Strength of Materials Practice Flashcards

100 flashcards covering Strength of Materials concepts from the lecture transcript, including material properties, stresses, beams, torsion, and columns.

What is a Homogeneous Material?

A material which have same elastic properties at any point in a given direction.

What is an Isotropic Material?

This material has same identical properties in all direction at a point.

What is an Anisotropic Material?

It has different properties in all direction at a point in the body.

What is an Orthotropic Material?

A material which has different properties in three mutually perpendicular planes.

What is the Poisson's Ratio for Cork?

$$0$$

What is the Poisson's Ratio range for Glass?

$$0.02 - 0.03$$

What is the Poisson's Ratio range for Cast Iron?

$$0.23 - 0.27$$

What is the Poisson's Ratio range for Elastic Material?

$$0.25 - 0.40$$

What is the Poisson's Ratio range for Steel?

$$0.27 - 0.33$$

What is the Poisson's Ratio for Rubber?

$$0.50$$

What is the Poisson's Ratio for Human Tissues?

$$-1$$

What is the Poisson's Ratio for Wrought Iron?

$$0.30$$

What is the Poisson's Ratio range for Concrete?

$$0.10 - 0.20$$

What is the formula for Young's Modulus (E)?

$$E = \frac{\text{Longitudinal Stress}}{\text{Longitudinal Strain}} = \frac{\sigma}{\epsilon}$$

How is Modulus of Rigidity or Shear Modulus (G) defined?

$$G = \frac{\text{Shear stress}}{\text{Shear strain}} = \frac{\tau}{\phi}$$

What is the formula for Poisson's Ratio (\mu)?

$$\mu = -\frac{\text{Lateral Strain}}{\text{Linear Strain}} = -\frac{\delta d / d}{\delta l / l}$$

What is the formula for Bulk Modulus (K)?

$$K = \frac{\text{Direct stress}}{\text{Volumetric Strain}} = \frac{\sigma}{\epsilon_v}$$

What is the stress formula for a Gradual load?

$$\sigma = \frac{P}{A}$$

How is stress calculated for a Sudden load?

$$\sigma_{\text{sudden}} = 2 \times \frac{P}{A}$$

What is the stress formula for an Impact load?

$$\sigma_i = \frac{W}{A} \left( 1 + \sqrt{1 + \frac{2hAE}{Wl}} \right)$$

What is the relation between E, G, and \mu?

$$E = 2G (1+\mu)$$

What is the relation between E, K, and \mu?

$$E = 3K(1-2\mu)$$

What is the formula for E in terms of K and G?

$$E = \frac{9KG}{3K + G}$$

What is the formula for \mu in terms of K and G?

$$\mu = \frac{3K - 2G}{6K + 2G}$$

How many independent elastic constants are there for Homogeneous and Isotropic materials?

$$2$$

How many total number of elastic constants are there for Homogeneous and Isotropic materials?

$$4$$

How many independent elastic constants are there for Orthotropic materials (like wood)?

$$9$$

How many total number of elastic constants are there for Orthotropic materials?

$$12$$

How many independent elastic constants are there for Anisotropic materials?

$$21$$

How many total number of elastic constants are there for Anisotropic materials?

$$\infty$$

What is the axial elongation formula for a Prismatic bar due to external load?

$$\delta l = \frac{Pl}{AE}$$

What is the elongation formula for a Circular tapered bar?

$$\delta l = \frac{4Pl}{\pi d_1 d_2 E}$$

What is the elongation formula for a Rectangular tapered bar?

$$\delta l = \frac{Pl}{(b_2 - b_1)Et} \ln \left( \frac{b_2}{b_1} \right)$$

In Composite bars, what is the relateion between total load P and segment loads?

$$P = P_1 + P_2$$

What is the elongation of a Prismatic bar due to self weight?

$$\delta l = \frac{\rho g l^2}{2E} = \frac{wl^2}{2E}$$

What is the elongation of a Uniform tapering or conical bar due to self weight?

$$\delta_c = \frac{\gamma l^2}{6E} = \frac{Wl}{2AE} \times \frac{1}{3}$$

What is the relationship between the elongation of a conical bar and a prismatic bar of same length and load?

