Types of Supports for Plane Structures

Supports are used to attach structures to the ground or other bodies, thereby restricting their movements under the action of applied loads. The loads tend to move the structures; but supports prevent the movements by exerting opposing forces or reactions to neutralize the effects of loads, thereby keeping the structures in equilibrium. The type of reaction a support exerts on a structure depends on the type of supporting device used and the type of movement it prevents. A support that prevents translation of the structure in a particular direction exerts a reaction force on the structure in that direction. Similarly, a support that prevents rotation of the structure about a particular axis exerts a reaction couple on the structure about that axis. The type of supports commonly used for plane structures are given below.

1) Fixed Support

A fixed support is the most rigid type of support or connection. It constrains the member in all translations and rotations (i.e. it cannot move or rotate in any direction). The easiest example of a fixed support would be a pole or column in concrete. The pole cannot twist, rotate or displace; it is basically restricted in all its movements at this connection. Fixed supports are extremely beneficial when you can only use a single support. The fixed support provides all the constraints necessary to ensure the structure is static. It is most widely used as the only support for a cantilever. The fixed support is also called rigid support. It provides greater stability to the structure as compared with all other supports. To provide good stability to the structure, at least one rigid support should be provided. Beam fixed in the wall is a good example of fixed support.

Fig. 1 Fixed Support

2) Pinned Support (Hinged Support)

The hinged support is also called pinned support. A pinned support is a very common type of support and is most commonly compared to a hinge in civil engineering. Like a hinge, a pinned support allows rotation to occur but no translation (i.e. it resists horizontal and vertical forces but not a moment). The horizontal and vertical components of the reaction can be determined using the equation of equilibrium. Pinned supports can be used in trusses. By linking multiple members joined by hinge connections, the members will push against each other; inducing an axial force within the member. The benefit of this is that the members contain no internal moment forces and can be designed according to their axial force only. Hinge support reduces sensitivity to an earthquake.

Fig. 2 Pinned Support

3) Roller Support

It is a support that is free to rotate and translate along the surface on which they rest. The surface on which the roller supports are installed may be horizontal, vertical and inclined to any angle. Roller supports can resist a vertical force but not a horizontal force. The roller supports has only one reaction, this reaction acts perpendicular to the surface and away from it. A roller support or connection is free to move horizontally as there is nothing constraining it. The most common use of a roller support is in a bridge. In civil engineering, a bridge will typically contain a roller support at one end to account for vertical displacement and expansion from changes in temperature. This is required to prevent the expansion causing damage to a pinned support. This type of support does not resist any horizontal forces. This obviously has limitations in itself as it means the structure will require another support to resist this type of force.

Fig. 3 Roller Support

4) Simple Support

A simple support is basically just where the member rests on an external structure. They are quite similar to roller supports in the sense that they are able to restrain vertical forces but not horizontal forces. The member simply rests on an external structure to which the force is transferred to. An example is a plank of wood resting on two concrete blocks. The plank can support any downward (vertical) force but if you apply a horizontal force, the plank will simply slide off the concrete blocks.

Fig. 4 Simple Support

5) Rocker Support

Rocker support is similar to roller support. It also resists vertical force and allows horizontal translation and rotation. But in this case, horizontal movement is due to the curved surface provided at the bottom as shown in Fig. 5. So, the amount of horizontal movement is limited in this case.

Fig. 5 Rocker Support

6) Link Support

A link has two hinges, one at each end. The link is supported and allows rotation and translation perpendicular to the direction of the link only. It does not allow translation in the direction of the link. It has a single linear resultant force component in the direction of the link which can be resolved into vertical and horizontal components. In other words, the reaction force of a link is in the direction of the link, along its longitudinal axis.

