Concrete

Introduction

Consider filming a version of the cylinder crush video and replacing the linked one here.

General description of concrete chemistry video.

Crushing a concrete cylinder video ($f'_c = 6,288 psi$).

Concrete is a composite material, consisting of cement and aggregate. Once fully cured, concrete is very strong in compression. High strength concrete can resist as much as 19 ksi (think six, 2013, Chevrolet Malibus being supported on one square inch of concrete). This is not as much as steel, but it weighs less, and is much cheaper. Additionally, it is fire resistant; so much so that it is sometimes used to protect steel. What you see in the video of a concrete cylinder being crushed (if available) is a sudden failure due to a cylinder being compressed. However the failure is technically due to tension developed via the Poisson's ratio effect. Again, this is something you'll see in other classes.

By the way, the organization that publishes the concrete building code is the American Concrete Institute (ACI). You will also be hearing more about them in other classes.

Brief History

Concrete was reportedly first used by the ancient Mycenaeans in Greece in between 1,400 and 1,200 BC (source), however less effective versions used earlier by the ancient Egyptians. When talking about ancient structures, the use of concrete by the Romans' is generally far better known. As such, examples of its use by them in structures that are still standing, like the Pantheon, the aqueducts and Colosseum (being made partly, or wholly made from concrete), and speak to concrete's durability.

Modern concrete was first developed in 1824 by Joseph Aspdin. The term "Portland cement" was coined by him since the hardened substance resembled limestone from the Isle of Portland in Dorset. Reinforced concrete was developed in the 1850s, with a patent being issued to a French gardener named F. Joseph Monier in 1857. The use of reinforcement provides a method to overcome the lack of tensile strength in unreinforced concrete.

Sustainability

Currently concrete manufacturing is one of the larger sources of CO$_2$. The production of concrete is responsible for $\approx$5% annually (377 million metric tons of carbon in 2007) (source). However there are several mix designs that reduce these emissions. The way this is accomplished is by either reducing the amount of energy consumed during the manufacturing process (using more fly ash, extreme example) and/or replacing calcium oxide (aka quicklime) with others that don't emit CO$_2$ during manufacturing. There is even research looking at incorporating biofuel waste to supplement the calcium oxide.

The concrete industry is very interested in being considered as producers of a sustainable product and provide documentation on how to use concrete in LEED certified projects.

Potential LEED points gained through the use of ready mixed concrete are discussed in the following areas: stormwater management; landscape paving; minimizing energy use; optimizing energy performance; managing construction waste; recycled content; use of regional materials; use of certified wood; innovation in design; and reduction in the use of portland cement.

...

Pervious Concrete Paving - Pervious paving allows water to percolate to the ground, filter contaminants and recharge groundwater and aquifers. Pervious concrete consists of specially formulated mixtures of portland cement, uniform, open-graded coarse aggregate, and water. It usually contains little or no sand. It has enough void space to allow rapid percolation of liquids through the pavement. ... It is placed over a layer of open-graded gravel and crushed stone. Voids in the aggregate layers act as a storage reservoir for stormwater.

Source - Ready Mixed Concrete Industry LEED Reference Guide

They go on to discuss the pollution removing aspects of the use of pervious concrete paving.

They provide contrast with asphalt in regards to the heat island effect. Gray concrete reflects 35-45% of solar energy back into the atmosphere and white concrete will reflect 70-80%. This is apposed to new and weathered asphalt which only reflects 5-15%. The energy that is not reflected is converted to heat which causes the temperature to rise.

The report does note that white concrete is more expensive. Additionally if concrete contains fly ash it may become darker and if it contains slag it may be lighter.

Overall there aspect of concrete which are very useful in terms of sustainability. There are even mixes of concrete that include titanium dioxide that are designed to remove smog from the air.

Self Healing Side Note

The concept of self healing concrete is base on the concept of embedding a repair mechanism within the mix itself. A student within this construction materials class was interested in the topic and did research upon it. This research yielded five different approaches, “chemical encapsulation, bacterial encapsulation, mineral admixtures, chemicals in glass tubing, and intrinsic self-healing”.

