Concept explainers
a)
Show that the maximum compressive stresses are in the ratio 4:5:7:9.
a)
Explanation of Solution
Given information:
The load act on the point of the bars is P.
Calculation:
At the point A:
Show the cross-sectional diagram of the square bar as in Figure 1.
Here,
Refer to Figure 1.
The maximum compressive stress of the square bar
Here, e is the eccentricity of the load and
The cross-sectional area of the square bar
The eccentricity of the load (e) is
The distance between the centroid from extreme fibre
The moment of inertia
Calculate the maximum compressive stress of the square bar
Substitute
Show the cross-sectional diagram of the circular bar as in Figure 2.
Here,
Refer to Figure 2.
The maximum compressive stress of the circular bar
The cross-sectional area of the circular bar
The eccentricity of the load (e) is
The distance between the centroid from extreme fibre
The moment of inertia
Calculate the maximum compressive stress of the circular bar
Substitute
Show the cross-sectional diagram of the diamond shape bar as in Figure 3.
Here,
Refer to Figure 3.
The maximum compressive stress of the diamond shape bar
The cross-sectional area of the diamond shape bar
The eccentricity of the load (e) is
The distance between the centroid from extreme fibre
The moment of inertia
Calculate the maximum compressive stress of the diamond shape bar
Substitute
Show the cross-sectional diagram of the triangular bar as in Figure 4.
Here,
Refer to Figure 4.
The maximum compressive stress of the triangular bar
The cross-sectional area of the triangular bar
The distance between the centroid from extreme fibre
The eccentricity of the load (e) is
The moment of inertia
Calculate the maximum compressive stress of the triangular bar
Substitute
Calculate the maximum compressive stresses are in the ratio:
Substitute
The four bars shown have the same cross-sectional area.
Hence the maximum compressive stresses are in the ratio 4:5:7:9 is proved.
b)
Show that the maximum tensile stresses are in the ratio 2:3:5:3.
b)
Explanation of Solution
Given information:
The load act on the point of the bars is P.
Calculation:
At the point B:
Refer to Figure 1.
The maximum tensile stress of the square bar
Here, the e is the eccentricity of the load and
The cross-sectional area of the square bar
The eccentricity of the load (e) is
The distance between the centroid from extreme fibre
The moment of inertia
Calculate the maximum tensile stress of the square bar
Substitute
Refer to Figure 2.
The maximum tensile stress of the circular bar
The cross-sectional area of the circular bar
The eccentricity of the load (e) is
The distance between the centroid from extreme fibre
The moment of inertia
Calculate the maximum tensile stress of the circular bar
Substitute
Refer to Figure 3.
The maximum tensile stress of the diamond shape bar
The cross-sectional area of the diamond shape bar
The eccentricity of the load (e) is
The distance between the centroid from extreme fibre
The moment of inertia
Calculate the maximum tensile stress of the diamond shape bar
Substitute
Refer to Figure 4.
The maximum tensile stress of the triangular bar
The cross-sectional area of the triangular bar
The distance between the centroid from extreme fibre
The eccentricity of the load (e) is
The moment of inertia
Calculate the maximum tensile stress of the triangular bar
Substitute
Calculate the maximum tensile stresses are in the ratio:
Substitute
The four bars shown have the same cross-sectional area.
Hence the maximum tensile stresses are in the ratio 2:3:5:3 is proved.
Want to see more full solutions like this?
