Truss

1. Introduction

In architecture and structural engineering, a truss is a structure comprising one or more triangular units constructed with straight members whose ends are connected at joints referred to as nodes. External forces and reactions to those forces are considered to act only at the nodes and result in forces in the members which are either tensile or compressive forces. Moments (torques) are explicitly excluded because, and only because, all the joints in a truss are treated as revolutes.[1] A structure that is composed of a number of bars pin connected at their ends to form a stable framework is called a truss. It is generally assumed that loads and reactions are applied to the truss only at the joints. A truss would typically be composed of triangular elements with the bars on the upper chord under compression and those along the lower chord under tension. Trusses are extensively used for bridges, long span roofs, electric tower, and space structures. Trusses are statically determinate when the entire bar forces can be determined from the equations of statics alone. Otherwise the truss is statically indeterminate. A truss may be statically (externally) determinate or indeterminate with respect to the reactions.[2]

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2. Characteristics of a Truss

A truss consists of straight members connected at joints. Trusses are composed of triangles because of the structural stability of that shape and design. A triangle is the simplest geometric figure that will not change shape when the lengths of the sides are fixed. In comparison, both the angles and the lengths of a four-sided figure must be fixed for it to retain its shape

Planar Truss Space Frame Truss 2.1 Planar Truss

The simplest form of a truss is one single triangle. This type of truss is seen in a framed roof consisting of rafters and a ceiling joist. and in other mechanical structures such as bicycles. Because of the stability of this shape and the methods of analysis used to calculate the forces within it, a truss composed entirely of triangles is known as a simple truss. The traditional diamond-shape bicycle frame, which utilizes two conjoined triangles, is an example of a simple trussA planar truss lies in a single plane. Planar trusses are typically used in parallel to form roofs and bridges.

2.2 Space Frame Truss

A space frame truss is a three-dimensional framework of members pinned at their ends. A tetrahedron shape is the simplest space truss, consisting of six members which meet at four joints. Large planar structures may be composed from tetrahedrons with common edges and they are also employed in the base structures of large freestanding power line pylons

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3. Truss Types

There are three basic types of truss:

•

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.[3]

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4. Truss Types used in Bridges

4.1 Allan Truss

The Allan Truss, designed by Percy Allan, is partly based on the Howe truss. The first Allan truss was completed on 13 August 1894 over Glennies Creek at Camberwell, New South Wales . completed in March 1895. The Hampden Bridge in Wagga Wagga, New South Wales, Australia, the first of the Allan truss bridges with overhead bracing, was originally designed as a steel bridge but was constructed with timber to reduce cost. In his design, Allan used Australian ironbark for its strength. A similar bridge also designed by Percy Allen is the Victoria Bridge on Prince Street Picton, New South Wales. Also constructed of ironbark, the bridge is still in use today for pedestrian and light traffic.

4.2 Bailey Bridge

Designed for military use, the prefabricated and standardized truss elements may be easily combined in various configurations to adapt to the needs at the site. In the image at right, note the use of doubled prefabrications to adapt to the span and load requirements. In other applications the trusses may be stacked vertically.

4.3 Baltimore Truss

The Baltimore truss is a subclass of the Pratt truss. A Baltimore truss has additional bracing in the lower section of the truss to prevent buckling in the compression members and to control deflection. It is mainly used for train bridges, boasting a simple and very strong design.

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4.4 Bollman Truss

The Bollman Truss Railroad Bridge at Savage, Maryland is the only surviving example of a revolutionary design in the history of American bridge engineering. The type was named for its inventor, Wendel Bollman, a self-educated Baltimore engineer. The design employs wrought iron tension members and cast iron compression members. The use of multiple independent tension elements reduces the likelihood of catastrophic failure. The structure was also easy to assemble.

4.5 Bowstring Arch Truss (Tied Arch Bridge)

Thrust arches transform their vertical loads into a thrust along the arc of the arch. At the ends of the arch this thrust may be resolved into two components, a vertical thrust equal to a proportion of the weight and load of the bridge section, and a horizontal thrust. In a typical arch this horizontal thrust is taken into the ground, while in a bowstring arch the thrust is taken horizontally by a chord member to the opposite side of the arch. This allows the footings to take only vertical forces, useful for bridge sections resting upon high pylons.

4.6 Cantilevered Truss

Most trusses have the lower chord under tension and the upper chord under compression. In a cantilever truss the situation is reversed, at least over a portion of the span. The typical cantilever truss bridge is a balanced cantilever, which enables the construction to proceed outward from a central vertical spar in each direction. Usually these are built in pairs until the outer sections may be anchored to footings. A central gap, if present, can then be filled by lifting a conventional truss into place or by building it in place using a traveling support.

[5] 4.7 Fink Truss

The Fink truss was designed by Albert Fink of Germany in the 1860s. This type of bridge was popular with the Baltimore and Ohio Railroad. The Appomattox High Bridge on the Norfolk and Western Railroad included 21 Fink deck truss spans from 1869 until their replacement in 1886.

4.8 Howe Truss

The relatively rare Howe truss, patented in 1840 by Massachusetts millwright William Howe, includes vertical members and diagonals that slope up towards the center, the opposite of the Pratt truss.[9] In contrast to the Pratt Truss, the diagonal web members are in compression and the vertical web members are in tension. Examples include Jay Bridge in Jay, New York, and Sandy Creek Covered Bridge in Jefferson County, Missouri.

4.9 Kingpost Truss

A king post (or crown post) extends vertically from a crossbeam to the apex of a triangular truss. King posts were used in roof construction in Medieval architecture in buildings. The truss consists of two diagonal members that meet at the apex of the truss, one horizontal beam that serves to tie the bottom end of the diagonals together, and the king post which connects the apex to the horizontal beam below. For a roof truss, the diagonal members are called rafters, and the horizontal member may serve as a ceiling joist. A bridge would require two king post trusses with the driving surface between them. A roof usually uses many side-by-side trusses depending on the size of the structure

[6] 4.10 Lattice Truss

A lattice bridge is a form of truss bridge that uses a large number of small and closely spaced diagonal elements that form a lattice. Originally a design to allow a substantial bridge to be made from planks employing lower–skilled labor, rather than heavy timbers and more expensive carpenters, this type of bridge has also been constructed using a large number of relatively light iron or steel members. The individual elements are more easily handled by the construction workers, but the bridge also requires substantial support during construction. A simple lattice truss will transform the applied loads into a thrust, as the bridge will tend to change length under load. This is resisted by pinning the lattice members to the top and bottom chords, which are more substantial than the lattice members, but which may also be fabricated from relatively small elements rather than large beams.

