Question 1

Deflection Comparison — Effect of Breadth on Beam Stiffness Two simply supported beams 'A' and 'B' have the same span length l and depth d. Beam 'A' has breadth b, while beam 'B' has breadth 2b. Both carry the same central point load W. The midspan …

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Introduction: This question evaluates your understanding of how the second moment of area influences deflection in beams under a central point load. Given Data / Assumptions: Both beams are simply supported with equal span l and depth d. Same central point load W and same material modulus E. Breadth of B is 2b; breadth of A is b. Concept / Approach: For a simply supported beam with a central point load: δ = W * l^3 / (48 * E * I) For a rectangle: I = b * d^3 / 12 Thus deflection is inversely proportional to breadth b if depth is constant. Step-by-Step Solution: I_A = b * d^3 / 12I_B = (2b) * d^3 / 12 = 2 * I_Aδ_B / δ_A = (1 / I_B) / (1 / I_A) = I_A / I_B = 1/2Hence δ_B = (1/2) * δ_A. Verification / Alternative check: Deflection scales as 1/I. Doubling I halves the deflection, confirming the result. Why Other Options Are Wrong: one-fourth, four times, double: These correspond to incorrect scaling with breadth. Common Pitfalls: Confusing changes in breadth with changes in depth (deflection is far more sensitive to depth, proportional to d^3). Final Answer: one-half

Question 2

Combined Stress — Minimum Principal (Normal) Stress A body is under a direct tensile stress of 300 MPa in one principal plane and a simple shear stress of 200 MPa. Determine the minimum normal (principal) stress.

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Introduction: Using principal stress transformation, we can find the minimum principal stress for a combined state of direct tension and shear. Given Data / Assumptions: σx = 300 MPa, σy = 0 MPa. τxy = 200 MPa. Plane stress; linear elastic behavior. Concept / Approach: Principal stresses are: σ1,2 = (σx + σy)/2 ± sqrt( ((σx - σy)/2)^2 + τxy^2 ) Step-by-Step Solution: Average = (300 + 0)/2 = 150 MPa.Radius = sqrt(150^2 + 200^2) = sqrt(62500) = 250 MPa.σ2 = 150 - 250 = -100 MPa (minimum principal stress). Verification / Alternative check: Sum of principal stresses equals σx + σy = 300 MPa, so σ1 + σ2 = 400 + (−100) = 300 MPa. Checks out. Why Other Options Are Wrong: 250 MPa: Radius, not a principal stress. 300 MPa: Applied direct stress, not the transformed minimum principal value. 400 MPa: This is the maximum principal stress. Common Pitfalls: Sign errors when subtracting the radius or interchanging σ1 and σ2 are frequent mistakes. Final Answer: -100 MPa

Question 3

Shear Stress Distribution in Beams — True/False When a rectangular beam is loaded longitudinally in bending and shear, the shear force develops on the top layer (extreme fiber). Choose the correct statement.

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Introduction: This checks understanding of shear stress distribution across a rectangular beam depth under transverse loading. Given Data / Assumptions: Rectangular cross-section; small-deflection, linear elastic bending. Beam experiences combined bending moment and shear force. Concept / Approach: In rectangular beams, transverse shear stress varies parabolically over the depth: zero at both top and bottom extreme fibers and maximum at the neutral axis. Step-by-Step Solution: Shear formula: τ = V * Q / (I * b)At the extreme fiber, the first moment of area Q = 0, so τ = 0 at the top and bottom surfaces.At the neutral axis, Q is maximum; hence τ is maximum at the mid-depth. Verification / Alternative check: Standard shear diagrams and textbook distributions confirm a parabolic curve peaking at the neutral axis for rectangles. Why Other Options Are Wrong: Agree / True near supports / under point loads / for non-prismatic beams: Shear magnitude may vary along the span, but at a given section the distribution across depth remains zero at the extreme fibers for a rectangular section. Common Pitfalls: Confusing shear force (a resultant) with shear stress distribution, or assuming maximum at outer fibers (true for bending stress, not shear). Final Answer: Disagree

Question 4

Maximum In-Plane Shear Stress under Combined σx and τxy A body is subjected to a direct tensile stress (sigma_x) in one plane and a simple shear stress (tau_xy). What is the expression for the maximum in-plane shear stress?

