Tuesday, June 28, 2005

What is Diaphragm Wall?

Diaphragm Walls (U.S. Slurry Walls)

The continuous diaphragm wall (also referred to as slurry wall) is a structure formed and cast in a slurry trench (Xanthakos, 1994). The trench is initially supported by either bentonite polymer based slurries. The term "diaphragm walls" refers to the final condition when the slurry is replaced by tremied concrete that acts as a structural system either for temporary excavation support or as part of the permanent structure. This construction sequence is illustrated in Figure 1.The term slurry wall is also applied to walls that are used as flow barriers (mainly in waste containment), by providing a low permeability barrier to contaminant transport.

Slurry wall technology hinges on specialized equipment for excavating slurry trenches. The simplest type of trenching equipment is the mechanical clamshell attached on a kelly bar. Individual contractors have developed their own specialized trenching equipment like hydraulic clamshells, fraise or hydromills (sample manufacturers: Icos, Bauer, Casagrande, Case Foundation, Rodio etc). Figure 2 shows selected pictures from construction of a new subway in Boston (MBTA South Boston Transit way).

The first diaphragm walls were tested in 1948 and the first full scale slurry wall was built by Icos in Italy in 1950 (Puller, 1996) with bentonite slurry support as a cut-off wall. Icos constructed the first structural slurry wall in the late 1950s for the Milan Metro (Puller, 1996). Slurry walls were introduced in the US in the mid 1960s by European contractors. The first application in the US was in New York City [1962] for a 7m diameter by 24m deep shaft (Tamaro, 1990), that was followed by the Bank of California in San Francisco (Clough and Buchignani, 1980), the CNA building in Chicago (Cunningham and Fernandez, 1972), and the World Trade Center in New York (Kapp, 1969, Saxena, 1974). The majority of diaphragm wall projects in the US are located in six cities Boston, Chicago, Washington DC, San Francisco and New York.

Diaphragm walls are extensively used in the Central Artery/Tunnel project (CA/T) in Boston, Massachusetts (Fig. 3). Work in the CA/T involves many cut and cover tunnels constructed under the existing artery. Some of the deepest T-slurry walls, extending 120' below the surface have been constructed for the Central Artery (Lambrechts et al., 1998).

Figure 1: Typical construction sequence of slurry walls: (A) Trenching under slurry, (B) End stop inserted (steel tube or other), (C) Reinforcement cage lowered into the slurry-filled trench, (D) Concreting by tremie pipes.

Figure 2: Trenching equipment, (A) Mechanical clamshell in front and hydraulic clamshell in the back, (B) Smaller size mechanical clamshell

Figure 4: Central Artery Tunnel Project, Boston MA (Ladd et al., 1999)

Monday, June 27, 2005

Finite Element Analysis (3) Buckling

Buckling Analyses: Sudden Collapse

    Buckling is a critical state of stress and deformation, at which a slight disturbance causes a gross additional deformation, or perhaps a total structural failure of the part. Structural behaviour of the part near or beyond 'buckling' is not evident from the normal arguments of statics. Buckling failures do not depend on the strength of the material, but are a function of the component dimensions & modulus of elasticity. Therefore, materials with a high strength will buckle just as quickly as low strength ones.

    If a structure has one or more dimensions that are small relative to the others (slender or thin-walled), and is subject to compressive loads, then a buckling analysis may be necessary.

    From an FE analysis point of view, a buckling analysis is used to find the lowest multiplication factor for the load that will make a structure buckle. The result of such an analysis is a number of buckling load factors (BLF). The first BLF (the lowest factor) is always the one of interest. If it is less than unity, then buckling will occur due to the load being applied to the structure. The analysis is also used to find the shape of the buckled structure.

Evaluating Linear Instabilities
    From a formal point of view, buckling is an eigenvalue problem that is a function of the material & geometric stiffness matrices. Consequently, there will be a number of buckling modes and corresponding mode shapes.

    As with a frequency analysis, eigenvalue extraction may be carried out using a number of available methods, the best choice depends on the form of the equations being solved. The main methods are the power, subspace, LR, QR, Givens, Householder & Lanczos methods.

    An important note is that the eigenvalue method does not take into account of any initial imperfections in the structure and so the results rarely correspond with practical tests. Eigenvalue solutions usually over estimate the buckling load and give no information about the post-buckling state of the structure. Sudden buckling simply does not occur in the real world.