4.11 Lenticular Truss

A lenticular truss bridge includes a lens-shape truss, with trusses between an upper arch that curves up and then down to end points, and a lower arch that curves down and then up to meet at the same end points. Where the arches extend above and below the roadbed, it is a lenticular pony truss bridge. The Royal Albert Bridge (United Kingdom) uses a single tubular upper chord. As the horizontal tension and compression forces are balanced these horizontal forces are not transferred to the supporting pylons .This in turn enables the truss to be fabricated on the ground and then to be raised by jacking as supporting masonry pylons are constructed. This truss has been used in the construction of a stadium with the upper chords of parallel trusses supporting a roof that may be rolled back.

[7] 4.12 Long Truss

Designed by Stephen H. Long in 1830; one surviving example is the Old Blenheim Bridge. The design resembles a Howe truss, but is entirely made of wood instead of a combination of wood and metal.

4.13 Parker Truss

A Parker truss bridge is a Pratt truss design with a polygonal upper chord. A "camelback" is a subset of the Parker type, where the upper chord consists of exactly five segments. An example of a Parker truss is the Traffic Bridge in Saskatoon.

4.14 Pegram Truss

The Pegram truss is a hybrid between the Warren and Parker trusses where the upper chords are all of equal length and the lower chords are longer than the corresponding upper chord. Because of the difference in upper and lower chord length, each panel was not square. The members which would be vertical in a Parker truss vary from near vertical in the center of the span to diagonal near each end (like a Warren truss). George H. Pegram, while the chief engineer of Edge Moor Iron Company in Wilmington, Delaware, patented this truss design in 1885.The Pegram truss consists of a Parker type design with the vertical posts leaning towards the center at an angle between 60 and 75°. The variable post angle and constant chord length allowed steel in existing bridges to be recycled into a new span using the Pegram truss design. This design also facilitated reassembly and permitted a bridge to be adjusted to fit different span lengths. There are ten remaining Pegram span bridges in the United States with seven in Idaho.

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4.15 Post Truss

A Post truss is a hybrid between a Warren truss and a double-intersection Pratt truss. Invented in 1863 by Simeon S. Post, it is occasionally referred to as a Post patent truss although he never received a patent for it.[14] The Ponakin Bridge and the Bell Ford Bridge are two examples of this truss.

4.16 Pratt truss

A Pratt truss includes vertical members and diagonals that slope down towards the center, the opposite of the Howe truss.It can be subdivided, creating Y- and K-shaped patterns. The Pratt Truss was invented in 1844 by Thomas and Caleb Pratt. This truss is practical for use with spans up to 250 feet and was a common configuration for railroad bridges as truss bridges moved from wood to metal. They are statically determinate bridges, which lend themselves well to long spans.

4.17 Queenpost Truss

The queenpost truss, sometimes queen post or queenspost, is similar to a king post truss in that the outer supports are angled towards the center of the structure. The primary difference is the horizontal extension at the center which relies on beam action to provide mechanical stability. This truss style is only suitable for relatively short spans

4.18 Truss Arch

A truss arch may contain all horizontal forces within the arch itself, or alternatively may be either a thrust arch consisting of a truss, or of two arcuate sections pinned at the apex. The latter form is common when the bridge is constructed as cantilever segments from each side as in the Navajo Bridge

[9] 4.19 Warren Truss

It consists of longitudinal members joined only by angled cross-members, forming alternately inverted equilateral triangle-shaped spaces along its length, ensuring that no individual strut, beam, or tie is subject to bending or torsional straining forces, but only to tension or compression. Loads on the diagonals alternate between compression and tension (approaching the center), with no vertical elements, while elements near the center must support both tension and compression in response to live loads. This configuration combines strength with economy of materials and can therefore be relatively light.

4.20 Whipple Pratt Truss

A whipple truss is usually considered a subclass of the Pratt truss because the diagonal members are designed to work in tension. The main characteristic of a whipple truss is that the tension members are elongated, usually thin, at a shallow angle and cross two or more bays (rectangular sections defined by the vertical members).An example of a Pratt Truss bridge is the Fair Oaks Bridge in Fair Oaks, California.

4.21 Vierendeel truss

The Vierendeel truss, unlike common pin-jointed trusses, imposes significant bending forces upon its members — but this in turn allows the elimination of many diagonal elements. While rare as a bridge type this truss is commonly employed in modern building construction as it allows the resolution of gross shear forces against the frame elements while retaining rectangular openings between column.[4]

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5. Statics of Truss

A truss that is assumed to comprise members that are connected by means of pin joints, and which is supported at both ends by means of hinged joints or rollers, is described as being statically determinate. Newton's Laws apply to the structure as a whole, as well as to each node or joint. In order for any node that may be subject to an external load or force to remain static in space, the following conditions must hold: the sums of all (horizontal and vertical) forces, as well as all moments acting about the node equal zero. Analysis of these conditions at each node yields the magnitude of the forces in each member of the truss. These may be compression or tension forces. Trusses that are supported at more than two positions are said to be statically indeterminate, and the application of Newton's Laws alone is not sufficient to determine the member forces. In order for a truss with pin-connected members to be stable, it must be entirely composed of triangles. In mathematical terms, we have the following necessary condition for stability: m = 2j - 3 where m is the total number of truss members, j is the total number of joints and r is the number of reactions (equal to 3 generally) in a 2-dimensional structure. When m = 2j − 3, the truss is said to be statically determinate, because the (m+3) internal member forces and support reactions can then be completely determined by 2j equilibrium equations. Given a certain number of joints, this is the minimum number of members, in the sense that if any member is taken out (or fails), then the truss as a whole fails. Their member forces depend on the relative stiffness of the members, in addition to the equilibrium condition described.[5]

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6. Truss Analysis

For the truss analysis, it is assumed that: Bars are pin-connected. Joints are frictionless hinges. Loads are applied at the joints only. Stress in each member is constant along its length. The objective of analyzing the trusses is to determine the reactions and member forces. The methods used for carrying out the truss analysis with the equations of equilibrium and by considering only parts of the structure through analyzing its free body diagram to solve the unknowns. There are 3 basic methods for determination of axial forces in members: Method of joints Methods of Sections Graphical Method

