Analysis and Design of Indoor Stadium

Elastic analysis of frames

Elastic analysis deals with the study of strength and behavior of the members and structure at working loads. Frames can be analyzed by various methods. However, the method of analysis adopted depends upon the types of frame, its configuration (portal bay or multibay) multistoried frame and Degree of indeterminacy.

It is based on the following assumptions:

1. Relation between force and displacement is linear. (i.e. Hook’s law is applicable).

2. Displacements are extremely small compared to the geometry of the structure in the sense that they do not affect the analysis.

The methods used for analysis of frame are:

1. Flexibility coefficient method.

2. Slope displacement method.

3. Iterative methods like

a. Moment distribution method(By Hardy Cross in 1930’s)

b. Kani’s method (by Gasper Kani in 1940’s)

4. Approximate methods like

a. Substitute frame method

b. Portal method

c. Cantilever method

1. FLEXIBILITY COEFFICIENT METHOD:

This method is called as force method or compatibility method. In this Redundant forces are chosen as unknowns. Additional equations are obtained by considering the geometrical conditions imposed on the formation of structures. This method is used for analyzing frames of lower D.O.R.

· Limitations:

1. This method involves long computations even for simple problems with small D.O.R.

2. This method becomes intractable for large D.O.R. (>3), when computed manually especially because of simultaneous equations involved.

This method is not ideal for computerizing, since a structure can be reduced to a statically determinate form in more than one way.

2. SLOPE DISPLACEMENT METHOD:

It is displacement or equilibrium or stiffness method. It consists of series of simultaneous equations, each expressing the relation between the moments acting at the ends of the members is written in terns of slope & deflection. The solution of slope deflection equations along with equilibrium equations gives the values of unknown rotations of the joints. Knowing these rotations, the end moments are calculated using slope deflection equations.

· Limitations:

1. This method is advantageous only for the structures with small Kinematic indeterminacy.

2. The solution of simultaneous equation makes the method tedious for annual computations.

The formulation of equilibrium conditions tends to be a major constraint in adopting this method.

Hence flexibility coefficients & slope displacement methods have limited applications in the analysis of frames. While other methods like iterative or approximate methods are used for analyzing frames containing larger indeterminacy.

3. APPROXIMATE METHODS:

Approximate analysis of hyper static structures provides a simple means of obtaining quick solutions for preliminary designs. It is a very useful process that helps to develop a suitable configuration for final (rigorous) analysis of a structure, compare alternative designs & provide a quick check on the adequacy of structural designs. These methods make use of simplifying assumptions regarding structural behavior so as to obtain a rapid solution to complex structures. However, these techniques should be applied with caution & not relied upon for final designs, especially complex structures.

The usual process comprises reducing the given indeterminate configuration to a structural system by introducing adequate number of hinges. It is possible to check the deflected profile of a structure for the given loading & there by locate the points of inflection.

Since each point of inflection corresponds to the location of zero moment in the structure, the inflection points can be visualized as hinges for purpose of analysis. The solution of the structure is rendered simple once the inflection points are located. In multistoried frames, two loading cases arise namely horizontal & vertical loading.

The analysis is carried out separately for these two cases:

· VERTICAL LOADS:

The stress in the structure subjected to vertical loads depends upon the relative stiffness of the beam & columns. Approximate methods either assumes adequate number of hinges to render the structure determinate or adopt simplified moment distribution methods.

· HORIZONTAL LOADS:

The behavior of a structure subjected to horizontal forces depends on its height to width ratio. The deformation in low-rise structures, where the height is smaller than its width, is characterized predominantly by shear deformations. In high rise building, where height is several times greater than its lateral dimensions, is dominated by bending action. There are two methods to analyze the structures subjected to horizontal loading.

3.1. PORTAL METHOD:

Since shear deformations are dominant in low rise structures, the method makes simplifying assumptions regarding horizontal shear in columns. Each bay of a structure is treated as a portal frame, & horizontal force is distributed equally among them.

The assumptions of the method can be listed as follows:

1. The points of inflection are located at the mid-height of each column above the first floor. If the base of the column is fixed, the point of inflection is assumed at mid height of the ground floor columns as well; otherwise it is assumed at the hinged column base.

2. Points of inflection occur at mid span of beams.

3. Total horizontal shear at any floor is distributed among the columns of that floor such that the exterior columns carry half the force carried by the inner columns.     

3.2. CANTILEVER METHOD:

This method is applicable to high rise structures. This is based on the simplifying assumptions regarding the Axial Force in columns.

