Stream Flow

Flow Characteristics of a Stream

The flow characteristics of a stream depend upon (i) the intensity and duration of rainfall besides spatial and temporal distribution of the rainfall, (ii) shape, soil, vegetation, slope and drainage network of the catchment basin and (iii) climatic factors influencing evapotranspiration. Based on the characteristics of yearly hydrograph (graphical plot of discharge versus time in chronological order), one can classify streams into the following three types.

i) Perennial streams which have some flow, at all times of a year due to considerable amount of base flow into the stream during dry periods of the year. The stream bed is lower than the ground water table in the adjoining aquifer (i.e., water bearing strata which is capable of storing and yielding large quantity of water).

Fig. 1 Temporal Variation of Discharge in Perennial Streams

ii) Intermittent streams have limited contribution from the ground water and that too during the wet season only when the ground water table is above the stream bed and there is base flow contributing to the stream flow. Excepting for some occasional storm that can produce short duration flow, such streams remain dry for most of the dry season periods of a year.

Fig. 2. Temporal Variation of Discharge in Intermittent Streams

iii) Ephemeral streams do not have any contribution from the base flow. The annual hydrograph of such a stream shows series of short duration hydrographs indicating flash flows in response to the storm and the stream turning dry soon after the end of the storm. Such streams, generally found in arid zones and it do not have well defined channels.

Fig. 3 Temporal Variation of Discharge in Ephemeral Streams

Streams are also classified as effluent (streams receiving water from ground water storage) and influent (streams contributing water to the ground water storage) streams. Effluent streams are usually perennial while the influent streams generally remain dry during long periods of dry spell.

Graphical Representation of Stream Flow

The stream flow data are usually recorded in tabular form. For analyzing these data, one has to prepare graphical plots of the stream flow data such as hydrograph, flow-duration curve, flow-mass curve or simply mass curve etc. Hydrograph is a graphical plot between discharge (on y-axis) and the corresponding time (days or months or even hours).

Flow-Mass Curves

Flow-mass curve or runoff-mass curve or inflow mass curve or simply mass curve is cumulative flow volume ‘V’ versus time curve. The mass curve ordinate V (m³ or ha.m or cumec-day) at any time t (in days or weeks or months) is given as

where, t0 is the time at the beginning of the curve.

The mass curve is an integral (i.e., summation) curve of a given hydrograph. Also, slope of the mass curve at any point on the plot i.e., dV/dt equals the rate of stream flow (i.e., stream discharge) at that time. Mass curve is always a rising curve or horizontal (when there is no inflow or runoff added into the stream) and is a useful means by which one can calculate storage capacity of a reservoir to meet specified demand as well as safe yield of a reservoir of given capacity.

Fig. 4 Reservoir Capacity from Mass-Flow Curve

Slope of the cumulative demand curve (usually a line since the demand rate is generally constant) is the demand rate which is known. The reservoir is assumed to be full at the beginning of a dry period (i.e., when the withdrawal or demand rate exceeds the rate of inflow into the reservoir) such as A in Fig. 4. Draw line AD (i.e., demand line) such that it is tangential to the mass curve at A and has a slope of the demand rate, between A and B (where there is maximum difference between the demand line and the mass curve) the demand is larger than the inflow (supply) rate and the reservoir storage would deplete. Between B and D, the supply rate is higher than the demand rate and the reservoir would get refilled. The maximum difference in the ordinates of the demand line and mass curve between A and D (i.e., BC) represents the volume of water required as storage in the reservoir to meet the demand from the time the reservoir was full i.e., A in Fig. 4. If the mass curve is for a large time period, there may be more than one such duration of dry periods which obtain the storages required for those durations (EH and IL). The largest of these storages (BC, FG and JK) is the required storage capacity of the reservoir to be provided on the stream in order to meet the demand.

For determining the safe yield of (or maintainable demand by) a reservoir of given capacity one needs to draw tangents from the apex points (A, E and I of Fig. 4) such that the maximum difference between the tangent and the mass curve equals the given capacity of the reservoir. The slopes of these tangents equals to the safe yield for the relevant dry period. The smallest slope of these slopes is the firm dependable yield of the reservoir. It should be noted that a reservoir gets refilled only if the demand line intersects the mass curve. Non-intersection of the demand line with the mass curve indicates inflow which is insufficient to meet the given demand. Also, the vertical difference between points D and E represents the spilled volume of water over the spillway.