[12] 6.1 Method of Joints

The first to analyze a truss by assuming all members are in tension reaction. A tension member is when a member experiences pull forces at both ends of the bar and usually denoted as positive sign. When a member experiencing a push force at both ends, then the bar was said to be in compression mode and designated as negative sign. In the joints method, a virtual cut is made around a joint and the cut portion is isolated as a Free Body Diagram (FBD). Using the equilibrium equations of ∑ Fx = 0 and ∑ Fy = 0, the unknown member forces could be solve. It is assumed that all members are joined together in the form of an ideal pin, and that all forces are in tension (+ve) of reactions. An imaginary section may be completely passed around a joint in the truss. The joint has become a free body in equilibrium under the forces applied to it. The equations ∑ H = 0 and ∑ V = 0 may be applied to the joint to determine the unknown forces in members meeting there. It is evident that no more than two unknowns can be determined at a joint with these two equations. A simple truss model supported by pinned and roller support at its end. Each triangle has the same length, L and it is equilateral where degree of angle, θ is 60° on every angle. The support reactions, Ra and Rc can be determine by taking a point of moment either at point A or point C, whereas Ha = 0 (no other horizontal force). Here are some simple guidelines for this method of truss analysis:

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1. Solve the reactions of the given structure, 2. Select a joint with a minimum number of unknown (not more than 2) and analyze it with ∑ Fx = 0 and ∑ Fy = 0, 3. Proceed to the rest of the joints and again concentrating on joints that have very minimal of unknowns, 4. Check member forces at unused joints with ∑ Fx = 0 and ∑ Fy = 0, 5. Tabulate the member forces whether it is in tension (+ve) or compression (-ve) reaction From fig.21, the forces in each member can be determine from any joint or point. The best way to start by selecting the easiest joint like joint C where the reaction Rc is already obtained and with only 2 unknown, forces of FCB and FCD. Both can be evaluate with ∑ Fx = 0 and ∑ Fy = 0 rules. At joint E, there are 3 unknown, forces of FEA, FEB and FED, which may lead to more complex solution compare to 2 unknown values. For checking purposes, joint B is selected to shown that the equation of ∑ Fx is equal to ∑ Fy which leads to zero value, ∑ Fx = ∑ Fy = 0. Each value of the member’s condition should be indicate clearly as whether it is in tension (+ve) or in compression (-ve) state. Trigonometric Functions: Taking an angle between member x and z…

• • •

Cos θ = x / z Sin θ = y / z Tan θ = y / x

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6.2 Method of Sections

The section method is an effective method when the forces in all members of a truss are being able to determine. Often we need to know the force in just one member with greatest force in it, and the method of section will yield the force in that particular member without the labor of working out the rest of the forces within the truss analysis. If only a few member forces of a truss are needed, the quickest way to find these forces is by the method of sections. In this method, an imaginary cutting line called a section is drawn through a stable and determinate truss. Thus, a section subdivides the truss into two separate parts. Since the entire truss is in equilibrium, any part of it must also be in equilibrium. Either of the two parts of the truss can be considered and the three equations of equilibrium ∑ Fx = 0, ∑ Fy = 0, and ∑ M = 0 can be applied to solve for member forces.Using the same model of simple truss, the details would be the same as previous figure with 2 different supports profile. Unlike the joint method, here we only interested in finding the value of forces for member BC, EC, and ED. Few simple guidelines of section truss analysis: 1. Pass a section through a maximum of 3 members of the truss, 1 of which is the desired member where it is dividing the truss into 2 completely separate parts, 2. At 1 part of the truss, take moments about the point (at a joint) where the 2 members intersect and solve for the member force, using ∑ M = 0, 3. Solve the other 2 unknowns by using the equilibrium equation for forces, using ∑ Fx = 0 and ∑ Fy = 0.

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In fig.23,a virtual cut is introduce through the only required members which is along member BC, EC, and ED. Firstly, the support reactions of Ra and Rd should be determine. Again a good judgment is require to solve this problem where the easiest part would be consider either on the left hand side or the right hand side. Taking moment at joint E (virtual pint) on clockwise for the whole RHS part would be much easier compare to joint C (the LHS part). Then, either joint D or C can be consider as point of moment, or else using the joint method to find the member forces for FCB, FCE, and FDE. Note: Each value of the member’s condition should be indicate clearly as whether it is in tension (+ve) or in compression (-ve) state.

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6.3 Graphical Method

The method of joints could be used as the basic for a graphical analysis of trusses. The graphical analysis was developed by force polygons drawn to scale for each joint, and then the forces in each member were measured from one of these force polygons. The number of lines which have to be drawn can be greatly reduced, however, if the various force polygons are superimposed. The resulting diagram of truss analysis is known as the Maxwell’s Diagram. In order to draw the Maxwell diagram directly, here are the simple guidelines: 1. Solve the reactions at the supports by solving the equations of equilibrium for the entire truss, 2. Move clockwise around the outside of the truss; draw the force polygon to scale for the entire truss, 3. Take each joint in turn (one-by-one), then draw a force polygon by treating successively joints acted upon by only two unknown forces, 4. Measure the magnitude of the force in each member from the diagram, 5. Lastly, note that work proceed from one end of the truss to another, as this use for checking of balance and connect to other end. In conclusion, the truss internal reaction as well as its member forces could be determine by either of this 3 methods especially in mechanics of structures.[6]

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7. Design of Members

A truss can be thought of as a beam where the web consists of a series of separate members instead of a continuous plate. In the truss, the lower horizontal member (the bottom chord) and the upper horizontal member (the top chord) carry tension and compression, fulfilling the same function as the flanges of an I-beam. Which chord carries tension and which carries compression depends on the overall direction of bending. The diagonal and vertical members form the truss web, and carry the shear force. Individually, they are also in tension and compression, the exact arrangement of forces is depending on the type of truss and again on the direction of bending. In addition to carrying the static forces, the members serve additional functions of stabilizing each other, preventing bucklingThe inclusion of the elements shown is largely an engineering decision based upon economics, being a balance between the costs of raw materials, off-site fabrication, component transportation, on-site erection, the availability of machinery and the cost of labor. In other cases the appearance of the structure may take on greater importance and so influence the design decisions beyond mere matters of economics. Modern materials such as prestressed concrete and fabrication methods, such as automated welding, have significantly influenced the design of modern bridges.