    So how should we know if a linear buckling analysis is sufficient ?? Carry out both a linear static analysis and a linear (eigenvalue) buckling analysis. If the max stress is significantly less than yield, and the buckling load factor is greater than 1.0, then buckling will probably not occur. If however the BLF is less than 1.0, then the buckling analysis will be linear provided that the max stress is far below yield. In all other cases, a non-linear buckling analysis should be carried out. If the component is critical to the safe operation of a system, full displacememnt analyses should be carried out.

Non-Linear Buckling
    A more practical approach is to carry out a large displacement analysis, where buckling can be detected by the change of displacement in the model. A large displacement problem is non-linear in nature. Geometric non-linearity arises when deformations are large enough to significantly alter the way load is applied, or load is resisted by the structure.

    The approach to a non-linear buckling solution is achieved by applying the load slowly (dividing it into a number of small loads increments). The model is assumed to behave linearly for each load increment, and the change in model shape is calculated at each increment. Stresses are updated from increment to increment, until the full applied load is reached. The solution becomes an iterative procedure rather than one of matrix factorisation alone, and consequently is computationally expensive.

    An interesting variation arises in the case of automotive applications. In the case of front end collision, the hood is expected to crumple (buckle) in order to absorb the energy of collision, as well as to save the passenger compartment. In such cases, we are not designing against, but for buckling.

Avoiding Instabilities
    Any structure is most efficient when subjected to evenly distributed tensile or compressive stress, such as occurring in cables, strings etc. Evidently, such modes of loading makes the best use of the material, and its strength. On the other hand bending (flexing) is the least efficient way of loading a structure. A high flexural stiffness of the structure means high resistance to buckling. This is true even if the load is entirely in-plane, since when buckling is imminent, the only stiffness that counts is flexural.

    Eccentricity of loading promotes buckling. Eccentricity means that the resultant load does not pass through the centroid of the load bearing cross section. It is safe to assume that in 100% of practical applications, loads are eccentric.

    When buckling occurs, symmetry of the part does not apply. There is no symmetry of the buckled shape, although both the part, and the loading may be symmetric. Correspondingly, when carring out an FE buckling investigations, it is advisable to implement a full 3D analysis of the structure under inspection.

    The non-linear stress strain behavior of the material reduces the stiffness at higher stress (load) levels, and hence elastic formulas from the handbooks tend to be highly unconservative.

    If a component is structurally slender, and is made of plastic, then the component faces buckling from three directions; from the low material stiffness, the large deflections producing eccentricity during deformation, and from the non-linearity of the material itself.

    By and large it is true that buckling usually occurs when compressive stress is present. But what is not evident that compressive stress can prevail in un-expected places. Shallow domes under internal pressure can develop local compressive stress regions, and make it vulnerable to instabilities.

Bifurcation & Snap Through Buckling
    In many systems a smooth change in a control parameter (the load) can lead to an abrupt change in the behaviour of the system. A simple example is the buckling of a rod. If a straight rod is compressed by a small load, it shrinks to some extent, but remains straight. For larger loads, however, it starts to buckle. Mathematically, the solution corresponding to a straight rod still exists, but it is unstable for the large load applied and very small transverse perturbations make the rod buckle. The transition from the unbuckled to the buckled state occurs via a bifurcation, that is, at the onset of the instability a new solution corresponding to the buckled rod comes into existence. In bifurcation buckling, there are two equilibrium solutions at the bifurcation point, the ordinary static strength of materials solution and the instable (buckling) solution.

    Snap through buckling occurs when a structure is subject to an increasing load that at some point causes the structure to undergo a gross deformation. Subseqent to this deformation, the structure regains sufficient stability to carry load, usually in a configuration that changes the structural load from being initially compressive to tensile. An example of this is a shallow dome in compression. If the load becomes too great, it buckles and snaps through so that the load is supported in tension.

Sunday, June 26, 2005

When would you be better off not wearing your seatbelt instead of airbag provision?

My friend Eddie's Uncle Joe heard about this truck driver who narrowly avoided a fiery wreck by plunging his truck through the gaurdrail and down a 100 meter cliff. If he had had his seatbelt on, he would never have been able to jump out at the last second and hang on to the gaurdrail.

Maybe so.

But the work-energy principle must be satisfied in every collision, and it dictates that the work done in stopping the driver must be equal to the driver's kinetic energy. The shorter the stopping distance, the greater the impact force. And cases where the seatbelt would not lengthen your stopping distance and decrease your impact force are about as rare as this kind of accident.