The losses from reservoir (such as due to evaporation and seepage into the ground or leakage) in a known duration can either be included in the demand rates or deducted from inflow rates. In practice, demand rates for irrigation, power generation or water supply vary with time. For such situations, mass curve of demand is superposed over the flow-mass curve with proper matching of time. If the reservoir is full at the first intersection of the two curves, the maximum intercept between the two curves represents the required storage capacity of the reservoir to meet the variable demand.

Flow-Duration Curve

Flow-duration curve (or discharge-frequency curve) of a stream is a graphical plot of stream discharge against the corresponding per cent of time the stream discharge was equaled or exceeded. The flow-duration curve describes the variability of the stream flow and is useful for

1. Determining dependable flow which information is required for planning of water resources and hydropower projects
2. Designing a drainage system
3. Flood control studies

For preparing a flow-duration curve, the stream flow data (individual values or range of values) are arranged in a descending order of stream discharges. If the number of such discharges is very large, one can use range of values as class intervals. Percentage probability Pp of any flow (or class value) magnitude Q being equaled or exceeded is given as

in which ‘m’ is the order number of the discharge (or class value) and N is the number of data points in the list. The discharge Q is plotted against Pp to get the yield flow-duration curve.

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Runoff

Runoff can be defined as the portion of the precipitation that makes its way towards rivers or oceans etc., as surface or subsurface flow. Portion which is not absorbed by the deep strata. Runoff occurs only when the rate of precipitation exceeds the rate at which water may infiltrate into the soil. Precipitation (or rainfall), after satisfying the requirements of evapotranspiration, interception, infiltration into the ground and detention storage, drains off or flows off from a catchment basin as an overland flow (or surface runoff which includes precipitation falling on the stream system too) into a stream channel. Some part of the infiltrating water moves laterally through the upper layers of the soil and returns to the ground surface as interflow or subsurface runoff at some place away from the point of infiltration into the soil. Part of the infiltrating water percolates deep into the ground and joins the ground water storage. When water table intersects the stream channels of the catchment basin, some ground water may reach the surface or join the stream as ground water runoff, also called base flow or dry-weather flow. Thus, the runoff from a catchment includes surface runoff, subsurface runoff and base flow.

The surface runoff starts soon after the precipitation and is the first to join the stream flow. Subsurface runoff is slower and joins the stream later. Depending upon the time taken by the subsurface runoff between the infiltration and joining the stream channel, it may be termed as prompt subsurface runoff or delayed subsurface runoff. The groundwater runoff is the slowest in joining the stream channel but, is responsible in maintaining low flows in the stream during dry season. Based on the time interval between the precipitation and runoff, the runoff is categorized as direct runoff (that enters the stream immediately after precipitation i.e., surface runoff, subsurface runoff) and base flow (i.e., ground water runoff). Runoff is the response of a catchment to the precipitation reflecting the combined effects of the nature of precipitation, other climatic characteristics of the region and the physiographic characteristics of the catchment basin.

Type, intensity, duration and areal distribution of precipitation over the catchment are the chief characteristics of the precipitation that affect the stream flow. Precipitation in the form of rainfall is quicker to appear as stream flow than when it is in the form of snow. For the surface runoff to start, the intensity of rainfall (or precipitation) must exceed the infiltration capacity of the soil which decreases with the increase in the duration of rainfall. It is obvious that a longer duration rainfall may produce higher runoff even if the intensity of rainfall is less but exceeding the infiltration capacity of the soil. Heavy rainfalls in the downstream region of the catchment will cause rapid rise in the stream levels and early peaking of the discharge. A rare occurrence of uniformly distributed rainfall may result in increased infiltration and therefore, increased subsurface runoff and base flow resulting in slow rise in levels and delayed peaking of the discharge. Likewise, antecedent higher soil moisture conditions at the time of precipitation would hasten the rise in the stream levels. Other climatic characteristics influencing the runoff are temperature, wind velocity and relative humidity. These characteristics affect the evapotranspiration and thus influence the availability of the precipitation for runoff.

