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

See similar textbooks

Concept explainers

Thermal Properties of Materials

In mechanical engineering, the thermal property is represented as the physical quantity that is relevant to the application of heat. The thermal properties are evaluated by variations in any specific material due to low heat or high heat. The quantities …

Videos

Question

Chapter 14.4, Problem 71P

To determine

Find the normal and shear stresses at point D.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 14.4, Problem 70P

Chapter 14.4, Problem 72P

Students have asked these similar questions

The average normal stress and maximum in-plane shear stress components are determined from the circle as the coordinates of either point. true or false?

50 mm The machine part has a 20 mm x 40 mm rectangular cross section where points a, b, and e are located. 150 mm For the loading shown, determine the state of stress at cach of those points. Write cach state of stress in 40 mm 0.5 kN the form of a tensor, observing the r-y-z coordinate system shown, and sketch the stresses acting on a differential element at each point. 20 mm 3 kN 160 mm 2.5 kN

Determine the normal strains at the point associated with the x and y axes. Determine the shear strain at the point associated with the x and y axes. Determine the normal stresses at the point associated with the x and y axes. Determine the shear stress at the point. Determine the normal stress that acts along an axis rotated at θ= 45° counterclockwise from the positive x-axis.

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...

Knowledge Booster

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, mechanical-engineering and related others by exploring similar questions and additional content below.

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.

Concept explainers

Videos

Find the normal and shear stresses at point D.

Want to see the full answer?

Chapter 14 Solutions