Graphical analysis 3 column equation9/22/2023 ![]() Taking the vertical reaction at support B and the reactive moments at support A as the redundant reactions, the primary structures that remain are in a state of equilibrium. A careful observation of the structure being considered will show that there are two possible redundant reactions and two possible primary structures (see Fig. A primary structure must always meet the equilibrium requirement. The structure that remains after the removal of the redundant reaction is called the primary structure. This means that there is one reaction force that can be removed without jeopardizing the stability of the structure. The beam has four unknown reactions, thus is indeterminate to the first degree. Once the redundant forces are known, the structure becomes determinate and can be analyzed completely using the conditions of equilibrium.įor an illustration of the method of consistent deformation, consider the propped cantilever beam shown in Figure 10.1a. This method entails formulating a set of compatibility equations, depending on the number of the redundant forces in the structure, and solving these equations simultaneously to determine the magnitude of the redundant forces. A redundant force can be an external support reaction force or an internal member force, which if removed from the structure, will not cause any instability. In this method, the unknowns are the redundant forces. The force method of analysis, also known as the method of consistent deformation, uses equilibrium equations and compatibility conditions to determine the unknowns in statically indeterminate structures. You figured out that the intercepts of the line this equation represents are (0,2) and (3,0). Once you have found the two intercepts, draw a line through them. ![]() You can use intercepts to graph linear equations. To find the x– and y-intercepts of a linear equation, you can substitute 0 for y and for x respectively.įor example, the linear equation 3y+2x=6 has an x intercept when y=0, so 3\left(0\right)+2x=6\\. Notice that the y-intercept always occurs where x=0, and the x-intercept always occurs where y=0. The y-intercept above is the point (0, 2). The x-intercept above is the point (−2,0). Every point on this line is a solution to the linear equation. The arrows at each end of the graph indicate that the line continues endlessly in both directions. ![]() Then you draw a line through the points to show all of the points that are on the line. However, it’s always a good idea to plot more than two points to avoid possible errors. Two points are enough to determine a line. One way is to create a table of values for x and y, and then plot these ordered pairs on the coordinate plane. There are several ways to create a graph from a linear equation. A linear equation is an equation with two variables whose ordered pairs graph as a straight line. There are multiple ways to represent a linear relationship-a table, a linear graph, and there is also a linear equation. ![]() In this case, the relationship is that the y-value is twice the x-value. You can think of a line, then, as a collection of an infinite number of individual points that share the same mathematical relationship. Look at how all of the points blend together to create a line. You have likely used a coordinate plane before. The coordinate plane consists of a horizontal axis and a vertical axis, number lines that intersect at right angles. ![]() This system allows us to describe algebraic relationships in a visual sense, and also helps us create and interpret algebraic concepts. The coordinate plane can be used to plot points and graph lines. In his honor, the system is sometimes called the Cartesian coordinate system. The coordinate plane was developed centuries ago and refined by the French mathematician René Descartes. (1.3.1) – Plotting points on a coordinate plane (1.3.5) – Graphing other equations using a table or ordered pairs.(1.3.4) – Recognizing and using intercepts.(1.3.3) – Determine whether an ordered pair is a solution of an equation.(1.3.2) – Create a table of ordered pairs from a two-variable linear equation and graph.(1.3.1) – Plotting points on a coordinate plane. ![]()
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