1. The basic assumption of the method can be stated as “the axial force in the column at any floor is linearly proportional to its distance from the centroid of all the columns at that level.

Assumptions 1&2 of the portal are also applicable to the cantilever method.

3 POINTS OF INFLECTION METHOD:

The frame is reduced to a statically determinate form by introducing adequate number of points of inflection. The loading on the frames usually comprises uniformly distributed dead loads & live loads.

The following are assumptions made:-

1. The beams of each floor act as continuous beams, with the points of inflection at a distance of one-tenth of the span from the joints.

2. The unbalanced beam moment at each joint is distributed equally among the columns at the joint.

3. Axial forces & deformations in beams are negligible.

3.4 SUBSTITUTE FRAME METHOD:

The method assumes that the moments in the beams of any floor are influenced by loading on that floor alone. The influence of loading on the lower or upper floors is ignored altogether. The process involves the division of multi-storied structure into smaller frames. These sub frames are known as equivalent frames or substitute frames.

The sub frames are usually analyzed by the moment distribution method, using only one cycle of distribution. The substitute frames are formed by the beams at the floor level under consideration, together with the columns above & below with their far ends fixed. The distributed B.M are not carried over far ends of the columns in this process; the moments in the columns are computed at each floor level independently & retained at that floor irrespective of further analysis.

4. ITERATIVE METHOD:

Iterative procedures form a powerful class of methods for analysis of indeterminate structures. These methods after elegant & simple procedure of analysis, that are adequate for usual structures.

These methods are based on the distribution of joint moments among members connected to a joint. The accuracy of the solution depends upon the number of iterations performed; usually three or five iterations are adequate for most of the structures.

The moment distribution methods were developed by Hardy Cross in 1930’s & by Gasper Kani in 1940’s. These methods involve distributing the known fixed moments of the structural members to the adjacent members at the joints, in order to satisfy the conditions of the continuity of slopes & displacements.

Though these methods are iterative in nature, they converge in a few iterations to give correct solution.

4.1 MOMENT DISTRIBUTION METHOD:

This method was first introduced by Prof. Hardy Cross is widely used for the analysis of intermediate structures. In this method first the structural system is reduced to its kinematically determinate form, this is accomplished by assuming all the joints to be fully restrained. The fixed end moments are calculated for this condition of structure. The joints are allowed to deflect rotate one after the other by releasing them successively. The unbalanced moment at the joint shared by the members connected at the joint when it is released.

LIMITATIONS:

1. This method is eminently suited to analyze continuous beams including non-prismatic members but it presents some difficulties when applied to rigid frames, especially when frames are subjected to side sway.

2. Unsymmetrical frames have to be analyzed more than once to obtain FM (fixed moments) in the structures.

3. This method can not be applied to structures with intermediate hinges.

4.2 KANI’S METHOD:

This method was introduced by Gasper Kani in 1940’s. It involves distributing the unknown fixed end moments of structural members to adjacent joints, in order to satisfy the conditions of continuity of slopes and displacements.

ADVANTAGES:

1. Hardy Cross method distributed only the unbalanced moments at joints, whereas Kani’s method distributes the total joint moment at any stage of iteration.

2. The more significant feature of Kani’s method is that the process is self corrective. Any error at any stage of iteration is corrected in subsequent steps.

Framed structures are rarely symmetric and subjected to side sway, hence Kani’s method is best and much simpler than pther methods like moment distribution method and slope displacement method.

PROCEDURE:

1. Rotation stiffness at each end of all members of a structure is determined depending upon the end conditions.

a. Both ends fixed

K_ij= K_ji= EI/L

b. Near end fixed, far end simply supported

K_ij= ¾ EI/L; K_ji= 0

2. Rotational factors are computed for all the members at each joint it is given by

      U_ij= -0.5 (K_ij/ ?K_ji)

{THE SUM OF ROTATIONAL FACTORS AT A JOINT IS -0.5}

(Fixed end moments including transitional moments, moment releases and carry over moments are computed for members and entered. The sum of the FEM at a joint is entered in the central square drawn at the joint).

3. Iterations can be commenced at any joint however the iterations commence from the left end of the structure generally given by the equation

               M?ij = Uij [(Mfi + M??i) + ? M?ji)]

4. Initially the rotational components ? Mji (sum of the rotational moments at the far ends of the joint) can be assumed to be zero. Further iterations take into account the rotational moments of the previous joints.

5. Rotational moments are computed at each joint successively till all the joints are processed. This process completes one cycle of iteration.

