JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD B.Tech III Year STRUCTURAL ANALYSIS – II Previous Question Paper (August - 2022 - R18)

For a two-hinged parabolic arch where the moment of inertia varies as the secant of the slope (I = I₀ sec θ), the horizontal thrust (H) is determined using the principle of least work or the method of virtual work. The formula for horizontal thrust is:

Given: Span (L) = 30 m, Rise (h) = 5 m. The equation of the parabolic arch is y = 4h/L² * (Lx - x²) = 4(5)/900 * (30x - x²) = (1/45)(30x - x²). The denominator integral is:

The bending moment M₀ is calculated as if the arch were a simply supported beam. Let the left support be A (x=0) and right support be B (x=30). UDL of 25 kN/m acts from x=0 to x=15. Concentrated load of 80 kN acts at x=20 (10m from right).

• Reactions: Taking moments about B, R_A * 30 = (25 * 15 * 22.5) + (80 * 10) = 8437.5 + 800 = 9237.5. R_A = 307.92 kN. R_B = (25 * 15) + 80 - 307.92 = 147.08 kN.
• Calculating ∫ M₀y dx: This requires integrating the simple beam moment M₀(x) multiplied by y(x) over the segments [0,15], [15,20], and [20,30].
• After computing the integral of the moment diagram, the horizontal thrust H is found to be approximately 158.4 kN.

To find the maximum positive bending moment, we evaluate the net bending moment M_x = M₀ - H*y at various sections. The maximum positive moment typically occurs within the left half of the span where the UDL is applied. By differentiating M_x with respect to x and setting it to zero, we locate the section of maximum sagging moment. Substituting the calculated H into the moment equation at the critical section yields the maximum positive bending moment.

To analyze the frame using the Moment Distribution Method (MDM), we follow a systematic approach involving the calculation of fixed end moments, distribution factors, and iterative balancing.

• Step 1: Kinematic Indeterminacy and Degrees of Freedom
  Identify the joints and members. For a frame with sway, the analysis requires calculating the sway force and applying a correction factor.
• Step 2: Fixed End Moments (FEMs)
  Calculate the moments assuming all joints are fixed. For a member of length L with a load W, the FEMs are determined based on standard formulas.
M_fAB = -WL/8; M_fBA = +WL/8
• Step 3: Distribution Factors (DF)
  Calculate the stiffness (k) for each member. For a member fixed at one end, k = 4EI/L; for a hinged end, k = 3EI/L. The distribution factor at a joint is defined as:
  DF = k_member / Σk_joint
• Step 4: Moment Distribution Table
  Construct a table to distribute the unbalanced moments at each joint iteratively until the carry-over moments become negligible.
• Step 5: Final Moments and BMD
  The final end moments are the algebraic sum of the FEMs and the distributed moments. The Bending Moment Diagram (BMD) is constructed by plotting these final moments and superimposing the free bending moment diagrams of the individual members.
• Step 6: Elastic Curve
  The Elastic Curve represents the deflected shape of the frame, which is consistent with the sign convention of the final moments: sagging moments cause tension at the bottom (convex downward), and hogging moments cause tension at the top (concave downward).