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Introduction: The goal is to identify the correct formula for maximum in-plane shear stress when a member has one normal stress and an in-plane shear stress. Given Data / Assumptions: Plane stress with σx given and σy assumed zero unless stated. Non-zero in-plane shear τxy. Linear elasticity; Mohr’s circle relations apply. Concept / Approach: For plane stress, the Mohr’s circle radius equals the maximum in-plane shear stress. With σy = 0: tau_max = sqrt( ((σx - σy)/2)^2 + τxy^2 )Since σy = 0, tau_max = sqrt( (σx/2)^2 + τxy^2 ) Step-by-Step Solution: Start from general radius: R = sqrt( ((σx - σy)/2)^2 + τxy^2 ).Set σy = 0 to get R = sqrt( (σx/2)^2 + τxy^2 ).Therefore maximum in-plane shear stress equals R, giving the required expression. Verification / Alternative check: Alternative form sometimes shown is: tau_max = 0.5 * sqrt( (σx - σy)^2 + 4 * τxy^2 ) Substituting σy = 0 reduces it to 0.5 * sqrt( σx^2 + 4 * τxy^2 ) which is algebraically identical to the chosen expression after simplifying under the square root. Why Other Options Are Wrong: (σx + τxy)/2 and (σx − τxy): Not principal-shear relations; incorrect dimensionally. σx/2: Only valid when τxy = 0 and σy = 0. 0.5 * sqrt(σx^2 + 4 τxy^2): Equivalent form but missing the explicit σy assumption; preferred canonical answer is the radius form shown in option B. Common Pitfalls: Confusing principal normal stresses with principal shear and omitting the division by 2 within the radius term. Final Answer: tau_max = sqrt( (sigma_x / 2)^2 + tau_xy^2 )

Question 5

Point of Contraflexure in a Simply Supported Beam with Uniformly Distributed Load For a simply supported beam carrying a uniformly distributed load w per unit length over the entire span, the point of contraflexure (where bending moment changes sign…

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Introduction: Contraflexure is where the bending moment diagram crosses zero (changes sign). This checks understanding of bending moment shapes under distributed loading. Given Data / Assumptions: Simply supported beam. UDL w over entire span. No intermediate point loads or couples. Concept / Approach: For a simply supported beam under full-span UDL, shear is linear and bending moment is a concave-down parabola, positive throughout the span (zero only at the supports). Step-by-Step Solution: Bending moment at supports = 0.Maximum positive moment at midspan: M_max = w * l^2 / 8.Since the moment does not become negative anywhere, the diagram does not cross zero inside the span. Verification / Alternative check: Drawing the SFD and BMD confirms a purely positive parabolic BMD for UDL on simple supports. Why Other Options Are Wrong: Centre or ends: These suggest specific locations, but the definition requires a sign change which does not occur. Depends on length: Length scales the magnitude, not the sign pattern. Common Pitfalls: Confusing zero bending moment at supports (boundary condition) with an internal sign change point. Final Answer: does not exist

Question 6

Riveted Joints — Possible Modes of Failure In a riveted joint, which of the following failure modes can occur under load?

