Deflection of Beams

  • Calculate deflection using the equation of the elastic curve.

  • Utilize the direct determination of the elastic curve.

  • Derive the deflection and slope curves for a beam through integration of the moment-curvature relationship.

  • Apply discontinuity functions and standardized solutions to simplify the calculation of deflection and slope curves for beams.

Differential Equation of deflection curve,

EI = M

Integrating once, we get \[EI \dfrac{dy}{dx} = \int M dx\] Integrating once more, we get \[EI y = \int \int M dx dx\] The constants of integration can be eliminated by applying boundary conditions, such as: At (unyielding) supports, \[y = 0\] At fixed supports \[\dfrac{dy}{dx} = \mathrm{slope\ of\ deflected\ curve} = 0\]

Solved Examples

Solved Example:

47-2-01

Slope at a point in a beam is the:

Correct Answer: B

Solved Example:

47-2-02

Which of the following is an elastic curve equation for shear force? (EI = flexural rigidity)

Correct Answer: C

Solved Example:

47-2-03

Which of the following statements is/are true for a simply supported beam?

Correct Answer: B

Solved Example:

47-2-04

Which of the following is a differential equation for deflection?

Correct Answer: C

Solved Example:

47-2-05

The equation of deformation is derived to be $y = x^3 - xL$ for a beam as shown in the figure. Curvature of the beam at the mid-span (in units, in integer) will be: (GATE Civil 2021)

47.2-05



Solution:

\[y = x^2 - xL\] Curvature at the midsection is: \[\dfrac{1}{R} = \dfrac{d^2y}{dx^2}\] \[\dfrac{dy}{dx} = 2x - L\] \[\dfrac{d^2y}{dx^2} = 2\]

Correct Answer: A