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Another typical phone call. An owner wants to know how long it will take for the lightweight concrete in the elevated slabs of his new building to dry before he can place the floor covering. The slabs were placed 4 months ago, but tests still show a moisture-vapor-emission rate of 8 pounds per 1000 square feet in 24 hours. The floor-covering manufacturer requires the concrete to be at 3 lbs/1000 sf/24 hrs. Delaying floorcovering installation will delay building occupancy. The owner has never had this problem in other buildings he has constructed. Why won’t this concrete dry? Concrete is concrete, right?

Well, unfortunately it’s not. Many owners and contractors have told us they’ve experienced project delays while waiting for lightweight concrete to dry. Though we couldn’t find any data regarding the drying time of lightweight concrete, field experience tells us that lightweight concrete takes longer to dry than normal-weight concrete. To help fill this information gap, CONCRETE CONSTRUCTION devised a testing program to find out how long it takes lightweight concrete to dry.

Three normal-weight concretes with water-cement ratios of 0.31, 0.37, and 0.40 were delivered to a testing lab in 1-cubic-yard loads. A lightweight concrete mix with a water cement ratio of 0.40 was also delivered to the testing lab. The proprietary mixes were supplied by the ready-mix division of CAMAS Colorado, Denver. The normal-weight concrete moisture-vapor-emission test results were reported in THE CONCRETE PRODUCER (Ref. 1). Here we compare the lightweight concrete moisture-vapor-emission test results with those for normal-weight concrete having the same water-cement ratio. The producer tightly controlled water content and water-cement ratio by closely monitoring aggregate moisture and water left in the drum. The fresh and hardened properties of the lightweight concrete were as follows:

■ 3-inch slump

■ 4.5% air content

■ 124-pound-per-cubic-foot unit weight

■ 48° F temperature

■ 6850-psi 28-day compressive strength

Workers placed and vibrated the concrete in 3-foot-square test slabs 2, 4, 6, and 8 inches thick. After striking off the surface with a 2x4 and floating it by hand, they covered the slabs with plastic sheeting for 3 days. Calciumchloride moisture-emission testing began after the sheeting was removed.

Moisture-emission tests were conducted in accordance with the test manufacturer’s instructions. Technicians ran one test on each slab at the 3-day age but thereafter ran two tests on each test slab and reported an average test value. 

The calcium-chloride test kits were left in place for 72 hours on slabs stored inside the test lab at a 70±3° F and a relative humidity of 28±5%. This is the normal indoor environment during the winter in the Rocky Mountain region, where the tests were conducted, and represents the environment found in many buildings without relative-humidity controls. 

The table shows moisture-vapor emission rates for the test slabs after drying for up to 183 days. Drying times are measured from the end of the 3-day curing period to the end of the 72-hour moisture-emission test. Thus the 3-day measurement was taken 6 days after concrete placement. The test data support the following conclusions.

Lightweight concrete dries slower. Regardless of the test slab thickness, the lightweight concrete took about 6 months to dry to a moisture-vapor emission rate of 3 lbs/1000 sf/24 hrs. Normal-weight concrete of the same water-cement ratio took only 6 weeks of drying in laboratory air to reach the same level (Ref. 1). From previous work (Ref. 2), we know that laboratory drying represents the fastest drying time. So field conditions that include wet-dry cycles will increase the actual time for the slab to reach the specified moisture-vapor-emission limits. 

As with normal-weight concrete, the moisture-vapor-emission rates were unaffected by slab thickness. 

Lightweight concrete offers many advantages. However, any owner using lightweight concrete that’s to be covered by a moisture-sensitive floor covering or any architect/engineer specifying this combination should consider the slower drying time as an important part of the construction schedule. Once lightweight concrete is specified, the contractor can’t change its drying characteristics. If owners want the benefits of lightweight concrete and a fast-track schedule, they may need to consider applying a polymer coating or sheet product to reduce moisture emissions.

Reinforced concrete structural systems can be formed into virtually any geometry to meet any requirement. Regardless of the geometry, standardized floor and roof systems are available that provide cost-effective solutions in typical situations. The most common types are classified as one-way systems and two-way systems. Examined later are the structural members that make up these types of systems.