Fig. 6 Link Support

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Equilibrium of Structures

A structure is considered to be in equilibrium if, initially at rest, it remains at rest when subjected to a system of forces and couples. If a structure is in equilibrium, then all its members and parts are also in equilibrium. In order for a structure to be in equilibrium, all the forces and couples (including support reactions) acting on it must balance each other and there must neither be a resultant force nor a resultant couple acting on the structure. For a space (three-dimensional) structure subjected to three-dimensional systems of forces and couples (Fig. 1), the conditions of zero resultant force and zero resultant couple can be expressed in a Cartesian coordinate system (xyz) as

     Σ Fx = 0, Σ Fy = 0, Σ Fz = 0

     Σ Mx = 0, Σ My = 0, Σ Mz = 0

These six equations are called the equations of equilibrium of space structures and are the necessary and sufficient conditions for equilibrium. The first three equations ensure that there is no resultant force acting on the structure and the last three equations express the fact that there is no resultant couple acting on the structure.

Fig. 1 Space Structure Subjected to Three-Dimensional Systems of Forces and Couples

Fig. 2 Plane Structure Subjected to a Coplanar System of Forces and Couples

For a plane structure lying in the 𝑥y plane and subjected to a coplanar system of forces and couples (Fig. 2), the necessary and sufficient conditions for equilibrium can be expressed as

      Σ Fx = 0, Σ Fy = 0, Σ Mz = 0

These three equations are referred to as the equations of equilibrium of plane structures. The first two of the three equilibrium equations express that the algebraic sums of the 𝑥 components and y components of all the forces are zero, thereby indicating that the resultant force acting on the structure is zero. The third equation indicates that the algebraic sum of the moments of all the forces about any point in the plane of the structure and the moments of any couples acting on the structure is zero, thereby indicating that the resultant couple acting on the structure is zero. All the equilibrium equations must be satisfied simultaneously for the structure to be in equilibrium.

It should be realized that if a structure (e.g., an aerospace vehicle) initially in motion is subjected to forces that satisfy the equilibrium equations, it will maintain its motion with a constant velocity, since the forces cannot accelerate it. Such structures may also be considered to be in equilibrium. However, the term equilibrium is commonly used to refer to the state of rest of structures and is used in this context herein.

Concurrent Force Systems

When a structure is in equilibrium under the action of a concurrent force system; that is, the lines of action of all the forces intersect at a single point - the moment equilibrium equations are automatically satisfied and only the force equilibrium equations need to be considered. Therefore, for a space structure subjected to a concurrent three-dimensional force system, the equations of equilibrium are

      Σ Fx = 0, Σ Fy = 0, Σ Fz = 0

Similarly, for a plane structure subjected to a concurrent coplanar force system, the equilibrium equations can be expressed as

      Σ Fx = 0, Σ Fy = 0

• If a structure is in equilibrium under the action of only two forces, the forces must be equal, opposite and collinear.
• If a structure is in equilibrium under the action of only three forces, the forces must be either concurrent or parallel.

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Analytical Models

An analytical model is a simplified representation or an ideal of a real structure for the purpose of analysis. The objective of the model is to simplify the analysis of a complicated structure. The analytical model represents the behavioural characteristics of the structure of interest to the analyst, while discarding much of the detail about the members, connections and so on, that is expected to have little effect on the desired characteristics. Establishment of the analytical model is one of the most important steps of the analysis process; it requires experience and knowledge of design practices in addition to a thorough understanding of the behaviour of structures. Remember that the structural response predicted from the analysis of the model is valid only to the extent that the model represents the actual structure. Development of the analytical model generally involves consideration of the following factors.

Plane and Space Structure

If all the members of a structure as well as the applied loads lie in a single plane, the structure is called a plane structure. The analysis of plane or two-dimensional structures is considerably simpler than the analysis of space or three-dimensional structures. Many actual three-dimensional structures can be subdivided into plane structures for analysis.