Rather than making judgements about safety from anecdotes like the one above, it is wise to consider the evidence from the large database on traffic fatalities.

While the driver with an airbag may experience the same average impact force as the driver with a good seatbelt, the airbag exerts an equal pressure on all points in contact with it according to Pascal's principle. The same force is distributed over a larger area, reducing the maximum pressure on the body.

The presence of an airbag should not be used as justification for not wearing the seatbelt! The seatbelt keeps the driver from moving out of the position where the airbag is effective in capturing the driver and cushioning the impact.

Saturday, June 25, 2005

Does the Berlin Wall still Exist?

Soldier Pile and Lagging Walls

Soldier pile and lagging walls are some of the oldest forms of retaining systems used in deep excavations. These walls have successfully being used since the late 18th century in metropolitan cities like New York, Berlin, and London. The method is also commonly known as the "Berlin Wall" when steel piles and timber lagging is used. Alternatively, caissons, circular pipes, or concrete piles can also be used as soldier piles but at an increased cost. Timber lagging is typically used although reinforced concrete panels can be also utilized for permanent conditions. Soldier pile and lagging walls are formed by:

1. Constructing soldier piles at regular intervals (6 ft to 12 ft, typical)
2. Excavating in small stages and installing lagging.
3. Backfilling and compacting the void space behind the lagging.

Moment resistance in soldier pile and lagging walls is provided solely by the soldier piles. Passive soil resistance is obtained by embedding the soldier piles beneath the excavation grade. The lagging bridges and retains soil across piles and transfers the lateral load to the soldier pile system.

Soldier pile and lagging walls are the most inexpensive systems compared to other retaining walls. They are also very easy and fast to construct. The major disadvantages of soldier pile and lagging systems are:

1. They are primarily limited to temporary construction.
2. Cannot be used in high water table conditions without extensive dewatering.
3. Poor backfilling and associated ground losses can result in significant surface settlements.
4. They are not as stiff as other retaining systems.
5. Because only the flange of a soldier pile is embedded beneath subgrade, it is very difficult to control basal soil movements.

Friday, June 24, 2005

Why should we use a seatbelt?

Forces in Car Crashes

Example of Force on Car

Using seat belt makes it different

For the car crash scenario where a car stops in 1 foot from a speed of 30 mi/hr, what is the force on the driver? Assume a 160 lb (mass = 5 slugs) driver.

If firmly held in non-stretching seatbelt harness: Stopping distance 1 ft.

  • Deceleration = 967 ft/s2 = 294 m/s2 = 30 g's
  • Force = 4813 lb = 21412 N = 2.4 tons
Non-stretching seatbelt
If not wearing seatbelt, stopping distance determined by nature of collision with windshield, steering column, etc. : stopping distance 0.2 ft.

  • Deceleration = 4836 ft/s2 = 1474 m/s2 = 150 g's
  • Force = 24068 lb = 107059 N = 12 tons!!
No seatbelt!
If seat belt harness stretches, increasing stopping distance by 50%: 1.5 ft.

  • Deceleration = 645ft/s2 = 197 m/s2 = 20 g's
  • Force = 3209 lb = 14274 N = 1.6 tons
Stretching seatbelt
These calculated numbers assume constant deceleration, and are therefore an estimate of the average force of impact.

Why does a Bigger Truck always win?

Truck Collision

In a head-on collision:
Which truck will experience the greatest
Which truck will experience the greatest
Which truck will experience the greatest
change in momentum?
Which truck will experience the greatest change in velocity?
Which truck will experience the greatest acceleration?
Which truck would you rather be in during the collision?


Comparison of the collision variables for the two trucks:

In a head-on collision:

Newton's Third Law dictates that the forces on the trucks are equal but opposite in direction.

Impulse is force multiplied by time, and time of contact is the same for both, so the impulse is the same in magnitude for the two trucks. Change in momentum is equal to impulse, so changes in momenta are equal. With equal change in momentum and smaller mass, the change in velocity is larger for the smaller truck. Since acceleration is change in velocity over change in time, the acceleration is greater for the smaller truck.

Ride in the bigger truck! There are good physical reason!

Truck Collision

In a head-on collision the forces on the two vehicles are constrained to be the same by Newton's Third Law. But from both Newton's Second Law and the Work Energy's Principles it becomes evident that it is safer to be in the bigger truck.

The change in velocity of the driver will be the same as the truck in which he/she is riding. A greater change in velocity implies a greater change in kinetic energy and therefore more work done on the driver.

