(a) Determine the horizontal component of deflection of joint D of the truss loaded as shown in the figure. The cross-sectional area of each member is tabulated below. Take E = 200 kN/mm². Length of the members are indicated in the figure. Use Castig

Question

(a) Determine the horizontal component of deflection of joint D of the truss loaded as shown in the figure. The cross-sectional area of each member is tabulated below. Take E = 200 kN/mm². Length of the members are indicated in the figure. Use Castigliano's theorems. (15 marks)

Table : Area of cross-section
| S.No | Member | Area of cross-section |
|------|--------|----------------------|
| 1. | AB | 765 mm² |
| 2. | AD | 390 mm² |
| 3. | DB | 575 mm² |
| 4. | BC | 765 mm² |
| 5. | CD | 390 mm² |

(b) Determine the collapse load in case of propped cantilever of span 'l' and subjected to uniformly distributed load 'P' per metre length as shown in the figure. Take the plastic moment capacity of beam as M_P. (15 marks)

(c) A circular water tank with flexible base is to be designed for a capacity of 450 kL. The depth of water is to be 4 m including a free board of 250 mm. Find the dimensions of the tank and design and detail the wall of the tank. Use M 20 concrete and Fe 250 steel. (20 marks)

Given : Tensile stress in steel under direct tension for plain mild steel bars, σ_s = 115 MPa
Permissible direct tensile stress in concrete (M 20), σ_ct = 1·2 MPa
Unit weight of water, γ = 9800 N/m³

UPSC Answer Check · Accepted Answer

Solve all three sub-parts systematically, allocating time proportional to marks: approximately 35% for part (a) truss deflection using Castigliano's theorem, 30% for part (b) plastic collapse analysis of propped cantilever, and 35% for part (c) complete water tank design including dimensioning, wall thickness calculation, and reinforcement detailing. Begin each part with clear identification of given data, show complete derivations with formulae, and conclude with boxed final answers. For part (c), present design calculations followed by a neat sketch showing reinforcement arrangement.
- Part (a): Correct application of Castigliano's first theorem by placing unit horizontal load at D, calculation of member forces P due to actual loading and ∂P/∂Q due to dummy load, summation of (P·L/AE)·(∂P/∂Q) for all members
- Part (b): Identification of correct collapse mechanism for propped cantilever with UDL—formation of plastic hinge at fixed end and within span, use of virtual work or equilibrium method to relate external work to internal energy dissipation
- Part (c): Calculation of tank diameter from capacity (450 kL) and effective depth (3.75 m), determination of hoop tension and bending moment coefficients from IS 3370 tables for H²/Dt ratio, design of vertical and horizontal reinforcement
- Part (c): Check for tensile stress in concrete against permissible σ_ct = 1.2 MPa, calculation of steel area using σ_s = 115 MPa, provision of minimum reinforcement as per IS 3370
- Part (c): Detailing showing distribution steel, vertical steel on both faces, development lengths, and proper lap splices for circular tank wall

Dimension	Weight	Max marks	Excellent	Average	Poor
Concept correctness	20%	10	Correctly identifies Castigliano's theorem application for (a), recognizes the two-hinge collapse mechanism for propped cantilever in (b), and applies IS 3370 provisions for circular tank with flexible base in (c); no conceptual errors in any sub-part	Minor errors in theorem application or collapse mechanism identification; tank design follows correct approach but with some incorrect coefficient selection or misapplication of H²/Dt ratio	Fundamental misunderstanding of Castigliano's method, wrong collapse mechanism assumed, or treats tank as fixed base instead of flexible base; major conceptual gaps
Numerical accuracy	20%	10	All calculations precise with correct unit conversions (mm to m, kN to N where needed); final deflection in mm, collapse load P in terms of Mp/l², and tank dimensions/reinforcement quantities accurately computed	Minor arithmetic errors or unit conversion mistakes in one sub-part; final answers slightly off but method sound; significant figures handled reasonably	Multiple calculation errors, incorrect unit handling (e.g., mixing mm and m), or order-of-magnitude mistakes; final answers unreliable or missing
Diagram quality	20%	10	Clear truss diagram with member lengths and load positions for (a); collapse mechanism diagram with hinge locations and rotation angles for (b); neat cross-section of tank wall showing reinforcement layout, cover, and dimensions for (c)	Diagrams present but lacking clarity or missing some labels; truss geometry or collapse mechanism sketch incomplete; tank section shows reinforcement but without proper dimensioning	Missing diagrams or poorly drawn sketches that hinder understanding; no attempt to illustrate collapse mechanism or reinforcement detailing
Step-by-step derivation	20%	10	Systematic presentation: for (a) table of member forces with P, ∂P/∂Q, L, A values; for (b) clear virtual work equation with external work and internal dissipation; for (c) complete design steps from capacity to final reinforcement with all IS code references	Derivations present but with skipped steps or missing intermediate calculations; some logical jumps that require examiner inference; tabular format for truss but incomplete	Disorganized working with no clear flow; missing essential steps; no tables or systematic presentation; appears as random calculations without derivation structure
Practical interpretation	20%	10	For (a), comments on significance of deflection magnitude for serviceability; for (b), discusses ductility requirement and practical moment redistribution; for (c), justifies bar diameter and spacing choices, mentions crack control and water-tightness provisions per IS 3370	Brief mention of practical relevance in one or two sub-parts; some code references but limited insight into why specific choices were made; standard concluding remarks	Purely mathematical treatment with no engineering judgment; no code references, no discussion of serviceability, durability, or constructability; answer ends with numbers only

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