As an example, consider the framing system of a bridge shown in Fig. 1. The main members of the system designed to support vertical loads are shown by solid lines whereas the secondary bracing members which is necessary to resist lateral wind loads and to provide stability are represented by dashed lines. The deck of the bridge rests on beams called stringers; these beams are supported by floor beams, which in turn are connected at their ends to the joints on the bottom panels of the two longitudinal trusses. Thus the weight of the traffic deck, stringers and floor beams is transmitted by the floor beams to the supporting trusses at their joints; the trusses, in turn transmit the load to the foundation. Because this applied loading acts on each truss in its own plane the trusses can be treated as plane structures.

Fig. 1 Framing System of Bridge

As another example, the framing system of a multi-story building is shown in Fig. 2. At each story, the floor slab rests on floor beams which transfer any load applied to the floor, the weight of the slab and their own weight to the girders of the supporting rigid frames. This applied loading acts on each frame in its own plane, so each frame can be analyzed as a plane structure. The loads thus transferred to each frame are further transmitted from the girders to the columns and then finally to the foundation.

Fig. 2 Framing System of Multi-storeyed Building

Although a great majority of actual three-dimensional structural systems can be subdivided into plane structures for the purpose of analysis, some structures such as latticed domes, aerospace structures and transmission towers cannot be subdivided into planar components due to their shape, arrangement of members or applied loading. Such structures, called space structures are analyzed as three-dimensional bodies subjected to three-dimensional force systems.

Line Diagram

The analytical model of the two or three-dimensional body selected for analysis is represented by a line diagram. On this diagram, each member of the structure is represented by a line coinciding with its centroidal axis. The dimensions of the members and the size of the connections are not shown on the diagram. The line diagrams of the bridge truss of Fig. 1 and the rigid frame of Fig. 2 are shown in Fig. 4 and 5 respectively.

Fig. 3 Line Diagram of Bridge and Connection

Fig. 4 Line Diagram of Multi-storeyed Building

Connections

Two types of connections are commonly used to join members of structures: rigid connections and flexible or hinged connections. A rigid connection or joint prevents relative translations and rotations of the member ends connected to it; that is, all member ends connected to a rigid joint have the same translation and rotation. In other words, the original angles between the members intersecting at a rigid joint are maintained after the structure has deformed under the action of loads. Such joints are capable of transmitting forces as well as moments between the connected members. Rigid joints are usually represented by points at the intersections of members on the line diagram of the structure, as shown in Fig. 3.

A hinged connection or joint prevents only relative translations of member ends connected to it; that is, all member ends connected to a hinged joint have the same translation but may have different rotations. Such joints are thus capable of transmitting forces but not moments between the connected members. Hinged joints are usually depicted by small circles at the intersections of members on the line diagram of the structure.

The perfectly rigid connections and the perfectly flexible frictionless hinges used in the analysis are merely idealizations of the actual connections, which are seldom perfectly rigid or perfectly flexible. However, actual bolted or welded connections are purposely designed to behave like the idealized cases. For example, the connections of trusses are designed with the centroidal axes of the members concurrent at a point to avoid eccentricities that may cause bending of members. For such cases, the analysis based on the idealized connections and supports generally yields satisfactory results.

Supports

Supports for plane structures are commonly idealized as either fixed supports- which do not allow any movement; hinged supports - which can prevent translation but permit rotation; roller or link - supports which can prevent translation in only one direction.

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Classification of Structures

The most important decision made by a structural engineer in implementing an engineering project is the selection of the type of structure to be used for supporting or transmitting loads. Commonly used structures can be classified into five basic categories, depending on the type of primary stresses that may develop in their members under major design loads. It should be realized that any two or more of the basic structural types described in the following may be combined in a single structure, such as a building or a bridge, to meet the structure’s functional requirements. The different type of structures is given below.