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Introduction: This question checks recognition of the common failure modes in riveted joints used in structural and mechanical connections. Given Data / Assumptions: Ductile plates and rivets under static loading. Standard lap or butt joints. Concept / Approach: Riveted joints can fail by plate tearing at the net section, tearing at the edge due to insufficient margin, or by rivet shear. Bearing (crushing) is another known mode, though not listed here. Step-by-Step Solution: Assess each listed mode against standard joint mechanics.All three listed modes are recognized failure mechanisms depending on geometry and load. Verification / Alternative check: Design codes compute joint efficiency considering each mode and adopt the governing (least capacity) case, confirming multiple possible modes. Why Other Options Are Wrong: Each single mode alone is possible, but the comprehensive and correct answer is that any one of these may govern. Common Pitfalls: Forgetting edge distance and pitch requirements which prevent premature plate tearing. Final Answer: any one of these

Question 7

Stress–Strain Curve — Correct Sequence of Key Points Which of the following lists the proper chronological sequence of characteristic points as load increases in a ductile material under a standard tensile test?

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Introduction: Understanding the order of key points on the stress–strain curve is essential for material selection and design safety. Given Data / Assumptions: Ductile metal specimen; standard tension test. Nominal engineering stress–strain curve. Concept / Approach: As load increases: linearity holds up to proportional limit; elastic behavior (recoverable strain) continues up to the elastic limit; yielding starts thereafter; ultimate and finally failure occur later. The listed options stop at failure initiation. Step-by-Step Solution: Proportional limit → end of strictly linear stress–strain relation.Elastic limit → largest stress with fully recoverable strain on unloading.Yielding → significant plastic strain starts (upper/lower yield in mild steel).Failure → fracture of the specimen. Verification / Alternative check: Reference curves in materials texts consistently show this order for ductile metals. Why Other Options Are Wrong: Option B and C place limits out of order. Option D is incorrect because A matches standard definitions. Common Pitfalls: Confusing elastic limit with proportional limit; they are close but not identical. Final Answer: proportional limit, elastic limit, yielding, failure

Question 8

Unwin's Formula — Rivet Diameter vs Plate Thickness According to Unwin's empirical rule for riveted joints, the relation between the rivet (or rivet-hole) diameter d (in mm) and the plate thickness t (in mm) is best given by which of the following?

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Introduction: Unwin's formula is a classic empirical rule to select rivet diameter from plate thickness in traditional riveted construction. Given Data / Assumptions: t in mm (plate thickness), d in mm (rivet or hole size, typically hole slightly larger than shank). Rule-of-thumb sizing, not a code-calibrated limit state design. Concept / Approach: Unwin's rule relates rivet diameter to the square root of the plate thickness, capturing the fact that rivet size grows sublinearly with thickness. Step-by-Step Solution: Unwin's recommended relation: d = 6 * sqrt(t)Example: for t = 9 mm, d ≈ 6 * 3 = 18 mm.Hole diameter is usually taken slightly larger than the rivet nominal diameter (e.g., +1 mm), but the core sizing follows this relation. Verification / Alternative check: Many classic design texts present d = 6 * sqrt(t) (with d and t in mm) as the selection guideline. Why Other Options Are Wrong: Linear rules like d = t, 1.6 t, 2 t, or 6 t greatly over/under-size rivets across the thickness range and do not reflect Unwin's square-root dependence. Common Pitfalls: Confusing Unwin's formula with edge distance and pitch rules; they are separate detailing requirements. Final Answer: d = 6 * sqrt(t)

Question 9

For a simply supported beam carrying a single concentrated load P at the mid-span (centre), sketch and identify the bending moment diagram (BMD). Explain which standard geometric shape best represents the BMD for this loading case, given that bendin…