Types of Runoff

1) Surface Runoff

Portion of rainfall (after all losses such as interception, infiltration, depression storage etc. are met) that enters streams immediately after occurring rainfall. After laps of few time, overland flow joins streams; sometime termed prompt runoff (as very quickly enters streams).

2) Subsurface Runoff

Amount of rainfall first enter into soil and then flows laterally towards stream without joining water table. It also takes little time to reach stream.

3) Base Flow

It is a delayed flow. Water that meets the groundwater table and join the stream or ocean. It is very slow movement and take months or years to reach streams.

Factors affecting runoff

• Climatic factors
• Rain and snow fall
• Duration of rainfall
• Rainfall distribution
• Direction of prevailing wind
• Other climatic factors like Temperature, wind velocity, relative humidity, annual rainfall etc.

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Interpretation of Precipitation Data

Precipitation data must be checked for the continuity and consistency before they are analysed for any significant purpose. This is essential when it is suspected that the gauge site (or its surroundings) might have changed appreciably during the period for which the average is being computed.

Estimation of Missing Data

The continuity of a record of precipitation data may have been broken with missing data due to several reasons such as damage (or fault) in a rain gauge during a certain period. The missing data is estimated using the rainfall data of the neighbouring rain gauge stations. The missing annual precipitation Px at a station x is related to the annual precipitation values, P1, P2, P3 ...... Pm and normal annual precipitation, N1, N2, N3 ...... Nm at the neighbouring M stations 1, 2, 3, … M respectively. The normal precipitation (for a particular duration) is the mean value of rainfall on a particular day or in a month or year over a specified 30-year period.

The 30-year normals are computed every decade. The term normal annual precipitation at any station is, therefore, the mean of annual precipitations at that station based on 30-year record.

The missing annual precipitation Px is simply given as

If the normal annual precipitations at various stations are within about 10% of the normal annual precipitation at station x i.e., Nx. Otherwise, one uses the normal ratio method which gives

This method works well when the precipitation regimes of the neighbouring stations and the station x are similar (or almost the same).

Multiple linear regression (amongst precipitation data of M stations and the station x, excluding the unknown missing data of station x and the concurrent (or corresponding) data of the neighbouring M stations) will yield an equation of the form

The regression method allows for some weighting of the stations and adjusts, to some extent, for departures from the assumption of the normal ratio method.

Test for Consistency of Precipitation Data

Changes in relevant conditions of a rain gauge (such as gauge location, exposure, instrumentation or observation techniques and surroundings) may cause a relative change in the precipitation catchment of the rain gauge. The consistency of the precipitation data of such rain gauges needs to be examined. Double-mass analysis, also termed double-mass curve technique, compares the accumulated annual or seasonal precipitation at a given station with the concurrent accumulated values of mean precipitation for a group of the surrounding stations (i.e., base stations). Since the past response is to be related to the present conditions, the data (accumulated precipitation of the station x, i.e., ΣPx and the accumulated values of the average of the group of the base stations, i.e., ΣPav) are usually assembled in reverse chronological order. Values of ΣPx are plotted against ΣPav for the concurrent time periods and is given in Fig. 1. A definite break in the slope of the resulting plot points to the inconsistency of the data indicating a change in the precipitation regime of the station x. The precipitation values at station x at and beyond the period of change is corrected using the relation,

Where,

     Pcx = corrected value of precipitation at station x at any time t

     Px = original recorded value of precipitation at station x at time t.

     Sc = corrected slope of the double-mass curve

     Sa = original slope of the curve

Fig. 1 Double-Mass Curve

Thus, the older records of station x have been corrected so as to be consistent with the new precipitation regime of the station x.

Presentation of Precipitation Data

Precipitation (or rainfall) data are presented as either a mass curve of rainfall (accumulated precipitation v/s time plotted in chronological order, Fig. 2) or a hyetograph (rainfall intensity v/s time). Mass curves of rainfall provide the information on the duration and magnitude of a storm. Intensities of rainfall at a given time can be estimated by measuring the slope of the curve at the specified time. The hyetograph derived from the mass curve is usually represented as a chart. The area of a hyetograph represents the total precipitation received during the period.

