E0045 conicity & ply steer 18 – moment of area#6 : Mohr’s circle, principle and proof - EulSeoggy Ko скачать в хорошем качестве

E0045 conicity & ply steer 18 – moment of area#6 : Mohr’s circle, principle and proof - EulSeoggy Ko

Substituting the angle increased at regular intervals such as every 1 or every 2 degrees from zero to 180 degrees into 𝜃 of the three equations Iz'z,' Iy'y', and Iz’y’ and displaying that result in a coordinate system with the second moment of area as the horizontal axis and the product moment of area as the vertical axis, it appears as a circle as shown in the figure on the right, which is called a Mohr circle. The principle of how to express the second moment of area Iz’z’, Iy’y’ and the product moment of area Iz’y’ of a new coordinate axis as a function of Izz, Iyy, Izy and θ with known coordinates is investigated and proved through geometric analysis. If the coordinate axis rotates by θ in the actual cross-section, it rotates in the same direction by 2θ, which is twice in the Mohr circle. In the Mohr circle, the second moment of area appears on the horizontal axis and the product moment of area on the vertical axis. The order of drawing the Mohr’s circle is as follows. 1. Draw the axis of Mohr's circle. The positive horizontal axis direction must be in the same direction as the positive horizontal axis direction (z) of the actual cross-section. 2. Calculate the origin of the Mohr’s circle. The origin of the Mohr’s circle is the average value of the second moment of area with respect to each coordinate axes. 3. Calculate the half value of the difference of the second moment of area with respect to each coordinate axis, and draw a vertical line at the horizontal distance corresponding to the half value from the origin. 4. The product moment of area is drawed by a horizontal line. 5. The intersection where the lines 3 and 4 above meet becomes a point on the circumference of the Mohr's circle, and the radius of Mohr's circle can be obtained from this using the identities of a right triangle. This point represents the second and product moment of area for the existing coordinate system. 6. Draw a more circle using the radius obtained in 5 above, and indicate the major and minor axis values. Among the characteristic values of the second and product moment of area, there are values that do not change even when the coordinate system is rotated, which is called invariant. The invariants include the average value, the second moment of area of the major and minor axes, the maximum product moment of area, and the radius of the Mohr’s circle. Using these values, it is possible to prove the principle and relationship between the Mohr’s circle and the second and product moments of area.