# Example Problem #6

Using sum of forces in the X direction and in the Y direction at B finds AB (with linear algebra techniques) which then gives BC. Then using sum of forces in the X direction and in the Y direction at C with the known value of BC allows us to find CD and then ultimately find the weight of the other object at C.

# Expansive Soils

Expansive soil, also known by such names as expansive clays, expandable soils, shrink-swell soils, and heavable soils, can produce significant forces on slabs, foundations, retaining walls, underground pipes and other structures resulting in property damage. These soils contain some proportion of clay minerals such as smectite, bentonite, montmorillonite, or others that have a propensity to adsorb water. The chemistry of these materials is such that they can adsorb significant quantities of water thus swelling or expanding the volume of the soil mass. Likewise, if the soil mass were to lose this water, the soil would collapse or shrink and experience a notable decrease in volume. As is evident from this discussion, the proportion or percentage of the soil mass that is composed of clay minerals, as well as the type of clay minerals, will determine the potential for more or less volume change.

Expansion

One of the primary means by which expansive soils can influence a structure is through this expansion process due to an increase in moisture content. Oftentimes this influence is called “heave” as it induces an upward force on any structure supported by the soil. Additionally, it can create increased lateral earth pressure forces on retaining walls, basement walls, and foundations.

Contraction

When an expansive soil is dried out it will shrink, pull away from, and remove support from the overlying structure. This subsidence can result in settlement of the structure.

Water

This shows how significant variation in the moisture content of the soil is to these issues. It is not so much a high moisture content or a low moisture content as it is the moisture content fluctuation or differential over time and/or over an area. Two types of fluctuation should be considered. First is a uniform global fluctuation. This is often due to seasonal changes in rain patterns like a drought or a rainy season. These changes will affect the structure, site, and soil mass as a whole. Second is localized fluctuations. These are due to things like topography and site drainage, structure rainwater runoff patterns, a water leak, or irrigation systems. These issues can affect a particular area of the supporting soil mass while not affecting another area depending on the situation. These issues should be considered in both the analysis of damage patterns as well as designs for remediation. [2]

Confinement

Another useful fact in the analysis of potential expansive soil influence on a structure is confinement. Using the basic principles of mechanics, the downward pressure provided by a structure will tend to confine the upward heave of an expansive soil mass. Since the downward pressure at every point in a structure is not equivalent, the areas with greater load should heave upward less than the areas with less load applied. Put another way, “Soil movement will be minimized where confining pressures are the largest while movement will be greatest where the magnitude of the confining pressure is the smallest.” [1]

Identification / Classification

In order to identify and classify a soil with regard to its expansion potential, a soils map, field observations, and/or laboratory tests must be considered. Geology.com provides a general U.S. soils map showing the geographic distribution of the swelling potential of area soils. However, as discussed on the site, this is very general information and is not meant to provide detailed information for individual properties. With regard to field observations, a good indicator of an expansive soil is visible desiccation cracks. These result from and show the dimensional shrinkage in the soil mass. Also, the damage patterns observed and their agreement with the previously discussed mechanics can indicate an expansive soil condition. Numerous lab tests and correlations with soil properties can be performed to determine the categorization of soils with regard to swelling potential. In general, a liquid limit in excess of 40 and a plasticity index in excess of 15 can be considered an expansive clay. [3] Also, a simple mold test can give an idea of swelling potential. This is done by placing a moist sample in a mold of known volume, drying the sample, and measuring the volume of the dried sample which has shrunk away from the mold. The change in volume divided by the dry volume will give the percent of expansion. Soils having a percent expansion in excess of 10% are at least somewhat expansive.

Active Zone

In a soil mass, there is an upper layer where the moisture content tends to vary over time. Deeper into the soil column the variation diminishes until at a certain depth there is minimal and uniform variability. This upper layer is known as the active zone. The depth of this zone can be determined through testing. Within this region, an expansive soil will tend to expand and shrink. Below this region, the volume will be relatively stable. The depth of the active zone is an important piece of information as it is used in determining the amount of heave and is the depth beyond which drilled shaft isolated footings need to be founded in order to function properly outside the influence of the expansive soil volume changes.

Magnitude of Swell

The amount of heave or swell (vertical distance the free surface will rise) can be determined through two common soil tests. These are the unrestrained swell test and the swelling pressure test. The details of these tests are beyond the scope of this introduction.

