Baricentro di una sezione a cassone

Center of Gravity of a Box-Box Section Calculating the XG and YG coordinates of the center of gravity of a box-box section is a fundamental operation in structural analysis for determining the section's center of gravity, or center of mass. This case is particularly interesting because it involves one of the most common structures in the area: the box-box bridge. In this case, we immediately note the presence of a vertical axis of symmetry coinciding with the Y-axis of the XY reference system we have adopted. This system originates at the center point of the base of the box-box itself. The formulas used. The box-box section is typically a hollow trapezoid, therefore composed of a "solid" part and an "empty" part. This is described in the video lesson, in 3D graphics. We will divide the section, both the solid and the empty parts, into simpler shapes: eight triangles, four solid and four empty, and five rectangles, three solid and two empty. Thirteen simple shapes in total. To calculate the centroid coordinates XG and YG, formulas based on a weighted average of the coordinates of the centroids of the individual parts that make up the section are used. The formulas are: XG=Sy/Atot and YG=Sx/Atot where: Sy is the static moment of the cross-section about the y-axis Sx is the static moment of the cross-section about the x-axis Atot is the total area of ​​the cross-section The same formulas can be written in terms of integrals: XG=Integral(x.dA)/Integral(dA) and YG=Integral(y.dA)/Integral(dA) Specifically, it is best to use the discretized formulas in terms of summation: XG=Sum(xi.Ai)/Sum(Ai) and YG=Sum(yi.Ai)/Sum(Ai) The summations range from 1 to n depending on the number of rectangles (or other simple shapes) into which the cross-section can be divided. In this case, there will be thirteen simple shapes (rectangles and triangles), seven "solid" and six "hollow." The specifications for each simple shape (rectangles and triangles) are as follows: ci shape coefficient (it is 1 for rectangles and 0.5 for triangles, positive for solids and negative for hollows) bi base of the i-th shape hi height of the i-th shape Ai area of ​​the i-th shape xi x coordinate of the centroid of the i-th shape yi y coordinate of the centroid of the i-th shape The i-th area is calculated with the formula Ai=ci.bi.hi Calculation procedure. First, note that the section in question has a vertical axis of symmetry coinciding with the Y-axis and that, consequently, the XG coordinate will be zero since the centroid always lies on the axes of symmetry. Which leads us to conclude that we only need to calculate YG. So, to perform the calculations, we adopt this procedure: (1) Subdivision into simpler shapes: filled rectangles with areas A2, A5, and A7, empty rectangles with areas A9, A12, filled triangles with areas A1, A3, A4, and A6, and empty triangles with areas A8, A10, A11, and A13. (2) Identify the barycentric coordinates of each shape at the origin, xi and yi. (2) Apply the above formula to obtain YG. In this case, we will make extensive use of the spreadsheet, which will greatly simplify the calculations. You will find the details in the video lesson. Conclusions. The barycentric coordinate YG of the proposed bridge section will be 2.58 meters. Below are links to access previous lessons. Centers of gravity of simple plane figures    • Baricentri di figure piane semplci   Centers of double-T sections    • Baricentri di sezioni a doppio T   Center of gravity of a trapezoid without integrals    • Calcolo delle coordinate baricentriche del...   Center of gravity of a trapezoid with integrals    • Calcolo del Baricentro del Trapezio con gl...   Center of gravity of a parabolic segment    • Baricentro di un segmento parabolico   Center of gravity of a half-parallel segment Parabolic    • Baricentro di mezzo segmento parabolico   Center of a generic parabolic segment    • Baricentro di un segmento parabolico generico   Center of a bridge section https://www.staticafacile.it/baricent...