Depth – Area - Duration (DAD) Analysis

Depth-area-duration (DAD) curves are plots of accumulated average precipitation versus area for different durations of a storm period. Depth-area-duration analysis of a storm is performed to estimate the maximum amounts of precipitation for different durations and over different areas. A storm of certain duration over a specified basin area seldom results in uniform rainfall depth over the entire specified area. The difference between the maximum rainfall depth over an area P0 and its average rainfall depth (P-bar) for a given storm, i.e., (P0 – P-bar) increases with increase in the basin area and decreases with increase in the storm duration. The depth-area-duration curve is obtained as given in the following example figure.

Fig. 2 Example Figure for DAD Curves

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Average Depth of Precipitation Over an Area

The information on the average depth of precipitation (or rainfall) over a specified area on either the storm basis or seasonal basis or annual basis is often required in several types of hydrologic problems. The depth of rainfall measured by a rain gauge is valid for that rain gauge station and in its immediate vicinity. Over a large area like watershed (or catchment) of a stream, there will be several such stations and the average depth of rainfall over the entire area can be estimated by one of the following methods.

1) Arithmetic Mean Method

This is the simplest method in which average depth of rainfall is obtained by obtaining the sum of the depths of rainfall (say P1, P2, P3, P4 .... Pn) measured at stations 1, 2, 3…. n and dividing the sum by the total number of stations i.e. n. Thus, 

This method is suitable if the rain gauge stations are uniformly distributed over the entire area and the rainfall variation in the area is not large.

2) Theissen Polygon Method

The Theissen polygon method takes into account the non-uniform distribution of the gauges by assigning a weightage factor for each rain gauge. In this method, the enitre area is divided into number of triangular areas by joining adjacent rain gauge stations with straight lines, as shown in Fig. 1 (a and b). If a bisector is drawn on each of the lines joining adjacent rain gauge stations, there will be number of polygons and each polygon, within itself, will have only one rain gauge station. Assuming that rainfall Pi recorded at any station ‘i’ is representative rainfall of the area Ai of the polygon i within which rain gauge station is located, the weighted average depth of rainfall P-bar for the given area is given as

Here, Ai/ A is termed the weightage factor for i^th  rain gauge.

Fig. 1 Areal averaging of precipitation a) Rain gauge network, b) Theissen polygons, c) Isohyets

This method is, obviously, better than the arithmetic mean method since it assigns some weightage to all rain gauge stations on area basis. Also, the rain gauge stations outside the catchment can also be used effectively. Once the weightage factors for all the rain gauge stations are computed, the calculation of the average rainfall depth P-bar is relatively easy for a given network of stations.

While drawing Theissen polygons, one should first join all the outermost rain gauge stations. Thereafter, the remaining stations should be connected suitably to form quadrilaterals. The shorter diagonals of all these quadrilaterals are then drawn. The sides of all these triangles are then bisected and thus, Theissen polygons for all rain gauge stations are obtained.

3) Isohyetal Method

An isohyet is a contour of equal rainfall. Knowing the depths of rainfall at each rain gauge station of an area and assuming linear variation of rainfall between any two adjacent stations, one can draw a smooth curve passing through all points indicating the same value of rainfall, Fig. 1 (c). The area between two adjacent isohyets is measured with the help of a planimeter.

The average depth of rainfall P-bar for the entire area A is given as

Since this method considers actual spatial variation of rainfall, it is considered as the best method for computing average depth of rainfall.

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Measurement of Precipitation - Rain Gauges

Rainfall may be measured by a network of rain gauges which may either be of non-recording or recording type. Rainfall can be measured with a weather radar also.

Non-Recording Rain gauge

The non-recording rain gauge used in India is the Symon’s rain gauge (Fig. 1). It consists of a funnel with a circular rim of 12.7 cm diameter and a glass bottle as a receiver. The cylindrical metal casing is fixed vertically to the masonry foundation with the level rim 30.5 cm above the ground surface. The rain falling into the funnel is collected in the receiver and is measured in a special measuring glass graduated in mm of rainfall; when full it can measure 1.25 cm of rain. The rainfall is measured every day at 08.30 hours IST. The funnel shank is inserted in a bottle which receives the rainwater. The water collected in the bottle is measured by pouring it into a measuring cylinder which gives the depth of rainfall in mm.

