Question 1

A rectangular plate in plane stress is subjected to normal stresses $\sigma_{x}=35 \mathrm{MPa}, \sigma_{y}=26 \mathrm{MPa}$ , and shear stress $\tau_{x y}=14 \mathrm{MPa}$ . The ratio of the magnitudes of the principal stresses $\left(\sigma_{1} / \sigma_{2}\right)$ is approximately: (A) $ 0.8 $
(B) $ 1.5 $
(C) $ 2.1 $
(D) $ 2.9 $

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Question 2

A drive shaft resists torsional shear stress of 45 MPa and axial com- pressive stress of 100 MPa. The ratio of the magnitudes of the principal $\left(\sigma_{1} / \sigma_{2}\right)$ approximately:
A) 0.15
B) 0.55
C) 1.2
D) 1.9

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Question 3

A rectangular plate (a = 120 mm, b = 160 mm) is subjected to compressive stress $\sigma_{x}=-4.5 \mathrm{MPa} \text { and tensile stress } \sigma_{y}=15 \mathrm{MPa}$ . The
Ratio of the normal stress acting perpendicular to the weld to the shear
Stress acting along the weld is approximately:
A) 0.27
B) 0.54
C) 0.85
D) 1.22

Accepted Answer

The normal stress ($\sigma_{n}$) and shear stress ($\tau$) acting on the weld can be calculated using the given stresses ($\sigma_{x}$ and $\sigma_{y}$) and the orientation of the weld. Since the problem does not specify the orientation of the weld, we assume the most common scenario where the weld is oriented at 45 degrees to the direction of the principal stresses for maximum shear stress calculation. At this orientation, the normal and shear stresses on the weld can be calculated using the equations for stresses on an oblique plane:$$\sigma_{n} = \frac{\sigma_{x} + \sigma_{y}}{2}$$$$\tau = \frac{\sigma_{y} - \sigma_{x}}{2}$$Substituting the given values:$$\sigma_{n} = \frac{-4.5 + 15}{2} = \frac{10.5}{2} = 5.25 \, \text{MPa}$$$$\tau = \frac{15 - (-4.5)}{2} = \frac{19.5}{2} = 9.75 \, \text{MPa}$$The ratio of the normal stress to the shear stress is:$$\frac{\sigma_{n}}{\tau} = \frac{5.25}{9.75} \approx 0.54$$However, my initial calculation led to an incorrect interpretation. Correcting this:The ratio should be calculated correctly based on the given scenario and the correct formulas. Given the mistake in the calculation process, the correct approach involves understanding that the ratio of normal to shear stress on the weld depends on the orientation and the correct application of stress transformation equations. Without specific orientation details or a mistake in the calculation, the correct answer should be directly derived from the problem's context and the formulas provided.

Question 4

A cylindrical tank is assembled by welding steel sections circumfer- entially. Tank diameter is 1.5 m, thickness is 20 mm, and internal pressure
Is 2.0 MPa. The maximum tensile stress in the cylindrical part of the tank
Is approximately: (A) $ 45 \mathrm{MPa} $
(B) $ 57 \mathrm{MPa} $
(C) $ 62 \mathrm{MPa} $
(D) $ 75 \mathrm{MPa} $

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Question 5

A bimetallic beam of aluminum $\left(E_{a}=70 \mathrm{GPa}\right)$ and copper $\left(E_{c}=110 \mathrm{GPa}\right)$ strips has a width of b = 25 mm; each strip has a thick- ness t = 1.5 mm. A bending moment of 1.75 N $1.75 \mathrm{~N} \cdot \mathrm{m}$ m is applied about
The z axis. The ratio of the maximum stress in the aluminum to that in the
Copper is approximately: (A) $ 0.6 $
(B) $ 0.8 $
(C) $ 1.0 $
(D) $ 1.5 $

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Question 6

A cylindrical tank is assembled by welding steel sections in a heli- cal pattern with angle $\alpha=50^{\circ}$ . Tank diameter is 1.6 m, thickness is
20 mm, and internal pressure is 2.75 MPa. Modulus E = 210 GPa and
Poisson's ratio $v=0.28$ . The normal stress acting perpendicular to the
Weld is approximately: (A) $ 39 \mathrm{MPa} $
(B) $ 48 \mathrm{MPa} $
(C) $ 78 \mathrm{MPa} $
(D) $ 84 \mathrm{MPa} $

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Question 7

A cylindrical tank is assembled by welding steel sections circumfer- entially. Tank diameter is 1.5 m, thickness is 20 mm, and internal pressure
Is 2.0 MPa. The maximum shear stress in the heads is approximately: (A) $ 19 \mathrm{MPa} $
(B) $ 23 \mathrm{MPa} $
(C) $ 33 \mathrm{MPa} $
(D) $ 35 \mathrm{MPa} $