$$\delta_c = \frac{1}{3} \delta_P$$

What is the thermal stress in Case 1 (Free expansion or contraction)?

$$\sigma_{\text{th}} = 0$$

What is the thermal stress in Case 2 (Fully prevented)?

$$\sigma_{\text{th}} = E\alpha \Delta T$$

What is the thermal stress formula for Case 3 (Partially prevented by distance x)?

$$\sigma_{\text{th}} = \frac{E}{l} (l\alpha \Delta T - x)$$

What is the thermal stress formula for a Taper section?

$$\sigma_{\text{th}} = \frac{d_2}{d_1} E \alpha \Delta T$$

What is the formula for True stress ($\sigma_T$)?

$$\sigma_T = \frac{P}{A_i}$$

What is the formula for True strain ($\epsilon_T$)?

$$\epsilon_T = \ln \left( \frac{l_i}{l_o} \right)$$

What is the relationship between True stress and Engineering stress ($\sigma_e$)?

$$\sigma_T = \sigma_e ( 1 + e )$$

What is the relationship between True strain and Engineering strain (e)?

$$\epsilon_T = \ln ( 1 + e )$$

What is the Young's Modulus (E) of Steel?

$$2 \times 10^5 \, \text{MPa}$$

What is the Young's Modulus (E) of Copper?

$$1.17 \times 10^5 \, \text{MPa}$$

What is the Young's Modulus (E) of Aluminium?

$$0.70 \times 10^5 \, \text{MPa}$$

What is the definition of Proof resilience?

Energy absorbed by body upto elastic limit.

What is the formula for Modulus of resilience?

$$U = \frac{\sigma_{max}^2}{2E}$$

Which theory of failure was given by Rankine and is suitable for Brittle materials?

Maximum Principal Stress or normal stress theory.

Which theory of failure was given by St. Venants?

Maximum Principal Strain theory.

Guest & Trasca's theory is suitable for which type of material?

Ductile

What is the graphical representation for the Maximum Principal Strain theory?

Rhombus

Which theory of failure is associated with Vonmises and Hencky?

Maximum Shear Strain Energy theory.

What is the formula for Normal stress ($\sigma_n$) on any plane?

$$\sigma_n = \frac{\sigma_x + \sigma_y}{2} + \frac{\sigma_x - \sigma_y}{2} \cos(2\theta) + \tau_{xy} \sin(2\theta)$$

What is the formula for Tangential or shear stress (\tau) on any plane?

$$\tau = -\frac{\sigma_x - \sigma_y}{2} \sin(2\theta) + \tau_{xy} \cos(2\theta)$$

What is the formula to find the Principal Plane ($\theta$)?

$$\tan(2\theta) = \frac{2\tau_{xy}}{\sigma_x - \sigma_y}$$

In Case-1 (Uniaxial load), what is the formula for Shear stress (\tau)?

$$\tau = \frac{P}{2A} \sin(2\alpha)$$

How is the Radius of Mohr's circle (\tau_{max}) calculated?

$$R = \frac{\sigma_1 - \sigma_2}{2} = \sqrt{ \left( \frac{\sigma_x - \sigma_y}{2} \right)^2 + \tau_{xy}^2 }$$

What is the coordinates for the Center of Mohr's circle?

$$\left( \frac{\sigma_x + \sigma_y}{2}, 0 \right)$$

What does the rate of change of shear force ($dV/dx$) equal?

$-W$ (load)

What does the rate of change of bending moment ($dM/dx$) equal?