1) Tension Structures

The members of tension structures are subjected to pure tension under the action of external loads. Because the tensile stress is distributed uniformly over the cross-sectional areas of members, the material of such a structure is utilized in the most efficient manner. Tension structures composed of flexible steel cables are frequently employed to support bridges and long- span roofs. Because of their flexibility, cables have negligible bending stiffness and can develop only tension. Thus, under external loads, a cable adopts a shape that enables it to support the load by tensile forces alone. In other words, the shape of a cable changes as the loads acting on it change. As an example, the shapes that a single cable may assume under two different loading conditions are shown in Fig. 1.

Fig. 1

Fig. 2 shows the cable structure of the suspension bridge. In a suspension bridge, the roadway is suspended from two main cables by means of vertical hangers. The main cables pass over a pair of towers and are anchored into solid rock or a concrete foundation at their ends. Because suspension bridges and other cable structures lack stiffness in lateral directions, they are susceptible to wind-induced oscillations. Bracing or stiffening systems are therefore provided to reduce such oscillations. Besides cable structures, other examples of tension structures include vertical rods used as hangers (for example, to support balconies or tanks) and membrane structures such as tents and roofs of large-span domes.

Fig. 2 Suspension Bridge

2) Compression Structures

Compression structures develop mainly compressive stresses under the action of external loads. Two common examples of such structures are columns and arches (Fig. 3). Columns are straight members subjected to axially compressive loads, as shown in Fig. 3. When a straight member is subjected to lateral loads and/or moments in addition to axial loads, it is called a beam-column.

Fig. 3 Column

An arch is a curved structure, with a shape similar to that of an inverted cable, as shown in Fig. 4. Such structures are frequently used to support bridges and long-span roofs. Arches develop mainly compressive stresses when subjected to loads and are usually designed so that they will develop only compression under a major design loading. However, because arches are rigid and cannot change their shapes as can cable, other loading conditions usually produce secondary bending and shear stresses in these structures, which should be considered in their designs. Because compression structures are susceptible to buckling or instability, the possibility of such a failure should be considered in their designs; if necessary, adequate bracing must be provided to avoid such failures.

Fig. 4 Arch

3) Trusses

Trusses are composed of straight members connected at their ends by hinged connections to form a stable configuration (Fig. 5). When the loads are applied to a truss only at the joints, its members either elongate or shorten. Thus, the members of an ideal truss are always either in uniform tension or in uniform compression. Real trusses are usually constructed by connecting members to gusset plates by bolted or welded connections. Although the rigid joints thus formed cause some bending in the members of a truss when it is loaded, in most cases such secondary bending stresses are small and the assumption of hinged joints yields satisfactory designs.

Because of their light weight and high strength, trusses are the most commonly used types of structures. Such structures are used in a variety of applications, ranging from supporting roofs of buildings to serving as support structures in space stations and sports arenas.

Fig. 5 Plane Truss

4) Shear Structures

Shear structures, such as reinforced concrete shear walls (Fig. 6), are used in multistory buildings to reduce lateral movements due to wind loads and earthquake excitations. Shear structures develop mainly in-plane shear, with relatively small bending stresses under the action of external loads.

Fig. 6 Shear Wall

5) Bending Structures

Bending structures develop mainly bending stresses under the action of external loads. In some structures, the shear stresses associated with the changes in bending moments may also be significant and should be considered in their designs. Some of the most commonly used structures such as beams, rigid frames, slabs and plates, can be classified as bending structures. A beam is a straight member that is loaded perpendicular to its longitudinal a𝑥is. The bending (normal) stress varies linearly over the depth of a beam from the maximum compressive stress at the fiber farthest from the neutral axis on the concave side of the bent beam to the maximum tensile stress at the outermost fiber on the convex side.

For example, in the case of a horizontal beam subjected to a vertically downward load, as shown in Fig. 7, the bending stress varies from the maximum compressive stress at the top edge to the maximum tensile stress at the bottom edge of the beam. To utilize the material of a beam cross section most efficiently under this varying stress distribution, the cross sections of beams are often I-shaped, with most of the material in the top and bottom flanges. The I-shaped cross sections are most effective in resisting bending moments.

