Question
Download Solution PDFIn an otherwise symmetrical portal frame with one end fixed and the other end hinged, the hinged, the hinge support sinks by an amount Δ. The fixed end bending moment induced at fixed end of the horizontal member of the frame due to the sinking of the support will be (given that ‘L’ is the length of the members and EI is the flexural stiffness)
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFExplanation:
For a symmetrical portal frame with one end fixed and the other end hinged, when the hinged support sinks by an amount Δ, the system behaves as a statically indeterminate structure. However, we can still analyze the behavior using principles of structural analysis and compatibility conditions.
In such a case, because of the symmetry and the specific boundary conditions, the sinking of the hinge induces a bending moment at the fixed end of the frame. The calculation of this moment generally involves compatibility conditions and equilibrium equations that take into account the geometry of the frame, the properties of the materials, and the specifics of the boundary conditions.
Given the simplicity that the problem is aiming for (not considering horizontal displacement or axial loads), and aiming to provide a direct answer from the options given, for a vertical displacement (sinking) Δ at the hinged support in such a frame, the induced bending moment (M_f) at the fixed end of the horizontal member can be approximated using principles of structural analysis applicable to such setups.
The correct formula, for the induced bending moment at the fixed end due to a vertical sinking Δ of a hinged support in a symmetrical portal frame considering its length (L) and flexural stiffness (EI), is:
\(M_f = \frac{6EIΔ}{L^2}\)
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