Q: A steel rod with a length of $$L = 2.0\ \mathrm{m}$$ and cross-sectional area $$A = 400\ \mathrm{mm}^2 is $$subjected to a tensile force of $$P = 50,000\ \mathrm{N}. If $$the Young's modulus for steel is $$E = 200\ \mathrm{GPa}$$, what is the axial deformation $$\Delta L of $$the rod?

$$\Delta L = \dfrac{PL}{AE} = \dfrac{50,000 \times 2.0}{400 \times 10$$^{-6}$$ \times 200 \times 10^9$} = 1.25\ \mathrm{mm}$$

Q: A 3-meter aluminum bar ($$E = 70\ \mathrm{GPa}$$, $$A = 300\ \mathrm{mm}^2) is $$compressed by a force of $$P = 70,000\ \mathrm{N}. $$What is the change in length $$\Delta L$$? (Assume compression is negative.)

$$\Delta L = -\dfrac{PL}{AE} = -\dfrac{70,000 \times 3}{300 \times 10$$^{-6}$$ \times 70 \times 10^9$} = -10\ \mathrm{mm}$$

Q: A steel bar ($$E = 200\ \mathrm{GPa}$$, $$\alpha = 12 \times 10$$^{-6}$$/^{\circ}$$\mathrm{C}$$, L_0$ = 2\ \mathrm{m}) is $$fixed at both ends. If the temperature increases by $$\Delta T = 50$$^{\circ}$$\mathrm{C}$, what is the thermal stress developed in the bar?

$$\sigma = E \alpha \Delta T = 200 \times 10^9$ \times 12 \times 10$$^{-6}$$ \times 50 = 120\ \mathrm{MPa} ($$compression)

Q: A steel bridge ($$\alpha = 12 \times 10$$^{-6}$$/^{\circ}$$\mathrm{C}$$, L_0$ = 100\ \mathrm{m}) $$experiences a temperature increase of $$\Delta T = 30$$^{\circ}$$\mathrm{C}$. If the bridge is free to expand, what is the change in length?

$$\Delta L = \alpha L_0$ \Delta T = 12 \times 10$$^{-6}$$ \times 100 \times 30 = 0.036\ \mathrm{m}$$

Q: A rod with a sudden change in diameter has a hole in the middle. If the nominal stress is $$\sigma_{nom} = 100\ \mathrm{MPa}$$ and the stress concentration factor is $$K_t = 2.5$$, what is the maximum local stress at the hole?

$$\sigma_{max} = K_t \sigma_{nom} = 2.5 \times 100 = 250\ \mathrm{MPa}$$

Question 1

A steel rod with a length of $$L = 2.0\ \mathrm{m}$$ and cross-sectional area $$A = 400\ \mathrm{mm}^2 is $$subjected to a tensile force of $$P = 50,000\ \mathrm{N}. If $$the Young's modulus for steel is $$E = 200\ \mathrm{GPa}$$, what is the axial deformation $$\Delta L of $$the rod?

Accepted Answer

$$\Delta L = \dfrac{PL}{AE} = \dfrac{50,000 	imes 2.0}{400 	imes 10$$^{-6}$$ 	imes 200 	imes 10^9$} = 1.25\ \mathrm{mm}$$

Question 2

A 3-meter aluminum bar ($$E = 70\ \mathrm{GPa}$$, $$A = 300\ \mathrm{mm}^2) is $$compressed by a force of $$P = 70,000\ \mathrm{N}. $$What is the change in length $$\Delta L$$? (Assume compression is negative.)

Accepted Answer

$$\Delta L = -\dfrac{PL}{AE} = -\dfrac{70,000 	imes 3}{300 	imes 10$$^{-6}$$ 	imes 70 	imes 10^9$} = -10\ \mathrm{mm}$$

Question 3

Which of the following statements about statically indeterminate members is correct?

Accepted Answer

Both equilibrium and compatibility of deformations must be used to solve for internal forces.

Question 4

A steel bar ($$E = 200\ \mathrm{GPa}$$, $$\alpha = 12 	imes 10$$^{-6}$$/^{\circ}$$\mathrm{C}$$, L_0$ = 2\ \mathrm{m}) is $$fixed at both ends. If the temperature increases by $$\Delta T = 50$$^{\circ}$$\mathrm{C}$, what is the thermal stress developed in the bar?

Accepted Answer

$$\sigma = E \alpha \Delta T = 200 	imes 10^9$ 	imes 12 	imes 10$$^{-6}$$ 	imes 50 = 120\ \mathrm{MPa} ($$compression)

Question 5

A steel bridge ($$\alpha = 12 	imes 10$$^{-6}$$/^{\circ}$$\mathrm{C}$$, L_0$ = 100\ \mathrm{m}) $$experiences a temperature increase of $$\Delta T = 30$$^{\circ}$$\mathrm{C}$. If the bridge is free to expand, what is the change in length?

