can withstand neither compression nor bending. A cable carrying a concentrated load at midspan would deform as indicated in Figure 1-12(a). The whole structure is in tension. If this exact shape were simply inverted and loaded precisely in the same way, it is evident by analogy that the resultant structure would be in a state of pure compression. If the loading condition is changed to a continuous load, a flexible cable carrying this load would naturally deform into the parabolic shape indicated in Figure 1-12(b). Again, the whole structure is in tension. If this exact shape were inverted and loaded with the same continuous load, the resultant structure would be in a state of compression [Figure 1-12(d)]. The Common Arch is predominantly a structure of this type. It is interesting to note that despite the fact that loads are applied transversely to the length of the members, as typically occurs in a beam, only tension or compression exists in these structures-not bending. Why a linear beam is in bending and a cable is not is, of course, partly a function of the material used. The cable cannot withstand bending. Therefore, it deforms under the action of the load. The rigid linear beam can take bending and therefore resists being deformed. Material alone, however, does not explain why some structures are funicular and others are not. The arch, which is predominantly in a state of compression, is often made from rigid materials. Obviously, the shape of the structure is the prime determinant of whether a structure is in pure tension or compression or is subject to bending. The easiest way to determine the funicular response for a particular loading condition is by determining the exact shape a flexible string would deform to under the load. This is the Tension Funicular. Inverting this shape exactly yields a Compression Funicular. A modern rigid arch is often shaped in response to a primary loading condition and carries loads in axial compression when this loading is actually present but is also designed to have sufficient bending resistance to carry load variations. The structure, however, cannot be funicular for the new loading. It is in this way that a rigid arch is fundamentally different from a flexible cable. A flexible cable must always change shape under changes in loading and thus essentially be momentless under all loadings, while a rigid arch is momentless only under one loading condition and is capable of carrying some bending due to load variations. DIAGRAM FOR REFERENCE ONLY.