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8. Design of Joints

After determining the minimum cross section of the members, the last step in the design of a truss would be detailing of the bolted joints, e.g., involving shear of the bolt connections used in the joints, see also shear stress. Based of the needs of the project, truss internal connections (joints) can be designed as rigid, semi rigid, or hinged. Rigid connections can allow transfer of bending moments leading to development of secondary bending moments in the members.

9.Applications

Component connections are critical to the structural integrity of a framing system. In buildings with large, clearspan wood trusses, the most critical connections are those between the truss and its supports. In addition to gravity-induced forces , these connections must resist shear forces acting perpendicular to the plane of the truss and uplift forces due to wind. Depending upon overall building design, the connections may also be required to transfer bending moment. Wood posts enable the fabrication of strong, direct, yet inexpensive connections between large trusses and walls. Exact details for post-to-truss connections vary from designer to designer. Solid-sawn timber and glulam posts are generally notched to form a truss bearing surface. The truss is rested on the notches and bolted into place. A special plate/bracket may be added to increase connection load transfer capabilities.[7]

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Index

Topics

1. Introduction 2. Characteristics of a Truss 2.1 Planar Truss 2.2 Space Frame Truss 3. Truss Types 4. Truss Types used in Bridges 4.1 Allan Bridge 4.2 Bailey Bridge 4.3 Baltimore Truss 4.4 Bollman Truss 4.5 Bowstring Arch Truss 4.6 Cantilevered Truss 4.7 Flink Truss 4.8 Howe Truss 4.9 Kingpost Truss

Pg. No.

1 2 2 2 3 4 4 4 4 5 5 5 6 6 6

4.10 Lattice Truss 4.11 Lenticular Truss 4.12 Long Truss 4.13 Parker Truss 4.14 Pegram Truss 4.15 Post Truss 4.16 Pratt Truss 4.17 Queenpost Truss 4.18 Truss Arch 4.19 Warren Truss 4.20 Whipple Pratt Truss 4.21 Vierendeel Truss 5. Statics of truss 6. Truss Analysis 6.1 Method of Joints 6.2 Method of Sections 6.3 Graphical Method

7 7 8 8 8 9 9 9 9 10 10 10 11 12 13 15 17

7. Design of Members 8. Design of Joints 9. Applications

18 19 19

Bibliography

1. http://en.wikipedia.org/wiki/Truss 2. http://www.civilcraftstructures.com/civil-subjects/3-methods-fortruss-analysis/ 3. http://en.wikipedia.org/wiki/Truss#Truss_types 4. http://en.wikipedia.org/wiki/Truss_bridge#Truss_types_used_in_bri dges 5. http://en.wikipedia.org/wiki/Truss#Statics_of_trusses 6. http://www.civilcraftstructures.com/civil-subjects/3-methods-fortruss-analysis/ 7. http://en.wikipedia.org/wiki/Truss#Applications

Pictures Reference

Fig.1 Railway Track http://en.wikipedia.org/wiki/File:RRTrussBridgeSideView.jpg Fig. 2 Planar Truss http://en.wikipedia.org/wiki/File:Trusses_008.jpg Fig.3 Space Frame Truss http://en.wikipedia.org/wiki/File:SpaceFrame02.png Fig.4 Allan Truss http://en.wikipedia.org/wiki/File:Hampden_Bridge_Wagga_design.jpg Fig.5 Bollman Bridge http://en.wikipedia.org/wiki/File:Bollman-bridge-1.jpg Fig.6 Bowstring Arch Truss http://en.wikipedia.org/wiki/File:FortPittBridge.jpg Fig.7 Cantilevered Truss http://en.wikipedia.org/wiki/File:CooperRiverBridge.svg Fig.8 Fink Truss http://en.wikipedia.org/wiki/File:Bridges_20.png Fig.9 Howe Truss http://en.wikipedia.org/wiki/File:Howe_truss.PNG Fig.10 Kingpost Truss http://en.wikipedia.org/wiki/File:King_post_truss.png Fig.11 Lattice Truss http://en.wikipedia.org/wiki/File:Lattice_truss.png Fig. 12 Long Truss http://en.wikipedia.org/wiki/File:Long_Truss_Bushing_CB_(Versailles_SP)_00003 r.jpg Fig.14 Post Truss http://en.wikipedia.org/wiki/File:Post_truss.svg

Fig.15 Pratt Truss http://en.wikipedia.org/wiki/File:Pratt_truss.PNG Fig.16 Queenpost Truss http://en.wikipedia.org/wiki/File:Queen_post_truss.png Fig.17 Warren Bridge http://en.wikipedia.org/wiki/File:Warren_truss.PNG Fig.18 Vierendeel Truss http://en.wikipedia.org/wiki/File:Vierendeel_truss.png Fig.19 Statics of Truss http://www.civilcraftstructures.com/civil-subjects/3methods-for-truss-analysis/ Fig.20 Method of Joints http://www.civilcraftstructures.com/civil-subjects/3methods-for-truss-analysis/ Fig.21 Joint B,C and E http://www.civilcraftstructures.com/civil-subjects/3methods-for-truss-analysis/ Fig.22 Method of Sections http://www.civilcraftstructures.com/civil-subjects/3methods-for-truss-analysis/ Fig.23 Cutting Section http://www.civilcraftstructures.com/civil-subjects/3methods-for-truss-analysis/ Fig.24 Graphical Method http://www.civilcraftstructures.com/civil-subjects/3methods-for-truss-analysis/

Acknowledgement

We are greatly indebted to Mr. Sukhvinder Singh, our worthy and highly respected Mechanics Teacher who inspired us to undertake this project and provided us with very valuable guidance in the preparation of the report. We express our deep and sincere gratitude to him Abhishek Sharma (12811502810) Ashish Kumar Singh (12911502810) Gaurav Nigam (13011502810) Mridul Malik (13111502810)

Bharti Vidyapeeth’s College of Engineering

A-4 , Paschim Vihar, New Delhi-110063

CERTIFICATE

This is to certify that the reportt entitled “Truss” submitted by ,Abhishek Sharma,Ashish Kumar Singh,Gaurav Nigam and Mridul Malik, of branch ECE-2 is a record of the project was carried by them. They have worked under my guidance and supervision and has fulfilled the requirements for the submission of the report, which to my knowledge has reached a requisite standard.