Fig. 1 Symon’s Rain Gauge

During heavy rains, it must be measured three or four times in the day in which the receiver fill and overflow, but the last measurement should be at 08.30 hours IST and the sum total of all the measurements during the previous 24 hours entered as the rainfall of the day in the register. Usually, rainfall measurements are made at 08.30 hr IST and sometimes at 17.30 hr IST also. Thus the non-recording or the Symon’s rain gauge gives only the total depth of rainfall for the previous 24 hours (i.e., daily rainfall) and does not give the intensity and duration of rainfall during different time intervals of the day. It is often desirable to protect the gauge from being damaged by cattle and for this purpose a barbed wire fence may be erected around it.

Recording Rain Gauge

This is also called self-recording, automatic or integrating rain gauge. This type of rain gauge has an automatic mechanical arrangement consisting of a clockwork, a drum with a graph paper fixed around it and a pencil point, which draws the mass curve of rainfall. From this mass curve, the depth of rainfall in a given time, the rate or intensity of rainfall at any instant during a storm and time of onset and cessation of rainfall can be determined. The gauge is installed on a concrete or masonry platform 45 cm square in the observatory enclosure by the side of the ordinary rain gauge at a distance of 2-3 m from it. The gauge is so installed that the rim of the funnel is horizontal and at a height of exactly 75 cm above ground surface. The self-recording rain gauge is generally used in conjunction with an ordinary rain gauge exposed close by, for use as standard, by means of which the readings of the recording rain gauge can be checked and if necessary adjusted. There are three types of recording rain gauges—tipping bucket gauge, weighing gauge and float gauge.

1) Tipping Bucket Rain Gauge

This consists of a cylindrical receiver 30 cm diameter with a funnel inside (Fig. 2). Just below the funnel a pair of tipping buckets is pivoted such that when one of the bucket receives a rainfall of 0.25 mm it tips and empties into a tank below, while the other bucket takes its position and the process is repeated. The tipping of the bucket actuates on electric circuit which causes a pen to move on a chart wrapped round a drum which revolves by a clock mechanism. This type of rain gauge cannot record snowfall.

Fig. 2 Tipping Bucket Rain Gauge

2) Weighing Type Rain Gauge

In this type of rain-gauge, when a certain weight of rainfall is collected in a tank, which rests on a spring-lever balance, it makes a pen to move on a chart wrapped round a clock driven drum (Fig. 3). The rotation of the drum sets the time scale while the vertical motion of the pen records the cumulative precipitation. The record thus gives the accumulation of rainfall with time.

Fig. 3 Weighing Type Rain Gauge

3) Float Type Rain Gauge

In this type, as the rain is collected in a float chamber, the float moves up which makes a pen to move on a chart wrapped round a clock driven drum. When the float chamber fills up, the water siphons out automatically through a siphon tube kept in an interconnected siphon chamber. The float rises with the rise of water level in the chamber and its movement is recorded on a chart through a suitable mechanism. The clockwork revolves the drum once in 24 hours. The clock mechanism needs rewinding once in a week when the chart wrapped round the drum is also replaced. A siphon arrangement is also provided to empty the chamber quickly whenever it becomes full. The weighing and float type rain gauges can store a moderate snow fall which the operator can weigh or melt and record the equivalent depth of rain.

Fig. 4 Float Type Rain Gauge

Bureau of Indian Standards has laid down the following guidelines for selecting the site for rain gauges (IS : 4897-1968).

• The rain gauge shall be placed on a level ground, not upon a slope or a terrace and never upon a wall or roof.
• On no account, the rain gauge shall be placed on a slope such that the ground falls away steeply in the direction of the prevailing wind.
• The distance of the rain gauge from any object shall not be less than twice the height of the object above the rim of the gauge.
• Great care shall be taken at mountain and coast stations so that the gauges are not unduly exposed to the sweep of the wind. A belt of trees or a wall on the side of the prevailing wind at a distance exceeding twice its height shall form an efficient shelter.
• In hills where it is difficult to find a level space, the site for the rain gauge shall be chosen where it is best shielded from high winds and where the wind does not cause eddies.
• The location of the gauge should not be changed without taking suitable precautions. Description of the site and surroundings should be made a matter of record.