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Question 8

A drive shaft resists torsional shear stress of 45 MPa and axial com- pressive stress of 100 MPa. The maximum shear stress is approximately: (A) $ 42 \mathrm{MPa} $
(B) $ 67 \mathrm{MPa} $
(C) $ 71 \mathrm{MPa} $
(D) $ 93 \mathrm{MPa} $

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Question 9

A cylindrical tank is assembled by welding steel sections circum- ferentially. Tank diameter is 1.5 m, thickness is 20 mm, and internal pres-
Sure is 2.0 MPa. The maximum shear stress in the cylindrical part of the
Tank is approximately: (A) $ 17 \mathrm{MPa} $
(B) $ 26 \mathrm{MPa} $
(C) $ 34 \mathrm{MPa} $
(D) $ 38 \mathrm{MPa} $

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Question 10

A steel pipe $\left(d_{3}=104 \mathrm{~mm}, d_{2}=96 \mathrm{~mm}\right)$ has a plastic liner with inner diameter $d_{1}=82 \mathrm{~mm}$ . The modulus of elasticity of the steel is 75
Times that of the modulus of the plastic. Allowable stresses in steel and
Plastic are 40 MPa and 550 kPa, respectively. The allowable bending
Moment for the composite pipe is approximately: (A) $ 1100 \mathrm{~N} \cdot \mathrm{m} $
(B) $ 1230 \mathrm{~N} \cdot \mathrm{m} $
(C) $ 1370 \mathrm{~N} \cdot \mathrm{m} $
(D) $ 1460 \mathrm{~N} \cdot \mathrm{m} $

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Question 11

A cylindrical tank is assembled by welding steel sections circumfer- entially. Tank diameter is 1.5 m, thickness is 20 mm, and internal pressure
Is 2.0 MPa. The maximum stress in the heads of the tank is approximately: (A) $ 38 \mathrm{MPa} $
(B) $ 45 \mathrm{MPa} $
(C) $ 50 \mathrm{MPa} $
(D) $ 59 \mathrm{MPa} $

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Question 12

A drive shaft resists torsional shear stress of $\tau_{x y}=40 \mathrm{MPa}$ and axial compressive stress $\sigma_{x}=-70 \mathrm{MPa}$ . One principal normal stress is
Known to be 38 MPa (tensile). The stress $\sigma_{y}$ is approximately: (A) $ 23 \mathrm{MPa} $
(B) $ 35 \mathrm{MPa} $
(C) $ 62 \mathrm{MPa} $
(D) $ 75 \mathrm{MPa} $

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Question 13

A rectangular plate in plane stress is subjected to normal stresses $\sigma_{x}$ and $\sigma_{y}$ and shear stress $\tau_{\mathrm{xy}^{+}}$ . Stress $\sigma_{\mathrm{x}}$ is known to be 15 MPa, but $\sigma_{y}$ and $\tau_{x y}$ are unknown. However, the normal stress is known to be 33 MPa at counterclockwise angles of 35° and 75° from the x axis. Based on this, the
Normal stress $\sigma_{y}$ on the element in the figure is approximately: (A) $ 14 \mathrm{MPa} $
(B) $ 21 \mathrm{MPa} $
(C) $ 26 \mathrm{MPa} $
(D) $ 43 \mathrm{MPa} $

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Question 14

A composite beam is made up of a 200 mm ⫻ 300 mm core $\left(E_{c}=14 \mathrm{GPa}\right)$ and an exterior cover sheet (300 mm ⫻ 12 mm, $\left.E_{c}=100 \mathrm{GPa}\right)$ on each side. Allowable stresses in core and exterior sheets are 9.5 MPa and 140 MPa, respectively. The ratio of the maximum
Permissible bending moment about the z axis to that about the y axis is
Most nearly: (A) $ 0.5 $
(B) $ 0.7 $
(C) $ 1.2 $
(D) $ 1.5 $

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Question 15

A composite beam is made up of a 90 mm ⫻ 160 mm wood beam . Allowable stresses in wood and steel are 6.5 MPa and $\left.E_{s}=190 \mathrm{GPa}\right)$ $\left(E_{w}=11 \mathrm{GPa}\right)$ and a steel bottom cover plate (90 mm ⫻ 8 mm,
110 MPa, respectively. The allowable bending moment about the z axis of
The composite beam is most nearly: (A) $ 2.9 \mathrm{kN} \cdot \mathrm{m} $
(B) $ 3.5 \mathrm{kN} \cdot \mathrm{m} $
(C) $ 4.3 \mathrm{kN} \cdot \mathrm{m} $
(D) $ 9.9 \mathrm{kN} \cdot \mathrm{m} $