It is common for one type of floor or roof system to be specified on one entire level of building; this is primarily done for cost savings. However, there may be cases that warrant a change in framing system. The feasibility of using more than one type of floor or roof system at any given level needs to be investigated carefully.

A one-way reinforced concrete floor or roof system consists of members that have the main flexural reinforcement running in one direction. In other words, reactions from supported loads are transferred primarily in one direction. Because they are primarily subjected to the effects from bending (and the accompanying shear), members in one-way systems are commonly referred to as flexural members.

FIGURE 1 One-way slab system.

Members in a one-way system are usually horizontal but can be provided at a slope if needed. Sloped members are commonly used at the roof level to accommodate drainage requirements.

Illustrated in Fig. 1 is a one-way slab system. The load that is supported by the slabs is transferred to the beams that span perpendicular to the slabs. The beams, in turn, transfer the loads to the girders, and the girders transfer the loads to the columns.

Individual spread footings may carry the column loads to the soil below. It is evident that load transfer between the members of this system occurs in one direction.

FIGURE 2 Standard one-way joist system.

Main flexural reinforcement for the one-way slabs is placed in the direction parallel to load transfer, which is the short direction. Similarly, the main flexural reinforcement for the beams and girders is placed parallel to the length of these members. Concrete for the slabs, beams, and girders is cast at the same time after the forms have been set and the reinforcement has been placed in the formwork. This concrete is also integrated with columns. In addition, reinforcing bars are extended into adjoining members. Like all cast-in-place systems, this clearly illustrates the monolithic nature of reinforced concrete structural members.

A standard one-way joist system is depicted in Fig. 2. The one-way slab transfers the load to the joists, which transfer the loads to the column-line beams (or, girders). This system utilizes standard forms where the clear spacing between the ribs is 30 in. or less. Because of its relatively heavy weight and associated costs, this system is not used as often as it was in the past.

FIGURE 3 Wide module joist system.

Similar to the standard one-way joist system is the wide-module joist system shown in Fig. 3. The clear spacing of the ribs is typically 53 or 66 in., which, according to the Code, technically makes these members beams instead of joists. Load transfer follows the same path as that of the standard joist system.

Reinforced concrete stairs are needed as a means of egress in buildings regardless of the number of elevators that are provided. Many different types of stairs are available, and the type of stair utilized generally depends on architectural requirements. Stair systems are typically designed as one-way systems.

FIGURE 4 Two-way beam supported slab system.

A two-way beam supported slab system is illustrated in Fig. 4. The slab transfers the load in two orthogonal directions to the column-line beams, which, in turn, transfer the loads to the columns. Like a standard one-way joist system, this system is not utilized as often as it once was because of cost.

Materials Used in Bridge Construction

Stones, Timber, Concrete and Steel are the traditional materials that are used to carry out bridge construction. During the initial period, timber and stones were used in the construction, as they are directly obtained from nature and easily available.

Brick was used as a subgroup construction material along with stone construction. Stones as construction materials were very popular because of its durable properties. Many historic bridges made from stones are still present as a symbol of past architectural culture.

But some of the timber bridge have been washed away or are in the stage of degradation due to their exposure to the environmental conditions.

As time passed, the bridge construction has undergone more development in terms of materials used for construction than based on the bridge technology.

The concrete and steel are manmade refined materials. The bridge construction with these artificial materials can be called the second period of the bridge engineering. This hence was the start of modern bridge engineering technology.

Modern bridges make use of concrete or steel or in combination. Different other innovative materials are being developed so that they can well suit with the bridge terminologies.

Incorporation of fibers which comes in the category of high strength gaining materials is now incorporated for the construction of bridges. These materials are also used in order to strengthen the existing bridges.

Stones for Bridge Construction

For a long time in the history, the stone has been used in and as a single form. They are mainly used in the form of arches. This is because they possess higher compressive strength.

The use of stones gave the engineers ease of constructing bridges that are aesthetically top and high in durability.

When considering the history of bridge construction with stones, the Romans were the greatest builders of bridges with stones. They had a clear idea and understanding of the load over bridge, the geometry as well as the material properties. This made them construct very larger span bridges when compared with any other bridge construction during that period.