Fig. 7 Beam

Rigid frames are composed of straight members connected together either by rigid (moment-resisting) connections or by hinged connections to form stable configurations. Unlike trusses, which are subjected only to joint loads, the external loads on frames may be applied on the members as well as on the joints. The members of a rigid frame are subjected to bending moment, shear and axial compression or tension under the action of external loads. The design of horizontal members or beams of rectangular frames is often governed by bending and shear stresses only, since the axial forces in such members are usually small.

Frames, like trusses, are among the most commonly used types of structures. Structural steel and reinforced concrete frames are commonly used in multistory buildings, bridges and industrial plants. Frames are also used as supporting structures in airplanes, ships, aerospace vehicles and other aerospace and mechanical applications. The generic term framed structure is frequently used to refer to any structure composed of straight members, including a truss.

Fig. 8 Rigid Frame

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Phases of Structural Engineering Projects

Structural engineering is the science and art of planning, designing and constructing safe and economical structures that will serve their intended purposes. Structural analysis is an integral part of any structural engineering project, its function being the prediction of the performance of the proposed structure. A flowchart showing the various phases of a typical structural engineering project is presented in Fig. 1.

Fig. 1 Phases of a Typical Structural Engineering Project

The process is an iterative one, and it generally consists of the following steps.

1) Planning Phase

The planning phase usually involves the establishment of the functional requirements of the proposed structure, the general layout and dimensions of the structure, consideration of the possible types of structures (e.g. rigid frame or truss) that may be feasible and the types of materials to be used (e.g., structural steel or reinforced concrete). This phase may also involve consideration of non-structural factors, such as aesthetics, environmental impact of the structure etc.

The outcome of this phase is usually a structural system that meets the functional requirements and is expected to be the most economical. This phase is perhaps the most crucial one of the entire project and requires experience and knowledge of construction practices in addition to a thorough understanding of the behavior of structures.

2) Preliminary Structural Design

In the preliminary structural design phase, the sizes of the various members of the structural system selected in the planning phase are estimated based on approximate analysis, past experience and code requirements. The member sizes thus selected are used in the next phase to estimate the weight of the structure.

3) Estimation of Loads

Estimation of loads involves determination of all the loads that can be expected to act on the structure.

4) Structural Analysis

In structural analysis, the values of the loads are used to carry out an analysis of the structure in order to determine the stresses or stress resultants in the members and the deflections at various points of the structure.

5) Safety and Serviceability Checks

The results of the analysis are used to determine whether or not the structure satisfies the safety and serviceability requirements of the design codes. If these requirements are satisfied, then the design drawings and the construction specifications are prepared, and the construction phase begins.

6) Revised Structural Design

If the code requirements are not satisfied, then the member sizes are revised and phases 3 through are repeated until all the safety and serviceability requirements are satisfied.

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Type of Loads on Structures

The objective of a structural engineer is to design a structure that will be able to withstand all the loads to which it is subjected while serving its intended purpose throughout its intended life span. In designing a structure, an engineer must consider all the loads that can realistically be expected to act on the structure during its planned life span. The loads that act on civil engineering structures can be grouped according to their nature and source into three classes: (1) dead loads due to the weight of the structural system itself and any other material permanently attached to it; (2) live loads, which are movable or moving loads due to the use of the structure; and (3) environmental loads, which are caused by environmental effects, such as wind, snow, and earthquakes.

In addition to estimating the magnitudes of the design loads, an engineer must also consider the possibility that some of these loads might act simultaneously on the structure. The structure is finally designed so that it will be able to withstand the most unfavorable combination of loads that is likely to occur in its lifetime.

1) Dead Loads

Dead loads are gravity loads of constant magnitudes and fixed positions that act permanently on the structure. Such loads consist of the weights of the structural system itself and of all other material and equipment permanently attached to the structural system. For example, the dead loads for a building structure include the weights of frames, framing and bracing systems, floors, roofs, ceilings, walls, stairways, heating and air conditioning systems, plumbing, electrical systems etc. The calculation of dead loads of each structure are calculated by the volume of each section and multiplied with the unit weight.

