Example 1-4
EXAMPLE 1-4 Calculate the maximum shear stress τmax in a simply supported wood beam carrying a uniform load of (w 18 kN/m) if the length is 1.75 m and the cross section is rectangular with width 150 mm and height 250 mm. Solution: wL 18 1.75 15.75kN 2 2 V 15.75 1000 max 1.5 1.5 0.63MPa A 150 250 V EXAMPLE 1-5 A cantilever beam is made of wood with cross-sectional dimensions as shown in the Figure. Calculate the shear stresses due to the load P at points located 25 mm, 50 mm, 75 mm, and 100 mm from the top surface of the beam. From these results, plot a graph showing the distribution of shear stresses from top to bottom of the beam. Solution: V P I bh 3 120 2003 80 106 mm4 12 12 VQ V h 2 ( y2 ) Ib 2 I 4 V h2 8 1000 2002 ( y2 ) ( y2 ) 6 2I 4 2 80 10 4 6 2 50 10 (10000 y ) 1
EXAMPLE 1-6 A laminated plastic beam of square cross section is built up by gluing together three strips, as shown in the figure. The beam has a total weight of 3.2 N and is simply supported with span length L 320 mm. Considering the weight of the beam, calculate the maximum permissible load P that may be placed at the midpoint if: (a) the allowable shear stress in the glued joints is 0.3 MPa. (b) the allowable bending stress in the plastic is 8 MPa. Solution: 3.2 10 N / m 320 bh 3 30 303 I 67500mm4 12 12 allow 0.3MPa (a) w P wL P 1. 6 2 2 2 Q 30 10 10 3000mm3 VQ ( P / 2 1.6) 3000 Ib 67500 30 ( P / 2 1.6) 3000 0.3 67500 30 0.3 67500 30 P 2 1.6 401.8 N 3000 V (b) allow 8MPa PL wL2 0.08 P 0.128 4 8 My (0.08 P 0.128) 15 I 67500 (0.08 P 0.128) 103 15 8 67500 1 8 67500 P 128 448.4 N 80 15 M max 2
EXAMPLE 1-7 A wood pole of solid circular cross section is subjected to a horizontal force P 450 lb. The length of the pole is L 6 ft, and the allowable stresses in the wood are 1900 psi in bending and 120 psi in shear. Determine the minimum required diameter of the pole based upon the allowable shear stress. Solution: r4 4 b 2r I 4r 1 2 2 3 r r 3 2 3 2 V r3 VQ 4V 3 4 r Ib 3A 2r 4 16V 16Vmax d2 6.366 2 3 d 3 allow Q d min 2.52in EXAMPLE 1-8 A hollow steel box beam has the rectangular cross section shown in the figure. Determine the maximum allowable shear force V that may act on the beam if the allowable shear stress is 36 MPa. Solution: bh 3 200 4503 180 4103 I 484.9 106 mm4 12 12 12 450 450 410 410 Q 200 180 1.28 106 mm3 2 4 2 4 V Q allow allow Ib Ib 36 484.8 106 20 Vallow allow 273kN Q 1.28 106 3
EXAMPLE 1-9 A box beam of wood is constructed of two (260 mm x 50 mm) boards and two 260 mm x 25 mm) boards as shown in the figure. The boards are nailed at a longitudinal spacing (S 100 mm). If each nail has an allowable shear force F 1200 N, what is the maximum allowable shear force V max? Solution: 4
EXAMPLE 1-10 Two wood box beams (beams A and B) have the same outside dimensions (200 mm x 360 mm) and the same thickness (t 20 mm) throughout, as shown in the figure. Both beams are formed by nailing, with each nail having an allowable shear load of 250 N. The beams are designed for a shear force V 3.2 kN. (a) What is the maximum longitudinal spacing (SA) for the nails in beam A? (b) What is the maximum longitudinal spacing (SB) for the nails in beam B? (c) Which beam is more efficient in resisting the shear force? Solution: (c) BEAM B IS MORE EFFICIENT because the shear flow on the contact surfaces is smaller and therefore fewer nails are needed. 5
EXAMPLE 1-11 A steel cantilever beam is made of two structural tee- section beams welded together as shown in the figure below. Determine the allowable safe load (P) that the beam can carry. The allowable stresses are: σ 150MPa in tension and compression, τ 100MPa in shear, and q 2000N/mm on the welded joint. M P 6
EXAMPLE 1-12 A wood box beam shown in the figure is constructed of two boards, each 180x40mm in cross section, that serve as flanges and two plywood webs, each 15mm thick. The total height of the beam is 280mm. The plywood is fastened to the flanges by wood screws having an allowable load in shear of F 800N each. If the shear force V acting on the cross section is 10.5kN, determine the maximum permissible longitudinal spacing (S) of the screws. For practical fabrication of the beam, use spacing between screws S 45mm 7
EXAMPLE 1-13 A beam is loaded so that the moment diagram of it varies as shown in the figure. a) Find the maximum longitudinal shearing force in the 12mm diameter bolts spaced 300mm apart. b) Find the maximum shearing stress in the glued joint. 8
H.W.1 A metal beam with span is simply supported at points A and B. The uniform load on the beam (including its own weight) is 28 kN/m. Determine the normal stress σc and shear stress τc at point C, which is located 25 mm below the top of the beam and 200 mm from the right support. Show these stresses on a sketch of a stress element at point C. H.W.2 A wood beam AB supporting two concentrated loads P has a rectangular cross section of width b 100 mm and height h 150 mm. The distance from each end of the beam to the nearest load is a 0.5 m. Determine the maximum permissible value Pmax of the loads if the allowable stress in bending is σ 11 MPa (for both tension and compression) and the allowable stress in horizontal shear is τ 1.2 MPa. 9
Solution: EXAMPLE 1-10 Two wood box beams (beams A and B) have the same outside dimensions (200 mm x 360 mm) and the same thickness (t 20 mm) throughout, as shown in the figure. Both beams are formed by nailing, with each nail having an allowable shear load of 250 N. The beams are designed for a shear force V 3.2 kN.
Aug 16, 2017 · Factoring out GCF and by Grouping[In Class Version][Algebra 1].notebook 1 September 17, 2017 Sep 17 4:14 PM Homework Assignment The following examples have to be copied for next class Example 1 Example 2 Example 3 Example 4 Example 5 Example 6 Example 10 Example 12 Example 9
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