تحليل العزم والدعائم

Summary of Beam Analysis

• Support Reactions: Reactions at supports are calculated using equilibrium equations (ΣFₓ = 0, ΣFᵧ = 0, ΣM = 0).
• Shear Diagram: Plotted by considering the forces acting on either side of a sectioned beam. Vertical forces are considered. Discontinuities occur at concentrated loads.
• Moment Diagram: Plotted by considering the moments acting on sections of the beam. Slope of the moment diagram is related to shear. Changes in the moment diagram occur at concentrated loads and changes in the shear diagram.
• Relationships between load, shear, and moment: The slope of the shear diagram corresponds to distributed load intensity. The slope of the moment diagram is equal to the shear. Change in shear is the area under the load curve. Change in moment is the area under the shear diagram.

Summary of Moments of Inertia

• Definition: Moments of inertia are a measure of a shape's resistance to rotational motion about a specific axis. It is a second-moment area.
• Calculation: Moments of inertia are calculated using the integral I =∫ y² dA (or I =∫x² dA).Where dA is an infinitesimal area element and "y" or "x" is the perpendicular distance from the axis of interest to the area element.
• Parallel Axis Theorem: Moment of inertia about an axis parallel to a centroidal axis is equal to the centroidal moment of inertia plus the product of the area and the square of the distance between the axes. (I = Ic + Ad²)
• Composite Shapes: For composite shapes, the moment of inertia is calculated for each individual part and then summed.
• Polar Moment of Inertia (Jo): The polar moment of inertia is the sum of the moments of inertia about two perpendicular axes (Jo = Ix + Iy).