Module 7: Deflection of Beams - Examples & Applications

STEP-BY-STEP

Steps for Double Integration:

Find the Moment Equation: Determine the algebraic equation for the bending moment $M(x)$ along the beam segment of interest.
First Integration (Yields the slope equation): Integrate $M(x)$ once. This gives the equation for the slope or angular rotation ( $\theta$ or $\frac{dy}{dx}$ ) of the beam at any point.

EI \theta = EI \frac{dy}{dx} = \int M(x) \, dx + C_1

Second Integration (Yields the deflection equation): Integrate the slope equation again. This gives the final equation for the vertical deflection ( $y$ ) at any point.

EI y = \int \left( \int M(x) \, dx + C_1 \right) dx + C_2

Apply Boundary Conditions: Determine the constants of integration ( $C_1$ and $C_2$ ) using known physical constraints of the beam's supports.
At a pinned or roller support, the deflection must be exactly zero ( $y = 0$ ).
At a perfectly fixed wall support (cantilever), both the deflection and the slope must be exactly zero ( $y = 0$ and $\theta = 0$ ).

A cantilever beam of length

L

is perfectly fixed at a wall on its right end (

x=L

) and subjected to a concentrated downward load

P

at its free left end (

x=0

). Using Double Integration, find the equation for its maximum deflection.

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Find the maximum deflection of a cantilever beam of length

L

with a point load

P

at the free end. The bending moment diagram is a triangle with maximum moment

-PL

at the fixed support.

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A simply supported beam of length

L

carries both a uniform distributed load

w

over its entire length and a concentrated point load

P

directly at its midspan. Using standard deflection formulas, find the total maximum deflection.

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