Direct Integration Method for Deflection

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SUMMARY

The discussion focuses on calculating the slope and deflection of a 4m long beam fixed at one end and subjected to a 50 kN·m clockwise moment at the free end. The governing equation used is the second derivative of deflection, expressed as d²y/dx² = M/EI. The final results indicate a slope of 0.0174 radians and a deflection of 34.8 mm at the free end, confirming the importance of knowing the product EI for accurate calculations.

PREREQUISITES
  • Understanding of beam mechanics and deflection theory
  • Familiarity with the moment-curvature relationship in structural analysis
  • Knowledge of material properties, specifically Young's modulus (E) and moment of inertia (I)
  • Ability to apply differential equations in engineering contexts
NEXT STEPS
  • Study the derivation of the moment-curvature relationship in beam theory
  • Explore the calculation of deflection using the Euler-Bernoulli beam theory
  • Learn about the significance of the product EI in structural analysis
  • Investigate numerical methods for solving beam deflection problems
USEFUL FOR

Civil engineers, structural analysts, and students studying mechanics of materials will benefit from this discussion, particularly those focused on beam deflection and moment analysis.

craig22
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Homework Statement


I'm given a 4m long beam that is fixed to a wall at the left end and is free on the other end. There is an external 50KN*m clockwise moment acting on the free end. I need to know the slope and deflection of the free end.


Homework Equations


d2y/dx2=M/EI



The Attempt at a Solution


I know that the reaction moment at the fixed support is 50KN*m. There are no x or y direction components, so all I have to work with are the moments. When I pass a section through the beam, how do I get my equation together from there?

The answers are supposed to be .0174rad and 34.8 mm
 
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Presumably, you know E and I, or the product EI.
Let x=0 at the fixed support, and x=4 at the free end. Can you write down the function for M at x?
 
I figured it out, thanks
 

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