4. Consider the rectangular beam in Fig. 1. We wish to confirm the relation between the...
Fantastic news! We've Found the answer you've been seeking!
Question:
Transcribed Image Text:
4. Consider the rectangular beam in Fig. 1. We wish to confirm the relation between the shear force in the beam and shear stresses that may exist in the beam. We will work within the general framework of elementary beam theory, a topic covered in detail later in this course. For now we will just anticipate some results from that theory. We assume our beam is in a state of plane stress where 33 = 13 = 023 = 0; we also recall of course that the Cauchy stress is symmetric so that = ji. Hence we are concerned with 2 (a) NA = Neutral axis of beam V = shear force -P/2 (c) h NA 011(b) 021-07 NA P/2 (d) X3 012 Beam cross sectional area Figure 1: (a) A rectangular beam simply supported at its ends subject to a concentrated force P measured per unit distance through its thickness b; the total height of the beam is h. (b) Sketch of the beam's bent configuration indicating compressive stresses 11 <0 at the beams top (22 > 0) and tensile stresses 11 > 0 at the beam's bottom (x2 < 0). Note the neutral aris NA at position 2 = 0 for a rectangular beam at which 110. (c) Plot of the shear force along the beam. (d) Cross section of rectangular beam indicating coordinate system 21 along beam's length, x2 normal to beam along the line of application of P, and 23 pointing outward. only 11, 22, and 12. As it happens, elementary beam theory leads to the following result for the shear stress of Fig. 1, viz. V 012= (-4) (9) Here I33 is the so-called moment of inertia of the beam's cross section and for the rectangular beam is 133 = 1/12 bh, and x2 is measured from the beam's so-called neutral axis where 11 = 0; for the rectangular beam the neutral axis lies midway in the beam's depth, h. Clearly 12 = 0 at the top and bottom of the beam where x2 = h/2 as it should. But, what is the shear force V as shown in Fig. 1c? Is it not just 012 dA, (10) where A is the cross sectional area of the rectangular beam? That is, V is simply the resultant of the shear stress, 12, acting on the cross sectional area of the beam. Confirm eq. 10 vis--vis Fig. 1c. A note of interest and caution. The student has undoubtedly studied the subject of statics and hence it is possible that the definition of shear force may have been inverted in algebraic sign so that, for example, the shear force diagram of Fig. 1c may have a reversed sign. This is the case in many textbooks. Unfortunately such a convention is inconsistent with the more rigorous conventions of the theory of elasticity. In this course we follow the conventions of elasticity as in this problem. An explanation follows below by way of an extra credit problem listed below. Topic: Interpretation of stress states 4. Consider the rectangular beam in Fig. 1. We wish to confirm the relation between the shear force in the beam and shear stresses that may exist in the beam. We will work within the general framework of elementary beam theory, a topic covered in detail later in this course. For now we will just anticipate some results from that theory. We assume our beam is in a state of plane stress where 33 = 13 = 023 = 0; we also recall of course that the Cauchy stress is symmetric so that = ji. Hence we are concerned with 2 (a) NA = Neutral axis of beam V = shear force -P/2 (c) h NA 011(b) 021-07 NA P/2 (d) X3 012 Beam cross sectional area Figure 1: (a) A rectangular beam simply supported at its ends subject to a concentrated force P measured per unit distance through its thickness b; the total height of the beam is h. (b) Sketch of the beam's bent configuration indicating compressive stresses 11 <0 at the beams top (22 > 0) and tensile stresses 11 > 0 at the beam's bottom (x2 < 0). Note the neutral aris NA at position 2 = 0 for a rectangular beam at which 110. (c) Plot of the shear force along the beam. (d) Cross section of rectangular beam indicating coordinate system 21 along beam's length, x2 normal to beam along the line of application of P, and 23 pointing outward. only 11, 22, and 12. As it happens, elementary beam theory leads to the following result for the shear stress of Fig. 1, viz. V 012= (-4) (9) Here I33 is the so-called moment of inertia of the beam's cross section and for the rectangular beam is 133 = 1/12 bh, and x2 is measured from the beam's so-called neutral axis where 11 = 0; for the rectangular beam the neutral axis lies midway in the beam's depth, h. Clearly 12 = 0 at the top and bottom of the beam where x2 = h/2 as it should. But, what is the shear force V as shown in Fig. 1c? Is it not just 012 dA, (10) where A is the cross sectional area of the rectangular beam? That is, V is simply the resultant of the shear stress, 12, acting on the cross sectional area of the beam. Confirm eq. 10 vis--vis Fig. 1c. A note of interest and caution. The student has undoubtedly studied the subject of statics and hence it is possible that the definition of shear force may have been inverted in algebraic sign so that, for example, the shear force diagram of Fig. 1c may have a reversed sign. This is the case in many textbooks. Unfortunately such a convention is inconsistent with the more rigorous conventions of the theory of elasticity. In this course we follow the conventions of elasticity as in this problem. An explanation follows below by way of an extra credit problem listed below. Topic: Interpretation of stress states
Expert Answer:
Related Book For
Posted Date:
Students also viewed these mechanical engineering questions
-
answer all questions as instructed below. attend all questions. 4 Computer Vision (a) Explain why such a tiny number of 2D Gabor wavelets as shown in this sequence are so efficient at representing...