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Introduction: In strength of materials and structural analysis, the bending moment diagram (BMD) shows how bending moment varies along the beam length for a given loading. For a simply supported beam with a central point load, understanding the qualitative BMD shape is essential for design and checks. Given Data / Assumptions: Simply supported beam of span L. Single concentrated load P applied at mid-span. Supports provide no moment; hence M = 0 at both ends. Material is linearly elastic; small deflection theory applies. Concept / Approach: The shear force is constant between loads and changes value only at a point load. The bending moment is the area under the shear force diagram. With symmetric loading, the BMD must also be symmetric and linear between load application points. Step-by-Step Solution: 1) Reactions at supports are equal: RA = RB = P/2 (by symmetry).2) Shear on left half is +P/2 up to the load; on right half it is -P/2 after the load.3) Bending moment varies linearly with the integral of shear: M(x) increases linearly from 0 at A to a maximum at mid-span, then decreases linearly to 0 at B.4) The straight-line rise and fall on two symmetric halves form a V-shaped diagram with equal slopes on each side — an isosceles triangle. Verification / Alternative check: M_max at mid-span = (P * L) / 4, occurring where shear changes sign (at the point load). Linear segments confirm a triangular BMD. Why Other Options Are Wrong: Right-angled triangle: would imply M ≠ 0 at one support; not true for simply supported ends. Equilateral triangle: shape is triangular but 'equilateral' is a geometric side-length property, not applicable to BMD axes. Rectangle: requires constant bending moment, which occurs only under pure couple loading, not a single mid-span point load. Common Pitfalls: Confusing the shear force diagram (which is rectangular here) with the BMD; also assuming curvature implies a parabolic BMD — parabolas arise under uniformly distributed loads, not point loads. Final Answer: Isosceles triangle

Question 10

In the mechanics of materials, a shear stress acting across a plane is always accompanied by an equal shear stress on a plane normal (perpendicular) to it, forming a pair of complementary shears that maintain rotational equilibrium. State whether th…

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Introduction: The statement tests the concept of complementary shear stresses, a fundamental requirement for rotational equilibrium in a stressed element in solid mechanics. Given Data / Assumptions: Continuum, static equilibrium, small deformations. No body couples or special anisotropic effects that violate standard Cauchy stress assumptions. Concept / Approach: For an elemental square (or cube) under shear stress on one face, rotational equilibrium requires a balancing shear on the adjacent, mutually perpendicular face. These are called complementary shear stresses and have equal magnitude and opposite sense to prevent net moment on the element. Step-by-Step Solution: 1) Consider a tiny element with shear tau acting on the horizontal plane.2) The shear produces a couple tending to rotate the element.3) To maintain zero net moment, an equal shear tau must act on the vertical (normal) plane, in the opposite sense.4) Hence, shear across one plane is necessarily accompanied by an equal shear on the plane normal to it. Verification / Alternative check: From equilibrium of moments on the differential element: tau_xy = tau_yx. Why Other Options Are Wrong: False: contradicts moment equilibrium. Depends on material only: this arises from equilibrium, not constitutive law. True only at yield: validity is independent of yielding; it holds in elastic and plastic ranges provided equilibrium applies. Common Pitfalls: Assuming complementary shear arises from Hooke's law; it actually comes from balance of moments and the symmetry of the Cauchy stress tensor in absence of body couples. Final Answer: True

Question 11

In strength of materials, the total strain energy stored in a body is commonly termed strain energy. The term 'proof resilience' specifically refers to the maximum strain energy stored up to the elastic limit (not merely any strain energy). State wh…

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Introduction: This question distinguishes between strain energy, proof resilience, and modulus of resilience — closely related but different terms in materials engineering. Given Data / Assumptions: Body subjected to gradually applied loading. Elastic limit is well defined for the material. Concept / Approach: Strain energy is the work done by internal forces stored as recoverable energy during deformation. Proof resilience is the maximum strain energy a body can store up to the elastic limit. Modulus of resilience is this maximum per unit volume. Step-by-Step Solution: 1) Let U be strain energy at any load within the elastic range.2) Increase load up to the elastic limit; the maximum stored energy is U_max.3) Definition: proof resilience = U_max (value at elastic limit).4) Therefore, calling “the total strain energy stored in a body” proof resilience is inaccurate unless it is specifically the maximum at the elastic limit. Verification / Alternative check: If stress–strain is linear: U_max = 0.5 * sigma_y * epsilon_y * Volume; modulus of resilience = 0.5 * sigma_y * epsilon_y. Why Other Options Are Wrong: Agree: ignores the crucial qualifier “up to elastic limit”. Agree only for elastic bodies: still misses the “maximum at elastic limit” definition. Agree if volume is unit: that corresponds to modulus of resilience, not proof resilience. Common Pitfalls: Confusing proof resilience (total at elastic limit) with modulus of resilience (per unit volume), and with generic strain energy at arbitrary load levels. Final Answer: Disagree

Question 12

In the below figure, curve D represents mild steel.

