Concept explainers
Bar AB has a cross-sectional area of 1200 mm2 and is made of a steel that is assumed to be elastoplastic with E = 200 GPa and σY = 250 MPa. Knowing that the force F increases from 0 to 520 kN and then decreases to zero, determine (a) the permanent deflection of point C, (b) the residual stress in the bar.
Fig. P2.122
(a)
The permanent deflection of point C.
Answer to Problem 122P
The permanent deflection of point C is
Explanation of Solution
Given information:
The cross sectional area A of section AB is
The modulus of elasticity E is
The yield stress
The force F is
Calculation:
Determine the force at yield portion AC using the relation:
Substitute
Sketch the bar ACB as shown in Figure 1.
Find the load
Substitute
Find the length
Refer to Figure 1.
Find the deflection at point C using the relation:
Here,
Substitute
Find the stress in rod along CB using the relation:
Substitute
Show the expression of deflection at point C for unloading to find the load
Here,
Substitute
Find the load
Substitute
Calculate the stress at point along AC using the relation:
Substitute
Calculate the stress at point along BC using the relation:
Substitute
Determine the deflection at point C using the relation:
Substitute
Determine the permanent deflection at point C using the relation:
Substitute
Thus, the permanent deflection of point C is
(b)
Find the residual stress in bar AC and CB.
Explanation of Solution
The residual stress in bar AC is
The residual stress in bar CB is
Calculation:
Find the residual stress in bar AC using the relation:
Substitute
Thus, the residual stress in bar AC is
Find the residual stress in bar BC using the relation:
Substitute
Thus, the residual stress in bar CB is
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Chapter 2 Solutions
Mechanics of Materials, 7th Edition
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