-
Let A, B be sets. Define: (a) the Cartesian product (A B) (b) the set of relations R between A and B (c) the identity relation A on the set A [3 marks] Suppose S, T are relations between A and B, and...
-
Comprehensive variance analysis review. Sonnet Inc. has the following budgeted standards for the month of March 2010: Sales of 2,000,000 units are budgeted for March. Actual March results are: Unit...
-
Lisa Sizemore, a taxpayer in the 10-15 percent tax bracket, purchased stock as an investment on July 11, 2011. She sold the stock on July 9, 2012, 2 days before qualifying for the long-term holding...
-
How could the owners of Willow Springs have ensured the safety of their customers and prevented the infections from occurring? Explain.
-
Using only the factor formulas given in Table 2.6, derive Equation 7.5 starting with Equation 7.3. TABLE 2.6 Summary of Discrete Compounding Interest Factors. To Find Given Factor Symbol Name P F...
-
1. Why is the operations function important in implementing the strategy of an organization? Explain why the changes put in place by Victoria Chen and her team could either hurt or help the bank. 2....
-
Consider the inverse demand function below for a market with 2 dominant firms: P=600-3(Q 1 +Q 2 ) Assume that both firms have identical marginal costs: $300, yet one of them, Firm 1, is the leader...
-
1. How multithread threads of a process can be lightweight when they are consuming same processor as required by a whole process? 2. Give two reasons why caches are useful. What problems do they...
-
3. Here is the declaration of a linked list node and head pointer. struct Node }; Node nextPtr_; int value_; Node headPtr; Write the code for the following function that searches the list for a node...
-
In 1995, Able purchased a parcel of real property described as "Lot 1 in Block 'A' in the City of Grover." In November 1996, Able conveyed a portion of Lot 1 to Baker by deed, which provided that he...
-
We often see the news report where hackers/scammers proliferate various company's' servers and gain personal data of customers. And we all have seen these class actions with these companies. Just...
-
Question 1 An investor, a writer, an actor, a professional athlete, and a food critic formed a general partnership to own and operate a high-end bistro. Each partner made a capital contribution. The...
-
On January 1, 2016, Titans Company issued 1,000 shares of 10%, $200 par value, cumulative preferred stock for $300,000. No preferred dividends were declared in 2016 and 2017. On December 30, 2018,...
-
Corporations that issue financial securities such as stock or debt obligations to the public do so primarily to: a. become profitable b. increase sales c. avoid double taxation of their profits d....
-
The Home Depot is the leading retailer in the home improvement industry and one of the 10largest retailers in the United States. The company included the following on its January 29, 2012, balance...
-
A circular aluminum tube with pinned ends supports a load P = 18 kN acting at distance e = 50 mm from the center (see figure). The length of the tube is 3.5m and its modulus of elasticity is 73 GPa....
-
A square aluminum bar with pinned ends carries a load P = 25 k acting at distance e = 2.0 in. from the center (see figure on the previous page). The bar has length L = 54 in. and modulus of...
-
The polyethylene liner of a settling pond is subjected to stresses (x = 350 psi, (y = 112 psi, and Txy= -120 psi, as shown by the plane-stress element in the first part of the figure. Determine the...
-
Which of the following types of epigenetic changes may promote cancer? a. DNA methylation b. Covalent modification of histones c. Chromatin remodeling d. All of the above may promote cancer.
-
The underlying cause(s) of epigenetic changes associated with cancer may be a. mutations in genes that encode chromatin-modifying proteins. b. environmental agents that alter the function of...
-
Normal (nonmutant) tumor-suppressor genes often function a. as negative regulators of cell division. b. in the maintenance of genome integrity. c. in the stimulation of cell division. d. as both a...
Study smarter with the SolutionInn App