Math Problem Statement
A cantilever beam of cross section 150 mm x 200 mm is 2 m long. If deflection at free end is 2 mm, calculate u.d.l. that the beam can carry over whole span. Also calculate maximum slope of the beam. Take E = 210 GPa.
Solution
To solve this problem, we will use the principles of beam deflection and slope for a cantilever beam subjected to a uniformly distributed load (UDL). Let's break down the problem into the required steps.
Given Data:
- Cross-sectional dimensions of the beam:
- Width, mm
- Height, mm
- Length of the beam, m
- Deflection at the free end, mm
- Young's Modulus, GPa = N/mm²
Step 1: Calculate the moment of inertia (I)
The moment of inertia for a rectangular cross-section is given by:
Substituting the given values:
Step 2: Relate deflection to UDL
For a cantilever beam subjected to a UDL , the deflection at the free end is given by:
Rearranging to solve for :
Substituting the known values:
Simplifying:
Thus, the UDL that the beam can carry is kN/m.
Step 3: Calculate the maximum slope of the beam
The maximum slope in a cantilever beam subjected to a UDL occurs at the fixed end and is given by:
Substituting the known values:
Simplifying:
Summary:
- The UDL that the beam can carry is kN/m.
- The maximum slope of the beam is mrad.
Would you like more details or explanations on any of these steps?
Relative Questions:
- How would the deflection change if the length of the beam were increased?
- What would be the effect of changing the material's Young's modulus on the deflection and slope?
- How does the moment of inertia influence the deflection of a beam?
- What happens to the maximum slope if the UDL is doubled?
- How would the deflection and slope change if the beam were simply supported instead of cantilevered?
Tip: When dealing with beam deflection problems, always double-check the units of the inputs to ensure consistent and accurate calculations.
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Math Problem Analysis
Mathematical Concepts
Beam Deflection
Moment of Inertia
Young's Modulus
Slope of Beams
Formulas
Moment of Inertia for Rectangular Section
Deflection due to UDL in Cantilever Beam
Maximum Slope in Cantilever Beam
Theorems
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Suitable Grade Level
Undergraduate Engineering
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