Accepted Answer

$$\Delta L = \alpha L_0$ \Delta T = 12 	imes 10$$^{-6}$$ 	imes 100 	imes 30 = 0.036\ \mathrm{m}$$

Question 6

A composite bar is made of steel ($$E_s = 200\ \mathrm{GPa}$$, $$\alpha_s = 12 	imes 10$$^{-6}$$/^{\circ}$$\mathrm{C}$$) and aluminum (E_a = 70$$\ \mathrm{GPa}$$, $$\alpha_a = 24 	imes $$10^{-6}$$/^{\circ}$$\mathrm{C}$$), both $$L_0 = 1$$\ \mathrm{m}$$. If the temperature increases by $$\Delta T = $$40^{\circ}$$\mathrm{C}$$ and the bar is fixed at both ends, which material will experience greater thermal stress and why?

Accepted Answer

Steel, because it has a lower coefficient of thermal expansion and higher modulus, so it resists more expansion.

Question 7

A rod with a sudden change in diameter has a hole in the middle. If the nominal stress is $$\sigma_{nom} = 100\ \mathrm{MPa}$$ and the stress concentration factor is $$K_t = 2.5$$, what is the maximum local stress at the hole?

Accepted Answer

$$\sigma_{max} = K_t \sigma_{nom} = 2.5 	imes 100 = 250\ \mathrm{MPa}$$

Question 8

Why is it important to use fillets (rounded transitions) instead of sharp corners in mechanical design?

Accepted Answer

Fillets reduce the stress concentration factor $$K_t$$, lowering the risk of premature failure.

Question 9

A steel pipe is fixed at both ends and subjected to a temperature drop of $$\Delta T = -20$$^{\circ}$$\mathrm{C}. $$What type of stress develops in the pipe?

Accepted Answer

Tensile stress, because the pipe contracts but is restrained.

Question 10

Which of the following best describes the relationship between material stiffness and deformation under axial load?

Accepted Answer

A material with a higher Young's modulus $$E$$ will deform less under the same load.

Question 11

A steel rod ($$E = 200\ \mathrm{GPa}$$, $$A = 500\ \mathrm{mm}^2$$, $$L = 2\ \mathrm{m}) is $$loaded in tension with $$P = 100,000\ \mathrm{N}. $$What is the axial stress in the rod?

Accepted Answer

$$\sigma = \dfrac{P}{A} = \dfrac{100,000}{500 	imes 10$$^{-6}$$} = 200\ \mathrm{MPa}$$

Question 12

A steel bar ($$E = 200\ \mathrm{GPa}$$, $$A = 400\ \mathrm{mm}^2$$, $$L = 2\ \mathrm{m}) is $$subjected to a compressive force of $$P = 80,000\ \mathrm{N}. $$What is the resulting strain?

Accepted Answer

$$\varepsilon = \dfrac{\Delta L}{L} = \dfrac{PL}{AE \cdot L} = \dfrac{P}{AE} = \dfrac{80,000}{400 	imes 10$$^{-6}$$ 	imes 200 	imes 10^9$} = 0.001$$

Question 13

A steel and aluminum bar are rigidly connected end-to-end and subjected to a temperature increase. Which statement is correct regarding the internal stresses if the assembly is constrained?

Accepted Answer

Steel will be in compression and aluminum in tension due to their different thermal expansions.

Question 14

Which of the following is a key reason why stress concentrations can lead to premature failure in real structures?

Accepted Answer

Local stresses can greatly exceed the nominal stress, initiating cracks or yielding.

Question 15

A steel rod ($$E = 200\ \mathrm{GPa}$$, $$A = 400\ \mathrm{mm}^2$$, $$L = 2\ \mathrm{m}) is $$loaded in tension with $$P = 80,000\ \mathrm{N}. If $$the rod is replaced with one of the same dimensions but made of aluminum ($$E = 70\ \mathrm{GPa}$$), what happens to the deformation under the same load?

Accepted Answer

The aluminum rod will deform more because it has a lower Young's modulus.

Question 16

A steel bar ($$E = 200\ \mathrm{GPa}$$, $$A = 400\ \mathrm{mm}^2$$, $$L = 2\ \mathrm{m}) is $$subjected to a tensile force of $$P = 80,000\ \mathrm{N}. If $$the bar is replaced with one of half the cross-sectional area, what happens to the axial deformation?

Accepted Answer

The deformation doubles because $$\Delta L = \dfrac{PL}{AE}$$ and $$A is $$halved.

Question 17

A steel rod ($$E = 200\ \mathrm{GPa}$$, $$A = 400\ \mathrm{mm}^2$$, $$L = 2\ \mathrm{m}) is $$subjected to a tensile force of $$P = 80,000\ \mathrm{N}. If $$the length is doubled, what happens to the axial deformation?

Accepted Answer

The deformation doubles because $$\Delta L = \dfrac{PL}{AE}$$ and $$L is $$doubled.

Question 18

Which of the following is NOT a correct practice when analyzing real-world structures for axial loads?

Accepted Answer

Ignoring stress concentrations in regions with holes or notches.

Axial Loads, Thermal Deformation, and Stress Concentrations

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