Chapter 4 Solutions
Mechanics of Materials, 7th Edition
- M = 500 Nm PROBLEM 4.2 В Knowing that the couple shown acts in the vertical plane, determine the stress at (a) point A, and (b) point B. [Ans. (a) -116.4 MPa (b) -87.3 MPa] 30 mm 40 mm Fig. P4.2arrow_forwardEXERCISE 4.2 1. Three long parallel wires equal in length are supporting a rigid bar connected at their bottoms as shown in Fig. 4.15. If the cross-sectional area of each wire is 100 mm, calculate the stresses in cach wire. Take E, = 100 GPa and E, = 200 GPa. (Ans. o, 25 MPa ; 0, = 50 MPa] 10 kN Fig. 4.15 Brassarrow_forward4.17. Determine the components of stress from the results obtained in (a) v=rsin 0, ve = 2r cos 0 (b) VT = cos 0, 1/4 = 0 (c) v = V₁ = 0 (d) v = (1 - 4) cos 0, Ve= - - (1 + 4/4) sin 0 - Barrow_forward
- The member having a rectangular cross-section, Fig. a, is designed to resist a moment of 40 N # m. In order to increase its strength and rigidity, it is proposed that two small ribs be added at its bottom, Fig. b. Determine the maximum normal stress in the member for both cases.arrow_forwardFig. 2 4. A steel shaft of diameter 50 mm and length 1.2 m (E = 210 GPa and v = 0.3) is loaded with multiple force system. At a point in the shaft, the state of stress relative to the x, y, z coordinate system was found to be: [600 0 T = 0 320 MPa -480 (a) Draw a cube element showing the stress components on each coordinate face (Hint: No vector lines for zero stresses; Warning: A stress element without reference axes will receive zero point). (b) From the given stress tensor, determine the values of (i) octahedral normal stress (Goct) and (ii) octahedral shear stress (toct). (c) From your answer in (b), determine (i) dilatational strain energy Udilat '); and (ii) deviatoric strain energy (Udist). (d) Find the total strain energy at the point.arrow_forward4.9 plzarrow_forward
- 4.99 plzarrow_forward20 40 20 Dimensions in mm PROBLEM 4.1 20 M = 15 kNm Knowing that the couple 80 shown acts in the vertical plane, determine the stress at (a) point A, and (b) point B. [Ans. (a) -61.2 MPa (b) 91.8 MPa] 20 В Fig. P4.1arrow_forwardThree forces, each of magnitude P = 3 kN, are applied to the structure shown. Determine the cross-sectional area of the uniform portion of rod BE for which the normal stress in that portion is +100 MPa. The cross-sectional area of the uniform portion of rod BE is mm2.arrow_forward
- Que 5.6. A crane hook trapezoidal horizontal cross-section is 50 mm wide inside and 30 mm wide outside. Thickness of the section is 60 mm. The crane hook carries a vertical load of 20 kN whose line of action is 50 mm from the inside edge of the section. The center of curvature is 60 mm from the inside edge. Determine the maximum tensile and compressive stresses in the section.arrow_forward2 of 4 ... QUESTION 2 Forces are applied at points A and B of the solid cast-iron (E = 67 GPa) bracket as shown in Figure. Knowing that the bracket has a diameter of 20 mm, at point C, by neglecting the effect of transverse shear, determine: i. the states of stresses and sketch the stress element, ii. the principal planes, the magnitude of the principal stresses and sketch the orientation of the element, iii. the maximum in-plane shear stress planes, the magnitude of the normal and shear stresses on that planes, and sketch the orientation of the element, iv. the magnitude of the absolute maximum shear stress 10 kN 50 mm 25 mm A 100 mm 2.5 kNarrow_forward2.13 A steel plate, which is 1.5 m by 1.5 m and 30 mm thick, is lifted by four cables attached to its corners that meet at a point that is 2 m above the plate. Determine the required cross-sectional area of the cables if the stress in them is not to exceed 20 MPa. Steel plate Prob. 2.13 Cablesarrow_forward
- Elements Of ElectromagneticsMechanical EngineeringISBN:9780190698614Author:Sadiku, Matthew N. O.Publisher:Oxford University PressMechanics of Materials (10th Edition)Mechanical EngineeringISBN:9780134319650Author:Russell C. HibbelerPublisher:PEARSONThermodynamics: An Engineering ApproachMechanical EngineeringISBN:9781259822674Author:Yunus A. Cengel Dr., Michael A. BolesPublisher:McGraw-Hill Education
- Control Systems EngineeringMechanical EngineeringISBN:9781118170519Author:Norman S. NisePublisher:WILEYMechanics of Materials (MindTap Course List)Mechanical EngineeringISBN:9781337093347Author:Barry J. Goodno, James M. GerePublisher:Cengage LearningEngineering Mechanics: StaticsMechanical EngineeringISBN:9781118807330Author:James L. Meriam, L. G. Kraige, J. N. BoltonPublisher:WILEY