The period was also competitive for Chinese. China had also developed large bridge called the famous Zhuzhou Bridge. The Zhuzhou bridge is the world’s known oldest open-spandrel, stone and segmental arch bridge. Nihonbashi is the most famous stone bridge in Japan. This is called as the Japan Bridge.

Fig.1: The Zhuzhou bridge, China

With time, the stone bridges have proved most efficient and economical due to the durability and low maintenance guaranty it provides throughout its life period.

Timber or Wood for Bridge Construction

The wood material was used highly in the construction of bridges, unlike today, where it is used for the construction of building works and related. Nowadays, steel and concrete grant a higher range of work flexibility, that the use of wood and timber for mega works diminished.

But, there are innovations related to the preservation of wood, which has helped to increase the demand of wood in structures.

Wood as an engineering material has the advantage of high toughness and renewable in nature. They are obtained directly from nature and hence are environmentally friendly.

The low density of wood makes it gain high specific strength. They have an appreciable strength value with a lower value of density. This property makes them be transported easily.

Some of the disadvantages related to wood as a construction material are that it is:

• Highly Anisotropic in Nature
• Susceptible to termites, infestations, and woodworm
• Highly combustible
• Susceptible to rot and disease
• Cannot be used for High temperature

There are a variety of timber bridges around the world. Figure-2 shows the Mathematical Bridge located in Cambridge. Another bridge is the Togetsu-Kyo Bridge over the Katsura River in Kyoto.

Fig.2: The Mathematical Bridge, Cambridge

Fig.3. The Togetsu-Kyo Bridge, Japan

Steel for Bridge Construction

 

Steel gain high strength when compared with any other material. This makes its suitable for the construction of bridges with longer span. We know that steel is a combination of alloys of iron and other elements, mainly carbon.

Based on the amount and variation of the elements, the properties of the same is altered accordingly. The properties of tensile strength, ductility and hardness are influenced by the change in its constitution.

The steel used for normal construction have several hundred Mega Pascal strength. This strength is almost 10 times greater than the compressive and the tensile strength obtained from a normal concrete mix.

The major inbuilt property of steel is the ductility property. This is the deformation capability before the final breakage tends to happen. This property of steel is an important criterion in the design of structures.

Fig.4. The Hachimanbashi Bridge

The first iron bridge, Danjobashi Bridge which was built in 1878 in Japan. The figure-4 below shows the Danjobashi Bridge. Danjobashi Bridge was relocated to the present location and was named as Hachimanbashi Bridge in 1929.

It has great historical and technical value as a modern bridge. The bridge was honored by the American Society of Civil Engineers in the year 1989.

The chemical composition and the method of manufacture determines the properties of structural steel. The main properties that are to be specified by the bridge designers when it is required to specify the products are:

• Strength
• Toughness
• Ductility
• Durability
• Weldability

When we mention the steel strength, it implies both the yield and the tensile strength. As the structures are more designed in the elastic stage, it is very essential to know the value of yield strength.

Yield strength is mostly used as it is more specified in the design codes. In Japan, the code recommended is designed for ultimate strength. For example, SS400 designated by the ultimate strength of 400MPa. This is an exception.

The property of ductility is very much relied on by designers and engineers for the design aspects related to the bolt group designs and the distribution of stress at the ultimate limit state conditions. Another important property is the corrosion resistance by the use of weathering steel.

Concrete for Bridge Construction

Most of the modern bridge construction make use of concrete as the primary material. The concrete is good in compression and weak in tensile strength. The reinforced concrete structures are the remedy put forward for this problem.

The concrete tends to have a constant value of modulus of elasticity at lower stress levels. But this value decreases at a higher stress condition. This will welcome the formation of cracks and later their propagation.

Other factors to which concrete is susceptible are the thermal expansion and shrinkage effects. Creep is formed in concrete due to long time stress on it.

The mechanical properties of concrete are determined by the compressive strength of concrete.

The reinforced or the prestressed concrete is used for the construction of bridges. The reinforcement in R.C.C provides the ductility property to the structure. Nowadays, ductility reinforcement is provided as an additional requirement mainly in the earthquake resistant construction.