First let's discuss the motivation for self healing concrete. As I will mention later, concrete placed in tension is generally one tenth as strong as concrete in compression. To make up for this structural deficiency, concrete is typically designed in such a way that steel, placed within the concrete, carries the tension load. Tension is commonly encountered in structures, in bending of beams for instance. The reinforced concrete member, once placed in tension, will develop cracks. Additionally, concrete shrinks as it ages, which can result in shrinkage cracks if the edges of the concrete are restrained. The point is that cracks will occur in most concrete members during their regular usage. When these cracks occur, if the concrete is exposed to water then the reinforcing steel in the concrete can become corroded. Corroding steel will expand, which will damage the concrete further, as well as reduce the tension carrying capacity of the reinforced concrete member.

None of these self healing approaches have yet lead to a widely adopted material, however research done using the bacterial encapsulation in the Netherlands has made some head way, and wound up in the news (CNN report). This approach is similar to the other approaches in that it's goal is to prevent damage from being done to the reinforcing steel to start with. To accomplish this, all of the approaches seek to fill cracks with a material that seals the steel off from the water. The Netherlands approach use capsules filled with bacteria, and calcium lactate (a food source). When a crack occurs, capsules that intersect the crack break releasing their contents. If water then penetrates the crack then the bacteria then become active, consuming the calcium lactate, and producing limestone. If the crack is small enough then this waste product can fill the crack and block any further flow of water.

Most of the self healing approaches attempt to tackle to the problem of corrosion prevention in a similar fashion. Cracking triggers a reaction that hopefully fills the very crack that started it.

This technology will be beneficial in extending the lifespan of concrete structures, and in the future, as the techniques become more sophisticated, it will become more widely available. Once it is, it can begin to reduce the production of green house gases by reducing the need to replace damaged concrete structures. It should be noted that there are also potential benefits in the area of hazardous material containment using self healing concrete.

Toxicity and the Environment

Concrete is generally not toxic, but before it has cured it's a very caustic liquid, with a pH level about the same as bleach (12-13). Exposure for an extended period of time can even lead to third degree chemical burns. For this reason both eye and skin protection is required when pouring concrete. (Source: MSDS) It is also important to prevent the water used to wash out concrete mixing equipment from getting into local storm water facilities and aquatic habitats.

Caustic washwater can harm fish gills and eyes and interfere with reproduction. The safe pH ranges for aquatic life habitats are 6.5 – 9 for freshwater and 6.5 – 8.5 for saltwater. (Source: EPA publication)

Recycling and Reuse

Concrete is commonly recycled by breaking it up and using it as gravel or stone. As gravel it can be used in fresh concrete, or bedding for roads, sidewalks or parking lots. In larger pieces, in the form of riprap, it can be used to control erosion. As the process of recycling concrete is becoming more widely used, the recycled material is becoming a more valuable commodity. Larger chunks of concrete can even be shaped and reused in landscaping.

Material Properties

Concrete is a composite material that is formed as a liquid and cures forming a solid. The solid finished product is basically rock. It has a large thermal mass. It is very durable, and highly reflective. It is also impervious to liquids unless specifically designed to be otherwise. However its most important quality is its compressive strength.

The compressive strength of concrete, denoted as $f'_c$, typically ranges from 2,500 psi to 6,000 psi, but there are commercially available mixes with strengths up to 18.9 ksi. The strength is adjusted by adjusting the ratio of ingredients in the mix. Basic concrete is made by just using cement, aggregate, and water, in varying amounts. To get specific properties like high strength, or increased fluidity, other ingredients are added.

An empirical relationship has been developed for determining modulus of elasticity of each mix design by simply using the mixes compressive strength and unit weight.

$$E_c = w_c^{1.5} 33 \sqrt{f'_c}$$

Where:

• $E_c$ is the modulus of elasticity of concrete in psi
• $w_c$ is the unit weight of concrete in pounds per cubic foot ($lb/ft^3$)
• $f'_c$ is the compressive strength of concrete in psi

If you don't have the unit weight, then you are allowed to use the following.

$$E_c = 57,000 \sqrt{f'_c}$$

In this case the unit weight is assumed to be approximately 145 pcf.

