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  • 1. EDUCATION HOLE PRESENTS Engineering Mechanics Unit-II
  • 2. Basic Structural Analysis.......................................................................................................... 2 Plane Truss.......................................................................................................................................2 Truss types............................................................................................................................................................2 Pratt truss .............................................................................................................................................................3 Bow string roof truss.............................................................................................................................................3 Difference between truss and frame ................................................................................................3 Perfect truss.....................................................................................................................................4 Imperfect truss......................................................................................................................................................4 Deficient Truss ......................................................................................................................................................4 Analysis of Plane trusses.......................................................................................................... 5 Method of joints ..............................................................................................................................5 Method of Sections..........................................................................................................................6 Zero force members.........................................................................................................................8 Reasons for Zero-force members in a truss system..............................................................................................8 Beams ..................................................................................................................................... 8 Types of beams ................................................................................................................................9 Statically Determinate Beams...............................................................................................................................9 Statically Indeterminate Beams........................................................................................................9 Shear force and bending moment in beams ................................................................................... 10 Shear force and bending moment diagrams ......................................................................................................11 Relationship between load, shear and bending moment................................................................ 12 Basic Structural Analysis Plane Truss A plane truss is one where all the members and joints lie within a 2-dimensional plane, while a space truss has members and joints extending into 3 dimensions. Truss types There are two basic types of truss:
  • 3. • The pitched truss, or common truss, is characterized by its triangular shape. It is most often used for roof construction. Some common trusses are named according to their web configuration. The chord size and web configuration are determined by span, load and spacing. • The parallel chord truss, or flat truss, gets its name from its parallel top and bottom chords. It is often used for floor construction. A combination of the two is a truncated truss, used in hip roof construction. A metal plate- connected wood truss is a roof or floor truss whose wood members are connected with metal connector plates. Pratt truss The Pratt truss was patented in 1844 by two Boston railway engineers; Caleb Pratt and his son Thomas Willis Pratt. The design uses vertical beams for compression and horizontal beams to respond to tension. What is remarkable about this style is that it remained popular even as wood gave way to iron, and even still as iron gave way to steel. The Southern Pacific Railroad bridge in Tempe, Arizona is a 393 meter (1291 foot) long truss bridge built in 1912. The structure is composed of nine Pratt truss spans of varying lengths. The bridge is still in use today. Bow string roof truss Named for its distinctive shape, thousands of bow strings were used during World War II for aircraft hangars and other military buildings. Difference between truss and frame Truss structure is designed to support huge loads in comparison to its weight. In a truss, the joints are of pin type and consists of 2 force members (tensile and compression - axial forces). Here the the members are free to rotate about the pin. A frame is a structure in which at least one of its individual member is a multi-force member. The members of frames are connected rigidly at joints by means of welding and bolting. The joints can transfer moments in addition to the axial loads. In a plane frame all the moments are perpendicular to a single plane Frames can be of 2 types - rigid noncollapsible and nonrigid collapsible frames.
  • 4. Perfect truss A truss is said to be perfect when the number of members in the truss is just sufficient to prevent distortion of its shape when loaded with an external load. A perfect truss has to satisfy the following equation: where m is the number of members. Imperfect truss An imperfect truss is that which does not satisfy the equation, m = 2j – 3. Or in other words, a truss in which the number of members is more or less than 2j – 3. The imperfect truss may be further classified into the following two types: 1. deficient truss 2. redundant truss. Deficient Truss A deficient truss is an imperfect truss, in which the number of members is less than 2j – 3.