Accepted Answer

Yes

Question 13

According to Euler's column theory, the crippling (buckling) load is given by P = (π^2 * E * I) / (C * l^2). For a column with one end fixed and the other end hinged (pinned), state whether the value of C is 1/2.

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Introduction: Euler's column theory relates the critical buckling load to end conditions through an effective length factor. This question checks consistency between the constant C and end condition “fixed–hinged”. Given Data / Assumptions: Long, slender column (Euler regime). Modulus E and second moment I are constants. End conditions: one end fixed, the other hinged (pinned). Concept / Approach: General Euler form: P_cr = π^2 * E * I / (L_e^2), where L_e = K * l. If the problem is written as P_cr = π^2 * E * I / (C * l^2), then C = (L_e / l)^2 = K^2. For fixed–hinged, K ≈ 0.7, so C ≈ 0.7^2 ≈ 0.49 ≈ 1/2. Step-by-Step Solution: 1) Take K (fixed–hinged) ≈ 0.7.2) Compute C = K^2 ≈ 0.49.3) Approximating 0.49 as 1/2 is standard in classroom design tables for Euler's ideal case.4) Therefore, the statement that C = 1/2 is correct within Euler theory's conventional approximation. Verification / Alternative check: Using numerator form P_cr = C_n * π^2 * E * I / l^2 with C_n values: for fixed–hinged, C_n ≈ 2. Hence in denominator form C = 1 / C_n ≈ 1/2. Why Other Options Are Wrong: False: conflicts with the effective length relation. True only for very short/very long columns: K depends on end restraint, not directly on length regime (though Euler validity requires slenderness). Common Pitfalls: Mixing the two conventions (C in numerator vs denominator) and forgetting that C here equals K^2. Also confusing fixed–fixed (C ≈ 1/4) with fixed–hinged (C ≈ 1/2). Final Answer: True

Question 14

In a balanced reinforced concrete (R.C.C.) rectangular beam section (neutral axis within the section), the design moment of resistance is governed by the compatible stresses developed in which material(s)? Choose the most appropriate basis used in l…

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Introduction: A balanced R.C.C. section reaches permissible/limit stresses in both concrete (in compression) and steel (in tension) simultaneously. The moment of resistance is then based on the combined stress resultants of both materials. Given Data / Assumptions: Balanced section: steel just reaches its design stress at the same time as concrete attains its allowable/limit compressive stress. Plane sections remain plane; perfect bond between steel and concrete. Concept / Approach: The internal compressive force in the concrete block and the tensile force in the steel must be equal and form a couple whose arm determines the moment of resistance. Thus both materials’ stresses determine the design capacity. Step-by-Step Solution: 1) Compute compressive force C in concrete (equivalent stress block).2) Compute tensile force T in steel (A_s * f_s).3) For equilibrium: C = T.4) Moment of resistance M_r = C * z = T * z, where z is the lever arm depending on neutral axis depth. Verification / Alternative check: Balanced condition confirmed when both concrete and steel reach their respective design stresses simultaneously. Why Other Options Are Wrong: Steel only / Concrete only: ignores the equilibrium couple; capacity depends on both. Neither: contradicts basic R.C.C. theory of composite action. Common Pitfalls: Assuming steel governs always; in a balanced section, both govern. Also confusing under-reinforced/over-reinforced behavior with the balanced case. Final Answer: Steel and concrete both

Question 15

For a mild steel column with both ends hinged (pinned), select the standard Rankine's constant (a) used in the Rankine–Gordon buckling formula. Choose the value typically adopted in design handbooks.

