4.55 and 4.56 Five metal strips, each 0.5 × 1.5-in. cross section, are bonded together to form the composite beam shown. The modulus of elasticity is 30 × 106 psi for the steel, 15 × 106 psi for the brass, and 10 × 106 psi for the aluminum. Knowing that the beam is bent about a horizontal axis by a couple of moment 12 kip∙in., determine (a) the maximum stress in each of the three metals, (b) the radius of curvature of the composite beam.
Fig. P4.55
(a)
Find the maximum stress in Aluminum, Brass, and Steel.
Answer to Problem 55P
The maximum stress in Aluminum is
The maximum stress in Brass is
The maximum stress in Steel is
Explanation of Solution
Given information:
The dimension of each strip is
The modulus of elasticity of aluminum is
The modulus of elasticity of brass is
The modulus of elasticity of steel is
The beam is bent about a horizontal axis by a couple of moment
Calculation:
Consider aluminum as the reference material.
Calculate the modular ratio
For steel.
Substitute
For brass.
Substitute
For aluminum.
Substitute
Sketch the transformed cross section as shown in Figure 1.
Refer to Figure 1.
The moment of inertia of the cross sections
Calculate the moment of inertia for each section as shown below.
For section (1).
Substitute
Hence,
For section (2).
Substitute
Hence,
For section (3).
Substitute
Calculate the moment of inertia
Substitute
Calculate the maximum stress
For steel.
Substitute
Hence, maximum stress in steel is
For brass.
Substitute
Hence, maximum stress in brass is
For aluminum.
Substitute
Therefore, maximum stress in aluminum is
(b)
The radius of curvature of the composite beam.
Answer to Problem 55P
The radius of curvature of the composite beam is
Explanation of Solution
Given information:
The dimension of each strip is
The modulus of elasticity of aluminum is
The modulus of elasticity of brass is
The modulus of elasticity of steel is
The beam is bent about a horizontal axis by a couple of moment
Calculation:
Refer to part (a).
The moment of inertia of the beam is
Calculate the radius of curvature
Substitute
Therefore, the radius of curvature of the composite beam is
Want to see more full solutions like this?
Chapter 4 Solutions
Mechanics of Materials, 7th Edition
- A steel pipe and an aluminum pipe are securely bonded together to form the composite beam shown. The modulus of elasticity is 200 GPa for the steel and 70 GPa for the aluminum. Knowing that the composite beam is bent by a couple of moment 500 N?m, determine the maximum stress (a) in the aluminum, (b) in the steel.arrow_forwardA copper strip (E = 105 GPa) and an aluminum strip (E = 75 GPa) are bonded together to form the composite beam shown. Knowing that the beam is bent about a horizontal axis by a couple of moment M = 35 N.m, determine the maximum stress in (a) the aluminum strip, (b) the copper strip. Fig. P4.39 Aluminum Copper 24 mm 6 mm 6 mmarrow_forwardA 6-x 10-in. timber beam has been strengthened by bolting two - x 2-in. steel straps to it as shown below. The moduli of elasticity are 1.5 x 106 psi for the wood and 30 x 106 psi for the steel. Knowing that the beam is bent about a horizontal axis by a bending moment of 200 kip in, determine (a) The maximum flexural stress in the wood. (b) The maximum flexural stress in the steel. 10 in. T 6 in.. wood 2 x 3/8 in. steelarrow_forward
- The 6x12-in timber beam has been strengthened by bolting to it the steel reinforcement shown. The modulus of elasticity for wood is 1.8x10° psi and for steel, 29x10° psi. Knowing that the beam is bent about a horizontal axis by a couple of moment M = 450 kip -in, detemine the maximum stress in (a) the wood, (b) the steel. M in.arrow_forward4.37 A W 200 x 31.3 rolled-steel beam is subjected to a couple M of moment 45 kN-m. Knowing that E= Z00GPA, v=0.29, determine (a) the radius of curvature P. (b) the radius of curvature p' of a transverse cross section. SOLUTIONarrow_forward5.86 The cast iron inverted T-section supports two concentrated loads of magni- tude P. The working stresses are 48 MPa in tension, 140 MPa in compression, and 30 MPa in shear. (a) Show that the neutral axis of the cross section is located at d = 48.75 mm and that the moment of inertia of the cross-sectional area about this axis is I = 11.918 x 106 mm“. (b) Find the maximum allowable value of P. 1.0 m 1.0 m 15 mm 3 m 150 mm NA- d 15 mm 150 mm FIG. P5.86arrow_forward
- A 1600-lb-in. couple is applied to a wooden beam, of rectangular cross section 1.5 by 3.5 in., in a plane forming an angle of 308 with the vertical (Fig. ). Determine (a) the maximum stress in the beam and (b) the angle that the neutral surface forms with the horizontal planearrow_forward5.86 The cast iron inverted T-section supports two concentrated loads of magnitude P. The working stresses are 48 MPa in tension, 140 MPa in compression, and 30 MPa in shear. (a) Show that the neutral axis of the cross section is located at d ¼ 48:75 mm and that the moment of inertia of the cross-sectional area about this axis is I ¼ 11:918 106 mm4. (b) Find the maximum allowable value of P.arrow_forwardQuestion 4: A beam of the cross section shown is made of a steel that is assumed to be elastoplastic with E=29×10° psi and oy = 42 ksi. A bending couple is applied to the beam about the z axis, causing plastic zones 2 in. thick to develop at the top and bottom of the beam. After the couple has been removed, determine (a) the residual stress at y=2 in., (b) the points where the residual stress is zero, (c) the radius of curvature corresponding to the permanent deformation of the beam. 1 in. 2 in. |1 in. 1 in. in. 1 in.arrow_forward
- 2.5m 3.5m 4.0m- The 4-mm diameter cable BC is made of steel with E = 200 GPa. Knowing that the maximum stress in the cable must not exceed 190 MPa and that the elongation of the cable must not exceed 6mm, find the maximum load P that can be applied as shown.arrow_forwardTwo wooden boards, measuring 15 mm thick and 225 mm wide, are joined by the socket joint shown. Knowing that the wood employed will break when the average shear stress reaches 10 MPa, determine the intensity P of the axial force that will break the joint.arrow_forward+3 in- B in. A vertical force P of magnitude 20 kips is applied at point C located on the axis of symmetry of the cross section of a short column. Knowing that y = 5 in., determine (a) the stress at point A, (b) the stress at point B, (c) the location of the neutral axis. 2 in. 4 in. A 2 in. 2 in. 1 in. (a) (b)arrow_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