The weight of the structure is not known in advance of design and is usually assumed based on past experience. After the structure has been analyzed and the member sizes determined, the actual weight is computed by using the member sizes and the unit weights of materials. The actual weight is then compared to the assumed weight and the design is revised if necessary. The unit weights of some common construction materials are given in Table 1. The weights of permanent service equipment, such as heating and air-conditioning systems, are usually obtained from the manufacturer.

Table 1: Unit Weights of Construction Materials

Sl. No.	Material	Unit Weight (kN/m^3)
1	Aluminum	25.9
2	Brick	18.8
3	Plain Cement Concrete	24
4	Reinforced Cement Concrete	25
5	Structural Steel	77.0
6	Wood	6.3

2) Live Loads/ Imposed loads 

Live loads are loads of varying magnitudes and positions caused by the use of the structure. Live loads are either movable or moving loads without any acceleration or impact. These loads are assumed to be produced by the intended use or occupancy of the building including weights of movable partitions or furniture etc. Live loads keep on changing from time to time. Sometimes, the term live loads are used to refer to all loads on the structure that are not dead loads, including environmental loads, such as snow loads or wind loads. However, since the probabilities of occurrence for environmental loads are different from those due to the use of structures, the current codes use the term live loads to refer only to those variable loads caused by the use of the structure. 

The magnitudes of design live loads are usually specified in building codes. In India, the minimum values of live loads to be assumed are given in IS 875 (part 2)–1987. It depends upon the intended use of the building. The position of a live load may change, so each member of the structure must be designed for the position of the load that causes the maximum stress in that member. Different members of a structure may reach their maximum stress levels at different positions of the given load. For example, as a truck moves across a truss bridge, the stresses in the truss members vary as the position of the truck changes.

Table 2: Minimum Floor Live Loads for Buildings

Sl. No.	Occupancy or Use	Live Load (kPa)
1	Hospital patient rooms, residential dwellings, apartments, hotel guest rooms, school classrooms	1.92
2	Library reading rooms, hospital operating roomsand laboratories	2.87
3	Dance halls and ballrooms, restaurants, gymnasiums	4.79
4	Light manufacturing, light storage warehouses,wholesale stores	6.00
5	Heavy manufacturing, heavy storage warehouses	11.97

Buildings Subjected to Environmental Loads

Because of the inherent uncertainty involved in predicting environmental loads that may act on a structure during its lifetime, the consequences of the failure of the structure are usually considered in estimating design environmental loads, such as due to wind, snow and earthquakes. In general, the more serious the potential consequences of the structural failure, the larger the magnitude of the load for which the structure should be designed.

a) Wind Loads 

Wind loads are produced by the flow of wind around the structure. Wind load is primarily horizontal load caused by the movement of air relative to earth. The magnitudes of wind loads that may act on a structure depend on the geographical location of the structure, obstructions in its surrounding terrain, such as nearby buildings, and the geometry and the vibration characteristics of the structure itself. Although the procedures described in the various codes for the estimation of wind loads usually vary in detail, most of them are based on the same basic relationship between the wind speed ‘V’ and the dynamic pressure ‘q’ induced on a flat surface normal to the wind flow, which can be obtained by applying Bernoulli’s principle and is expressed as

q = 1/ 2 rV^2

where ‘r’ is the mass density of the air 

Wind load is required to be considered in structural design especially when the height of the building exceeds two times the dimensions transverse to the exposed wind surface. For low rise building up to four to five stories, the wind load is not critical because the moment of resistance provided by the continuity of floor system to column connection and walls provided between columns are sufficient to accommodate the effect of these forces. The horizontal forces exerted by the components of winds are to be kept in mind while designing the building. The calculation of wind loads depends on the two factors, namely velocity of wind and size of the building. In India, calculation of wind load on structures is given below by the IS-875 (Part 3) -1987. Using colour code, basic wind pressure ‘Vb’ is shown in a map of India. Designer can pick up the value of Vb depending upon the locality of the building. To get the design wind velocity Vz the following expression shall be used: 

