13.1 Elastic curve equation and boundary conditions

The elastic curve equation is a powerful tool for understanding beam deflection. It relates a beam's deformation to applied loads, material properties, and support conditions. This equation forms the foundation for analyzing how beams respond to various forces in structural engineering.

Boundary conditions are crucial in solving beam deflection problems. They describe how a beam is supported and constrained, allowing engineers to determine integration constants and find specific solutions for deflection, slope, moment, and shear force along the beam's length.

Elastic Curve Equation for Beams

Derivation and Assumptions

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• The elastic curve is the deformed shape of the neutral axis of a beam under load
  • Relates the deflection of the beam to the applied load, material properties, and boundary conditions
• The Euler-Bernoulli beam theory assumes:
  • Plane sections remain plane and normal to the neutral axis after deformation
  • The material is linearly elastic, homogeneous, and isotropic
• The elastic curve equation is a fourth-order linear differential equation: $\frac{d^4y}{dx^4} = \frac{q(x)}{EI}$
  • $y$ is the deflection
  • $x$ is the position along the beam
  • $q(x)$ is the distributed load
  • $E$ is the elastic modulus
  • $I$ is the moment of inertia

Simplification and Loading Conditions

• For a beam with a constant flexural rigidity ($EI$), the elastic curve equation can be simplified to: $\frac{d^2y}{dx^2} = \frac{M(x)}{EI}$
  • $M(x)$ is the bending moment
• The elastic curve equation can be derived for various loading conditions by applying the appropriate load functions and boundary conditions
  • Concentrated loads (point loads)
  • Distributed loads (uniform, linearly varying, or non-uniform)
  • Moments (concentrated or distributed)

Boundary Conditions for Beam Deflection

Types of Boundary Conditions

• Boundary conditions describe the support conditions and constraints at the ends of a beam
  • Necessary to solve the elastic curve equation and determine the deflection
• Common boundary conditions for beams include:
  • Simply supported: zero deflection ($y = 0$) and zero moment ($M = 0$) at the supports
  • Clamped (fixed): zero deflection ($y = 0$) and zero slope ($\frac{dy}{dx} = 0$) at the supports
  • Free end: zero moment ($M = 0$) and zero shear force ($V = 0$) at the end
  • Guided end: zero deflection ($y = 0$) and zero moment ($M = 0$), but non-zero slope ($\frac{dy}{dx} \neq 0$)

Application of Boundary Conditions

• Boundary conditions are used to determine the constants of integration that arise when solving the elastic curve equation
• The number of boundary conditions required depends on the order of the differential equation and the number of integration constants
  • For the fourth-order elastic curve equation, four boundary conditions are needed
• Examples of applying boundary conditions:
  • Cantilever beam (fixed at one end, free at the other): $y(0) = 0$, $\frac{dy}{dx}(0) = 0$, $M(L) = 0$, $V(L) = 0$
  • Simply supported beam: $y(0) = 0$, $M(0) = 0$, $y(L) = 0$, $M(L) = 0$

Curvature, Moment, and Rigidity Relationship

Beam Curvature

• Beam curvature ($\kappa$) is a measure of how much the beam deforms under loading
  • Defined as the reciprocal of the radius of curvature ($\rho$) at a given point along the beam: $\kappa = \frac{1}{\rho}$
• For small deflections, the curvature can be approximated as the second derivative of the deflection with respect to the position along the beam: $\kappa \approx \frac{d^2y}{dx^2}$

Moment-Curvature Relationship

• The curvature of a beam is related to the bending moment ($M$) and the flexural rigidity ($EI$) through the moment-curvature relationship: $M = (EI)\kappa$
• The flexural rigidity ($EI$) is a measure of a beam's resistance to bending
  • $E$ is the elastic modulus
  • $I$ is the moment of inertia
  • Depends on the material properties and cross-sectional geometry of the beam
• The moment-curvature relationship is a fundamental concept in the analysis of beam deflection
  • Used to derive the elastic curve equation
• Examples of moment-curvature relationship:
  • For a rectangular cross-section: $I = \frac{bh^3}{12}$, where $b$ is the width and $h$ is the height
  • For a circular cross-section: $I = \frac{\pi r^4}{4}$, where $r$ is the radius

Integration Constants for Beam Deflection

Solving the Elastic Curve Equation

• When solving the elastic curve equation, integration constants arise due to the integration process
  • These constants represent the beam's initial conditions and must be determined using the boundary conditions
• The number of integration constants depends on the order of the differential equation
  • For the fourth-order elastic curve equation, there will be four integration constants ($C_1$, $C_2$, $C_3$, and $C_4$)

Determining Integration Constants

• To determine the integration constants, substitute the boundary conditions into the general solution of the elastic curve equation, which includes the integration constants
• Create a system of equations by applying the boundary conditions and solve for the integration constants simultaneously
• Once the integration constants are determined, substitute their values back into the general solution to obtain the specific solution for the beam deflection problem
• The specific solution describes the deflection, slope, moment, and shear force along the beam as functions of the position ($x$) and the applied loads and boundary conditions
• Example of determining integration constants:
  • For a cantilever beam with a point load at the free end, the general solution is: $y(x) = \frac{Px^2}{6EI}(3L - x) + C_1x^3 + C_2x^2 + C_3x + C_4$
  • Applying the boundary conditions: $y(0) = 0$, $\frac{dy}{dx}(0) = 0$, $\frac{d^2y}{dx^2}(L) = 0$, $\frac{d^3y}{dx^3}(L) = \frac{P}{EI}$
  • Solving the system of equations yields: $C_1 = 0$, $C_2 = 0$, $C_3 = 0$, $C_4 = 0$