### 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 Example:

#### 47-2-01

Slope at a point in a beam is the:

### Solved Example:

#### 47-2-02

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

### Solved Example:

#### 47-2-03

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

### Solved Example:

#### 47-2-04

Which of the following is a differential equation for deflection?

### 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)

Solution:

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