Vz = k1.k2.k3.Vb 

Where 

k1 = Risk coefficient 

k2 = Coefficient based on terrain, height and structure size

k3 = Topography factor 

The design wind pressure is given by

pz = 0.6 V^2 * z 

where pz is in N/m^2 at height Z and Vz is in m/sec. 

Up to a height of 30 m, the wind pressure is considered to act uniformly. Above 30 m height, the wind pressure increases.

Table 3: Risk Categories of Buildings for Environmental Loads

Risk category	Occupancy or Use	Importance Factor
		Snow Loads (Is)	Earthquake Loads (Ie)
I	Buildings representing low risk to human life in the case of failure, such as agricultural and minor storage facilities.	0.8	1.0
II	All buildings other than those listed in Risk Categories I, III, and IV. This risk category applies to most of the residential, commercial and industrial buildings (except those which have been specifically assigned to another category).	1.0	1.0
III	Buildings whose failure would pose a substantial risk to human life, and/or could cause a substantial economic impact or mass disruption of everyday public life. This category contains buildings such as: theaters, lecture and assembly halls where a large number of people congregate in one area; elementary schools; small hospitals; prisons; power generating stations; water and sewage treatment plants; telecommunication centers; and buildings containing hazardous and explosive materials.	1.1	1.25
IV	Essential facilities, including hospitals, fire and police stations, national defense facilities and emergency shelters, communication centers, power stations and utilities required in an emergency, and buildings containing extremely hazardous materials.	1.2	1.5

b) Snow Loads

In many parts of the world, snow loads must be considered in designing structures. The design snow load for a structure is based on the ground snow load for its geographical location, which can be obtained from building codes or meteorological data for that region. Once the ground snow load has been established, the design snow load for the roof of the structure is determined by considering such factors as the structure’s exposure to wind and its thermal, geometric, and functional characteristics. In most cases, there is less snow on roofs than on the ground. In India, the code IS 875 (Part-4):1987 deals with snow loads on roofs of the building. 

c) Earthquake Loads

An earthquake is a sudden undulation of a portion of the earth’s surface. Although the ground surface moves in both horizontal and vertical directions during an earthquake, the magnitude of the vertical component of ground motion is usually small and does not have a significant effect on most structures. It is the horizontal component of ground motion that causes structural damage and that must be considered in designs of structures located in earthquake-prone areas. 

Fig.1 Deformation during an earthquake

During an earthquake, as the foundation of the structure moves with the ground, the above-ground portion of the structure, because of the inertia of its mass, resists the motion, thereby causing the structure to vibrate in the horizontal direction. These vibrations produce horizontal shear forces in the structure. For an accurate prediction of the stresses that may develop in a structure, in the case of an earthquake, a dynamic analysis, considering the mass and stiffness characteristics of the structure, must be performed. The response of the structure to the ground vibration is a function of the nature of foundation soil, size and mode of construction and the duration and intensity of ground motion. In India, IS 1893– 2014 gives the details of calculations for structures standing on soils which will not considerably settle or slide appreciably due to earthquake.

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Introduction to Structural Analysis

A structure comprises several components that are connected to one another and function to transfer the loads to the soil successfully. The structure is a collection of elements linked together in such a way that serves a meaningful purpose. Thus, a structure is an arrangement and organization of interrelated elements in an object or system, with the load affecting structural components vertically or laterally. Different types of structures are concrete, framed, shell, membrane, truss, cables and arches, surface structure, etc. 

In other terms “structure” refers to a building or any other artificial object designed to support a load in construction. Structures can be made of various materials such as wood, steel, concrete or brick and can range from simple structures such as a shed or fences to complex structures such as bridges, skyscrapers or dams. 