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Introduction: The Rankine–Gordon formula blends Euler buckling and crushing to predict column strength. It employs a material constant a (Rankine's constant) that depends primarily on material type, not directly on end conditions. Given Data / Assumptions: Material: mild steel. Empirical constant a required for Rankine equation. Concept / Approach: A typical form is: P = sigma_c * A / (1 + a * (L_e / r)^2). For mild steel, standard references adopt a ≈ 1/7500; for cast iron, a ≈ 1/1600 to 1/9000 depending on source and grade. The choice 1/7500 is widely used for mild steel in exams and handbooks. Step-by-Step Solution: 1) Identify material: mild steel.2) Recall standard a values from design data.3) Select a = 1/7500. Verification / Alternative check: Cross-check with common data books where mild steel is tabulated with a ≈ 1/7500. Why Other Options Are Wrong: 1/750: excessively large; would under-predict strength. 1/1600 and 1/9000: values often associated with cast iron ranges; not the usual choice for mild steel. Common Pitfalls: Mixing a for different materials or confusing a with end-condition factor K (which affects L_e). Final Answer: 1/7500

Question 16

The given figure shows the Mohr's circle of stress for two unequal and like principal stresses (σx and σy) acting at a body across two mutually perpendicular planes. The tangential stress is given by

Accepted Answer

PQ

Question 17

Define the modulus of rigidity (shear modulus, G). Select the correct ratio that defines G in terms of basic stress–strain quantities.

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Introduction: The modulus of rigidity, also known as the shear modulus G, quantifies a material's resistance to shear deformation and is one of the three fundamental elastic constants alongside E and K (or ν). Given Data / Assumptions: Small strains; linear elastic behavior. Uniform shear state in the considered element. Concept / Approach: By definition, an elastic modulus is a stress–strain ratio. For shear, the relevant quantities are shear stress tau and shear strain gamma. Thus G = tau / gamma. Step-by-Step Solution: 1) Apply a pure shear tau to a small element.2) Measure resulting shear strain gamma (angular distortion).3) Define G = tau / gamma.4) Relate to other constants when needed: E = 2 * G * (1 + ν). Verification / Alternative check: Check units: tau in N/m^2; gamma is dimensionless; hence G has units of N/m^2, consistent with other elastic moduli. Why Other Options Are Wrong: Linear stress to lateral strain and lateral to linear strain: unrelated ratios for G. Linear stress to linear strain: that defines Young's modulus E, not G. Common Pitfalls: Confusing shear modulus G with bulk modulus K or Young's modulus E; also mixing up lateral strain definitions used with Poisson's ratio ν. Final Answer: Shear stress to shear strain

Question 18

Helical spring stiffness under axial load For a closely-coiled helical spring of mean coil diameter D subjected to an axial load W, the stiffness k (load per unit deflection) is:

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Introduction / Context: Stiffness of a helical spring tells how much load is required to cause a unit deflection. For closely-coiled springs, the coils are nearly perpendicular to the spring axis, so the wire primarily experiences torsion rather than pure bending. Designers use stiffness to size the wire diameter and count the number of active coils for the target deflection under service load. Given Data / Assumptions: Closely-coiled helical spring, mean coil diameter D. Wire diameter d; number of active coils n. Material shear modulus G; small deformations; linear elasticity. Axial load W causes an axial deflection delta. Concept / Approach: The axial deflection of a closely-coiled spring under W is delta = 8 * W * D^3 * n / (G * d^4). Stiffness is k = W / delta, so invert the deflection expression to obtain k in terms of geometry and G. Step-by-Step Solution: Start with delta = 8 * W * D^3 * n / (G * d^4).Rearrange for k: k = W / delta.Substitute delta: k = W / [8 * W * D^3 * n / (G * d^4)].Cancel W and invert: k = (G * d^4) / (8 * D^3 * n).Hence stiffness increases strongly with d and decreases with D and n. Verification / Alternative check: Dimensional check: G has units of stress, d^4 has length^4, denominator has length^3, giving stress * length = force per deflection, consistent with stiffness units N/m. Why Other Options Are Wrong: Option b swaps numerator/denominator (that is the deflection expression inverted incorrectly). Option c incorrectly uses E (Young’s modulus) instead of G (shear modulus). Option d has wrong powers of D and d. Option e is Euler buckling, not spring stiffness. Common Pitfalls: Confusing E with G; forgetting that only active coils count; misreading wire vs coil diameters (d vs D). Final Answer: k = (G * d^4) / (8 * D^3 * n)