One thing to be aware of is that concrete is a brittle material, and is extremely weak in tension. Some work has been done in an effort to create mixes that behave ductilely. These are know as Engineered Cementitious Composites (ECC), or Strain-Hardening Cementitious Composites (SHCC), and are made with the addition of fine polymer fibers. However, they have not become used widely yet due to higher initial costs.

Strength

Concrete is not only strong but inexpensive. This makes it a popular choice in construction. The problem with plain concrete construction is that when elements are placed in bending or tension they will crack and fail. The reason that concrete became so popular is the development of using reinforcement (rebar). Rebar is typically steel, and comes in a variety of sizes and grades. These grades, like noted in the steel lecture, refer to the yield strength of the steel in units of ksi. The ones that are typically available are 40, 60, and 70.

In terms of sizes, reinforcing bars have single number designations. These numbers are approximately equal to the bar's diameter expressed in eighths of an inch, i.e. a number 3 bar has a diameter of 3/8".

Rebar Sizes

Bar No.	Diameter (in)	Area (in$^2$)
#2	0.250 = 2/8"	0.05
#3	0.375 = 3/8"	0.11
#4	0.500 = 4/8"	0.2
#5	0.625 = 5/8"	0.31
#6	0.750 = 6/8"	0.44
#7	0.875 = 7/8"	0.6
#8	1.000 = 8/8"	0.79
#9	1.128 = 9.02/8"	1.0
#10	1.270 = 10.16/8"	1.27
#11	1.410 = 11.28/8"	1.56
#14	1.693 = 13.54/8"	2.25
#18	2.257 = 18.06/8"	4.0

The steel reinforcing bars allow for beams to be able to crack and still be strong since the steel continues to carried the tension. Additional techniques include using steel cables to prestress the concrete. Prestressing refers to a member being placed in compression before being used in service. If the prestressing is done carefully, then the side of the member that was going to be in tension will be placed in compression. Once the service loading is applied, the tension will just lessen the pre-applied compression, and the member will still remain in compression. This will prevent any cracking from forming since the member never goes into tension. The ultimate effect is that the concrete element is stiffer and stronger making it unnecessary to use as large an element as would be necessary without prestressing.

The limit to the strength of reinforced concrete is set to the load at which any portion of a member undergoes 0.003 in/in strain. This amount of strain is defined as when concrete fails. It is a fixed value. This isn't even where the peak strength of concrete occurs at, but rather this is where concrete is considered to have lost its internal cohesion due to turning to ruble. It leads to some complications in design, but through clever analysis these have been simplified. The design process actually becomes easier by picking a fixed level of strain.

However, the details of designing concrete are not what we are covering in this class so let's leave it at that.

Additionally, the stress vs. strain diagram is only showing compression. If that plot included tension stress you would see a small line that ends at about 1/10 the peak compression stress and that would be at $f_t$. This value can be estimated with the modulus of rupture ($f_r$).

$$f_r = 7.5 \sqrt{f'_c}$$

Where both $f_r$ and $f'_c$ are in psi. This is an empirical equation as apposed to a theoretical one and the units do not work out correctly since the constant 7.5 does not have the correct units shown. There equations like this out there, and you'll just need to remember the units, or look them up.

One last note, the strength of concrete will depend on how long it is kept moist while curing. If it isn't kept moist the concrete will dry out, and there won't be sufficient water for the curing reactions to complete. If concrete is steam cured, it will actually reach the maximum 28 day strength much sooner, due to the continuous moisture and additional heat. This is one of the reasons precast plants often use steam curing, to increase production.

Structural Failure Modes

Concrete will fail if it is subjected to tension. This occurs when concrete is placed in tension or in bending. Reinforced concrete provides a work-around, but there are old buildings that were built without proper reinforcing. Additionally, if the reinforcing has become damaged due to corrosion, accidents, or other reasons, then the effect of tension is unrestrained and the member can fail.