RCC is nowadays made from steel, polymer or other combination of composite materials. Much sustainable materials is available that can take the role of cement. This is a new innovation in sustainable bridge construction.

When compared with RCC bridge construction, prestressed concrete is the most preferred and employed. A pre-compressive force is induced in the concrete with the help of high strength steel tendons before the actual service load.

Hence this compressive stress will resist the tensile stress that is coming during the actual load conditions. The prestress is induced in concrete either by means of post tensioning or by means of pretensioning the steel reinforcement.

Many disadvantages of normal reinforced concrete like strength limitations, heavy structures, building difficulty is solved using prestressed concrete.

Also Read: Dutch inaugurate the 3D Printed Reinforced Concrete Bridge Designed by Technical University of Eindhoven 

Composite Materials in Bridge Construction

 

Composite materials are developed and used for both the construction of new bridges as well as for the rehabilitation purposes.

Fiber reinforced plastic is one such material which is a polymer matrix. This is reinforced with fibers which can be either glass or carbon. These materials are light in weight, durable, high strength giving and ductile in nature.

New solution and materials are encouraged due to the problems of deterioration the steel and concrete bridges are facing.

Another material is the reactive powder concrete (RPC ) that was developed in Korea. This material is a form of high performance concrete that is reinforced with steel fibers. This mix will help to make slender columns for bridges of a longer span. This also guarantees durability extensively.

Composite materials are used in the repair of bridge columns and any other supporting elements to improve the ductility and the resistance against the seismic force.

Epoxy impregnated fiberglass are used to cover the column (columns that are non-ductile in nature). This is an alternative for the steel jacket technique.

INVESTIGATIONS FOR STRUCTURAL DEFECTS

1. What is a failure?

A failure can be considered as occurring in a component when that component can no longer be relied upon to fulfill its principal functions. Limited deflection in a floor which caused a certain amount of cracking/distortion in partitions could reasonably be considered as a defect but not a failure. While excessive deflection resulting in serious damage to partitions, ceiling and floor finishes could be classed as a failure.

2. Introduction

This section considers the situation if the initial inspection/investigation detailed in Chapter 4 indicated that some parts of the structure may require strengthening. This can arise for three basic reasons:

a. serious deterioration of some of the structural members;

b. serious overloading of members.

c. proposed change of use involving substantial increase in floor loading.

3. Indications of structural defects

What are the likely signs of structural distress? No precise answer can be given to this question, but the following brief notes are relevant:

a. Diagonal cracks in beams and walls usually denote high shear stress and should be investigated.

b. Excessive deflexion in beams and floor slabs indicates that the members are over-loaded. This is also likely to show as cracking in the soffit at right angles to the main reinforcement (flexural cracking).

c. Bowing in columns and load-bearing walls is likely to cause cracking parallel to the main reinforcement.

4. Bowing in wall panels may be due to differential shrinkage/thermal effects between one face and the other.

5. Errors in the location, design and/or execution of movement joints, isolation joints, stress relief joints and sliding joints can result in cracking, spalling and distortion. This type of defect can be very difficult to rectify.

4. Investigation procedure

It will be seen from the previous section that visible cracking plays an important part in indicating that the structure or parts of the structure are suffering from structural distress. In other words, the members affected were unable to carry the loads imposed on them with an adequate factor of safety.

Such a state of affairs may be brought about by:

a. error(s) in design;

b. errors in construction (workmanship and/or materials);

c. actual loading significantly in excess of the design load;

d. physical damage, impact, explosion fire etc.

e. serious corrosion of reinforcement, which may be the result of many factors.

The engineer should make every effort to obtain copies of the structural calculations and as-built drawings. Unfortunately, this important information is often not available, in which case a ‘structural appraisal’ would be needed and this is time-consuming and expensive.

Assuming that adequate background information is available the general procedure is basically the same but with the emphasis on obtaining information for a practical diagnosis of the structural shortcomings. Additional methods of investigation may include:

a. Ultrasonic Pulse Velocity (UPV) survey

b. an impulse radar survey

c. core testing for strength

d. load tests (seldom used)

It should be noted that the above methods are supplementary to normal investigation techniques, and often used in combination.