6. Steps 4 and 5 are repeated till the difference in the values of rotation moments from successive cycles is neglected.

7. Final moments in the members at each joint are computed from the rotational members of the final iterations step.

               M_ij = (Mfij + M??ij) + 2 M?ij + M?jii

The lateral translation of joints (side sway) is taken into consideration by including column shear in the iterative procedure.

8. Displacement factors are calculated for each storey given by

               Uij = -1.5 (Kij/?Kij)

PPT on ANALYSIS AND DESIGN OF AMPHITHEATRE

WRE: Surface and Ground Water Resources (NPTEL)

Cross drainage works.

Why I-section beam is preferred for heavy loading?

Why I-section beam is preferred for heavy loading?

Cross sectional shape I, giving many benefits. It is very good for giving stiffness (less deformation on loading) and to withstand higher bending moments (as a result of heavy loading) on comparison with other cross-sectional shapes of same area. Also, it is very easy to manufacture. It will have more moment of inertia.

STONE MASTIC ASPHALT

 IINTRODUCTION

            Stone Mastic asphalt (SMA), otherwise known as Stone Matrix Asphalt / Split Mastic Asphalt, was developed in Germany in the mid of 1960's and it has spread throughout Europe and across the world in 1980's and 1990's respectively. The excellent performances include resistant to mechanical and temperature deformation, cracking, and particularly rutting, resistant to weathering actions such as aging and low temperature cracking. Durability is excellent even under slow moving heavy traffic. The textured surface increases skid resistance and provides environmental and driving comfort by reduced noise level, and improved visibility in rainy days.

SMA provides a deformation resistant, durable, surfacing material, suitable for heavily trafficked roads. SMA has found use in Europe, Australia and the United States as a durable asphalt surfacing option for residential streets and highways. SMA has a high coarse aggregate content that interlocks to form a stone skeleton that resist permanent deformation. The stone skeleton is filled with mastic of bitumen and filler to which fibres are added to provide adequate stability of bitumen and to prevent drainage of binder during transport and placement. Typical SMA composition consists of 70−80% coarse aggregate, 8−12% filler, 6.0−7.0% binder, and 0.3 per cent fibre. The deformation resistant capacity of SMA stems from a coarse stone skeleton providing more stone-on-stone contact than with conventional dense graded asphalt (DGA) mixes. Improved binder durability is a result of higher bitumen content, a thicker bitumen film and, lower air voids content. This high bitumen content also improves of flexibility. Addition of a small quantity of cellulose or mineral fibre prevents drainage of bitumen during transport and placement. The essential features, which are the coarse aggregate skeleton and mastic composition, and the consequent surface texture and mixture stability, are largely determined by the selection of aggregate grading and the type and proportion of filler and binder.

SMA is characterized by a stone-on-stone structure. SMA uses a high proportion of larger stones or aggregate that contacts each other. This skeleton of larger stones resists heavy loads by transmitting them to the pavement below. If the under laying pavement is sufficiently strong then the SMA will resist the heavier loads effectively. (A surfacing cannot compensate for a weak pavement).

3.     PERFORMANCE CHARACTERISTICS OF SMA

The development of modern pavement technology is needed to accelerate significant improvement of pavement quality of highways, airport runways and urban roads.

SMA meets the following demands upon an asphalt pavement:

·         Good stability at high temperatures

·         Good flexibility at low temperatures

·         High wearing resistance

·         High adhesive capacity between the stone granules and the bitumen

·         A mix with no tendency to separate

·         Good skid resistance

·         Reduced water spray

·         Lower traffic noise

Good stability at high temperatures

SMA mix has a self-supporting stone skeleton of crushed high quality coarse aggregate, which provides an increase in internal friction and shear resistance and hence its extremely high stability.

Good flexibility at low temperatures

SMA mix has a binder rich mastic mortar which has superior properties over dense graded asphalt in resisting thermal cracking.

High wearing resistance

SMA mix has low air voids, which make the mix practically impermeable, and provide satisfactory ageing resistance, moisture susceptibility and durability.

High adhesive capacity between the stone granules and the bitumen

With the increase of the amount of filler, cellulose fibres are added as stabiliser. The three dimensional structure of cellulose fibre assists the bitumen to maintain a high viscosity, thickens the bituminous film and improves the bitumen/aggregate adhesion.

A mix with no tendency to separate

            An efficient stabilisation of the mastic in order to prevent its segregation from the coarse particles.

Good skid resistance

Because of the macro-texture of the road surface and the use of coarse aggregates with a high Polished Stone Value, SMA pavement achieves a better level of skid resistance.

Reduced water spray

Because of its greater texture depth, there is less water spray, and at night there is fewer glares reflected from the road surface and better visibility of road markings.