This is particularly bad if the member is a supporting column. The effect of Poisson's ratio are mitigated in columns with the use of reinforcement. This can come in the form of loops of steel rebar or even spiral coils of rebar. The steel carries the tension as hoop stress and prevents the concrete from expanding and tearing itself apart.

Tension can also develop due to shear ($\tau$) (see Mohr's circle) as can be seen in the image on the right. The upper right corner is being pulled away from the lower left corner; they are in tension with respect to each other. Additionally, the upper left corner is being pushed towards the lower right corner. The line connecting these two corners is in compression. Without getting into it further (again this will be covered in more detail in following classes) all failure modes of concrete can be thought of as due to its poor capacity in tension.

Environment Induced Failure Modes

Concrete can be damaged by repeated exposure to extreme temperature ranges in the presents of ice and salt. If water manages to seep into cracks in a piece of concrete, and it freezes, then the concrete will break apart. For similar reasons if saltwater seeps into cracks in a piece of concrete and then evaporates the salt will crystallize, expanding, and break apart the concrete. One other process that can lead to expansion and spalling in concrete is rusting of the reinforcing steel. Rust has a greater volume than the original steel, and as it gets bigger, it will break apart the concrete exposing more reinforcement to the elements.

Fire can damage concrete by reversing the chemical process of hydration essentially driving off the water. Concrete can even erode if it exposed to fast flowing water.

Chemical Weaknesses

Concrete must be protected from soluble alkalis, sulfates, and acids. All three of these compounds can react chemically with cured concrete and weaken it. Though alkali salts are basic (not acidic) they can react with some aggregates in concrete forming a gel that takes up more volume that the original aggregate. Sulfates will also react with the concrete creating an internal pressure that can break it apart. Acid does not cause the concrete to swell but rather it reacts with the concrete by dissolving it, releasing carbon dioxide. If you have ever see a reaction between vinegar and baking soda then you are familiar with this type of reaction.

Safe Operating Temperatures

Solid concrete can be used in temperature ranges up to 750$^\circ$F without suffering too much damage. If it is going to be used at temperatures at or above this it will need to have some protection.

Concrete that is liquid has a much lower and controlled temperature range. It should only be placed in conditions between 40-90$^\circ$F otherwise there are adverse effects in curing.

Material Tests

The main material test that is relevant to concrete is the compression test. This results in the value for $f'_c$ and is used for determining shear strength, compressive strength, and even the modulus of elasticity. This test is typically conducted by taking a cylindrical sample and applying a compression load to it until it fails. This test is done for every batch of concrete used in a construction project to verify the strength. It is also done to determine the strength of a mix design before it is ever used. It is one of the most common tests done in construction.

There are two other tests that are commonly conducted and these are done in the field. The first is the slump test. The purpose of this test is to determine workability of the concrete and to verify consistency of water content between batches. To do this test concrete is placed in the test cone in three layers, tamping each layer after it is placed. At the end of placement the surface is smoothed and the cone is removed. The distance that the fresh concrete sags from its initial height is know as the slump. If the slump varies from one batch to the next this may be an indication of a change in water content which will substantially effect its strength.

The other test is for air content of the concrete. This to will have an effect on the strength of the concrete. This requires more specialized equipment, and familiarity with the ASTM C 231 standard.

There exists a test to determine the tensile strength of concrete, but since concrete is not supposed to be used in tension the cylinder splitting test is not particularly relevant to most construction.

Quality Levels

Concrete is specified by its compressive strength ($f'_c$). Normal weight concrete can range from 2,500 psi to 4,000 psi and even up to 19 ksi. Since there is a natural variation in each mix due to the variations in aggregate quality, mixing, and curing conditions, it is generally not specified to increments of less than 100 psi.

When ordering concrete there is something else you'll need to keep in mind. Every batch of concrete needs to have its quality verified. This is done by taking three samples at the time of pouring. These samples are typically prepared in cylinders, and need to be cured in either laboratory, or field conditions, depending on the testing purpose. If tests are being conducted to verify structural strength, then they need to be cured in conditions as close as possible to the concrete being used in the structure. These will be typically tested at 7 days, and 28 day to verify strength. The third cylinder is held in reserve if further testing is required.