Question 19

Simply supported beam with a symmetric, gradually varying load Load varies from zero at both ends to w per metre at mid-span. Shear force at the centre equals:

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Introduction / Context: Shear force V(x) and bending moment M(x) are related to the applied load intensity w(x) by dV/dx = -w(x) and dM/dx = V(x). For symmetric loading on a simply supported beam, the shear diagram is antisymmetric about the mid-span. Identifying where shear is zero helps locate extreme bending moments. Given Data / Assumptions: Simply supported span length l. Transverse load intensity increases from 0 at supports to w at mid-span, symmetrically. Linear elastic behavior; small deflection theory. Concept / Approach: By symmetry, reactions at supports are equal and the shear distribution about mid-span must change sign. The point where shear changes sign is exactly where V = 0, which in symmetric cases is at the centre. Since the question asks the shear at the centre, it must be zero. Step-by-Step Solution: Symmetry ⇒ R_A = R_B.Integrate w(x) over half span to see that left-half resultant equals the left reaction.At mid-span, the net effect of left loads and reaction balances, so V(centre) = 0.Therefore the shear force at the centre is zero. Verification / Alternative check: Because V = dM/dx, a zero shear indicates an extremum of bending moment at the centre for symmetric loading, consistent with known beam behavior. Why Other Options Are Wrong: w l / 4 or w l / 2 have dimensions of force but contradict symmetry; w l^2 / 2 has wrong dimensions; the pi-based option is arbitrary. Common Pitfalls: Confusing resultants of distributed loads with point shear values; forgetting symmetry arguments. Final Answer: zero

Question 20

Beam relationships: sign change in shear force and its effect on bending moment Statement: If the shear force at a point changes sign from positive to negative (or vice versa), then the bending moment at that point is zero.

Accepted Answer

Introduction / Context: Shear force V and bending moment M are linked through calculus: dM/dx = V. Understanding this relationship is critical for drawing correct shear force and bending moment diagrams and identifying locations of maximum or minimum bending moment. Given Data / Assumptions: Beam subject to transverse loading. V changes sign at some section x = x0. Linear elastic, small deflection theory. Concept / Approach: Because dM/dx = V, when V = 0 at a point, M has a stationary value (local maximum or minimum), not necessarily zero magnitude. The magnitude of M depends on boundary conditions and loading history up to that point. Zero bending moment occurs at free ends of simple beams or at internal hinges, not generically where V crosses zero. Step-by-Step Solution: At x0, V(x0) = 0 ⇒ dM/dx|_{x0} = 0.A zero derivative implies extremum of M, not M = 0.Therefore, the statement asserting M = 0 whenever V changes sign is false. Verification / Alternative check: For a simply supported beam with uniform load, V = 0 at mid-span but M is maximum, not zero. This counterexample confirms the statement is false. Why Other Options Are Wrong: The conditional True variants (cantilever, UDL, fixed supports) remain incorrect; the calculus relationship does not change for those cases. Common Pitfalls: Equating stationary bending moment with zero value; misapplying support boundary conditions. Final Answer: False

Strength of Materials Questions