Vehicle Mass and Accidents

The more massive vehicle in a two-vehicle collision would be presumed to be safer since it would undergo less change in velocity during the collision. Not so evident is the fact that the more massive car is safer in a single-car accident. Leonard Evans collected data from the FARS database for a Fatality vs Mass comparison. His problem in assessing the rate of incidence of fatal accidents was to have a base for comparison since non-fatal single car accidents are not included in the database, nor are the number of cars in a given mass range. He used the number of pedestrian fatalities as a base, a "surrogate" for the number of serious accidents of all types, since it was presumed that the ratio of collisions with pedestrians to other types of collisions, e.g. with trees, would be similar for any mass class. It was also presumed that the pedestrian fatality rate would be more-or-less independent of car mass since even the lightest car is so much more massive than a pedestrian. The self-consistency of the curves for different ages of driver offers some evidence of validity since their accident rates were much different.

Thursday, June 23, 2005

What is Ground Anchor?

Anchored walls have become popular in braced excavations because of a) the substantial progress in the technology and availability of high-capacity anchor systems, and b) the absence of interior obstructions that permit uninterrupted earth moving and thus improve the construction conditions of the underground portion of a building (Xanthakos, 1994). Figures 1 and 2 show a photo and a schematic of tied-back slurry wall excavations. In some projects tiebacks have been used in combination with rakers and soil berms and/or corner braces (Gnaedinger et al., 1975). Tieback anchors comprise a barrel anchorage located either in a bearing layer which is tensioned at the front face of the wall. The part of the anchor that transfers the force to the surrounding soil is frequently called the "fixed length", while the "free length" transmits forces from the fixed length through the anchor head to the slurry wall.

In order to minimize wall movement and ground settlement, tieback anchors are designed to achieve the highest stiffness possible within economical considerations. In urban cities like Boston, Chicago, New York, and Washington where land is precious such deep excavations are more common. Tieback capacity depends on the vertical and horizontal spacing of anchors and on surcharge conditions. Prestress levels typically range from 40 to 250 kips when the grouted portion of tiebacks is within soil, higher loads are used when the ties are located in bedrock. Typical tieback spacing ranges from 7ft to 13ft in the vertical, and from 5ft to 15ft in the horizontal direction (from the current database). Tieback capacity is reduced if the spacing is too close due to interference between adjacent grouted zones.

Often the tiebacks are used only for temporary excavation support, while the basement floors provide permanent lateral earth support. In such projects the tiebacks are detensioned when the basement floors have gained sufficient strength. The basement floors should be designed to resist permanent lateral earth pressures, since stress transfer from the tiebacks to the floor system will take place when the ties are detensioned. This stress transfer has reportedly caused long-term cracking of many the basement floors.
Tieback installation follows a predetermined sequence as to minimize soil movements and speed the excavation construction (Fig. 3). The excavation is carried a couple of feet below the tieback to enable access for the drill rig. Further excavation occurs only after prestressing and proof-testing of the anchors. As Figure 4 illustrates, the process can be repeated for additional levels of tiebacks. Building codes require that all tiebacks are proof-tested to an excess percentage of their final lock-off load, which usually ranges from 120 to 150% of the final lock-off load. Regroutable tiebacks are most commonly used because their capacity can be increased by regrouting (to meet test requirements) without having to drill a new anchor hole.

A tieback is made by first drilling a hole with an auger and then placing a bar (tendon) in the hole, concrete is then poured in the hole and the connection with wall is made (Figure 3). Different types of augers are used to drill the tieback holes. The choice of the drilling method depends on the soil/rock conditions on the site.

Drilling should be done carefully since inadequate procedures can cause significant soil losses. The biproduct of drilling is removed by flushing the hole with either air, water, or slurry. Air is most efficient in dry ground, but it requires special attention because it can become entrapped during drilling, building up zones of high pressure in the soil that can eject material for several feet and at high speeds (potentially injuring workers). Water flushing is best used in sticky clayey soil, and it also cleans the sides of the hole by its sweeping action, providing a stronger bond at the grout-anchor interface. Bentonite slurry flushing works the best since it keeps particles in suspension, while the sealing action keeps the hole from collapsing.

