(a) Draw the shearing force and bending moment diagrams for the beam loaded as shown in the figure below. (8 marks)

(b) During the design of a beam, an ISMB 550 @ 1·037 kN/m is selected for use as a simply supported beam of 7 m span carrying a reinf

Question

(a) Draw the shearing force and bending moment diagrams for the beam loaded as shown in the figure below. (8 marks)

(b) During the design of a beam, an ISMB 550 @ 1·037 kN/m is selected for use as a simply supported beam of 7 m span carrying a reinforced concrete floor capable of providing lateral restraint to the top compression flange. The total uniformly distributed load is made up of 100 kN dead load and 150 kN imposed load. In addition to this load, the beam also carries a point load at its midspan which is made up of 50 kN dead load and 50 kN imposed load. Check the adequacy of the section for the following :

(i) Shear strength
(ii) Bending strength
(iii) Deflection
(iv) Web buckling at support

Assume the section is plastic.

Given :
Stiff bearing length = 100 mm
f_y = 250 MPa, E = 2 × 10⁵ MPa
γ_mo = 1·1
For plastic section β_b = 1·0
For simply supported beam, ψ = 1·2

| KL/r | 90 | 100 | 110 | 120 |
|------|-----|------|------|------|
| f_cd (MPa) | 121 | 107 | 94·6 | 83·7 |

Properties of ISMB 550 :
Elastic section modulus, Zₑ = 2359·8 × 10³ mm³
Plastic section modulus, Zₚ = 2711·98 × 10³ mm³
Moment of Inertia about major axis, I₂₂ = 64900 × 10⁴ mm⁴

(All dimensions are in mm) (20 marks)

(c) Using the unit load method, determine horizontal and vertical components of deflection at point A for the frame loaded as shown in the figure below. Support C is fixed and B is a rigid joint. Take E as constant and same for both the members. (15 marks)

UPSC Answer Check · Accepted Answer

Solve this multi-part structural analysis problem by allocating approximately 20% time to part (a) for SFD/BMD construction, 45% to part (b) for comprehensive steel beam design checks (shear, bending, deflection, web buckling), and 35% to part (c) for unit load method application. Begin with clear free-body diagrams, proceed through systematic calculations with IS 800:2007 provisions, and conclude with adequacy statements for each check.
- Part (a): Correct determination of support reactions and construction of SFD and BMD with proper sign conventions and peak values marked
- Part (b)(i): Shear strength check using V_d = V_n/γ_mo where V_n = 0.577f_yA_v with A_v = h×t_w for ISMB 550
- Part (b)(ii): Bending strength calculation using M_d = β_bZ_pf_y/γ_mo with lateral restraint condition (ψ = 1.2) and comparison with factored moment
- Part (b)(iii): Deflection check under service loads comparing calculated δ = 5WL³/384EI + PL³/48EI against span/360 or span/240 limits
- Part (b)(iv): Web buckling check at support using F_w = (b_1+n_1)t_wf_cd with n_1 = 2.5d and interpolation from given KL/r vs f_cd table
- Part (c): Application of unit load method with proper virtual work integration, calculation of horizontal and vertical deflection components at A using M and m diagrams
- Correct use of partial safety factors: 1.5 for dead load, 1.5 for imposed load, and load combinations per IS 800:2007

Dimension	Weight	Max marks	Excellent	Average	Poor
Concept correctness	20%	9	Correctly identifies plastic section behavior, applies IS 800:2007 clauses for all four design checks, recognizes lateral restraint enables full plastic moment, and properly sets up virtual work equations for unit load method with correct sign conventions	Applies basic formulas with minor errors in code interpretation (e.g., confuses elastic and plastic modulus or misses lateral restraint benefit), some conceptual gaps in virtual work setup	Fundamental misconceptions such as using working stress method, ignoring partial safety factors, or incorrect virtual work principle application
Numerical accuracy	20%	9	All calculations precise to 3 significant figures, correct interpolation for f_cd at KL/r ≈ 98-102, accurate factored load combinations (1.5DL + 1.5IL), and correct deflection superposition for combined loading	Minor arithmetic errors in intermediate steps or slight inaccuracies in interpolation, but final conclusions remain valid; occasional unit conversion errors (mm to m)	Major calculation errors leading to wrong adequacy conclusions, incorrect load factors, or order-of-magnitude mistakes in section properties
Diagram quality	20%	9	Neat, scaled SFD/BMD with all critical values labeled, clear free-body diagrams for frame in part (c), proper M and m diagrams for unit load method with dimensions and sign conventions indicated, and well-organized tabular presentation for design checks	Diagrams present but lacking scale, missing some critical values, or poor labeling; adequate but not exemplary presentation	Missing diagrams, unrecognizable sketches, or diagrams contradicting calculations; no free-body diagrams for frame analysis
Step-by-step derivation	20%	8	Systematic presentation: reactions → internal forces → design checks with explicit formula statements, clear substitution of values, and logical flow from given data to final adequacy statement for each sub-part; proper virtual work integration steps shown	Some steps skipped or combined without clarity, adequate but not fully transparent derivation; missing intermediate checks	Disorganized calculations, unexplained jumps between steps, or missing essential derivations (e.g., direct final answers without showing work)
Practical interpretation	20%	8	Explicit adequacy statements for all four checks in (b) with percentage utilization, practical significance of web buckling check at supports noted, and real-world implications of deflection limits on floor serviceability; mentions if section is over-designed or requires revision	Basic adequacy statements present but lacking quantitative utilization ratios or practical context; generic conclusions	No conclusions on adequacy, missing interpretation of results, or incorrect practical implications (e.g., declaring unsafe section as safe)

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