Among the following, which beam can be classified as a statically indeterminate beam?

Explanation:

Statically Indeterminate Beam

Definition: A statically indeterminate beam is a structural element subjected to constraints that cannot be solely determined by static equilibrium equations. This means that the beam has more supports or restraints than required to maintain its equilibrium, leading to additional reactions that are not computable through basic statics alone. Instead, methods such as compatibility conditions, deflection calculations, or advanced structural analysis techniques are needed to solve for these reactions.

Fixed Beam: A fixed beam, also known as a clamped beam, is a type of beam that is rigidly fixed at both ends. This fixing condition imposes both rotational and translational constraints at the support points, resulting in a beam that cannot rotate or translate at its ends. This type of beam is inherently statically indeterminate because the number of unknown reactions exceeds the number of static equilibrium equations available. Specifically, a fixed beam has four unknown reactions: two vertical reactions, one horizontal reaction, and two moments (one at each end), while only three equilibrium equations (∑Fx = 0, ∑Fy = 0, and ∑M = 0) are available. Therefore, additional equations derived from the beam's deflection or compatibility conditions are required to solve for the unknown reactions.

Advantages:

• Increased stiffness and reduced deflection due to the fixed support conditions, making it suitable for applications requiring minimal deformation.
• Enhanced load-carrying capacity because the fixed ends provide additional moment resistance.

Disadvantages:

• Complexity in analysis and design due to the need for additional compatibility conditions or deflection calculations to determine the reactions.
• Potential for higher internal stresses at the fixed ends, which may require careful consideration in the design to avoid material failure.

Applications: Fixed beams are commonly used in construction and mechanical structures where high rigidity and minimal deflection are desired, such as in bridges, buildings, and machine frames.

Important information about other options:

Simply Supported Beam: A simply supported beam is supported at both ends, typically with a pin support at one end and a roller support at the other. This arrangement allows the beam to rotate but not translate at the supports, making it statically determinate. The reactions at the supports can be determined using only the static equilibrium equations, without the need for additional compatibility conditions.

Overhanging Beam: An overhanging beam extends beyond its supports on one or both ends. While it may appear more complex than a simply supported beam, it is still statically determinate if it has only two supports (one pin and one roller), as the reactions can be found using the basic equilibrium equations.

Cantilever Beam: A cantilever beam is fixed at one end and free at the other. It is statically determinate because the reactions (vertical reaction, horizontal reaction, and moment) at the fixed end can be determined using the equilibrium equations. Despite its simplicity in terms of analysis, cantilever beams are commonly used in applications requiring a free end, such as in balconies, overhangs, and certain types of bridges.