If you have a specified concrete strength that needs to be ordered from a batch plant, then you are going to be curing the samples in laboratory conditions. This sort of testing should be done prior to the start of construction. If you are ordering a mix that has previously been produced by the batch plant, then you can use their records to check to see if the mix strength meets the required design specifications.

To determine the required strength for a project, you need to use the standard deviation ($S$ or $\sigma$ in the plot above) of the mix strength.

$$S = \left[ \frac{\Sigma(X_i - \bar{X})^2}{(n-1)} \right]^{1/2}$$

Where:

• $X_i$ is individual strength test results
• $\bar{X}$ is average of the $n$ strength tests ($\mu$ in the plot above)
• $n$ is the number of consecutive strength tests (must be at least 15 to uses this method)

If $n$ is less than 30 then the value of $S$ needs to be increased by 3 to 16% depending on the number of tests available.

Next the required average compressive strength ($f'_{cr}$) needs to be calculated. This is done by finding the larger value of the two following equations.

$$f'_{cr} = f'_c + 1.34 S$$$$f'_{cr} = f'_c + 2.33 S - 500$$

Where:

• $f'_c$ is in psi and is less then or equal to 5,000 psi
• $S$ is the modified standard deviation

If $f'_c$ is greater than 5,000 psi then the second equation becomes:

$$f'_{cr} = 0.90 f'_c + 2.33 S $$

If and only if the batch plant's mix is of equal to or greater than the value of the largest value of $f'_{cr}$ then it can be used. The purpose of this check is to provide a safety factor in ordering concrete. This procedure will limit the probability of a batch strength failure to less then 1 in 100.

If a batch of concrete is poured on site and turns out to have a strength of more then 500 psi less than the specified $f'_c$, (or by more than 10% below the design strength if $f'_c$ is more than 5000 psi) it will need to be removed and replaced. Concrete also needs to be removed and replaced if the average of three consecutive compression tests is less than the specified strength, $f'_c$. This can be quite costly and time consuming.

(CIP 35 - Testing Compressive Strength of Concrete)

(See slide below for example.)

This table contains selectable text.

Standard Deviation Modifiers

No. of tests	Modification Factor
15	1.16
20	1.08
25	1.03
30 or more	1.00

Given the standard deviation of 15 samples is 250 psi then the standard deviation to use is, $$S = S_{15}\times 1.16$$ $$S = 250 psi\times 1.16 = 290 psi$$

If you need $f'_c = 3,000 psi$ then,

\begin{equation*} f'_{cr} = \max \left\{ \begin{array}{rl} f'_c +1.34 S = 3,389 psi\\ f'_c +2.33 S -500 = 3,176 psi \end{array} \right. \end{equation*}$$f'_{cr} = 3,389 psi$$

But since you only specify orders in 100 psi increments and you cannot order less than required you round up and order 3,400 psi concrete.

Note: When there is reasonable doubt, any estimated sample standard deviation used to calculate the required average strength should always be on the conservative (high) side. (source)

To few samples:

When there are not at least 15 samples available then you need to use the following equations.

\begin{equation*} f'_{cr} = \left\{ \begin{array}{rl} f'_c+1,000 & \text{if } f'_c < 3,000\\ f'_c+1,200 & \text{if } 3,000 \le f'_c\le 5,000\\ 1.10 f'_c+700 & \text{if } f'_c > 5,000 \end{array} \right. \end{equation*}

Where both $f'_c$ and $f'_{cr}$ are in $psi$.

Residential Applications

Concrete is used in the residential arena mainly in foundations and floors. However as insulated concrete forms (ICFs) become more popular it is being used more in walls as well. Walls can also be made from concrete using concrete masonry units (concrete blocks) though these come as pre-formed units rather than liquid concrete. Depending on the location of the residence it may be necessary to pour concrete in the open cells of these block walls for additional strength.

Concrete walkways, driveways and support pads are also very common in residential applications. Some of these can be pre-formed or even decorative but the durable nature of concrete means that it always economical long term.