Radar Measurement of Precipitation

In regions of difficult and inaccessible terrains, precipitation can be measured (within about 10% accuracy of the rain gauge measurements) with the help of a radar (radio detecting and ranging). A radar transmits a pulse of electromagnetic waves as a beam in a direction depending upon the position of the movable antenna. The wave travelling at a speed of light is partially reflected by cloud or precipitation particles and returns to the radar where it is received by the same antenna. The display of the magnitude of energy of the returned wave on the radarscope (i.e., radar screen) is called an echo and its brightness is termed echo intensity. The duration between the transmission of the pulse and appearance of the echo on the radarscope is a measure of the distance (i.e., range) of the target from the radar.
Direction of the target with respect to the radar is decided by the orientation of the antenna at the time the target signal is received. The echo is seen in polar coordinates. If there is no target (i.e., cloud or precipitation particles), the screen is dimly illuminated. A small target would appear as a bright point whereas an extended target (such as a rain shower) would appear as a bright patch. The radarscope being divided as per the coordinate system; the position of the target can be estimated. By having a proper calibration between the echo intensity and rainfall (or its intensity), one can estimate the rainfall (or rainfall intensity).

Satellite Measurement of Precipitation

It is a common experience that gauge network for measuring precipitation in a large and inaccessible area (such as in desert and hilly regions) is generally inadequate and non-existent in oceans. The satellite observation is the only effective way for continuous monitoring of precipitation events over a large or inaccessible area. Use of the meterological satellites for weather and water balance studies is, therefore, continuously increasing.

In satellite measurements, the precipitation is estimated by correlating the satellite- derived data and observed rainfall data. These relationships can be developed for a part of electromagnetic spectrum using cloud life history or cloud indexing approach. The first approach uses data from geo-stationary satellites that produce data at every half an hour interval. The second approach, based on cloud classification, does not require a series of consecutive observations of the same cloud system. Microwave remote sensing techniques that can directly monitor the rainfall characteristics have great potential in rainfall measurement.

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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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Tests on Bricks

I) Field Tests on Bricks

The field test on bricks gives an idea about its basic quality based on its shape, size and colour at first observation without any big appliances. They are very common and easiest way to check the quality of brick. Field tests of brick are very helpful on the site. Some very common tests of brick that is followed to find if brick is good at first observation are as follows.

1) Shape and Size

The clay bricks should have a uniform rectangular plane surface, as per standard size and sharp straight edges. BIS recommends the standard size of brick is 190 x 90 x 90 mm and constructional size is 200 x 100 x 100 mm.

2) Visual Inspection

In this test, bricks are closely inspected for its shape. The bricks of good quality should be uniform in shape and should have truly rectangular shape with sharp edges.

3) Hardness

The clay bricks should be sufficiently hard when scratched by a finger-nail and no impression should be left on the brick surface.

4) Colour

The clay bricks should have a uniform deep red colour throughout. It indicates the uniformity of chemical composition and the quality of burning of the bricks.

5) Texture and Compactness

The surfaces should not be so smooth to cause skidding of mortar. The clay brick should have a pre-compact, homogeneous and uniform texture. A broken surface should be free form cracks, holes grits or lumps of lime.

6) Soundness

When two clay bricks are stuck together, a metallic ringing sound should come.

7) Structure

A brick is broken and its structure is examined. It should be homogeneous, compact and free from any defects such as holes, lumps etc.

8) Basic Strength

When dropped flat on the hard ground from a height of about one meter, clay bricks should not break.

II) Laboratory Tests on Bricks

Laboratory tests on brick determine the mechanical properties of brick and give a scientific approach to ensure the quality of bricks. It is essential while purchasing the brick and examine the properties for the quality of construction. Followings brick tests are performed in the laboratory to determine the quality of brick.

