25.2:
Equation of the Elastic Curve
The plane curve's curvature at a point on the curve can be expressed using an expression involving the curve's first and second derivatives.
The slope is insignificant for the beam's elastic curve, so the governing equation for the elastic curve is expressed as a second-order linear differential equation, considering flexural rigidity.
If flexural rigidity varies along the beam, it is expressed as a function of x. However, for a prismatic beam, flexural rigidity remains constant.
Integration of this equation provides the angle formed by the tangent to the elastic curve at a point with the horizontal.
This small angle, when integrated, gives us the deflection of the beam at any point. The beam supports' boundary conditions determine the constants in the equations.
Supported, overhanging, and cantilever are the three types of beams that are primarily considered. For supported and overhanging beams, the deflection at support points is zero.
Both deflection and slope at the support point are zero for cantilever beams. These conditions enable the calculation of the constants.
25.2:
Equation of the Elastic Curve
The concept of curvature in plane curves, crucial in structural engineering, defines how sharply a beam bends under load. This curvature is determined using the curve's first and second derivatives.
Consider a cantilever beam with a point load at its free end (for instance, a diving board). When analyzing beam deflection with small slopes, the shape of the beam's elastic curve becomes key. The governing equation for this analysis involves the bending moment and the beam's flexural rigidity, which is a product of the modulus of elasticity and the moment of inertia of the beam's cross-section.
For prismatic beams, where the cross-section remains constant, the analysis simplifies, making the flexural rigidity constant along the beam's length. Integrating the governing equation allows the calculation of the angle formed by the tangent to the curve at any point, which, upon further integration, yields the beam's deflection at that point.
Boundary conditions at the beam supports are vital for completing these calculations. Supported, overhanging, and cantilever are common types of beams, each with distinct boundary conditions. For example, the deflection and slope at a cantilever beam's support point are zero, which is essential for calculating the constants of the deflection equations.
Accurately predicting beam deflection is crucial for ensuring structural safety and functionality. Excessive deflection can cause structural failures or serviceability issues, underscoring the importance of understanding beam behavior under load.