02-02_chap_gere

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 5 diameter of hole S HORTENING d OF THE BAR (Eq. 1) N UMERICAL VALUES ( DATA ): d 5 maximum allowable shortening of the bar 5 8.0 mm 5 PL p E ¢ 1 d 1 2 2 d 2 1 1 d 1 2 1 2 d 2 2 5 P E C L / 4 p 4 ( d 1 2 2 d 2 ) 1 L / 4 p 4 d 1 2 1 L / 2 p 4 d 2 2 S d 5 a N i L i E i A i 5 P E a L i A i P 5 110 kN L 5 1.2 m E 5 4.0 GPa d 1 5 100 mm d max 5 maximum allowable diameter of the hole d 2 5 60 mm S UBSTITUTE NUMERICAL VALUES INTO E Q . (1) FOR d AND SOLVE FOR d 5 d max : U NITS : Newtons and meters 3 d max 5 23.9 mm d 5 0.02387 m d 2 5 569.81 3 10 2 6 m 2 5 761.598 2 100 2 555.556 5 106.042 1 0.01 2 d 2 761.598 5 1 0.01 2 d 2 1 1 0.01 1 2 0.0036 B 1 (0.1) 2 2 d 2 1 1 (0.1) 2 1 2 (0.06) 2 R 0.008 5 (110,000)(1.2) p (4.0 3 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 d of the pile in terms of P , L , E , and A . (b) Draw a diagram showing how the compressive stress s 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 5 100 mm and segment BC has diameter d 2 5 60 mm. Both segments have length L /2 5 0.6 m. A longitudinal hole of diameter d is drilled through segment AB for one-half of its length (distance L /4 5 0.3 m). The bar is made of plastic having modulus of elasticity E 5 4.0 GPa. Compressive loads P 5 110 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 AB 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 d OF PILE : At distance y from the base: © F vert 5 0 Ê c 1 T 2 Ê fL 2 P 5 0 Ê f 5 P L N ( y ) 5 axial force N ( y ) 5 fy (Eq. 2) (b) C OMPRESSIVE STRESS s c IN PILE At the base ( y 5 0): s c 5 0 See the diagram above. At the top( y 5 L ): s c 5 P A s c 5 N ( y ) A 5 fy A 5 Py AL d 5 PL 2 EA d 5 # L 0 d d 5 f EA # L 0 ydy 5 fL 2 2 EA 5 PL 2 EA d d 5 N ( y ) dy EA 5 fy dy EA L y P f dy f = P L s 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 d C of point C , located at distance h from the lower end of the bar. (b) What is the elongation d B of the entire bar? (c) What is the ratio b of the elongation of the upper half of the bar to the elongation of the lower half of the bar?
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02-02_chap_gere - 80 CHAPTER 2 Axially Loaded Numbers...

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