Significant soil losses through the tiebacks cause significant settlements even if the retaining walls do not move towards the excavation. In granular soils the drilled hole must be cased to avoid collapse.
Some tieback creep can be expected especially if the ties are very short and the fixed length of the tie is within soft ground. For stability reasons, the fixed anchor should be located beyond the active zone of movements. As a result, tieback anchors may not be an option at sites congested where there are adjacent underground utilities or when adjacent owners do not grant permission to drill them under their properties.

Special attention should be given to the waterproofing details at the anchor heads and at the tieback holes. Significant leakage can be caused by inadequate waterstopping details at these locations.

Figure 1: Picture from a tieback slurry wall excavation (World Bank Project Washington)

Figure 2: Tieback slurry wall excavation (Dana Farber Tower, Boston).

Figure 3: Tieback configuration, free and fixed lengths (Adapted from Schnabel, 1982)

Figure 4: Steps in making a tieback: (a) hole drilled; (b) bar placed in hole; (c) concrete poured for anchor; (d) wall connection made (Adapted from Schnabel, 1982).

Figure 5: Steps in making a multilevel tieback excavation, (A) first level of tiebacks installed and second level of tiebacks drilled, (B) second level of tiebacks installed.

Refference: www.deepexcavation.com

Wednesday, June 22, 2005

What is Top/Down Construction?

Top/down or up/down construction methods are another method for constructing deep excavations. In this case the basement floors are constructed as the excavation progresses. The top/down method has been used for deep excavation projects where tieback installation was not feasible and soil movements had to be minimized. Figures 1 through 2 show construction photographs from two top/down excavations in Indonesia. The general top/down construction sequence is shown in Figure 3.. The Post Office Square Garage in Boston (7-levels deep) is one of the best-instrumented and documented top/down projects in the US (Whittle, et al., Whitman et al., 1991).

The sequence construction begins with retaining wall installation and then load-bearing elements that will carry the future super-structure. The basement columns (typically steel beams) are constructed before any excavation takes place and rest on the load bearing elements. These load bearing elements are typically concrete barrettes constructed under slurry (or caissons). The top few feet of a barrette with a steel beam can be seen in Figure 2. Then the top floor slab is constructed with at least on construction (glory) hole left open to allow removal of spoil material (Figs. 2, 3).

The excavation starting at the glory hole begins once the top floor has gained sufficient strength. Soil under the top basement floor is excavated around the basement columns to slightly lower than the first basement floor elevation in order to allow for the installation of the forms for the first level basement slab. Glory holes are left open within each newly formed basement floor slab and the procedure is repeated. Each floor rests on the basement columns that were constructed earlier (Fig. 2).

Figure 1:Top/Down Excavation (Pasar Baru Bandung, Project)

Figure 2: Menara Merdeka Project. Left: Looking up at a glory hole, Top right: Llowest most level note LBE on the left and the barrette, Bottom right: close up view the same barrette (LBE) and steel beam.

Figure 3: Top/down basic construction.
a) Slurry wall and basement column construction
b) Ground floor construction and pouring
c) Excavation and floor construction under and above the ground floor
d) Excavation & lowest basement floor completed

The Top-Down Construction in Indonesia

The top down construction in Indonesia was introduced for the first time in 1994. The project was the Bank Indonesia C-Building Projects (12 stories, and 3 basement), and PT. (Persero) Wijaya Karya was the main contractor.

In, 1995 another project was Menara Merdeka Building (30 stories, 3 basement) also built by top-down construction. The main contractor was joint operation of PT. (Persero) Pembangunan Perumahan and PT. (Persero) Wijaya Karya.

The developing of this method was introduced by Ir. Ign. Chen, and called the new concept as Up & Down Construction. The modification was the dismissing of diagprahm wall as the element of the basic top-down construction method. The first project that using the Up & Down Construction Company was Bandung Electronic Centre (BEC), 6 stories plus 2 basement, in 2002, and the main contractor was PT. (Persero) Pembangunan Perumahan.

The largest project in Indonesia that use the top-down construction was Pasar Baru Bandung Project , 2003. The huge building was consisted 120,000 sq.m, 10 stories plus 3 basement, and the time line for complishing the project was only 8 months. The main contractor was PT. (Persero) Pembangunan Perumahan.

The Project Manager of Pasar Baru Bandung, decided to deploy the Up & Down Construction with several modifications to take that challenge. The Pasar Baru Bandung was acomplished precisely at the time, and regarded as the fastest commercial building project at that time.