  • 5. Analysis of Plane trusses Because the forces in each of its two main girders are essentially planar, a truss is usually modeled as a two-dimensional plane frame. If there are significant out-of-plane forces, the structure must be modeled as a three-dimensional space. The analysis of trusses often assumes that loads are applied to joints only and not at intermediate points along the members. The weight of the members is often insignificant compared to the applied loads and so is often omitted. If required, half of the weight of each member may be applied to its two end joints. Provided the members are long and slender, the moments transmitted through the joints are negligible and they can be treated as "hinges" or 'pin-joints'. Every member of the truss is then in pure compression or pure tension – shear, bending moment, and other more complex stresses are all practically zero. This makes trusses easier to analyze. This also makes trusses physically stronger than other ways of arranging material – because nearly every material can hold a much larger load in tension and compression than in shear, bending, torsion, or other kinds of force. Structural analysis of trusses of any type can readily be carried out using a matrix method such as the direct stiffness method, the flexibility method or the finite element method. Method of joints This method uses the force balance in the x and y directions at each of the joints in the truss structure. At A, At D,
  • 6. At C, Although we have found the forces in each of the truss elements, it is a good practice to verify the results by completing the remaining force balances. At B, Method of Sections This method can be used when the truss element forces of only a few members wants to be known. This method is used by introducing a single straight line cutting through the member whose force wants to be calculated. However this method has a limit in that the cutting line can pass through a maximum of only 3 members of the truss structure. This restriction is because this method uses the force balances in the x and y direction and the moment balance, which gives us a maximum of 3 equations to find a maximum of 3 unknown truss element forces through which this cut is made. Let us try to find the forces FAB, FBD and FCD in the above example
  • 7. Method 1: Ignore the right side Method 2: Ignore the left side
  • 8. The truss elements forces in the remaining members can be found by using the above method with a section passing through the remaining members. Zero force members In the field of engineering mechanics, a zero force member refers to a member (a single truss segment) in a truss which, given a specific load, is at rest: neither in tension, nor in compression. In a truss a zero force member is often found at pins (any connections within the truss) where no external load is applied and three or fewer truss members meet. Recognizing basic zero force members can be accomplished by analyzing the forces acting on an individual pin in a physical system. NOTE: If the pin has an external force or moment applied to it, then all of the members attached to that pin are not zero force members UNLESS the external force acts in a manner that fulfills one of the rules below: • If only two members meet in an unloaded joint, both are zero-force members. • If three members meet in an unloaded joint of which two are in a direct line with one another, then the third member is a zero-force member. • If two members meet in a loaded joint and the line of action of the load coincides with one of the members, the other member is a zero-force member. Reasons for Zero-force members in a truss system • These members contribute to the stability of the structure, by providing buckling prevention for long slender members under compressive forces • These members can carry loads in the event that variations are introduced in the normal external loading configuration Beams A beam is a bar subject to forces or couples that lie in a plane containing the longitudinal section of the bar. According to determinacy, a beam may be determinate or indeterminate.
  • 9. Types of beams Statically Determinate Beams Statically determinate beams are those beams in which the reactions of the supports may be determined by the use of the equations of static equilibrium. The beams shown below are examples of statically determinate beams. Statically Indeterminate Beams If the number of reactions exerted upon a beam exceeds the number of equations in static equilibrium, the beam is said to be statically indeterminate. In order to solve the reactions of the beam, the static equations must be supplemented by equations based upon the elastic deformations of the beam. The degree of indeterminacy is taken as the difference between the umber of reactions to the number of equations in static equilibrium that can be applied. In the case of the propped beam shown, there are three reactions R1, R2, and M and only two equations (ΣM = 0 and ΣFv = 0) can
  • 10. be applied, thus the beam is indeterminate to the first degree (3 - 2 = 1). Shear force and bending moment in beams The deformation of a beam is usually expressed in terms of its deflection from its original unloaded position. The deflection is measured from the original neutral surface of the beam to the neutral surface of the deformed beam. The configuration assumed by the deformed neutral surface is known as the elastic curve of the beam.
  • 11. Methods of Determining Beam Deflections Numerous methods are available for the determination of beam deflections. These methods include: 1. Double-integration method 2. Area-moment method 3. Strain-energy method (Castigliano's Theorem) 4. Conjugate-beam method 5. Method of superposition Shear force and bending moment diagrams Consider a simple beam shown of length L that carries a uniform load of w (N/m) throughout its length and is held in equilibrium by reactions R1 and R2. Assume that the beam is cut at point C a distance of x from he left support and the portion of the beam to the right of C be removed. The portion removed must then be replaced by vertical shearing force V together with a couple M to hold the left portion of the bar in equilibrium under the action of R1 and wx.
  • 12. The couple M is called the resisting moment or moment and the force V is called the resisting shear or shear. The sign of V and M are taken to be positive if they have the senses indicated above. Relationship between load, shear and bending moment Since this method can easily become unnecessarily complicated with relatively simple problems, it can be quite helpful to understand different relations between the loading, shear, and moment diagram. The first of these is the relationship between a distributed load on the loading diagram and the shear diagram. Since a distributed load varies the shear load according to its magnitude it can be derived that the slope of the shear diagram is equal to the magnitude of the distributed load. The relationship between distributed load and shear force magnitude is: Some direct results of this is that a shear diagram will have a point change in magnitude if a point load is applied to a member, and a linearly varying shear magnitude as a result of a constant distributed load. Similarly it can be shown that the slope of the moment diagram at a given point is equal to the magnitude of the shear diagram at that distance. The relationship between distributed shear force and bending moment is
  • 13. A direct result of this is that at every point the shear diagram crosses zero the moment diagram will have a local maximum or minimum. Also if the shear diagram is zero over a length of the member, the moment diagram will have a constant value over that length. By calculus it can be shown that a point load will lead to a linearly varying moment diagram, and a constant distributed load will lead to a quadratic moment diagram.