Explanation Given information: The grains at point D make an angle of 30 ° with the horizontal. Calculation: Sketch the free body diagram of the beam AB as shown in Figure 1. Let the force and moment at point D be V and M . Sketch the free body diagram of the left cut segment as shown in Figure 2. Refer to Figure 2. Apply Equilibrium equations. Sum of vertical forces is equal to 0. Σ F y = 0 − V + 5 = 0 V = 5 kN Sum of moments is equal to 0. Σ M = 0 M − 5 ( 1 ) = 0 M = 5 kN ⋅ m Sketch the rectangular cross section as shown in Figure 3. Find the moment of inertia of the section ( I ) using Figure 3. I = 1 12 ( 0.1 m ) ( 0.3 m ) 3 = 2.25 × 10 − 4 m 4 Find the first moment of area at point D ( Q D ) using the relation: Q D = y ¯ ' A ' Here, y ¯ ' is the centroid distance to point D and A ' is the area from centroid to point D . Refer to Figure 3. Q D = 0.1 m ( 0.1 m ) ( 0.1 m ) = 0.001 m 3 Calculate the normal stress ( σ D ) acting at point D using the relation: σ D = M y I Here, M is the moment at point D , y is the distance from centroid to point D , and I is the moment of inertia. Substitute 5 kN ⋅ m for M , 0.05 m for y , and 2.25 × 10 − 4 m 4 for I . σ D = ( 5 kN ⋅ m ) ( 0.05 m ) 2.25 × 10 − 4 m 4 = ( 5 kN ⋅ m × 10 3 N 1 kN ) ( 0.05 m ) 2.25 × 10 − 4 m 4 = 1.111 × 10 6 Pa = 1.111 MPa Find the shear stress ( τ D ) at point D using the relation: τ D = V Q I t Here, t is the thickness. Substitute 5 kN for V , 0.001 m 3 for Q , 2.25 × 10 − 4 m 4 for I , and 0.1 m for t . τ D = ( 5 kN ) ( 0.001 m 3 ) ( 2.25 × 10 − 4 m 4 ) ( 0.1 m ) = ( 5 kN × 10 3 N 1 kN ) ( 0.001 m 3 ) ( 2.25 × 10 − 4 m 4 ) ( 0.1 m ) = 0.2222 × 10 6 Pa = 0.2222 MPa Sketch the state of stress on the differential element as shown in Figure 4. Refer to Figure 4. Apply the sign conventions: The value of normal stresses are σ x = 1.111 MPa and σ y = 0

Statics and Mechanics of Materials (5th Edition)

5th Edition

ISBN: 9780134382593

Author: Russell C. Hibbeler

Publisher: PEARSON

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Chapter 14.4, Problem 71P

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Find the normal and shear stresses at point D.

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Chapter 14 Solutions

Statics and Mechanics of Materials (5th Edition)

Ch. 14.3 - In each ease, the state of stress x, y, xy...Ch. 14.3 - Given the state of stress shown on the element,...Ch. 14.3 - Determine the normal stress and shear stress...Ch. 14.3 - Prob. 2FP Ch. 14.3 - Determine the equivalent state of stress on an...Ch. 14.3 - Prob. 4FP Ch. 14.3 - The beam is subjected to the load at its end....Ch. 14.3 - Prob. 6FP Ch. 14.3 - Prove that the sum of the normal stresses x+y=x+y...Ch. 14.3 - Determine the stress components acting on the...

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Find the normal and shear stresses at point D .

Solution Summary: The author calculates the normal and shear stresses at point D by drawing the free body diagram of the beam AB as shown in Figure 1.

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Find the normal and shear stresses at point D.

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Chapter 14 Solutions