Head of Department

Date

Professor Incharge

1. Introduction

In architecture and structural engineering, a truss is a structure comprising one or more triangular units constructed with straight members whose ends are connected at joints referred to as nodes. External forces and reactions to those forces are considered to act only at the nodes and result in forces in the members which are either tensile or compressive forces. Moments (torques) are explicitly excluded because, and only because, all the joints in a truss are treated as revolutes.[1] A structure that is composed of a number of bars pin connected at their ends to form a stable framework is called a truss. It is generally assumed that loads and reactions are applied to the truss only at the joints. A truss would typically be composed of triangular elements with the bars on the upper chord under compression and those along the lower chord under tension. Trusses are extensively used for bridges, long span roofs, electric tower, and space structures. Trusses are statically determinate when the entire bar forces can be determined from the equations of statics alone. Otherwise the truss is statically indeterminate. A truss may be statically (externally) determinate or indeterminate with respect to the reactions.[2]

[1]

2. Characteristics of a Truss

A truss consists of straight members connected at joints. Trusses are composed of triangles because of the structural stability of that shape and design. A triangle is the simplest geometric figure that will not change shape when the lengths of the sides are fixed. In comparison, both the angles and the lengths of a four-sided figure must be fixed for it to retain its shape

Planar Truss Space Frame Truss 2.1 Planar Truss

The simplest form of a truss is one single triangle. This type of truss is seen in a framed roof consisting of rafters and a ceiling joist. and in other mechanical structures such as bicycles. Because of the stability of this shape and the methods of analysis used to calculate the forces within it, a truss composed entirely of triangles is known as a simple truss. The traditional diamond-shape bicycle frame, which utilizes two conjoined triangles, is an example of a simple trussA planar truss lies in a single plane. Planar trusses are typically used in parallel to form roofs and bridges.

2.2 Space Frame Truss

A space frame truss is a three-dimensional framework of members pinned at their ends. A tetrahedron shape is the simplest space truss, consisting of six members which meet at four joints. Large planar structures may be composed from tetrahedrons with common edges and they are also employed in the base structures of large freestanding power line pylons

[2]

3. Truss Types

There are three basic types of truss:

•

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.[3]

.

[3]

4. Truss Types used in Bridges

4.1 Allan Truss

The Allan Truss, designed by Percy Allan, is partly based on the Howe truss. The first Allan truss was completed on 13 August 1894 over Glennies Creek at Camberwell, New South Wales . completed in March 1895. The Hampden Bridge in Wagga Wagga, New South Wales, Australia, the first of the Allan truss bridges with overhead bracing, was originally designed as a steel bridge but was constructed with timber to reduce cost. In his design, Allan used Australian ironbark for its strength. A similar bridge also designed by Percy Allen is the Victoria Bridge on Prince Street Picton, New South Wales. Also constructed of ironbark, the bridge is still in use today for pedestrian and light traffic.

4.2 Bailey Bridge

Designed for military use, the prefabricated and standardized truss elements may be easily combined in various configurations to adapt to the needs at the site. In the image at right, note the use of doubled prefabrications to adapt to the span and load requirements. In other applications the trusses may be stacked vertically.

4.3 Baltimore Truss

The Baltimore truss is a subclass of the Pratt truss. A Baltimore truss has additional bracing in the lower section of the truss to prevent buckling in the compression members and to control deflection. It is mainly used for train bridges, boasting a simple and very strong design.

[4]

4.4 Bollman Truss

The Bollman Truss Railroad Bridge at Savage, Maryland is the only surviving example of a revolutionary design in the history of American bridge engineering. The type was named for its inventor, Wendel Bollman, a self-educated Baltimore engineer. The design employs wrought iron tension members and cast iron compression members. The use of multiple independent tension elements reduces the likelihood of catastrophic failure. The structure was also easy to assemble.

4.5 Bowstring Arch Truss (Tied Arch Bridge)

Thrust arches transform their vertical loads into a thrust along the arc of the arch. At the ends of the arch this thrust may be resolved into two components, a vertical thrust equal to a proportion of the weight and load of the bridge section, and a horizontal thrust. In a typical arch this horizontal thrust is taken into the ground, while in a bowstring arch the thrust is taken horizontally by a chord member to the opposite side of the arch. This allows the footings to take only vertical forces, useful for bridge sections resting upon high pylons.

4.6 Cantilevered Truss

Most trusses have the lower chord under tension and the upper chord under compression. In a cantilever truss the situation is reversed, at least over a portion of the span. The typical cantilever truss bridge is a balanced cantilever, which enables the construction to proceed outward from a central vertical spar in each direction. Usually these are built in pairs until the outer sections may be anchored to footings. A central gap, if present, can then be filled by lifting a conventional truss into place or by building it in place using a traveling support.

[5] 4.7 Fink Truss

The Fink truss was designed by Albert Fink of Germany in the 1860s. This type of bridge was popular with the Baltimore and Ohio Railroad. The Appomattox High Bridge on the Norfolk and Western Railroad included 21 Fink deck truss spans from 1869 until their replacement in 1886.

4.8 Howe Truss

The relatively rare Howe truss, patented in 1840 by Massachusetts millwright William Howe, includes vertical members and diagonals that slope up towards the center, the opposite of the Pratt truss.[9] In contrast to the Pratt Truss, the diagonal web members are in compression and the vertical web members are in tension. Examples include Jay Bridge in Jay, New York, and Sandy Creek Covered Bridge in Jefferson County, Missouri.

4.9 Kingpost Truss

A king post (or crown post) extends vertically from a crossbeam to the apex of a triangular truss. King posts were used in roof construction in Medieval architecture in buildings. The truss consists of two diagonal members that meet at the apex of the truss, one horizontal beam that serves to tie the bottom end of the diagonals together, and the king post which connects the apex to the horizontal beam below. For a roof truss, the diagonal members are called rafters, and the horizontal member may serve as a ceiling joist. A bridge would require two king post trusses with the driving surface between them. A roof usually uses many side-by-side trusses depending on the size of the structure

[6] 4.10 Lattice Truss

A lattice bridge is a form of truss bridge that uses a large number of small and closely spaced diagonal elements that form a lattice. Originally a design to allow a substantial bridge to be made from planks employing lower–skilled labor, rather than heavy timbers and more expensive carpenters, this type of bridge has also been constructed using a large number of relatively light iron or steel members. The individual elements are more easily handled by the construction workers, but the bridge also requires substantial support during construction. A simple lattice truss will transform the applied loads into a thrust, as the bridge will tend to change length under load. This is resisted by pinning the lattice members to the top and bottom chords, which are more substantial than the lattice members, but which may also be fabricated from relatively small elements rather than large beams.