There were 6 top down construction projects until the year of 2005 in Indonesia:
1. Bank Indonesia C Building Projects - PT. (Persero) Wijaya Karya - Ir. Abbas Sudrajat
(Project Manager).
2. Menara Merdeka Building Project - PT. (Persero) Pembangunan Perumahan &
PT. (Persero) Wijaya Karya - Ir. H. Sugeng Haryono, MM (Project Manager)
3. Puri Mustika Hotel Building Project - PT. (Persero) Pembangunan Perumahan -
Ir. Andi Reman (Project Manager)
4. Bandung Electronic Centre (BEC) Building Project - PT. (Persero) Pembangunan
Perumahan - Ir. Anton Satyo Hendriatmo (Project Manager)
5. Pasar Baru Bandung Building Project - PT. (Persero) Pembangunan Perumahan -
Ir. Anton Satyo Hendriatmo (Project Manager)
6. St. Baromeus Hospital Bandung Project - PT. (Persero) Pembangunan Perumahan-Ir. Nurlistyo Hadi (Project Manager)

Finite Element (2) Dynamic Analysis

Time Dependant Dynamic Analyses: Modelling Impulse Problems

    If the excitation applied to a structure is impulsive rather than harmonic, many modes contribute to the response and it becomes more appropriate to use direct integration methods rather than modal analysis. There are a large number of applications where transient analyses are necessary. Many structures are subject to time varying loads such as impluse, blast, impact & seismic loadings. Transient dynamic analysis determines the time-response history of a structure subjected to a forced displacement function. The structure may behave linearly, or in some cases, friction, plasticity, large deflections or gaps may produce nonlinear behavior. Once the time response history is known, complete deflection and stress information can be obtained for specific times.

    The first step in any dynamic analysis should be the determination of the frequencies and shapes of the natural vibration modes. In a 3-D structure there are three dynamic degrees of freedom (DDOF) for every unrestrained node with non-zero mass and there is potentially a natural vibration mode for each DDOF. Thus, there are usually many potential vibration modes in a typical structure, but usually only a small number of vibration modes with the lowest frequencies that are of interest. In a multi-storey building, for example, it might be only a few in each of two horizontal directions, plus one or two torsional modes that have to be considered.

Frequency & Transient Analysis Differences
    While frequency analyses take place in the frequency domain, transient analyses are studies in the time domain. It is always possible to go from the time domain to the frequency domain via a fourier transform. Correspondingly, a change from the frequency domain to the time domain may be achieved by implementing an inverse fourier transform.

    Due to mathematical difficulties, solutions in the frequency domain can only be linear in nature. Therefore if the application requires a solution that is a non-linear function of time, then a time domain analysis must be carried out. The solution can subsequently be projected to the frequency domain if required. Frequency analyses can be solved using no boundary conditions, while transient analyses must be fully constrained.

Transient Solutions: Modal & Direct
    There are usually two approaches one can take when carrying out a transient analysis, modal solutions or purely direct solutions.

    The modal approach involves evaluating the relevant natural frequencies of a structure first. Once this is carried out, the response is converted to the time domain and is included in the evaluation of transient response of the structure. Modal analyses are usually used where there many natural frequencies within the operation range. It is important to evaluate the natural frequencies above and below that of the analysis range. This is due to the fact that, in practice, there is never just one distinct mode of vibration due to an excitation, but one dominant mode with a range of additional harmonics from the adjacent upper and lower modes.

    Direct solutions are used to evaluate the response of a structure within a very narrow frequency range of interest, and are usually used for models that subject to high frequency impulses. The solution is purely transient, no frequency extraction is carried out first.

The Solution Approach
    As a transient load is applied, the solution must follow the response of the structure. To achieve this, the overall time period being studied is divided into a number of linear time pieces, each one being referred to as a time step. The successful implementation of any time domain analysis is dependant on a suitable number of time steps being selected. If the time step is too large, portions of the response (such as spikes) could be missed or truncated. On the other hand, if the time step is too small, the analysis will become excessively long or even prohibitive.

    The progression of the solution from one time step to the next is achieved by implementing time integration techniques. Despite many packages providing automatic time stepping estimates, the full response of the structure may not be captured, and manual intervention will be required. If there is a discontinuity in your automatically time stepped results, chances are there is a spike in the response that is not being fully captured.

Stepping Schemes: Time Integration
    Many time stepping algorithms have been developed, each having their advantage over others under certain circumstances. However, three main types of solution dominate, backward difference (Implicit), central difference (Crank-Nicolson) & forward difference (Explicit). The explicit & implicit techniques are often referred to as Euler's rule & the backwards Euler's rule respectively.