4.2 Impulse radar survey

The author is indebted to GB Geotechnics for the information which follows. A transducer containing the transmit and receive antennae is drawn over the surface under investigation at a constant speed. Pulses of energy are transmitted into the material and are reflected from internal surfaces and objects, e.g. changes in density, voids, reinforcing steel. The data is recorded graphically or digitally thus enabling a preliminary assessment on site, followed, if considered necessary, by detailed processing in the laboratory.

Radar responds to changes; it can identify boundaries between layers, measure thicknesses and assess voids and relative moisture content. The radar profile is effectively continuous, radio pulses are transmitted at around 50 000 pulses per second.

Transducers can be hand-held or mounted below survey vehicles, and can be operated up to 200 m from the recording station.

4.3 Core testing for strength

The location of the cores should be carefully selected to provide the information required and for checking the results of UPV and radar surveys.

The cores should be cut, prepared and tested in accordance with the appropriate National Standard; in the UK this is BS 1881: Part 120.

Reference should also be made to BS 6089: Assessment of Concrete Strength in existing Structures, and to Concrete Society Technical Report No. 11.

Misunderstandings sometimes arise over the interpretation of the test results. The actual test on the core will give the compressive strength of the concrete in the core. BS 6089 refers to the ‘estimated in situ cube strength’ which is defined as ‘The strength of concrete at a location in a structural member estimated from indirect means and expressed in terms of specimens of cubic shape’. The Concrete Society Report refers to two types of strength; firstly: ‘Estimated Potential Strength’ which is defined as:

The strength of concrete sampled from an element and tested in accordance with this procedure, such that the result is an estimate of the strength of the concrete provided for manufacture of the element, expressed as the 28 day BS.1881 cube strength, allowance being made for differences in curing, history, age, and degree of compaction between core and BS.1881 cube.

The report also provides for a correction for the influence of included steel. When all these corrections have been made, the result is intended to give the 28-day cube strength of the concrete if cubes had been made and tested in accordance with BS 1881, at the time the member was cast. The intention is to provide an acceptable answer to the questions which arise in new construction when cubes fail. Many experienced engineers feel that with so many corrections only limited reliance can be placed on the results.

The second ‘type of strength’ referred to in the CS Report is ‘The Estimated Actual Strength’; this is defined as:

The strength of concrete sampled from an element and tested in accordance with this procedure, such that the result, expressed as an equivalent cube strength, is an estimate of the concrete strength as it exists at the sampling location, without correction for the effect of curing, history, age or degree of compaction.

The majority of investigations involving existing buildings are concerned with a reasonable assessment of actual strength as defined above, of the concrete in the load-bearing members.

4.4 Load tests

The testing described above should provide information on the general quality of the concrete and condition of the reinforcement. For the engineer to be able to predict with reasonable accuracy the load-carrying capacity of the various structural elements—beams, columns, floor slabs etc.—the following information would also be required:

a. original or, preferably, the as-built drawings of the structure;

b. similar information on any alterations made subsequently;

c. assessment of existing dead and live loads based on the present use;

d. assessment of dead and live loads which will arise from any proposed alterations.

When there is serious doubt about the value of the information available, consideration may have to be given to a load test on selected structural elements. It is accepted that design assumptions do not exactly match the as-built conditions; this is due mainly to the effects of composite action and load sharing. A load test on a beam or floor slab, if correctly carried out, will show how the element under test will react to the applied load under working conditions. During the test it is necessary to record deflections, recovery on removal of load, and details of any crack development.

Load tests must be carried out with great care by an experienced firm with an experienced engineer on site during the test. Provision must be made to deal with any unexpected collapse. All necessary safety precautions must be observed.

Load tests are time-consuming and expensive and should only be carried out after careful consideration of practical value of the results.

The moment of inertia of an area is the capacity of a cross section to resist bending or buckling. It represents a mathematical concept that is dependent on the size and shape of the section of the member. The bending axis of a member is also the centroidal axis; therefore, the ability to locate the centroid of a shape is closely associated with moment of inertia. Engineers use the moment of inertia to determine the state of stress in a section, and determine the amount of deflection in a beam.