4.11 Lenticular Truss

A lenticular truss bridge includes a lens-shape truss, with trusses between an upper arch that curves up and then down to end points, and a lower arch that curves down and then up to meet at the same end points. Where the arches extend above and below the roadbed, it is a lenticular pony truss bridge. The Royal Albert Bridge (United Kingdom) uses a single tubular upper chord. As the horizontal tension and compression forces are balanced these horizontal forces are not transferred to the supporting pylons .This in turn enables the truss to be fabricated on the ground and then to be raised by jacking as supporting masonry pylons are constructed. This truss has been used in the construction of a stadium with the upper chords of parallel trusses supporting a roof that may be rolled back.

[7] 4.12 Long Truss

Designed by Stephen H. Long in 1830; one surviving example is the Old Blenheim Bridge. The design resembles a Howe truss, but is entirely made of wood instead of a combination of wood and metal.

4.13 Parker Truss

A Parker truss bridge is a Pratt truss design with a polygonal upper chord. A "camelback" is a subset of the Parker type, where the upper chord consists of exactly five segments. An example of a Parker truss is the Traffic Bridge in Saskatoon.

4.14 Pegram Truss

The Pegram truss is a hybrid between the Warren and Parker trusses where the upper chords are all of equal length and the lower chords are longer than the corresponding upper chord. Because of the difference in upper and lower chord length, each panel was not square. The members which would be vertical in a Parker truss vary from near vertical in the center of the span to diagonal near each end (like a Warren truss). George H. Pegram, while the chief engineer of Edge Moor Iron Company in Wilmington, Delaware, patented this truss design in 1885.The Pegram truss consists of a Parker type design with the vertical posts leaning towards the center at an angle between 60 and 75°. The variable post angle and constant chord length allowed steel in existing bridges to be recycled into a new span using the Pegram truss design. This design also facilitated reassembly and permitted a bridge to be adjusted to fit different span lengths. There are ten remaining Pegram span bridges in the United States with seven in Idaho.

[8]

4.15 Post Truss

A Post truss is a hybrid between a Warren truss and a double-intersection Pratt truss. Invented in 1863 by Simeon S. Post, it is occasionally referred to as a Post patent truss although he never received a patent for it.[14] The Ponakin Bridge and the Bell Ford Bridge are two examples of this truss.

4.16 Pratt truss

A Pratt truss includes vertical members and diagonals that slope down towards the center, the opposite of the Howe truss.It can be subdivided, creating Y- and K-shaped patterns. The Pratt Truss was invented in 1844 by Thomas and Caleb Pratt. This truss is practical for use with spans up to 250 feet and was a common configuration for railroad bridges as truss bridges moved from wood to metal. They are statically determinate bridges, which lend themselves well to long spans.

4.17 Queenpost Truss

The queenpost truss, sometimes queen post or queenspost, is similar to a king post truss in that the outer supports are angled towards the center of the structure. The primary difference is the horizontal extension at the center which relies on beam action to provide mechanical stability. This truss style is only suitable for relatively short spans

4.18 Truss Arch

A truss arch may contain all horizontal forces within the arch itself, or alternatively may be either a thrust arch consisting of a truss, or of two arcuate sections pinned at the apex. The latter form is common when the bridge is constructed as cantilever segments from each side as in the Navajo Bridge

[9] 4.19 Warren Truss

It consists of longitudinal members joined only by angled cross-members, forming alternately inverted equilateral triangle-shaped spaces along its length, ensuring that no individual strut, beam, or tie is subject to bending or torsional straining forces, but only to tension or compression. Loads on the diagonals alternate between compression and tension (approaching the center), with no vertical elements, while elements near the center must support both tension and compression in response to live loads. This configuration combines strength with economy of materials and can therefore be relatively light.

4.20 Whipple Pratt Truss

A whipple truss is usually considered a subclass of the Pratt truss because the diagonal members are designed to work in tension. The main characteristic of a whipple truss is that the tension members are elongated, usually thin, at a shallow angle and cross two or more bays (rectangular sections defined by the vertical members).An example of a Pratt Truss bridge is the Fair Oaks Bridge in Fair Oaks, California.

4.21 Vierendeel truss

The Vierendeel truss, unlike common pin-jointed trusses, imposes significant bending forces upon its members — but this in turn allows the elimination of many diagonal elements. While rare as a bridge type this truss is commonly employed in modern building construction as it allows the resolution of gross shear forces against the frame elements while retaining rectangular openings between column.[4]

[10]

5. Statics of Truss

A truss that is assumed to comprise members that are connected by means of pin joints, and which is supported at both ends by means of hinged joints or rollers, is described as being statically determinate. Newton's Laws apply to the structure as a whole, as well as to each node or joint. In order for any node that may be subject to an external load or force to remain static in space, the following conditions must hold: the sums of all (horizontal and vertical) forces, as well as all moments acting about the node equal zero. Analysis of these conditions at each node yields the magnitude of the forces in each member of the truss. These may be compression or tension forces. Trusses that are supported at more than two positions are said to be statically indeterminate, and the application of Newton's Laws alone is not sufficient to determine the member forces. In order for a truss with pin-connected members to be stable, it must be entirely composed of triangles. In mathematical terms, we have the following necessary condition for stability: m = 2j - 3 where m is the total number of truss members, j is the total number of joints and r is the number of reactions (equal to 3 generally) in a 2-dimensional structure. When m = 2j − 3, the truss is said to be statically determinate, because the (m+3) internal member forces and support reactions can then be completely determined by 2j equilibrium equations. Given a certain number of joints, this is the minimum number of members, in the sense that if any member is taken out (or fails), then the truss as a whole fails. Their member forces depend on the relative stiffness of the members, in addition to the equilibrium condition described.[5]

[11]

6. Truss Analysis

For the truss analysis, it is assumed that: Bars are pin-connected. Joints are frictionless hinges. Loads are applied at the joints only. Stress in each member is constant along its length. The objective of analyzing the trusses is to determine the reactions and member forces. The methods used for carrying out the truss analysis with the equations of equilibrium and by considering only parts of the structure through analyzing its free body diagram to solve the unknowns. There are 3 basic methods for determination of axial forces in members: Method of joints Methods of Sections Graphical Method