    Explicit schemes, which are conditionally stable (stability of solution not guaranteed), find the response at the end of the time step in terms of the conditions at the start of the time step. In other words, the calculation of the solution at time (t+Dt) is obtained by considering the situation at time t. The advantage of this approach is that the underlying system of equations that comprise the model (stiffness matrix, capacitance matrix, flexibility matrix) does not have to be solved at each time step. Furthermore, the material & time matrices can be diagonalised to become uncoupled, and so the solution can be calculated explicitly. Very fast calculations of individual time steps can be achieved as no matrix factorisation is required. However, the technique is much less stable than the implicit method, so very small time steps must be used to ensure an appropriate solution.

    Implicit schemes, which are unconditionally stable, find the response at the end of the time step in terms of the conditions at the end of the time step. In other words, the calculation of the solution at time (t+Dt) is found by considering the response at time (t+Dt). An important point to note is that the solution at each time step involves matrix factorisation (evaluating the system of equations that comprise the model), which is a computationally intensive process. Despite this disadvantage, implicit schemes are often used, as the solution is inherently reliable & robust. Implicit analyses allow much larger time steps than the others, and so the solution can be obtained with fewer calculation increments. As implicit schemes are always stable, the time step length is governed by considerations of accuracy alone.

    The Crank-Nicolson approach evaluates the next step of the solution by using the prediction at the centre of the time step. As with the backward difference scheme, this is an implicit solution which is conditionally stable (results in an oscillatory solution if the critical time step for stability is exceeded). The central difference method is more accurate than both the purely implicit or explicit techniques since neither favours the response at the start or end of the time step.

Response Spectrum Analysis
    Response spectrum analysis (RSA) is a procedure for computing the statistical maximum response of a structure to a ground bourne excitation. Each vibration mode considered may be assumed to respond independently as a single-degree-of-freedom system. Design guideline codes specify response spectra that determine the base acceleration applied to each mode according to its period (the number of seconds required for a cycle of vibration). The design response spectrum is then usually obtained by multiplying the basic acceleration coefficient by a factor based on required structural performance, risk & location.

    Having determined the response of each vibration mode to the excitation, it is necessary to obtain the response of the structure by combining the effects of each vibration mode. Because the maximum response of each mode will not necessarily occur at the same instant, the statistical maximum response, where damping is zero, is taken as the square root of the sum of the squares of the individual responses.

    Response spectrum analysis produces a set of results for each excitation load case which is in the form of an envelope. All results are absolute values, each value represents the maximum absolute value of displacement, moment, shear, etc. that is likely to occur during the event which corresponds to the input response spectrum.

Concepts associated with Dynamic Strucural Analyses
    SHAKEDOWN ANALYSIS: If load intensities on a structure remain sufficiently low, the response of the body is purely elastic (with the exception of stress singularities). If the load intensities become sufficiently high, the instantaneous load-carrying capacity of the structure becomes exhausted (unconstrained plastic flow and damage evolution occurs) & collapses.
    If the plastic strain increments in each load cycle are of the same sign then, after a sufficient number of cycles, the total strains (and therefore displacements) become so large that the structure departs from its original form and becomes unserviceable. This phenomenon is called incremental collapse or ratchetting.
    If the strain increments change sign in every cycle, they tend to cancel each other and total deformation remains small leading to alternating plasticity. In this case, however, the material at the most stressed points begins to fails due to low-cycle fatigue.
    If, after some time plastic flow and damage evolution cease to develop further and the accumulated dissipated energy in the whole structure remains bounded such that the structure responds purely elastically to the applied variable loads, one says that the structure shakes down

    FLUTTER is a dynamic instability that involves coupling of aerodynamic forces and elastic and inertial forces of the structure. In a flow, an oscillating structure generates unsteady aerodynamic forces. These unsteady aerodynamic forces introduce coupling into the structure and cause phase shifts between the motions of the structure (degrees of freedom). The speed of the flow affects the amplitude ratios and phase shifts between the various degrees of freedom in such a way that energy is extracted from the airstream. At the critical airspeed, the energy dissipated is exactly equal to the available structural damping. At speeds greater than the critical speed, the extracted energy dissipated is less than available structural damping and the motion is divergent.

Tuesday, June 21, 2005

How is to build a Skyscraper in Earth Quake Zone Area?

Bad news: You just built this skyscraper in a major earthquake zone. If you don't strengthen this skyscraper before the next earthquake hits, it will whip back and forth and snap in pieces! You need to further brace this tall building before the earth moves or you're in deep trouble.