Structural engineering is the field of engineering that deals with the design, analysis and construction of structures. A properly designed and constructed structure must resist various forces and loads such as gravity, wind, earthquakes and other external factors to ensure the safety of the structure.

Structural analysis is the prediction of the performance of a given structure under prescribed loads and/or other external effects, such as support movements and temperature changes. The performance characteristics commonly used in the design of structures are

1. Stress or stress resultant, such as axial force, shear force and bending moment
2. Deflection
3. Support reaction

Thus, the analysis of a structure usually involves determination of these quantities as caused by a given loading condition. 

History of Structural Engineering

Since the dawn of history, structural engineering has been an essential part of human endeavor. However, it was not until about the middle of the seventeenth century that, engineers began applying the knowledge of mechanics (mathematics and science) in designing structures. Earlier engineering structures were designed by trial and error and by using rules of thumb based on past experience. The fact that some of the magnificent structures from earlier eras, such as Egyptian pyramids (about 3000 BC), Greek temples (500–200 BC), Roman coliseums and aqueducts (200 BC–AD 200), and Gothic cathedrals (AD 1000–1500), still stand today is a testimonial to the ingenuity of their builders.

Galileo Galilei (1564–1642) is generally considered to be the originator of the theory of structures. In his book entitled ‘Two New Sciences’, which was published in 1638, Galileo analyzed the failure of some simple structures, including cantilever beams. Although Galileo’s predictions of strengths of beams were only approximate, his work laid the foundation for future developments in the theory of structures and ushered in a new era of structural engineering, in which the analytical principles of mechanics and strength of materials would have a major influence on the design of structures.

Following Galileo’s pioneering work, the knowledge of structural mechanics advanced at a rapid pace in the second half of the seventeenth century and into the eighteenth century. Among the notable investigators of that period were Robert Hooke (1635–1703), who developed the law of linear relationships between the force and deformation of materials (Hooke’s law); Sir Isaac Newton (1642–1727), who formulated the laws of motion and developed calculus; John Bernoulli (1667–1748), who formulated the principle of virtual work; Leonhard Euler (1707–1783), who developed the theory of buckling of columns; and C. A. de Coulomb (1736–1806), who presented the analysis of bending of elastic beams. In 1826 L. M. Navier (1785–1836) published a treatise on elastic behavior of structures, which is considered to be the first textbook on the modern theory of strength of materials.

The development of structural mechanics continued at a tremendous pace throughout the rest of the nineteenth century and into the first half of the twentieth, when most of the classical methods for the analysis of structures described in this text were developed. The important contributors of this period included B. P. Clapeyron (1799–1864), who formulated the three-moment equation for the analysis of continuous beams; J. C. Maxwell (1831–1879), who presented the method of consistent deformations and the law of reciprocal deflections; Otto Mohr (1835–1918), who developed the conjugate-beam method for calculation of deflections and Mohr’s circles of stress and strain; Alberto Castigliano (1847–1884), who formulated the theorem of least work; C. E. Greene (1842–1903), who developed the moment-area method; H. Muller-Breslau (1851–1925), who presented a principle for constructing influence lines; G. A. Maney (1888–1947), who developed the slope-deflection method, which is considered to be the precursor of the matrix stiffness method; and Hardy Cross (1885–1959), who developed the moment-distribution method in 1924.

The moment-distribution method provided engineers with a simple iterative procedure for analyzing highly statically indeterminate structures. This method, which was the most widely used by structural engineers during the period from about 1930 to 1970, contributed significantly to their understanding of the behavior of statically indeterminate frames. Many structures designed during that period, such as high-rise buildings, would not have been possible without the availability of the moment-distribution method. The availability of computers in the 1950s revolutionized structural analysis. Because the computer could solve large systems of simultaneous equations, analyses that took days and sometimes weeks in the precomputer era could now be performed in seconds.

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