793_Mechanics SolutionInstructors_Sol.Manual-Mechanics_Materials_7e.book_Gere_light.1

793_Mechanics SolutionInstructors_Sol.Manual-Mechanics_Materials_7e.book_Gere_light.1

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SECTION 9.10 Deflections Produced by Impact 787 Problem 9.10-4 A weight W 20 kN falls through a height h 1.0 mm onto the midpoint of a simple beam of length L 3 m (see figure). The beam is made of wood with square cross section (dimension d on each side) and E 12 GPa. If the allowable bending stress in the wood is s allow 10 MPa, what is the minimum required dimension d ? A B h W d d L 2 L 2 Solution 9.10-4 Simple beam with falling weight W D ATA : W 20 kN h 1.0 mm L 3.0 m E 12 GPa s allow 10 MPa C ROSS SECTION OF BEAM ( SQUARE ) d dimension of each side M AXIMUM DEFLECTION (E Q . 9-94) d max d st ( d 2 st 2 h d st ) 1/2 M AXIMUM BENDING STRESS For a linearly elastic beam, the bending stress s is proportional to the deflection d . (1) S TATIC TERMS s st AND d st (2) (3) d st WL 3 48 EI WL 3 48 E a 12 d 4 b WL 3 4 Ed 4 s st M S a WL 4 b a 6 d 3 b 3 WL 2 d 3 s max s st d max d st 1 + a 1 + 2 h d st b 1/2 I d 4 12 S d 3 6 S UBSTITUTE (2) AND (3) INTO E Q . (1) S UBSTITUTE NUMERICAL VALUES : S QUARE BOTH SIDES , REARRANGE , AND SIMPLIFY S OLVE NUMERICALLY
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