How will you strengthen your skyscraper?

 Rigid backbone with diagonal braces
 Long, supporting arms
 Rigid backbone with solid, concrete walls
Rigid backbone with diagonal braces (A)

Long, supporting arms (B)

Rigid backbone, with solid concrete walls (C)

Answer: A
Build a rigid backbone through the core of the building with diagonal steel braces

You're on the right track!

 Build a rigid backbone through the core of the building with diagonal steel braces = Caution

A rigid core made of diagonal steel braces would do a good job holding the tall, wobbly building together while the ground shakes during an earthquake. But there is another material that you should add to the core of the building. It tends to be cheaper than steel, and it has great stiffness in the horizontal direction -- a great characteristic to have in an earthquake.

Answer: B
Build long, supporting arms

Watch out -- buttresses aren't very sturdy in an earthquake!

 Build long, supporting arms = Caution

Long, supporting arms, called buttresses, would be useless during an earthquake. They'd snap and break right off the tall building as soon as the earth started shifting back and forth. Besides, downtown Los Angeles is pretty crowded. There's not enough room for the humungous buttresses you'd need to brace a skyscraper this tall!

Answer: C
Build a rigid backbone through the core of the building with solid, concrete walls

Congratulations! Your skyscraper is ready for the next big earthquake!

 Build a rigid backbone through the core of the building with solid, concrete walls = Right

Solid concrete walls, also called shear-walls, have great stiffness in the horizontal direction. This means that when the ground moves back and forth during an earthquake, the shear backbone of your skyscraper will keep it standing strong and firm.

The First Interstate World Centre in Los Angeles, the tallest building in the world in a major earthquake zone, has a solid concrete core right up the center of the building. This design allows it to withstand an earthquake with a magnitude of 8.3 on the Richter scale! Great job!

Why could the Skyscraper collapse? How to prevent it?

Location: Los Angeles, California
Building Description: 75-story steel-reinforced concrete skyscraper
Problem: The building is in danger of collapsing from poor design
Task: Recommend a new structural design
Budget: $500 million

Special Notes: "This is going to be a state-of-the-art building. We're planning to have 100 deluxe apartments, two swimming pools, 50 shops, and one movie theater in this skyscraper. We want lots of open space and large, unobstructed windows on each floor. Can you do this for us?"
-- Building Owners

The pressure is on. You want to make the owners of the skyscraper happy, but you also have to decide which design is strong enough to support the tall and heavy structure.

Which structural design will you choose?

 Columns Throughout:
 Stiff Core
 Hollow Tube

Answer: A
Columns Throughout

Your skyscraper is standing tall -- but there are way too many columns inside!

 3-D Grid: beams and columns evenly spaced throughout skyscraper = Caution

This design is not ideal for this particular skyscraper because the owners want lots of open space on each floor. With this design, there will be too many columns in the way!

Engineers used this structural design to build one of the earliest American skyscrapers, the Empire State Building in New York City, which is also one of the heaviest buildings in the world. Builders used 60,000 tons of steel to construct the Empire State Building! Luckily, engineers have developed newer, more inexpensive designs that require less material and time to build.

Answer: B
Stiff Core

You've given the owners what they want. Congratulations!

Stiff Core: beams, columns and diagonal braces form a rigid backbone up the center of the skyscraper = Right

The stiff core forms a rigid backbone up the center of the skyscraper. This design gives the owners plenty of space to arrange apartments, movie theaters, and shops on each floor. Plus, the stiff inner core acts like the spine of the human body and provides extra structural support for the skyscraper against heavy winds. It also creates space for an elevator shaft. After all, who wants to climb 1,000 feet of stairs?

One of the world's tallest skyscrapers, the Chrysler Building in New York City, is strengthened by a stiff inner core. Super job!

Answer: C
Hollow Tube

You've just designed a good, strong building -- but the columns around the perimeter block the view!

 Hollow Tube: beams, columns and diagonal braces stiffen the outside walls and form a stiff, hollow tube = Caution

The hollow tube design is an inexpensive and very effective skyscraper structure. The fifth and sixth tallest buildings in the world, the World Trade Centre Towers in New York City, are supported by this hollow tube structural design.

Unfortunately, the columns around the perimeter significantly reduce the size of the windows on each floor. This design simply won't work for your skyscraper because the owners specifically asked for large, unobstructed windows.