# Mathematical applications in Civil Engineering

Beautiful structures make beautiful cities, there are works of civil engineering all around us, schools, libraries, houses, malls, halls, even your favorite restaurant. You are probably in one right now while reading this, when most people look at edifices all they see is the beauty. The Eiffel Tower, for instance, is one of the worldâ€™s most graceful civil engineering feats. Perhaps the last time you saw it, you were awed by its beauty and form. You may have even praised the civil engineering firm that designed and constructed the tower. What you may not have discerned, though, is that all structures including the magnificent Eiffel Tower, are created using mathematical concepts that have been applied to civil engineering.

Some students of a Civil Engineering institute were asked a couple of questions on the relevance of mathematics to civil engineering. One of the students stated “Mathematics is important in both the study and practice of civil engineering”. Another student observed that “math is used in both simple and complex analysis of civil engineering structures, without mathematics, it is simply impossible to stand a structure”. From the above statements, it is clear that mathematics plays a huge role in civil engineering. You may wonder, however, how experts apply mathematics in creating the places you love, there are countless concepts used in civil engineering. With specific construction example, the applications of three of such mathematical concepts, trigonometry, vector analysis and integration will be discussed.

Let us use an hypothetical example, the office you work in has just bought a measure of land. They have also hired a civil engineering firm to design and develop the site where the headquarters will be located. To make it a reality they must first start by surveying the site to make sure that the headquarters fits within the proposed site. While doing this, the surveyor sees an obstacle while measuring the position of the south wall, there are some trees getting in the way. You may be of the idea that the surveyor could just tie a rope from the start of the south wall to the end of the wall to measure the length. Basically, you are not wrong but this method will be difficult or even impossible because of the trees in the way.

The concept of trigonometry can be of great use in this case. Knowing his distance from the south wall and the angle between his eyes and both ends of the south wall. The surveyor can use the tangent function in Pythagoras theorem to compute the position of the south wall. He can then add the adjacent values to give the exact length of the wall. Surveying tools such as the clinometers can help measure the angles correctly.

Another application of this concept is in the calculations for the angle of a structural member. Structural members are the primary load bearing components of the structure and each have their own structural features which need to be considered. Therefore, when considering the properties of the such members and the exact position to place them, trigonometry is used by the civil engineer. The placements of roofs and staircase require the use of trigonometry as well.

Imagine now that the headquarters is completed, and it is your first day of work, you are sitting in your office and there is nice view outside. Suddenly, though, a strong wind starts blowing, and then you notice the whole workplace moving in the direction of the wind, very likely, you will run out of there. The office must be strong and firm, and as such, the firm must consider the loads acting on the building, this is key in knowing exactly how to design it.

To prevent it from moving, the civil engineer will consider the two major strains acting on the building, the lateral and vertical forces. The lateral forces include that of the wind, that is the force the wind exerts on the facility, while the vertical forces includes the dead loads and live loads. Dead load is the self weight of the edifice, that is, the materials used in the construction. These would include bricks, beams, cables, roofs, structural members and much more. Also, under the vertical forces are the live loads, these are weights that the structure is meant to carry such as people, chairs, computers and tables to name a few. All these factors need to be considered in the design and analysis of the structure.

In grasping their impact, the civil engineer employs yet another concept of mathematics known as vector analysis. He visually depicts these forces as vectors indicating both their magnitude and direction. The firm then designs and constructs the building ensuring the force acting on it is balanced to prevent it from moving. Structural members are key in maintaining this balance, an example of this are horizontal beams known as cantilevers, these cantilevers resists load from people and materials.

All materials including cantilevers are flexible to a certain point, picture yourself jumping on the first floor of the office building and the floors are acting like a spring or a diving board. Of course, no civil engineering firm would want to be linked with such a building. So to prevent this, the civil engineer will employ yet another principle, integration. Using integration, the deflection of the beams at any given point can be calculated to confirm whether the beams to be used are acceptable.

Deepak Chopra said “Mathematics expresses values that reflect the cosmos, including balance, harmony, logic, and abstract beauty”. Granted, mathematical concepts and guidelines are strong tools that every civil engineer must have in his arsenal. Without the use of basic mathematical processes such as addition and subtraction. To the complex processes like trigonometry, integration and vector analysis that have been discussed, it would be impossible to create that hypothetical office or any other structure for that matter.

So, when next you see a striking construction, most likely you would still value the aesthetics, but you just might tip your hat in respect for all the mathematical work that went into creating it.