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QUESTION: 1

In a closed coiled helical spring, the maximum shear stress occurs on the

Solution:

The major stresses in a helical spring are of two types, shear stress due to torsion and direct shear due to applied load. It is observed that for both tensile load as well as compressive load on the spring, maximum shear stress always occurs at the inner side of the spring. Hence, failure of the spring, in the form of crake, is always initiated from the inner radius of the spring.

QUESTION: 2

A rigid bar AB of length ‘L’ is supported by a hinge and two springs of stiffness ‘K’ as shown in figure. The buckling load P_{cr} for the bar will be-

Solution:

F = Kδ and F_{1} = K × δ/2

Now taking moment about A,

Pδ = F × L + F_{1} × L/2

⇒ P × δ = Kδ × L + K × δ/2 × L/2

⇒ P = 5KL/4

QUESTION: 3

If 10 coils of spring cut out from a spring of 25 coils, then the stiffness of new spring will be

Solution:

⇒k′ = 2.5 k

QUESTION: 4

A bar of length L is supported by springs of stiffness Coefficient k_{A}, k_{B, }and a Torsional spring of torsional stiffness coefficient k_{T} as shown in the figure below. The critical load is

Solution:

M_{A} = 0

P x Δ = (K_{A}Δ +K_{B}Δ) L + K_{T} θ

P = (K_{A}+K_{B})L+K_{T}/L

*Answer can only contain numeric values

QUESTION: 5

A column has a rectangular section of 10 mm x 20 mm & effective length of 1 m. The value of buckling stress (in MPa) is, given that E=200 GPa

Solution:

Buckling Load

*Answer can only contain numeric values

QUESTION: 6

A weight of 40 N falls from a height of 200 mm on a rigid massless bar supported by two identical springs as shown in the figure. If the maximum deflection is 20 mm then the stiffness of spring (in N/mm) is ………..

Solution:

mg(h+x) = 2 × 1/2 kx^{2}

40 ×(200+x) = kx^{2}

40 ×(200+x) = k ×(20)^{2}

k = 22 N/mm

QUESTION: 7

The euler’s crippling load for a column whose one end is fixed and other free is 60 kN. If one end is made fixed and other hinged, then the crippling load will

Solution:

End Condition - Effective length (L_{e})

Both end hinged - L

One end fixed other free - 2L

Both end fixed - 1/2

One end fixed and other hinged - L/√2

**Crippling load:**

**Calculation:**

where L_{e} = effective length of column

Effective length for 1^{st} case = 2L, effective length for 2^{nd} case = L/√2

QUESTION: 8

An a aluminium column of square cross-section (10 mm × 10 mm) and length 300 mm has both ends pinned. This is replaced by a circular cross-section (of diameter 10 mm) column of the same length and made of the same material with the same end conditions. The ratio of critical stresses for these two columns corresponding to Euler’s critical load, (σ_{critical})_{square} : (σ_{cirtical})_{circular’} is

Solution:

*Answer can only contain numeric values

QUESTION: 9

A column has a rectangular cross section of 40 cm × 15 cm and of flexural rigidity 225 Nm^{2}. If the column gets buckle under the axial load of 1110 N. Calculate the slenderness ratio of the column.

Solution:

Given A = 0.15 × 0.4 m^{2}

Flexural rigidity: EI =225 Nm^{2}

P_{e} = 1110 N

We know

I = 1/12 × 0.15^{3} × 0.4 = 1.125 × 10^{−4} m^{4}

Slenderness Ratio = S = L_{e}/k = 32.665

*Answer can only contain numeric values

QUESTION: 10

Two beams are connected by a linear spring as shown in the following figure. For a load P as shown in the figure, the percentage of the applied load P carried by the spring is _______________.

Solution:

Δ_{spring }= Δ_{B }− Δ_{D}

Compression of spring

(P – R) – R = 2R

P = 4R

R = P/4

% force carried by spring = 25%

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