Dealing with Expansive Soils

Several different measures can be taken to address placing structures in areas with expansive soils. First would be to remove the expansive soils to a certain depth and replace them with non-expansive soil which is then properly compacted. This is relatively straightforward but can be costly. Second would be to change the soil by such means as compaction, prewetting, installing some means of controlling the variation in soil moisture content, or the application of chemical treatments or stabilizers to the soil. Third would be to design the structure such that it is strong enough and/or flexible enough to compensate for the projected heave/shrinkage or circumvent the expansive soil by installing a deep foundation beyond the depth of the active zone. Any combination of these techniques could also be utilized.

Sources:

[1] “Expansive Soil Problems and Solutions.” Your Foundation Repair Guide-An Underground World Revealed. N.p., n.d. Web. 29 June 2017

[2]”Damage To Foundations From Expansive Soils.” ResearchGate (1985): 1-7. Web.

[3]Das, Braja M. Principles of foundation engineering. Boston, MA: Cengage, 2016. Print.

# Cold Weather & Condensation

From our experience, we know that when a glass of cold water (or another refrigerated item) is taken outside when it is warm, or placed on the table in the house at room temperature, condensation forms on the outside of the glass. As we were taught in science class, this is not because water is seeping through the walls of the glass, or because of some leak. It is the result of the water vapor in the air around the glass changing into liquid form. The water vapor changes state when it cools as a result of being in contact with the surface of the cold glass. This same simple phenomenon can occur in structures and can create moisture damage and promote mold and fungal growth.

Ideally the building envelope of a structure creates an insulated barrier between the outdoor and indoor environments. Consequently, there is a gradual temperature gradient from the outside surface of a wall to the interior surface of the wall (or window, etc) when there is a temperature difference between the outdoor and indoor environments. Each surface then is closer to thermal equivalence with the corresponding environment with which it is in contact. However, in some situations, the insulating properties have inadequacies that allow the interior surfaces’ temperature to trend toward equivalence with the exterior environment. If the interior surface temperature drops below the dew point for the given humidity level and atmospheric pressure, water will leave the air and condense on the wall surface.

In the image above, moisture has condensed on the interior surface of the window and is dripping down onto the window sill. Notice the fungal growth that is occurring on the sill and around the edge of the window.

In this photograph, moisture along with its dripping pattern can be seen on the wall to the right. Although the presence of moisture here is obvious and could be detected both visually and manually, moisture meter testing and infrared thermography can be used to investigate the area of influence as well as the pattern of exposure. (So as to differentiate between some kind of leak, etc.) Additionally, areas, where the insulation is inadequate, can be identified.

The infrared image above shows an area of low temperature located on an exterior wall. Below is shown the same area and the moisture damage and fungal growth that have resulted subsequent to the forming of condensation on the wall.

# Example Problem #5

To solve this graphically, draw a vertical vector to scale representing the 220 lb. weight. I used 1 inch = 100 lb. Then using a protractor draw the line of action for BC from the head of the weight vector and draw the line of action for AB from the tail of the weight vector. Where the two lines of action intersect shows where the head of BC and the tail of AB are. Then use a ruler and the scale to determine the magnitude of each.

Mathematically use vector components in X and Y directions and the equilibrium equations. This results in two equations with two unknowns. Different methods of linear algebra can be used to solve this system of equations. I used the method of elimination.

# Groundwater Seepage into the Crawl Space

Groundwater moisture vapor exposure over an extended period of time in a crawl space can cause significant damage to wood frame construction. One way in which water migrates into the crawlspace is through seepage. Seepage is the flow of water through the soil. The figure below shows a soil / foundation cross-sectional view with a superimposed grid called a flow net.

The blue lines are called flow lines or stream lines. They represent the path of the water flowing through the soil. The space in between each adjacent pair of flow lines is called a flow channel. The red lines are called equipotential lines. They represent places in the soil at the same head. In other words, if a series of piezometers were installed along an equipotential line, the water would rise to the same level in all of these piezometers. Flow nets can be used to calculate the rate a flow through the soil. So, in this case, it is the rate of flow of water from outside the foundation wall into the crawlspace. The two primary factors affecting flow rate are hydraulic conductivity and the difference in the head from outside to inside.