The definition of the moment of inertia of an area can be thought of as the sum of the products of all the small areas and the squares of their distances from the axis being considered. This gives

If we represent the moment of inertia by the letter I, then the moments of inertia with respect to the x and y axis axis are

Units

Moment of inertia is expressed in units of length to the fourth power. Although dimensionally speaking it seems unusual, it is just a mathematical abstract and is an important property in the design of beams and columns. We will see in the following examples the methods of calculating the moment of inertia for a given beam section subjected to bending. If we choose the unit of length as mm., then the unit of the moment of inertia is

mm² x mm² = mm⁴

When the ratio of the longer to the shorter side (L/S) of the slab is at least equal to 2.0, it is called one-way slab. Under the action of loads, it is deflected in the short direction only, in a cylindrical form. Therefore, main reinforcement is placed in the shorter direction, while the longer direction is provided with shrinkage reinforcement to limit cracking. When the slab is supported on two sides only, the load will be transferred to these sides regardless of its longer span to shorter span ratio, and it will be classified as one-way slab.

DOWNLOAD

Two way slabs are the slabs that are supported on four sides and the ratio of longer span (l) to shorter span (b) is less than 2. In two way slabs, load will be carried in both the directions. So, main reinforcement is provided in both direction for two way slabs.

ACI 318 Direct Design Method will be used in this example to design an interior bay of a flat plate slab system of  multi bay building.

 

Download  design example.

One way slab is supported on two opposite side only thus structural action is only at one direction. Total load is carried in the direction perpendicular to the supporting beam. If a slab is supported on all the four sides but the ratio of longer span (l) to shorten span (b)  is greater than 2, then the slab will be considered as one way slab. Because due to the huge difference in lengths, load is not transferred to the shorter beams. Main reinforcement is provided in only one direction for one way slabs.

Two way slabs are the slabs that are supported on four sides and the ratio of longer span (l) to shorter span (b) is less than 2. In two way slabs, load will be carried in both the directions. So, main reinforcement is provided in both direction for two way slabs.

There are some basic differences between one way slabs and two way slabs. To clear the concept of one way and two way slabs a table is shown below.

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Structural members are usually classified according to the types of loads that they support. For instance, an axially loaded bar supports forces having their vectors directed along the axis of the bar, and a bar in torsion supports torques (or couples) having their moment vectors directed along the axis.

Beams are structural members subjected to lateral loads, that is, forces or moments having their vectors perpendicular to the axis of the bar.

The beams shown in Fig. 1 are classified as planar structures because they lie in a single plane. If all loads act in that same plane, and if all deflections (shown by the dashed lines) occur in that plane, then we refer to that plane as the plane of bending.

FIG. 1 Examples of beams subjected to lateral loads

Beams are usually described by the manner in which they are supported. For instance, a beam with a pin support at one end and a roller support at the other (Fig. 2a) is called a simply supported beam or a simple beam. The essential feature of a pin support is that it prevents translation at the end of a beam but does not prevent rotation. Thus, end A of the beam of Fig.2a cannot move horizontally or vertically but the axis of the beam can rotate in the plane of the figure. Consequently, a pin support is capable of developing a force reaction with both horizontal and vertical components (HA and RA), but it cannot develop a moment reaction.

At end B of the beam (Fig.2a) the roller support prevents translation in the vertical direction but not in the horizontal direction; hence this support can resist a vertical force (RB) but not a horizontal force. Of course, the axis of the beam is free to rotate at B just as it is at A. The vertical reactions at roller supports and pin supports may act either upward or downward, and the horizontal reaction at a pin support may act either to the left or to the right.

The beam shown in Fig.2b, which is fixed at one end and free at the other, is called a cantilever beam. At the fixed support (or clamped support) the beam can neither translate nor rotate, whereas at the free end it may do both. Consequently, both force and moment reactions may exist at the fixed support.

The third example in the figure is a beam with an overhang (Fig.2c). This beam is simply supported at points A and B (that is, it has a pin support at A and a roller support at B) but it also projects beyond the support at B. The overhanging segment BC is similar to a cantilever beam except that the beam axis may rotate at point B.