Lower traffic noise

SMA road surfaces generally offer lower levels of noise due to the texture properties.

4.     COMPOSITION OF SMA

Stone Mastic Asphalt is characterised by its high stone content which forms a gap-graded skeleton-like stone structure. The voids of the structural matrix are filled with high viscosity bituminous mastic. The high stone content of at least 70% ensures stone-on-stone contact after compaction. The required degree of mastic stiffness is achieved through the addition of crushed sand.

SMA mixes have a bitumen content of minimum 6.5%. The bitumen in the gap-graded mix is stabilised during the mixing process, intermediate storage, transportation, surfacing and compaction through the addition of cellulose fibre stabilising additive.

Addition of cellulose fibre does not chemically modify bitumen, but rather enhances physical property of the finished product by allowing the use of higher bitumen contents. It tends to thicken or bulk the bitumen so that it does not run off the aggregate prior to compaction. The content of cellulose fibre is 0.3% by weight of mixture. If the technological requirements of SMA are fully met, good results can be obtained by just using standard bitumen and a cellulose fibre drainage inhibitor.

  

4.1 Materials Used

1.      Coarse and fine aggregate

2.      Bitumen

3.      Fibre

4.      Filler

4.1.1 Coarse and fine aggregate

The aggregates are crushed by using jaw pressure to get different size of aggregates varying from 16 mm to 75 micron. The coarse aggregate must be hard, durable, and roughly cubical in shape when crushed. Qualities of aggregates were check through various tests like Impact Value Test, Crushing Value Test, Los Angel’s Abrasion Value Test, Flakiness and Elongation Index Test.

4.1.2 Bitumen

Bitumen  act as a binder in SMA mix. Different grade of bitumen are used in different mix like hot-mix or gap-graded mix or dense-graded mix. For preparation of SMA mix we used 60/70 bitumen.

4.1.3 Fibre

Fibres are used as stabiliser in SMA mix. Fibres help to increase the strength and stability and decrease the drain down in SMA mix. There are different types of fibres are used in SMA mix like cellulose fibre, polymer fibre, natural fibre and mineral fibre.

4.1.4 Filler

Filler is used in SMA mix for better binding of materials. Rock dust, slag dust, hydrated lime, hydraulic cement, fly ash, mineral filler and cement are used as filler in SMA mix, also we can use the fine aggregate below 75 micron as filler.

5.     ADVANTAGES

·         20-30% increase in pavement life over conventional pavements

·         Good aggregate interlock

·         Low permeability

·         Improve in skid resistance due to the high percentage of fractured aggregate to motoring public particularly on wet pavement.

·         Surface texture characteristic may reduce sound from the tyre and pavement contact as well as water spray and glare.

·         Strength and stiffness derived from binder and aggregate structure

·         Relatively high binder contents provide good Durability

·         Durability (longer in-service life) of SMA should be equal to, or greater than, DGA and significantly greater than OGA.

·         It provides a textured, durable and rut resistant wearing course.

·         Surface texture characteristics are similar to OGA, so noise generated is lower than DGA but slightly higher than OGA.

·         It can be produced and compacted with the same plant and equipment as for normal hot mix DGA using procedural modifications.

·         SMA can be used on heavily trafficked roads where good deformation resistance is required.

·         Surfacing may reduce reflective cracking from underlying cracked pavements due to its flexible mastic.

·         At the end of its service life it is 100% recyclable.

6.     DISADVANTAGES

·         SMA mix requires higher mixing temperature.

·         Potential construction problem with SMA mixtures are drainage and bleeding.

·         Storage and placement temperatures cannot be lowered to control drainage and bleeding problem due to the difficulty in obtaining the required compacted.

·         Increased material cost associated with high bitumen and filler content.

·         Increased mixing time and time taken to add extra filler may result in reduced productivity.

·         Possible delays in openings (the road) as SMA should be cooled to 40°c to prevent early flushing of the binder to the surface.

·         Needs more carefully monitoring the composition at the mixing plant.

·         Moisture seeping from the SMA surface for long periods after rain.

·         White fines on the surface of the pavements.

·         Premature rutting

·         Stripping of asphalt layers below the SMA surfacing.

·         Potholing

7. APPLICATIONS

Stone Mastic Asphalt has proved superior on heavily trafficked roads and industrial applications:

·         with high lorry frequency

·         intense wheel tracking

·         at traffic lights

·         at intersections

·         on highways

·         on gradients

·         on bridges

·         in bus lanes

·         at bus-stops

·         in car parks

·         in harbours

·         on airport runways

·         on un/loading areas