Commercial Applications

Concrete is used in commercial buildings as foundations, slabs, floors, and roofs. It is cheap, and readily available making it ubiquitous in the commercial sector. It is also use to decorative effect. Being infinitely formable, concrete allows architects design with curves, and shapes that would be cost prohibitive with other materials. Disney's Tree of Life® in the Animal Kingdom is made from reinforced concrete. Due to the size, and location no other material would make sense to use.

Industrial Applications

Concrete is used in the industrial setting in many ways. Obviously foundations for any structures but also some of the structures themselves. Large holding tanks can be made from concrete. Piping can be concrete. Retention facilities like settling ponds can be concrete. Anything that needs to be durable and/or exceptionally stable and strong is likely to contain a substantial amount of concrete.

In settings like hydroelectric power plants the very facility would not be possible without the use of substantial amounts of concrete. Large bridges would be impossible without the use of concrete foundations and often the superstructure itself.

Cost Examples and Material Sources

Concrete is generally sold by the cubic yard or by the bag for small amounts. Depending on the strength and aggregate used concrete can vary from \$95 - \$105 per cubic yard. (source) Buying concrete by the cubic yard is done when purchasing premixed or batch concrete. This concrete is delivered from a batch plant via a ready mix truck, which is basically a large concrete mixer. The fact that concrete cures instead of drying means that considerations need to be taken when ordering from a batch plant since the cure time starts at the plant. Additives like curing retarders can be included in the mix, but they add cost, and there is a limit to what they can do. It is interesting to note that on days that the temperature is high the concrete cures more quickly. To limit the impact of high temperature some batch plants will add ice to the mix to reduce the temperature, and slow the curing process.

I've digressed though, and need to mention that batch plant will generally have a minimum order amount, and a delivery fee. Trucks hold approximately 10$~$yd$^3$s of concrete, and can be ordered to deliver over a period of time rather than all at once. Depending on the rate of construction, and size of the project this could be as little as a few minutes apart, to close to an hour.

If you buy concrete buy the bag, you will need to mix it on site, which means you will have to have access to water that is suitable for mixing. The mixing can be done via concrete mixer, or by hand if necessary. Either way, batch plant, or on site mixing, you will likely need to place the concrete with additional equipment such as a pump, or wheelbarrow. This will add additional costs to the project, and though the cost may be small, it should be considered. In some situations a separate pump truck will have to be used, and that is not a minor expense.

Examples of Installation Tools

It is possible that you might place small quantities of concrete yourself. For a typical small job you would need at least a shovel, and a wheelbarrow to move the concrete around. You would need finishing tools like a broom, concrete trowel and/or float, depending on the type of finish. But probably most important of all, you need water. Water for mixing, cleaning, and flushing you eyes, or skin if you got concrete on you. Concrete is very caustic, and you'll need to wear gloves, and eye protection. This is a substance that you are likely to trust too much. You can work with it without electricity so safety can become a secondary thought; it shouldn't be. If you are not careful when working with concrete you can develop a wound that can lead to blood poising, and even death. Wear your safety equipment.

Famous Failures

Concrete is one of those substances that you don't hear about in the news as a cause of failure. Typically, if it is, it's due to poor engineering, or corroded reinforcement. If you do hear about a concrete structure collapsing it is typically something under construction, somewhere that's hit by an earthquake, or the structure was impacted by a vehicle, or a bomb. Concrete is tough, and since it's cheap, concrete structures have a tendency to be over designed. Code requires an additional factor of safety in the design of concrete structures because of its brittle nature. This is why you are more likely to hear about a steel structure collapsing, because it can be built so much closer to the limit of the material's strength.

Placing Liquid Concrete

The fact that concrete cures instead of drying out has an effect on how it should be handled during placement. Generally when concrete is poured it needs a form to define its final shape. From your Physics course you should remember the concept of hydrostatic pressure; basically the pressure on a point on a wall is proportional to the depth, $h$, that that point is below the surface of the liquid and the liquid's unit weight $\gamma$.
$$P = \gamma h$$ This holds true for concrete as well, as long as it is liquid. The reason this is important to know is that the pressure the concrete exerts on the form containing it needs to be accounted in the form design, otherwise it can collapse. For normal weight concrete the unit weight is 145 pcf and a wall does not need to be very tall before that pressure can become quite large.

