(a) Explanation Given information: The initial temperature of the rod is 150 ° C . Refer to problem 2.118. The modulus of elasticity E is 200 GPa . The yield stress ( σ Y ) is 250 MPa . The coefficient of thermal expansion α is 11.7 × 10 − 6 / ° C . The area ( A A C ) of rod AC is 500 mm 2 . The area ( A C B ) of rod CB is 300 mm 2 . The length ( L A C ) of the rod AC is 150 mm The length ( L C B ) of the rod CB is 250 mm . Calculation: Sketch the rod ACB as shown in Figure 1. Calculate the temperature changes to cause yielding in portion CB using the relation as follows: − P L A C E A A C − P L C B E A C B = L A B α ( Δ T ) Δ T = P L A B E α ( L A C A A C + L C B A C b ) (1) Here, L A B is length of the rod AB and P y is yielding point. ( P = P Y ) . Find the load at yielding point ( P ) using the relation: P = P Y = A C B σ Y (2) Substitute 300 mm 2 for A C B and 250 MPa for σ Y in Equation (2). P = P Y = 300 mm 2 ( 1 m 2 10 3 mm ) 2 × 250 MPa ( 10 6 Pa 1 MPa ) = 300 × 10 − 6 × ( 2.50 × 10 6 ) = 75 , 000 N Substitute 75 , 000 N for P Y , 150 mm for L A C , 250 mm for L C B , 400 mm for L A B , 200 GPa for E , 11.7 × 10 − 6 / ° C for α , 300 mm 2 for A C B , and 500 mm 2 for A A C in Equation (1). Δ T = [ 75 × 10 3 400 mm ( 1 m 10 3 mm ) × 200 GPa ( 10 9 Pa 1 GPa ) × 11.7 × 10 − 6 ( 150 mm ( 1 m 10 3 mm ) 500 mm 2 ( 1 m 10 3 mm ) 2 + 250 mm ( 1 m 10 3 mm ) 300 mm 2 ( 1 m 10 3 mm ) 2 ) ] = 75 × 10 3 0 .04 × 200 × 10 9 × 11.7 × 10 − 6 ( 0.150 500 × 10 − 6 + 0.250 300 × 10 − 6 ) = 0.80128 ( 300 + 833.333 ) = 90.812 ° C Find the actual temperature ( Δ T ) using the relation: The initial temperature is 150 ° C and then returned to 150 ° C . 150 ° C − 25 ° C = 125 ° C > ( Δ T Y ) Since Δ T > ( Δ T ) Y yielding occurs (b) To determine Find the permanent deflection at point C .

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Author: BEER

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Chapter 2.13, Problem 118P

(a)

To determine

Find the stress in portion AC and CB of the rod.

(b)

To determine

Find the permanent deflection at point C.

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Chapter 2.13, Problem 117P

Chapter 2.13, Problem 119P

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Find the stress in portion AC and CB of the rod.

Solution Summary: The author calculates the stress in portion AC and CB of the rod using the relation as follows:

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Find the stress in portion AC and CB of the rod.

Find the permanent deflection at point C.

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