In a simply supported beam of span L subjected to central concentrated load, the central deflection is 24 mm. Then the slope at supports is:
(72/L) radians
Understanding Simply Supported Beams and Central Loads
A simply supported beam is a beam supported at both ends, allowing rotation but preventing vertical displacement. When a concentrated load is applied exactly at the center of the span (L), it causes the beam to deflect downwards, with the maximum deflection occurring at the point of load application (the center).
Along with deflection, the beam also undergoes rotation, resulting in slopes at various points. The slope is typically zero at the point of maximum deflection (the center for a symmetric load) and maximum at the supports.
Formulas for Deflection and Slope in Simply Supported Beams
For a simply supported beam of span L subjected to a central concentrated load P, the standard formulas for central deflection and slope at the supports are:
- Central Deflection (\(\delta_C\)): The maximum deflection occurs at the mid-span and is given by: \[ \delta_C = \frac{PL^3}{48EI} \] where E is the Young's modulus of the beam material and I is the moment of inertia of the beam's cross-section. EI is the flexural rigidity of the beam.
- Slope at Supports (\(\theta_{supports}\)): The slope at each support is equal in magnitude and is given by: \[ \theta_{supports} = \frac{PL^2}{16EI} \]
Relating Central Deflection and Slope at Supports
We are given the central deflection and need to find the slope at the supports. We can establish a relationship between the two formulas. From the central deflection formula:
\[ \delta_C = \frac{PL^3}{48EI} \]We can rearrange this formula to express \(\frac{P}{EI}\):
\[ \frac{P}{EI} = \frac{48 \delta_C}{L^3} \]Now, substitute this expression for \(\frac{P}{EI}\) into the formula for the slope at supports:
\[ \theta_{supports} = \frac{PL^2}{16EI} \] \[ \theta_{supports} = \left( \frac{P}{EI} \right) \times \frac{L^2}{16} \] \[ \theta_{supports} = \left( \frac{48 \delta_C}{L^3} \right) \times \frac{L^2}{16} \]Simplify the expression:
\[ \theta_{supports} = \frac{48}{16} \times \frac{\delta_C L^2}{L^3} \] \[ \theta_{supports} = 3 \times \frac{\delta_C}{L} \]This equation shows that the slope at the supports is 3 times the ratio of the central deflection to the span.
Calculating the Slope at Supports
The problem provides the central deflection (\(\delta_C\)) as 24 mm. We can substitute this value into the derived relationship:
\[ \theta_{supports} = 3 \times \frac{\delta_C}{L} \]Substitute \(\delta_C = 24\) mm:
\[ \theta_{supports} = 3 \times \frac{24 \text{ mm}}{L} \] \[ \theta_{supports} = \frac{72}{L} \text{ mm/L} \]Slope is an angle, and when derived using these small deflection formulas, it is inherently in radians, provided consistent units are used for \(\delta_C\) and L. Since the options are given in radians, the result is:
\[ \theta_{supports} = \frac{72}{L} \text{ radians} \]Therefore, the slope at the supports is \(\frac{72}{L}\) radians.
Revision Table: Simply Supported Beam Formulas
| Quantity | Formula (Central Load P) | Location |
|---|---|---|
| Maximum Deflection (\(\delta_{max}\) or \(\delta_C\)) | \( \frac{PL^3}{48EI} \) | Mid-span |
| Slope at Supports (\(\theta_{supports}\)) | \( \frac{PL^2}{16EI} \) | At supports |
Additional Information on Beam Deflection and Slope
Understanding beam deflection and slope is crucial in structural engineering for ensuring structures are safe and perform as intended. Excessive deflection can lead to damage to finishes or affect the function of elements supported by the beam, even if the beam itself is not yielding. The formulas used are based on the Euler-Bernoulli beam theory, which assumes small deflections and elastic material behavior.
- The product EI is known as the flexural rigidity, representing the beam's resistance to bending. A higher EI value means the beam is stiffer and will deflect less under the same load.
- The units of slope calculated using these formulas are typically radians. If \(\delta_C\) and L are in meters, the result is in radians. If \(\delta_C\) is in mm and L is in meters, you'd need to be careful with units, but here the relationship simplifies such that if \(\delta_C\) is in mm and L is just L, the slope is \(3 \times \frac{24}{L}\) which gives \(\frac{72}{L}\), and this value is in radians if 24 and L were in the same units originally. Since the options have radians, the value \(\frac{72}{L}\) is understood to be in radians.
- Different loading conditions (e.g., uniformly distributed load, multiple point loads) will result in different formulas for deflection and slope.
Important Questions from Deflection of Beam
A cantilever beam of length L has flexural rigidity EI up to length L/2 from the fixed end and EI/2 for the rest. It carries a moment M at the free end. The slope at the free end is given by-
The reaction of the prop of a propped cantilever beam of span I with UDL W kN/m is
Which of the following methods is NOT used for finding deflection of beam?
The maximum deflection occurs in a structural member when the slope is
The deflection of a simply supported beam at supports is generally
