Euler-Bernoulli’s equation is commonly used to describe the deflected shape of beam. The beam is assumed isotropic, elastic and having consistent cross section. The deflection is assumed small compared to beam’s geometry, to allow simple solution using Euler-Bernoulli’s equation.
Double integration is performed on the equation. To solve for unknown constants, boundary condition and continuity condition are used. With the exact equation for deflection and rotation of beam, we can determine them for any point on the beam.
The following introduces the calculation of beam deflection using double integration method. Watch the video above for full details.
Euler Bernoulli's equation
Assumptions to use Euler-Bernoulli's equation
Beam overview
Support reactions calculation
Derivation of internal moment function
Review of expression for slope and displacement of elastic curve
Double integration on the internal moment function
Boundary condition 1 to solve for C2
Boundary condition 2 to solve for C1
Simplification of expression for slope and displacement
Calculation of slope of elastic curve at supports
Determination of location where maximum displacement occurs
Derivation of simplified formula to calculate maximum deflection
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