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The moment required to produce a unit slope near the end of the beam is called the ___________ of the beam.
Flexibility
Stiffness
Rigidity
None of these
Stiffness
Stiffness is defined as the force or moment required to produce unit displacement or unit rotation at a specific point in a structural member. In the context of beam analysis, rotational stiffness (k) is the moment required to induce a unit slope (radian) at the end of the member while keeping other ends fixed or constrained.
Stiffness is defined as the force or moment required to produce unit displacement or unit rotation at a specific point in a structural member. In the context of beam analysis, rotational stiffness (k) is the moment required to induce a unit slope (radian) at the end of the member while keeping other ends fixed or constrained.
k=╬╕MтАЛ тАФ Fundamental definition of rotational stiffness
k=L4EIтАЛ тАФ Rotational stiffness for a beam with far end fixed
When a bending moment M is applied at the end of a beam of length L and flexural rigidity EI, it induces an end rotation ╬╕. By the Slope-Deflection Method or Stiffness Matrix Method, the relationship is established as M=k╬╕. If we set ╬╕=1 radian, then the moment M required is equal to the stiffness k. For a prismatic beam pinned at one end and fixed at the other, k=L4EIтАЛ.
Stiffness represents the resistance offered by a structure against deformation.
Flexibility is the reciprocal of stiffness (f=k1тАЛ), defined as displacement due to unit force/moment.
The units for rotational stiffness are NтЛЕm/rad or kNтЛЕm/rad.
Essential for calculating internal forces in statically indeterminate structures.
Forms the basis for the Stiffness Matrix Method used in finite element analysis.
Depends heavily on the support conditions at the far end of the beam.
Requires knowledge of the cross-sectional properties (I) and material modulus (E).
Analysis of continuous beams.
Design of rigid frames in structural engineering.
Calculation of natural frequencies in structural dynamics.
| Feature | Stiffness | Flexibility |
|---|---|---|
Definition | Force/Moment for unit deformation | Deformation per unit force/moment |
Flexibility is the inverse of stiffness. If stiffness is defined as k, then flexibility f=k1тАЛ.
Option A (Flexibility) is incorrect because it relates to displacement caused by unit force, not the force required to cause unit displacement.
B is correct тАФ The moment required to produce unit rotation is strictly defined as the rotational stiffness of the beam.
Remember: Stiffness (k) is force/rotation, Flexibility (f) is rotation/force. They are reciprocals; high stiffness structures undergo small deformations under load.