[12] 6.1 Method of Joints

The first to analyze a truss by assuming all members are in tension reaction. A tension member is when a member experiences pull forces at both ends of the bar and usually denoted as positive sign. When a member experiencing a push force at both ends, then the bar was said to be in compression mode and designated as negative sign. In the joints method, a virtual cut is made around a joint and the cut portion is isolated as a Free Body Diagram (FBD). Using the equilibrium equations of ∑ Fx = 0 and ∑ Fy = 0, the unknown member forces could be solve. It is assumed that all members are joined together in the form of an ideal pin, and that all forces are in tension (+ve) of reactions. An imaginary section may be completely passed around a joint in the truss. The joint has become a free body in equilibrium under the forces applied to it. The equations ∑ H = 0 and ∑ V = 0 may be applied to the joint to determine the unknown forces in members meeting there. It is evident that no more than two unknowns can be determined at a joint with these two equations. A simple truss model supported by pinned and roller support at its end. Each triangle has the same length, L and it is equilateral where degree of angle, θ is 60° on every angle. The support reactions, Ra and Rc can be determine by taking a point of moment either at point A or point C, whereas Ha = 0 (no other horizontal force). Here are some simple guidelines for this method of truss analysis:

[13]

1. Solve the reactions of the given structure, 2. Select a joint with a minimum number of unknown (not more than 2) and analyze it with ∑ Fx = 0 and ∑ Fy = 0, 3. Proceed to the rest of the joints and again concentrating on joints that have very minimal of unknowns, 4. Check member forces at unused joints with ∑ Fx = 0 and ∑ Fy = 0, 5. Tabulate the member forces whether it is in tension (+ve) or compression (-ve) reaction From fig.21, the forces in each member can be determine from any joint or point. The best way to start by selecting the easiest joint like joint C where the reaction Rc is already obtained and with only 2 unknown, forces of FCB and FCD. Both can be evaluate with ∑ Fx = 0 and ∑ Fy = 0 rules. At joint E, there are 3 unknown, forces of FEA, FEB and FED, which may lead to more complex solution compare to 2 unknown values. For checking purposes, joint B is selected to shown that the equation of ∑ Fx is equal to ∑ Fy which leads to zero value, ∑ Fx = ∑ Fy = 0. Each value of the member’s condition should be indicate clearly as whether it is in tension (+ve) or in compression (-ve) state. Trigonometric Functions: Taking an angle between member x and z…

• • •

Cos θ = x / z Sin θ = y / z Tan θ = y / x

[14]

6.2 Method of Sections

The section method is an effective method when the forces in all members of a truss are being able to determine. Often we need to know the force in just one member with greatest force in it, and the method of section will yield the force in that particular member without the labor of working out the rest of the forces within the truss analysis. If only a few member forces of a truss are needed, the quickest way to find these forces is by the method of sections. In this method, an imaginary cutting line called a section is drawn through a stable and determinate truss. Thus, a section subdivides the truss into two separate parts. Since the entire truss is in equilibrium, any part of it must also be in equilibrium. Either of the two parts of the truss can be considered and the three equations of equilibrium ∑ Fx = 0, ∑ Fy = 0, and ∑ M = 0 can be applied to solve for member forces.Using the same model of simple truss, the details would be the same as previous figure with 2 different supports profile. Unlike the joint method, here we only interested in finding the value of forces for member BC, EC, and ED. Few simple guidelines of section truss analysis: 1. Pass a section through a maximum of 3 members of the truss, 1 of which is the desired member where it is dividing the truss into 2 completely separate parts, 2. At 1 part of the truss, take moments about the point (at a joint) where the 2 members intersect and solve for the member force, using ∑ M = 0, 3. Solve the other 2 unknowns by using the equilibrium equation for forces, using ∑ Fx = 0 and ∑ Fy = 0.

[15]

In fig.23,a virtual cut is introduce through the only required members which is along member BC, EC, and ED. Firstly, the support reactions of Ra and Rd should be determine. Again a good judgment is require to solve this problem where the easiest part would be consider either on the left hand side or the right hand side. Taking moment at joint E (virtual pint) on clockwise for the whole RHS part would be much easier compare to joint C (the LHS part). Then, either joint D or C can be consider as point of moment, or else using the joint method to find the member forces for FCB, FCE, and FDE. Note: Each value of the member’s condition should be indicate clearly as whether it is in tension (+ve) or in compression (-ve) state.

[16]

6.3 Graphical Method

The method of joints could be used as the basic for a graphical analysis of trusses. The graphical analysis was developed by force polygons drawn to scale for each joint, and then the forces in each member were measured from one of these force polygons. The number of lines which have to be drawn can be greatly reduced, however, if the various force polygons are superimposed. The resulting diagram of truss analysis is known as the Maxwell’s Diagram. In order to draw the Maxwell diagram directly, here are the simple guidelines: 1. Solve the reactions at the supports by solving the equations of equilibrium for the entire truss, 2. Move clockwise around the outside of the truss; draw the force polygon to scale for the entire truss, 3. Take each joint in turn (one-by-one), then draw a force polygon by treating successively joints acted upon by only two unknown forces, 4. Measure the magnitude of the force in each member from the diagram, 5. Lastly, note that work proceed from one end of the truss to another, as this use for checking of balance and connect to other end. In conclusion, the truss internal reaction as well as its member forces could be determine by either of this 3 methods especially in mechanics of structures.[6]

[17]

7. Design of Members

A truss can be thought of as a beam where the web consists of a series of separate members instead of a continuous plate. In the truss, the lower horizontal member (the bottom chord) and the upper horizontal member (the top chord) carry tension and compression, fulfilling the same function as the flanges of an I-beam. Which chord carries tension and which carries compression depends on the overall direction of bending. The diagonal and vertical members form the truss web, and carry the shear force. Individually, they are also in tension and compression, the exact arrangement of forces is depending on the type of truss and again on the direction of bending. In addition to carrying the static forces, the members serve additional functions of stabilizing each other, preventing bucklingThe inclusion of the elements shown is largely an engineering decision based upon economics, being a balance between the costs of raw materials, off-site fabrication, component transportation, on-site erection, the availability of machinery and the cost of labor. In other cases the appearance of the structure may take on greater importance and so influence the design decisions beyond mere matters of economics. Modern materials such as prestressed concrete and fabrication methods, such as automated welding, have significantly influenced the design of modern bridges.