Hydraulic Conductivity

Hydraulic conductivity is a proportionality constant relating speed or velocity of flow to the hydraulic gradient. What this essentially means is that at a given head difference between two locations a soil with a high hydraulic conductivity will have a high velocity of flow through it while a soil with a low hydraulic conductivity will cause the water to flow through it more slowly. Generally, this property of the soil is fixed as the soil present on the site is typically utilized as is in most residential construction. Therefore the flow rate of water from outside the foundation to inside the crawlspace is mostly influenced by changes in head difference.

The head difference in this situation is the pressure and elevation head outside the foundation wall minus the pressure and elevation head inside the crawl space. Therefore as the pressure difference between inside and outside increases or the elevation difference between inside and outside increases, the head difference increases, and consequently the rate of flow of water into the crawlspace increases given the same parameters.

Pressure Difference

Pore water pressure develops and increases outside the structure when the soils become saturated and in particular when water accumulates on the surface of the ground. Consequently, to reduce the pressure difference between outside and inside, the accumulation of water adjacent to the foundation wall needs to be avoided. This is one of the reasons why building codes like the International Residential Code (IRC) require grading away from the structure to drain surface water away. The 2012 IRC, for instance, requires a minimum fall of 6 inches within the first 10 feet (R401.3). Additionally, roof system rainwater runoff can cause the accumulation of water in this area if the downspout deposits the runoff in this region. This is one of the reasons it is good practice to have the water carried away by non-perforated surface or subsurface downspout extensions or diverters.

Foundation Dampproofing and Waterproofing

This accumulation of water behind the foundation wall also creates hydrostatic pressure that can force the groundwater into and through the porous concrete foundation wall materials or the footing / foundation wall joint into the crawl space. This is often visible as water staining or efflorescence on the inside face of the foundation wall and water accumulation in the inside of the footing trench. This can be particularly problematic as water that migrates through the wall can potentially accumulate on top of the vapor retarder which will then effectively hold the moisture in instead of keeping it out of the crawl space. There are a number of code compliant methods called for to dampproof or waterproof the exterior surface of the foundation to keep this from happening. However, sometimes the application is minimal or discontinuous in areas and can have a design life much less than that of the structure. One potential solution to this issue is the attachment of the vapor retarder to the inside foundation wall face at an elevation above which the water might migrate through the wall. This keeps the water below the vapor retarder.

Elevation Difference

Many times during residential construction when the site is being excavated in order to install the footings and the foundation wall, the elevation in the crawlspace ends up appreciably lower than what the finished outside grade will be once construction is complete. This elevation difference also drives the migration of groundwater into the crawlspace. In areas with a high water table or poor surface drainage characteristics, the 2012 IRC states that the grade in the under-floor space shall be as high as the outside finished grade or equipped with an approved drainage system for this reason. Although it would be good to minimize or eliminate this elevation difference during excavation another alternative that can be used during the initial construction or as a remediation technique is to add a layer of gravel or granular material over the ground surface in the crawl space and under the vapor retarder. The granular material provides not only a way of raising the surface elevation in the crawl space but also a high void ratio or high porosity material that can contain a larger volume of water and does not provide for capillary action. This keeps the water in and below this layer and in particular below the vapor retarder which rests on top of the granular material. It is important that the granular material not be too angular and consequently cause damage to the vapor retarder.

Foundation Drainage

Finally, another helpful system component for eliminating or limiting the amount of water seeping into the crawl space is a perimeter foundation or footing drain. This usually includes a continuous run of perforated pipe or tile with an open end that runs out to daylight laid adjacent to the footings. The pipe is sometimes wrapped in a filter fabric sock and then lain in and encapsulated with clean, coarse rock. Also sometimes this rock is wrapped in a filter membrane. The filter membrane keeps fine soil particles from collecting and clogging the flow of water into and through the drainage system over time. These systems, when properly installed, help keep groundwater from accumulating outside the foundation wall and intercept and channel away water that might otherwise have seeped into the crawlspace.

As can be seen, a number of important details need to be considered when attempting to limit the amount of groundwater migration or seepage into the crawl space including exterior surface grading or slope, roof system rainwater runoff channelization, height of the water table, type of soil, foundation wall dampproofing or waterproofing, elevation difference between the interior and exterior of the foundation, and the perimeter foundation drain system.

# Example Problem #3

This problem resolves two vectors into their x and y component parts using trigonometry.