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Beam Deflection Equation:
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The beam deflection equation \( EI \frac{d^2v}{dx^2} = M(x) \) is a fundamental equation in structural engineering that relates the bending moment in a beam to its deflection. This differential equation is used to calculate how much a beam will bend under various loading conditions.
The calculator uses the beam deflection equation:
Where:
Explanation: The equation shows that the curvature of a beam (second derivative of deflection) is proportional to the bending moment and inversely proportional to the product of modulus of elasticity and moment of inertia.
Details: Calculating beam deflection is crucial in structural design to ensure that beams will not deflect excessively under load, which could lead to structural failure or serviceability issues in buildings and bridges.
Tips: Enter the modulus of elasticity in Pascals, moment of inertia in meters to the fourth power, and bending moment in Newton-meters. All values must be valid positive numbers.
Q1: What is modulus of elasticity?
A: Modulus of elasticity (E) is a material property that measures its stiffness or resistance to elastic deformation under load.
Q2: What is moment of inertia?
A: Moment of inertia (I) is a geometric property that measures how a cross-section resists bending. It depends on the shape and size of the cross-section.
Q3: What are typical values for these parameters?
A: For steel, E is typically around 200 GPa. Moment of inertia varies greatly with cross-sectional shape. Bending moments depend on loading conditions and span length.
Q4: How do I calculate deflection from the second derivative?
A: You need to integrate the equation twice with respect to x and apply appropriate boundary conditions based on support conditions.
Q5: Are there limitations to this equation?
A: This equation assumes linear elastic material behavior, small deflections, and applies specifically to Euler-Bernoulli beam theory.