02-02_chap_gere - 80 CHAPTER 2 Axially Loaded Numbers...

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Solution 2.3-8 Bar with a hole 80 CHAPTER 2 Axially Loaded Numbers d 2 d 1 L 4 P P L 4 L 2 d d diameter of hole S HORTENING OF THE BAR (Eq. 1) N UMERICAL VALUES ( DATA ): maximum allowable shortening of the bar 8.0 mm PL E ¢ 1 d 1 2 d 2 1 d 1 2 2 d 2 2 P E C L 4 4 ( d 1 2 d 2 ) L 4 4 d 1 2 L 2 4 d 2 2 S a N i L i E i A i P E a L i A i P 110 kN L 1.2 m E 4.0 GPa d 1 100 mm d max maximum allowable diameter of the hole d 2 60 mm S UBSTITUTE NUMERICAL VALUES INTO E Q . (1) FOR AND SOLVE FOR d d max : U NITS : Newtons and meters d max 23.9 mm d 0.02387 m d 2 569.81 10 6 m 2 761.598 100 555.556 106.042 1 0.01 d 2 761.598 1 0.01 d 2 1 0.01 2 0.0036 B 1 (0.1) 2 d 2 1 (0.1) 2 2 (0.06) 2 R 0.008 (110,000)(1.2) (4.0 10 9 ) Problem 2.3-9 A wood pile, driven into the earth, supports a load P entirely by friction along its sides (see figure). The friction force f per unit length of pile is assumed to be uniformly distributed over the surface of the pile. The pile has length L , cross-sectional area A , and modulus of elasticity E . (a) Derive a formula for the shortening of the pile in terms of P , L , E , and A . (b) Draw a diagram showing how the compressive stress c varies throughout the length of the pile. L P f Problem 2.3-8 A bar ABC of length L consists of two parts of equal lengths but different diameters (see figure). Segment AB has diameter d 1 100 mm and segment BC has diameter d 2 60 mm. Both segments have length L /2 0.6 m. A longitudinal hole of diameter d is drilled through segment AB for one-half of its length (distance L /4 0.3 m). The bar is made of plastic having modulus of elasticity E 4.0 GPa. Compressive loads P 110 kN act at the ends of the bar. If the shortening of the bar is limited to 8.0 mm, what is the maximum allowable diameter d max of the hole? d 2 d 1 L 4 P P A B C L 4 L 2
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Solution 2.3-9 Wood pile with friction SECTION 2.3 Changes in Lengths under Nonuniform Conditions 81 F ROM FREE - BODY DIAGRAM OF PILE : (Eq. 1) (a) S HORTENING OF PILE : At distance y from the base: © F vert 0 c T fL P 0 f P L N ( y ) axial force N ( y ) fy (Eq. 2) (b) C OMPRESSIVE STRESS c IN PILE At the base ( y 0): c 0 See the diagram above. At the top( y L ): s c P A s c N ( y ) A fy A Py AL PL 2 EA L 0 d f EA L 0 ydy fL 2 2 EA PL 2 EA d N ( y ) dy EA fy dy EA L y P f dy f = P L c = P y AL P A 0 Compressive stress in pile Friction force per unit length of pile Problem 2.3-10 A prismatic bar AB of length L , cross-sectional area A , modulus of elasticity E , and weight W hangs vertically under its own weight (see figure). (a) Derive a formula for the downward displacement C of point C , located at distance h from the lower end of the bar. (b) What is the elongation B of the entire bar? (c) What is the ratio of the elongation of the upper half of the bar to the elongation of the lower half of the bar? L h B A C Solution 2.3-10 Prismatic bar hanging vertically W Weight of bar (a) D OWNWARD DISPLACEMENT C Consider an element at distance y from the lower end.
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