[18]

8. Design of Joints

After determining the minimum cross section of the members, the last step in the design of a truss would be detailing of the bolted joints, e.g., involving shear of the bolt connections used in the joints, see also shear stress. Based of the needs of the project, truss internal connections (joints) can be designed as rigid, semi rigid, or hinged. Rigid connections can allow transfer of bending moments leading to development of secondary bending moments in the members.

9.Applications

Component connections are critical to the structural integrity of a framing system. In buildings with large, clearspan wood trusses, the most critical connections are those between the truss and its supports. In addition to gravity-induced forces , these connections must resist shear forces acting perpendicular to the plane of the truss and uplift forces due to wind. Depending upon overall building design, the connections may also be required to transfer bending moment. Wood posts enable the fabrication of strong, direct, yet inexpensive connections between large trusses and walls. Exact details for post-to-truss connections vary from designer to designer. Solid-sawn timber and glulam posts are generally notched to form a truss bearing surface. The truss is rested on the notches and bolted into place. A special plate/bracket may be added to increase connection load transfer capabilities.[7]

[19]

Index

Topics

1. Introduction 2. Characteristics of a Truss 2.1 Planar Truss 2.2 Space Frame Truss 3. Truss Types 4. Truss Types used in Bridges 4.1 Allan Bridge 4.2 Bailey Bridge 4.3 Baltimore Truss 4.4 Bollman Truss 4.5 Bowstring Arch Truss 4.6 Cantilevered Truss 4.7 Flink Truss 4.8 Howe Truss 4.9 Kingpost Truss

Pg. No.

1 2 2 2 3 4 4 4 4 5 5 5 6 6 6

4.10 Lattice Truss 4.11 Lenticular Truss 4.12 Long Truss 4.13 Parker Truss 4.14 Pegram Truss 4.15 Post Truss 4.16 Pratt Truss 4.17 Queenpost Truss 4.18 Truss Arch 4.19 Warren Truss 4.20 Whipple Pratt Truss 4.21 Vierendeel Truss 5. Statics of truss 6. Truss Analysis 6.1 Method of Joints 6.2 Method of Sections 6.3 Graphical Method

7 7 8 8 8 9 9 9 9 10 10 10 11 12 13 15 17

7. Design of Members 8. Design of Joints 9. Applications

18 19 19

Bibliography

1. http://en.wikipedia.org/wiki/Truss 2. http://www.civilcraftstructures.com/civil-subjects/3-methods-fortruss-analysis/ 3. http://en.wikipedia.org/wiki/Truss#Truss_types 4. http://en.wikipedia.org/wiki/Truss_bridge#Truss_types_used_in_bri dges 5. http://en.wikipedia.org/wiki/Truss#Statics_of_trusses 6. http://www.civilcraftstructures.com/civil-subjects/3-methods-fortruss-analysis/ 7. http://en.wikipedia.org/wiki/Truss#Applications

Pictures Reference

Fig.1 Railway Track http://en.wikipedia.org/wiki/File:RRTrussBridgeSideView.jpg Fig. 2 Planar Truss http://en.wikipedia.org/wiki/File:Trusses_008.jpg Fig.3 Space Frame Truss http://en.wikipedia.org/wiki/File:SpaceFrame02.png Fig.4 Allan Truss http://en.wikipedia.org/wiki/File:Hampden_Bridge_Wagga_design.jpg Fig.5 Bollman Bridge http://en.wikipedia.org/wiki/File:Bollman-bridge-1.jpg Fig.6 Bowstring Arch Truss http://en.wikipedia.org/wiki/File:FortPittBridge.jpg Fig.7 Cantilevered Truss http://en.wikipedia.org/wiki/File:CooperRiverBridge.svg Fig.8 Fink Truss http://en.wikipedia.org/wiki/File:Bridges_20.png Fig.9 Howe Truss http://en.wikipedia.org/wiki/File:Howe_truss.PNG Fig.10 Kingpost Truss http://en.wikipedia.org/wiki/File:King_post_truss.png Fig.11 Lattice Truss http://en.wikipedia.org/wiki/File:Lattice_truss.png Fig. 12 Long Truss http://en.wikipedia.org/wiki/File:Long_Truss_Bushing_CB_(Versailles_SP)_00003 r.jpg Fig.14 Post Truss http://en.wikipedia.org/wiki/File:Post_truss.svg

Fig.15 Pratt Truss http://en.wikipedia.org/wiki/File:Pratt_truss.PNG Fig.16 Queenpost Truss http://en.wikipedia.org/wiki/File:Queen_post_truss.png Fig.17 Warren Bridge http://en.wikipedia.org/wiki/File:Warren_truss.PNG Fig.18 Vierendeel Truss http://en.wikipedia.org/wiki/File:Vierendeel_truss.png Fig.19 Statics of Truss http://www.civilcraftstructures.com/civil-subjects/3methods-for-truss-analysis/ Fig.20 Method of Joints http://www.civilcraftstructures.com/civil-subjects/3methods-for-truss-analysis/ Fig.21 Joint B,C and E http://www.civilcraftstructures.com/civil-subjects/3methods-for-truss-analysis/ Fig.22 Method of Sections http://www.civilcraftstructures.com/civil-subjects/3methods-for-truss-analysis/ Fig.23 Cutting Section http://www.civilcraftstructures.com/civil-subjects/3methods-for-truss-analysis/ Fig.24 Graphical Method http://www.civilcraftstructures.com/civil-subjects/3methods-for-truss-analysis/

Acknowledgement

We are greatly indebted to Mr. Sukhvinder Singh, our worthy and highly respected Mechanics Teacher who inspired us to undertake this project and provided us with very valuable guidance in the preparation of the report. We express our deep and sincere gratitude to him Abhishek Sharma (12811502810) Ashish Kumar Singh (12911502810) Gaurav Nigam (13011502810) Mridul Malik (13111502810)

Bharti Vidyapeeth’s College of Engineering

A-4 , Paschim Vihar, New Delhi-110063

CERTIFICATE

This is to certify that the reportt entitled “Truss” submitted by ,Abhishek Sharma,Ashish Kumar Singh,Gaurav Nigam and Mridul Malik, of branch ECE-2 is a record of the project was carried by them. They have worked under my guidance and supervision and has fulfilled the requirements for the submission of the report, which to my knowledge has reached a requisite standard.

Head of Department

Date

Professor Incharge