When drawing sketches of beams, we identify the supports by conventional symbols, such as those shown in Fig.2. These symbols indicate the manner in which the beam is restrained, and therefore they also indicate the nature of the reactive forces and moments. However, the symbols do not represent the actual physical construction. For instance, consider the examples shown in Fig.3. Part (a) of the figure shows a wide-flange beam supported on a concrete wall and held down by anchor bolts that pass through slotted holes in the lower flange of the beam. This connection restrains the beam against vertical movement (either upward or downward) but does not prevent horizontal movement.

Also, any restraint against rotation of the longitudinal axis of the beam is small and ordinarily may be disregarded. Consequently, this type of support is usually represented by a roller, as shown in part (b) of the figure.

The last example (Fig.3e) is a metal pole welded to a base plate that is anchored to a concrete pier embedded deep in the ground. Since the base of the pole is fully restrained against both translation and rotation, it is represented as a fixed support (Fig.3f ).

The task of representing a real structure by an idealized model, as illustrated by the beams shown in Fig.2, is an important aspect of engineering work. The model should be simple enough to facilitate mathematical analysis and yet complex enough to represent the actual behavior of the structure with reasonable accuracy. Of course, every model is an approximation to nature. For instance, the actual supports of a beam are never perfectly rigid, and so there will always be a small amount of translation at a pin support and a small amount of rotation at a fixed support. Also, supports are never entirely free of friction, and so there will always be a small amount of restraint against translation at a roller support. In most circumstances, especially for statically determinate beams, these deviations from the idealized conditions have little effect on the action of the beam and can safely be disregarded.

Types of Loads

Several types of loads that act on beams are illustrated in Fig.2. When a load is applied over a very small area it may be idealized as a concentrated load, which is a single force. Examples are the loads P1, P2, P3, and P4 in the figure. When a load is spread along the axis of a beam, it is represented as a distributed load, such as the load q in part (a) of the figure. Distributed loads are measured by their intensity, which is expressed in units of force per unit distance (for example, newtons per meter or pounds per foot). A uniformly distributed load, or uniform load, has constant intensity q per unit distance (Fig.2a). A varying load has an intensity that changes with distance along the axis; for instance, the linearly varying load of Fig.2b has an intensity that varies linearly from q1 to q2. Another kind of load is a couple, illustrated by the couple of moment M1 acting on the overhanging beam (Fig.2c).

We assume in this discussion that the loads act in the plane of the figure, which means that all forces must have their vectors in the plane of the figure and all couples must have their moment vectors perpendicular to the plane of the figure. Furthermore, the beam itself must be symmetric about that plane, which means that every cross section of the beam must have a vertical axis of symmetry. Under these conditions, the beam will deflect only in the plane of bending (the plane of the figure).

Reactions

Finding the reactions is usually the first step in the analysis of a beam. Once the reactions are known, the shear forces and bending moments can be found, as described later in this chapter. If a beam is supported in a statically determinate manner, all reactions can be found from free-body diagrams and equations of equilibrium.

In some instances, it may be necessary to add internal releases into the beam or frame model to better represent actual conditions of construction that may have an important effect on overall structure behavior. For example, the interior span of the bridge girder shown in Fig.4 is supported on roller supports at either end, which in turn rest on reinforced concrete bents (or frames), but construction details have been inserted into the girder at either end to ensure that the axial force and moment at these two locations are zero. This detail also allows the bridge deck to expand or contract under temperature changes to avoid inducing large thermal stresses into the structure.

To represent these releases in the beam model, a hinge (or internal moment release, shown as a solid circle at each end) and an axial force release (shown as a C-shaped bracket) have been included in the beam model to show that both axial force (N) and bending moment (M), but not shear (V), are zero at these two points along the beam. (Representations of the possible types of releases for two-dimensional beam and torsion members are shown below the photo). As examples below show, if axial, shear, or moment releases are present in the structure model, the structure should be broken into separate free-body diagrams (FBD) by cutting through the release; an additional equation of equilibrium is then available for use in solving for the unknown support reactions included in that FBD.