(a) Analyse the continuous beam shown in the figure by slope-deflection method. Draw the Shear Force Diagram (SFD) and Bending Moment Diagram (BMD). Also calculate point of contraflexure in BMD. 20 marks

(b) A simply supported reinforced concrete be

Question

(a) Analyse the continuous beam shown in the figure by slope-deflection method. Draw the Shear Force Diagram (SFD) and Bending Moment Diagram (BMD). Also calculate point of contraflexure in BMD. 20 marks

(b) A simply supported reinforced concrete beam of size 300 × 500 mm (effective) is reinforced with 4 bars of 16 mm φ of Fe 500 grade steel. Determine the anchorage length of the bars at simply supported end if it is subjected to a factored shear force of 350 kN at the centre of 300 mm wide masonry support. The concrete mix of grade M25 is to be used.
Bond stress (τbd) for plain bar for M25 = 1·4 MPa
Es = 2 × 10⁵ N/mm² 10 marks

(c) A tension member consists of two angle-irons, back to back, of size ISA 75 × 75 × 8 and is connected to the same side of a gusset plate by a single row of six 20 mm diameter bolts as shown in the figure. Calculate the load carrying capacity when the two angles are tack-bolted.
Yield stress of steel (f_y) = 250 MPa and
Ultimate tensile stress (f_u) = 410 MPa.

Given :
β = 1·4 - 0·076 (w/t) (f_y/f_u) (b_s/L_c) ≤ (f_u γ_m0 / f_y γ_m1)
≥ 0·7

where, w = outstand leg width
b_s = shear lag width
L_c = length of the end connection
α = 0·6 for one or two bolts
α = 0·7 for three bolts
α = 0·8 for four or more bolts 20 marks

UPSC Answer Check · Accepted Answer

Analyse the continuous beam in part (a) using slope-deflection method, then solve the anchorage length calculation in part (b), and finally determine the tension member capacity in part (c). Allocate approximately 40% time to part (a) as it carries 20 marks and requires complete SFD/BMD with contraflexure point; 20% to part (b) for the anchorage calculation; and 40% to part (c) for the complex steel tension member design with β factor evaluation. Present each part separately with clear headings.
- Part (a): Correct application of slope-deflection equations, handling of boundary conditions (θ=0 at fixed ends, continuity at supports), calculation of fixed end moments, solution of simultaneous equations for unknown rotations, and derivation of final moments for SFD/BMD
- Part (a): Accurate plotting of SFD with proper sign convention and shear values at critical sections; BMD with correct maximum moments, inflection points, and explicit calculation of contraflexure point location
- Part (b): Correct determination of development length using Ld = (φσs)/(4τbd) with appropriate modification factors for Fe 500 steel and deformed bars (τbd = 1.4×1.6 = 2.24 MPa), check against IS 456:2000 clause 26.2.1
- Part (b): Verification of anchorage length against support width constraint (300 mm) and shear force consideration at support face; application of increased development length if required by bond conditions
- Part (c): Calculation of gross and net area of double angle section (ISA 75×75×8), determination of shear lag width bs and connection length Lc for six bolt arrangement
- Part (c): Correct evaluation of β factor using given formula with α = 0.8 for four or more bolts, check against limits (0.7 ≤ β ≤ fuγm0/fyγm1), and calculation of design strength based on yielding of gross section and rupture of net section as per IS 800:2007

Dimension	Weight	Max marks	Excellent	Average	Poor
Concept correctness	20%	10	Demonstrates flawless understanding of slope-deflection fundamentals for part (a) including correct sign convention and handling of continuous beam; applies IS 456:2000 provisions accurately for anchorage in part (b) with proper modification factors; correctly interprets IS 800:2007 clauses for tension member design in part (c) including shear lag and β factor concept	Shows basic understanding of methods but makes minor errors in sign convention for slope-deflection, applies standard formulas for anchorage without checking limits, or calculates tension capacity with some confusion between gross and net section concepts	Fundamental misconceptions in slope-deflection method application, incorrect use of bond stress values without modification factors, or complete misunderstanding of shear lag phenomenon and β factor evaluation in steel design
Numerical accuracy	20%	10	All calculations precise to appropriate significant figures: correct fixed end moments and final moments in part (a), accurate anchorage length with proper unit conversions in part (b), and exact β value with correct limiting checks in part (c); final answers match expected values within acceptable tolerance	Generally correct approach with minor arithmetic errors in one part, or correct final answers but with some intermediate calculation errors that partially cancel out; unit consistency maintained throughout	Significant calculation errors in multiple parts, wrong formula substitutions, incorrect unit handling (mm vs m, kN vs N), or answers that are orders of magnitude incorrect indicating conceptual-numerical confusion
Diagram quality	20%	10	Clear, well-labelled SFD and BMD for part (a) with all critical values marked, proper sign convention indicated, dimensions shown, and contraflexure point explicitly located; neat free body diagrams if used; steel connection sketch for part (c) showing bolt arrangement and dimensions	Diagrams present but with missing labels, unclear scaling, or omitted critical values; basic SFD/BMD shapes correct but numerical values not marked at key points; connection sketch present but lacking dimensional details	Missing diagrams for part (a), poorly drawn diagrams without scale or values, incorrect diagram shapes (e.g., parabolic where linear required), or complete absence of visual representation for the continuous beam analysis
Step-by-step derivation	20%	10	Systematic presentation: for (a) complete slope-deflection equations written, boundary conditions stated, simultaneous equations solved showing working; for (b) formula stated with code reference, substitution shown, check against support width; for (c) all parameters defined, β calculation steps explicit, final capacity determination with both limit states	Some steps shown but with gaps in derivation, or final formulas stated without showing intermediate working; attempts at systematic approach but with missing logical connections between steps	Only final answers stated without derivation, or disorganized working with no logical flow; jumps from problem statement to final answer without showing how results obtained; missing essential steps like boundary condition application or β limit checks
Practical interpretation	20%	10	Interprets contraflexure point significance for reinforcement curtailment in part (a); discusses anchorage failure modes and practical detailing for part (b); explains why β factor reduces effective area and implications for connection design in part (c); references relevant IS codes (456:2000, 800:2007) appropriately	Brief mention of practical significance without elaboration, or code references stated without explaining their relevance; understands that results have practical meaning but does not articulate engineering implications clearly	Purely mathematical treatment with no engineering context, no code references, or failure to recognize that calculated values must satisfy practical constraints like available support width or minimum connection efficiency

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