1) Water Absorption of Brick

The brick is porous by nature and Porosity is the ability to release and absorb moisture. Therefore, it tends to absorb the water or moisture. It’s an important and useful property of brick. But if brick absorbs more water than the recommended result, then it affects the strength of brick as well as durability of the structure and will damage plaster and paint over walls. Dry bricks are put in an oven at a temperature of 105° to 115°C till these attain constant mass. The weight (W1) of the bricks is recorded after cooling them to room temperature. The bricks are then immersed in water at a temperature of 27° ± 2°C for 24 hours. The specimens are then taken out of water and wiped with a damp cloth. Three minutes later it is weighed again and recorded as W2.

Water absorption test is performed to know the percentage of water absorption of bricks. Water absorption of bricks should not more than 20% by its dry weight. If brick fails in the water absorption test, possible reasons are like manufacturing error, insufficient burning, error in clay composition etc. If brick fails in water absorption as well as efflorescence than never use those bricks because it will cause in permanent problems and it will be very difficult to solve them. Water bath, weight balance and oven are required for performing this test.

2. Compressive Strength of Brick

The compressive strength of the brick is the most essential property of the bricks because in the construction, bricks are widely used in masonry and it also plays a significant role as a load bearing component. When bricks are used in any structure, the bottom-most layer of the brick will be subjected to the highest compressive stress. Therefore, it is essential to know that any particular brick will be able to withstand that load or not. This test is performed to know the strength of bricks because it affects the overall structure in the way of quality, durability and serviceability.

For testing bricks for compressive strength from a sample the two bed faces of bricks are ground to provide smooth, even and parallel faces. The bricks are then immersed in water at room temperature for 24 hours. These are then taken out of water and surplus water on the surfaces is wiped off with cotton or a moist cloth. The frog of the brick is flushed level with cement mortar and the brick is stored under damp jute bags for 24 hours followed by its immersion in water at room temperature for three days. The specimen is placed in the compression testing machine with flat faces horizontal and mortar filled face being upwards. Load is applied at a uniform rate of 14 N/m² per minute till failure. The maximum load at failure divided by the average area of bed face gives the compressive strength.

Recommended Result of Compressive Strength Test of Brick

Test result recommendations are as follows.

• For first class bricks, it should not less than 10 N/mm² (102 kg/cm²).
• For second class bricks, it should not less than 7 N/mm² (71 kg/cm²).
• For third class bricks, it should not less than 3.5 N/mm² (36 kg/cm²).

In India, the northern and the eastern region produce bricks having good compressive strength than the western region because the western region has black cotton soil, while the soil is good in Gangetic region. If the test result is not as per recommendation, there are many reasons behind it such as the clay composition, degree of burning like over burning or insufficient burning, error in the testing appliance or testing procedure etc. If bricks fail in strength as well as water absorption test, then do not use it. If bricks are irregular in some minor shape/size than it can be corrected with mortar.

3) Efflorescence Test

This test should be conducted in a well-ventilated room. The brick is placed vertically in a dish 30 x 20 cm approximately in size with 2.5 cm immersed in distilled water. The whole water is allowed to be absorbed by the brick and evaporated through it. After the bricks appear dry, a similar quantity of water is placed in the dish and the water is allowed to evaporate as before. The brick is to be examined after the second evaporation and reported as follows.

• Nil : When there is no perceptible deposit of salt
• Slight : When not more than 10% of the area of brick is covered with salt
• Moderate : When there is heavy deposit covering 50% of the area of the brick but unaccompanied by powdering or flaking of the surface.
• Heavy : When there is heavy deposit covering more than 50% of the area of the brick accompanied by powdering or flaking of the surface.
• Serious : When there is heavy deposit of salts accompanied by powdering and/or flaking of the surface and this deposition tends to increase in the repeated wetting of the specimen.

Bricks for general construction should not have more than slight to moderate efflorescence.

4) Dimension Tolerance

Twenty bricks are selected at random to check measurement of length, width and height. These dimensions are to be measured in one or two lots of ten each as shown in Fig. 1. Variation in dimensions is allowed only within narrow limits, ±3% for class one and ±8% for other classes.

Fig. 1

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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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