## Solution to Problem 616 | Double Integration Method

**Problem 616**

For the beam loaded as shown in Fig. P-616, determine (a) the deflection and slope under the load P and (b) the maximum deflection between the supports.

Solution to Problem 617 | Double Integration Method

**Problem 617**

Replace the load P in Prob. 616 by a clockwise couple M applied at the right end and determine the slope and deflection at the right end.

**Problem 616**

For the beam loaded as shown in Fig. P-616, determine (a) the deflection and slope under the load P and (b) the maximum deflection between the supports.

Method of Superposition | Beam Deflection
### Rotation and Deflection for Common Loadings

The slope or deflection at any point on the beam is equal to the resultant of the slopes or deflections at that point caused by each of the load acting separately.

**Case 1: Concentrated load at the free end of cantilever beam**

Maximum Moment

$M = -PL$

Slope at end

$\theta = \dfrac{PL^2}{2EI}$

Maximum deflection

$\delta = \dfrac{PL^3}{3EI}$

Deflection Equation ($y$ is positive downward)

$EI \, y = \dfrac{Px^2}{6}(3L - x)$

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Area-Moment Method | Beam Deflections

Another method of determining the slopes and deflections in beams is the area-moment method, which involves the area of the moment diagram.

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