Mechanics of Materials CIVL 3322 / MECH 3322

Shear Flow and Beam Construction II

Shear Flow and Beam Construction II 2

Excerpts from this work may be reproduced by instructors for distribution on a not-for-profit basis for testing or instructional purposes only to students enrolled in courses for which the textbook has been adopted. Any other reproduction or translation of this work beyond that permitted by Sections 107 or 108 of the 1976 United States Copyright Act without the permission of the copyright owner is unlawful.

9.53 A W310 × 60 steel beam (see Appendix B) in an existing structure is to be strengthened by adding a 250 mm wide by 16 mm thick cover plate to its lower flange, as shown in Fig. P9.53. The cover plate is attached to the lower flange by pairs of 24-mm-diameter bolts spaced at intervals of s along the beam span. Bending occurs about the z centroidal axis. (a) If the allowable bolt shear stress is 96 MPa, determine the maximum bolt spacing interval s required to support an internal shear force in the beam of V = 50 kN. (b) If the allowable bending stress is 150 MPa, determine the allowable bending moment for the existing W310 × 60 shape, the allowable bending moment for the W310 × 60 with the added cover plate, and the percentage increase in moment capacity that is gained by adding the cover plate.

Fig. P9.53

Solution Centroid location in y direction:

Shape Width b Height h Area Ai

yi (from bottom) yi Ai

(mm) (mm) (mm2) (mm) (mm3) W310 × 60 - - 7,550 167 1,260,850 cover plate 250 16 4,000 8 32,000 11,550 1,292,850

3

2

1,292,850 mm 111.935 mm (from bottom of shape to centroid)11,550 mm

206.065 mm (from top of shape to centroid)

i i

i

y Ay

A

Moment of inertia about the z axis: Shape IC d = yi – y d²A IC + d²A

(mm4) (mm) (mm4) (mm4) W310 × 60 128,000,000 55.065 22,892,764 150,892,764 cover plate 85,333.33 –103.935 43,209,937 43,295,270

Moment of inertia about the z axis (mm4) = 194,188,035 (a) Maximum bolt spacing Consider the cover plate, which is connected to the W310 × 60 shape with two bolts:

3

3

4

(250 mm)(16 mm)(111.935 mm 16 mm/2) 415,740 mm

(50,000 N)(415,740 mm ) 107.0457 N/mm194,188,035 mm

QV Qq

I

The cross-sectional area of a 24-mm-diameter bolt is:

2 2bolt (24 mm) 452.389 mm

4A

Relate the shear flow and the bolt shear stress with Eq. (9.14):

2 2(2 bolts)(96 N/mm )(452.389 mm )

107.0457 N/mm811 mm

f f f

f f f

q s n A

n As

q Ans.

Shear Flow and Beam Construction II 3

Excerpts from this work may be reproduced by instructors for distribution on a not-for-profit basis for testing or instructional purposes only to students enrolled in courses for which the textbook has been adopted. Any other reproduction or translation of this work beyond that permitted by Sections 107 or 108 of the 1976 United States Copyright Act without the permission of the copyright owner is unlawful.

9.50 A wooden beam is fabricated by bolting together three members, as shown in Fig. P9.50a. The cross-sectional dimensions are shown in Fig. P9.50b. The 8-mm-diameter bolts are spaced at intervals of s = 200 mm along the x axis of the beam. If the internal shear force in the beam is V = 7 kN, determine the shear stress in each bolt.

Fig. P9.50a Fig. P9.50b

Solution Centroid location in y direction:

Shape Width b Height h Area Ai

yi (from bottom) yi Ai

(mm) (mm) (mm2) (mm) (mm3) left board 40 90 3,600 255 918,000 center board 40 300 12,000 150 1,800,000 right board 40 90 3,600 255 918,000 19,200 3,636,000

3

2

3,636,000 mm 189.375 mm (from bottom of shape to centroid)19,200 mm

110.625 mm (from top of shape to centroid)

i i

i

y Ay

A

Moment of inertia about the z axis:

Shape IC d = yi – y d²A IC + d²A (mm4) (mm) (mm4) (mm4)

left board 2,430,000 65.625 15,503,906.25 17,933,906.25 center board 90,000,000 -39.375 18,604,687.50 108,604,687.50 right board 2,430,000 65.625 15,503,906.25 17,933,906.25

Moment of inertia about the z axis (mm4) = 144,472,500.00

Homework

¢ Problem P9.51 ¢ Problem P9.54 ¢ Problem P9.56

Shear Flow and Beam Construction II 4