In a continuous beam ABC, support A is fixed and supports B and C are simply supported. A uniformly distributed load of w per meter run is applied over the span AB and span BC is subjected to a point load W at mid-span. If the beam has uniform cross-section but differs in span lengths for AB and BC, the conditions to be used for the analysis of continuous beam by the slope deflection method are

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  1. θ= 0, MBA + MBC = 0, θ= 0
  2. θ= 0, MBA MBC = 0, MCB = 0
  3. θ= 0MBA MBCMCB = 0
  4. θ= 0MBA MBC , θ= 0

Answer (Detailed Solution Below)

Option 2 : θ= 0, MBA MBC = 0, MCB = 0
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Concept:

To analyze a continuous beam using the slope deflection method, specific boundary conditions and equilibrium equations must be considered at each support. The conditions will differ based on the type of support and load applied to each span.

Given Beam:

  • Support A is fixed, implying \( \theta_A = 0 \).
  • Support B and Support C are simply supported, meaning no moment resistance at C (\( M_{CB} = 0 \)).
  • Span AB has a uniformly distributed load \( w \) per meter run.
  • Span BC has a point load \( W \) at its mid-span.
  • The beam has a uniform cross-section but different span lengths for AB and BC.

Applying the Slope Deflection Method:

  • Fixed Support A: Since support A is fixed, the rotation \( \theta_A \) is zero.
  • Moment at Joint B: The sum of moments at B due to adjacent spans should be zero for equilibrium, giving \( M_{BA} + M_{BC} = 0 \).
  • Simply Supported Support C: The moment at support C is zero (\( M_{CB} = 0 \)) since it is simply supported.

Conditions for Analysis:

\( \theta_A = 0\)

\(M_{BA} + M_{BC} = 0 \)

\(M_{CB} = 0\)

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