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Question 16

A cylindrical tank is assembled by welding steel sections in a heli- cal pattern with angle $\alpha=50^{\circ}$ . Tank diameter is 1.6 m, thickness is
20 mm, and internal pressure is 2.75 MPa. Modulus $E=210 \mathrm{GPa}$ and
Poisson's ratio $v=0.28$ . The longitudinal strain in the the wall of the
Tank is approximately: (A) $ 1.2 \times 10^{-4} $
(B) $ 2.4 \times 10^{-4} $
(C) $ 3.1 \times 10^{-4} $
(D) $ 4.3 \times 10^{-4} $

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Question 17

A rectangular beam with semicircular notches has dimen- sions $h=160 \mathrm{~mm} \text { and } h_{1}=140 \mathrm{~mm}$ . The maximum allowable bending
Stress in the plastic beam is $\sigma_{\max }=6.5 \mathrm{MPa}$ , and the bending moment
Is $M=185 \mathrm{~N} \cdot \mathrm{m}$ . The minimum permissible width of the beam is: (A) $ 12 \mathrm{~mm} $
(B) $ 20 \mathrm{~mm} $
(C) $ 28 \mathrm{~mm} $
(D) $ 32 \mathrm{~mm} $

FIG. 5-50 Stress-concentration factor K for a notched beam of rectangular
Cross section in pure bending (h = height of beam; b = thickness of beam,
Perpendicular to the plane of the figure). The dashed line is for semicircular $\left(h=h_{1}+2 R\right)$

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Question 18

A cylindrical tank is assembled by welding steel sections in a heli- cal pattern with angle $\alpha=50^{\circ} \text {. Tank diameter is } 1.6 \mathrm{~m}$ 20 mm, and internal pressure is 2.75 MPa. Modulus E = 210 GPa and
Tank is approximately:
Poisson's ratio $v=0.28$ . The circumferential strain in the wall of the (A) $ 1.9 \times 10^{-4} $
(B) $ 3.2 \times 10^{-4} $
(C) $ 3.9 \times 10^{-4} $
(D) $ 4.5 \times 10^{-4} $

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Question 19

A cylindrical tank is assembled by welding steel sections circumfer- entially. Tank diameter is 1.5 m, thickness is 20 mm, and internal pressure
Is 2.0 MPa. The maximum tensile stress perpendicular to the welds is
Approximately: (A) $ 22 \mathrm{MPa} $
(B) $ 29 \mathrm{MPa} $
(C) $ 33 \mathrm{MPa} $
(D) $ 37 \mathrm{MPa} $

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Question 20

A composite beam of aluminum $\left(E_{a}=72 \mathrm{GPa}\right)$ and steel $\left(E_{s}=190 \mathrm{GPa}\right)$ has a width b = 25 mm and heights $h_{a}=42 \mathrm{~mm}$ and $h_{s}=68 \mathrm{~mm}$ , respectively. A bending moment is applied about the z axis resulting in a maximum stress in the aluminum of 55 MPa. The maximum
Stress in the steel is approximately:
A) 86 MPa
B) 90 MPa
C) 94 MPa
D) 98 MPa

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Quiz 1: Mechanics of Materials

A drive shaft resists torsional shear stress of 45 MPa and axial com- pressive stress of 100 MPa. The ratio of the magnitudes of the principal $\left(\sigma_{1} / \sigma_{2}\right)$ approximately:

A rectangular plate (a = 120 mm, b = 160 mm) is subjected to compressive stress $\sigma_{x}=-4.5 \mathrm{MPa} \text { and tensile stress } \sigma_{y}=15 \mathrm{MPa}$ . The Ratio of the normal stress acting perpendicular to the weld to the shear Stress acting along the weld is approximately:

A cylindrical tank is assembled by welding steel sections circumfer- entially. Tank diameter is 1.5 m, thickness is 20 mm, and internal pressure Is 2.0 MPa. The maximum shear stress in the heads is approximately: (A) $19 \mathrm{MPa}$ (B) $23 \mathrm{MPa}$ (C) $33 \mathrm{MPa}$ (D) $35 \mathrm{MPa}$

Quiz 1: Mechanics of Materials

A drive shaft resists torsional shear stress of 45 MPa and axial com- pressive stress of 100 MPa. The ratio of the magnitudes of the principal (σ1/σ2) \left(\sigma_{1} / \sigma_{2}\right) (σ1​/σ2​) approximately:

A drive shaft resists torsional shear stress of 45 MPa and axial com- pressive stress of 100 MPa. The maximum shear stress is approximately: (A) 42MPa 42 \mathrm{MPa} 42MPa (B) 67MPa 67 \mathrm{MPa} 67MPa (C) 71MPa 71 \mathrm{MPa} 71MPa (D) 93MPa 93 \mathrm{MPa} 93MPa

A drive shaft resists torsional shear stress of 45 MPa and axial com- pressive stress of 100 MPa. The ratio of the magnitudes of the principal $\left(\sigma_{1} / \sigma_{2}\right)$ approximately: