Simple Stresses · Strength of Materials Or Solid Mechanics · GATE CE

A hanger is made of two bars of different sizes. Each bar has a square cross-section. The hanger is loaded by three-point loads in the mid vertical plane as shown in the figure. Ignore the self-weight of the hanger. What is the maximum tensile stress in N/mm² anywhere in the hanger without considering stress concentration effects?

An elastic bar of length L, uniform cross sectional area A, coefficient of thermal expansion a, and Young’s modulus E is fixed at the two ends. The temperature of the bar is increased by T, resulting in an axial stress $$\sigma$$. Keeping all other parameters unchanged, if the length of the bar is doubled, the axial stress would be

In a material under a state of plane strain, a 10×10 mm square centered at a point gets deformed as shown in the figure. If the shear strain $$\gamma_{xy}$$ at this point is expressed as 0.001k (in rad), the value of k is

A tapered circular rod of diameter varying from 20 mm to 10 mm is connected to another uniform circular rod of diameter 10 mm as shown in the following figure. Both bars are made of same material with the modulus of elasticity, E = 2 $$ \times $$ 10⁵ MPa. When subjected to a load P = 30$$\pi $$ kN, the deflection at point A is _______________ mm.

The Poisson's ratio is defined as

The number of independent elastic constants for a linear elastic isotropic and homogeneous material is

For an isotropic material, the relationship between the young’s modulus (E), shear modulus (G) and Poisson’s ratio (μ) is given by

The shear modulus (G), modulus of elasticity (E) and the Poisson's ratio ($$\mu$$) of a material are related as

The stress-strain diagram for two materials A and B is shown below: The following statements are made based on this diagram:
I. Material A is more brittle than material B.
II. The ultimate strength of material B is more than that of A.
With reference to the above statements, which of the following applies?

Pick the incorrect statement from the following four statements:

A cantilever beam of tubular section consists of 2 materials copper as outer cylinder and steel as inner cylinder. It is subjected to a temperature rise of $$20^\circ C$$ and $$\alpha_{copper}\;>\;\alpha_{steel}$$ the stresses developed in the tubes will be

The maximum value of Poisson's ratio for an elastic material is

In a linear elastic structural element

The principle of superposition is made use of in structural computations when:

A 5 cm long metal rod AB was initially at a uniform temperature of T0°C. Thereafter, temperature at both the ends are maintained at 0°C. Neglecting the heat transfer from the lateral surface of the rod, the heat transfer in the rod is governed by the one-dimensional diffusion equation $\rm\frac{\partial T}{\partial t}=D\frac{\partial^2T}{\partial x^2}$, where D is the thermal diffusivity of the metal, given as 1.0 cm2/s.

The temperature distribution in the rod is obtained as

$\rm T(x,t)=\Sigma_{n=1,3,5...}^{\infty}C_n\sin\frac{n\pi x}{5}e^{-\beta n^2t}$,

where x is in cm measured from A to B with 𝑥 = 0 at A, t is in s, 𝐶𝑛 are constants in °C, T is in °C, and β is in s−1 .

The value of β (in 𝑠−1 , rounded off to three decimal places) is ________.

Consider two linearly elastic rods HI and IJ. Each of length b, as shown in the figure. The rods are co-linear, and confined between two fixed supports at if and J. Both the rods are initially stress free. The coefficient of linear thermal expansion is a for both the rods. The temperature of the rod IJ is raised by $$\Delta$$T whereas the temperature of rod HI remains unchanged. An external horizontal force P is now applied at node I. It is given that a = 10^$$-$$6 $$^\circ$$C^$$-$$1, $$\Delta$$T = 50$$^\circ$$C, b = 2m, AE = 10⁶N. The axial rigidities of the rods HI and U are 2 AE and AE, respectively.

To make the axial force in rod HI equal to zero, the value of the external force P (in N) is _________. (rounded off to the nearest integer).

An elastic isotropic body is in a hydrostatic state of stress as shown in the figure. For no change in the volume to occur, what should be its Poisson's ratio?

A box of weight 100 kN shown in the figure is to be lifted without swinging. If all forces are coplanar, the magnitude and direction ($$\theta$$) of the force (F) with respect to x-axis should be

A metal bar of length 100mm is inserted between two rigid supports and its temperature is increased by $$10^o$$C. If the coefficient of thermal expansion is 12 x$$10^{-6}$$ per $$^o$$C and the Young’s modulus is 2 x $$10^{5}$$ MPa, the stress in the bar is

The axial movement of top surface of stepped column as shown in figure is