$V$ (shear force)

State the governing Bending equation for a beam.

$$\frac{M}{I} = \frac{\sigma_b}{y} = \frac{E}{R}$$

What is the definition of Section modulus (Z)?

$$Z = \frac{I}{y}$$

What is the formula for Flexural Rigidity?

$$E \times I$$

What is the Moment of Inertia ($I_{XX}$) for a Rectangular section?

$$\frac{bd^3}{12}$$

What is the Section Modulus (Z) for a Rectangular section?

$$\frac{bd^2}{6}$$

What is the Moment of Inertia ($I_{base}$) for a Triangular section?

$$\frac{bh^3}{12}$$

What is the Section Modulus (Z) for a Solid Circular section?

$$\frac{\pi D^3}{32}$$

What is the Moment of Inertia ($I_{XX}$) for a Solid Circular section?

$$\frac{\pi D^4}{64}$$

What is the Moment of Inertia ($I_{XX}$) for a Square section?

$$\frac{a^4}{12}$$

What is the ratio of $\tau_{max}$ to $\tau_{avg}$ for a rectangular section?

$$\frac{3}{2}$$

What is the ratio of $\tau_{max}$ to $\tau_{avg}$ for a circular section?

$$\frac{4}{3}$$

State the Pure Torsion equation.

$$\frac{T}{J} = \frac{\tau}{R} = \frac{G\theta}{L}$$

What is the formula for Shear stress (\tau) in a shaft subjected to torque T?

$$\tau = \frac{16T}{\pi D^3}$$

What is the formula for Power (P) transmitted by a shaft in kW?

P = \frac{2\n̐ NT}{60 \times 1000}

What is the formula for Polar section modulus ($Z_P$)?

$$Z_p = \frac{J}{R}$$

What is the Polar Moment of Inertia (J) for a solid shaft?

$$\frac{\pi d^4}{32}$$

What are the conditions for shafts connected in Parallel?

$T = T_1 + T_2$ and $\theta_1 = \theta_2$

What are the conditions for shafts connected in Series?

$T_1 = T_2 = T$ and $\theta = \theta_1 + \theta_2$

According to maximum shear stress theory, what is the maximum shear stress ($\tau_{max}$) for combined loading?

$$\tau_{max} = \frac{1}{2} \sqrt{\sigma_b^2 + 4\tau^2}$$

What is the Equivalent Torque ($T_e$) formula?

$$T_e = \sqrt{M^2 + T^2}$$

What is the Equivalent Bending Moment ($M_e$) formula?

$$M_e = \frac{1}{2} \left( M + \sqrt{M^2 + T^2} \right)$$

Name one method used to determine Slope and Deflection.

Macaulay's Method (or Double Integration, Area Moment, Strain energy, Conjugate Beam, Superposition).

What is the maximum deflection ($y_{max}$) for a cantilever beam with a point load W at the free end?

$$\frac{WL^3}{3EI}$$

What is the maximum slope ($\theta_{max}$) for a cantilever beam with a point load W at the free end?

$$\frac{WL^2}{2EI}$$

What is the maximum deflection ($y_{max}$) for a cantilever beam with UDL (w) over the entire length?

$$\frac{wL^4}{8EI}$$

What is the maximum deflection ($y_{max}$) for a simply supported beam with a central point load W?

$$\frac{WL^3}{48EI}$$

What is the maximum deflection ($y_{max}$) for a simply supported beam with UDL (w)?

$$\frac{5wL^4}{384EI}$$

What is the formula for Strain energy (U) due to axial loading on a uniform bar?

$$U = \frac{P^2 L}{2AE}$$

What is the formula for Hoop stress ($\sigma_H$) in a thin cylinder?

$$\sigma_H = \frac{PD}{2t}$$

What is the formula for Longitudinal stress ($\sigma_L$) in a thin cylinder?

$$\sigma_L = \frac{PD}{4t}$$

What is the relation between Hoop stress and Longitudinal stress in a thin cylinder?

\sigma_h = 2 ̐ \sigma_L

What is the formula for Volumetric strain (\epsilon_v) in a thin sphere?

$$\epsilon_v = \frac{3PD}{4tE} (1 - \mu)$$

How is the Slenderness ratio (S.R.) defined?

$S.R. = \frac{l_e}{K_{min}}$ (Effective length / Minimum radius of gyration).

What is the failure mode for a Long column (S.R. > 120)?

Buckling

What is the Euler's critical load ($P_{cr}$) formula?

$$P_{cr} = \frac{\pi^2 EI_{min}}{l_e^2}$$

What is the effective length ($l_e$) for a column with one end fixed and other end free?

$$l_e = 2L$$

What is the effective length ($l_e$) for a column with both ends hinged?

$$l_e = L$$