The deflection of a simply supported beam at supports is generally
Understanding Beam Deflection at Supports
When studying the behavior of beams under load, two important concepts are deflection and slope. Deflection refers to the displacement of a point on the beam from its original position, while slope refers to the angle of the tangent to the deflected curve of the beam.
What is a Simply Supported Beam?
A simply supported beam is a beam that is supported by a pin support at one end and a roller support at the other end. These types of supports allow rotation but prevent vertical displacement at the support points.
- Pin Support: Prevents both vertical and horizontal displacement. However, for typical beam problems with only vertical loads, it mainly prevents vertical displacement. It allows rotation.
- Roller Support: Prevents vertical displacement but allows horizontal movement and rotation. This is important to prevent stresses due to thermal expansion or contraction.
Because these supports prevent vertical movement at their locations, the beam cannot move downwards or upwards at these points.
Deflection at the Supports of a Simply Supported Beam
Based on the definition of the supports in a simply supported beam, the vertical displacement (deflection) at the points where the supports are located is constrained to be zero. This is a fundamental boundary condition for this type of beam.
Imagine placing your hand under a flexible ruler at both ends and pushing down on the middle. Your hands are acting like supports preventing the ruler from moving down at those points. The maximum bending and downward movement will occur somewhere between your hands, not at your hands themselves.
Analyzing the Given Options
Let's consider the given options in the context of a simply supported beam and its deflection at the supports:
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twice of the slope: Slope and deflection are related, but deflection at the support is a specific boundary condition (zero), while the slope at the support is generally not zero (unless it's a special loading case or a fixed support). So, deflection is not typically twice the slope at the supports.
-
maximum: The maximum deflection of a simply supported beam under typical loading conditions (like a point load or uniformly distributed load) occurs somewhere between the supports, usually near the middle of the span, not at the supports. Deflection is constrained to be zero at the supports.
-
zero: As explained by the nature of pin and roller supports, they prevent vertical displacement. Therefore, the deflection at the support points of a simply supported beam is always zero.
-
half of slope: Similar to option 1, there is no general rule stating that deflection at the support is half of the slope. Deflection is fixed at zero at the supports.
Therefore, the deflection of a simply supported beam at supports is generally zero because the supports prevent vertical movement at those points.
| Location | Deflection (v) | Slope (dv/dx or $\theta$) |
|---|---|---|
| At Supports (Simply Supported) | Zero (v = 0) | Generally Not Zero |
| Mid-span (for symmetric loading) | Maximum (v = vmax) | Zero ($\theta$ = 0) |
Revision Table: Beam Deflection Key Concepts
| Concept | Definition | Simply Supported Beam Boundary Conditions at Supports |
|---|---|---|
| Deflection (v) | Vertical displacement from original position | v = 0 |
| Slope ($\theta$) | Angle of the tangent to the deflected curve | Generally $\theta \neq$ 0 (unless specific load cases) |
Additional Information: Support Types and Boundary Conditions
Different types of supports impose different constraints on a beam's movement, leading to different boundary conditions for deflection and slope.
- Fixed Support (Cantilever or Propped Cantilever end):
- Deflection is zero (v = 0).
- Slope is zero ($\theta$ = 0).
- Pin Support (Simply Supported end):
- Deflection is zero (v = 0).
- Rotation/Slope is allowed ($\theta \neq$ 0).
- Roller Support (Simply Supported end):
- Deflection is zero (v = 0).
- Rotation/Slope is allowed ($\theta \neq$ 0).
- Horizontal movement is allowed.
- Free End (Cantilever end):
- Deflection is generally not zero (can be maximum).
- Slope is generally not zero (can be maximum).
Understanding these boundary conditions is crucial for analyzing beam behavior and solving problems related to beam deflection and stress.
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-
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:
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