This is where the fact that concrete cures comes into play. If the wall is filled slowly enough, then at a certain depth the concrete will turn to a solid and no longer exert greater pressure with depth as the pour continues. There are a few equations that are used to calculate this maximum pressure. I'm not going to ask you to remember them now but I wanted you to see them.

\begin{equation*} P_{max} = \begin{cases} C_W C_C \left[150 +9,000 \frac{R}{T}\right] \le \gamma h& \text{for } R< 7 ft/hr \text{ and } h\le 14ft\\ C_W C_C \left[150 +\frac{43,400}{T}+2,800 \frac{R}{T}\right] \le \gamma h& \text{for } 7 ft/hr \le R \le 15 ft/hr \text{ and } h\le 14ft\\ 600 C_W & \text{minimum form pressure}\\ \gamma h & \text{maximum form pressure} \end{cases} \end{equation*}

Where:

• $P_{max}$ - The maximum expected form pressure in psf
• $C_W$ - Unit Weight Coefficient
• $C_C$ - Chemistry Coefficient
• $T$ - Ambient Temperature at Time of Pour in $^\circ$F
• $R$ - Rate of Filling (Change of depth with respect to time) in ft/hr

The values of $C_W$ and $C_C$ come from tables in ACI 347-04.

Unit Weight of Concrete	$C_W$
Less than 140 pcf	$ 0.5\times[1+\frac{\gamma}{145 pcf})] \ge 0.8$
140 to 150 pcf	$1.0$
More than 150 pcf	$\frac{\gamma}{145 pcf}$

Cement Type or Blend	$C_C$
Types I, II, and III without retarders*	1.0
Types I, II, and III with a retarder*	1.2
Other types of blends containing less than 70% slag or 40% fly ash without retarders*	1.2
Other types of blends containing less than 70% slag or 40% fly ash with a retarder*	1.4
Blends containing more than 70% slag or 40% fly ash	1.4

• Retarders include any admixture, such as a retarder, retarding water reducer, retarding mid-range water-reducing admixture, or retarding high-range water-reducing admixture (superplasticizer), that delays setting of concrete.

The reason these equations are useful is that they allow for lower form work design requirements than pure hydrostatic pressure. You should notice that the longer it takes to fill a form, the lower the maximum pressure is in that form. Again, this is due to the concrete curing (or setting up) and the pressure no longer increasing with the additional of more concrete.

I would like you to know how to calculate the fill rate of a form. This fill rate ($R$) is important and easy to calculate. As the contractor you specify the delivery schedule of the concrete trucks, which is in turn equivalent to the volumetric fill rate, $\left(vol \text{, in units of }\frac{ft^3}{hr}\right)$ of the form. Since the liquid concrete will always try to fill the form from the bottom up uniformly, the fill rate $R$ is found by simply dividing $vol$ by the horizontal cross-sectional area of the form. If your form changes area with height it is best to use the narrowest section to calculate the pressure. For an example of this calculation, if you have 3, 10 yd$^3$ trucks arriving per hour to pour a wall that is 40 ft long, 16 inches wide, and 13 ft tall, you can find the fill rate like this, $$vol = 10 yd^3 \times \frac{3}{hr} \times \frac{27 ft^3}{yd^3} = 30 \frac{yd^3}{hr} \times \frac{27 ft^3}{yd^3}$$ $$vol = 810 \frac{ft^3}{hr}$$ $$A = 16 in\times \frac{1 ft}{12 in} \times 40 ft = 53.33 ft^2$$ $$R = \frac{vol}{A} = \frac{810 ft^3/hr}{53.33 ft^2} = 15.2 \frac{ft}{hr}$$

References

UF Class website

Github.io version of course website (Do not use this link if you are taking this course in Summer A or B.)

IPython.org (IPython is the opensource software used in the development of much of this course.)

Complete Software List

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