# EXAMPLE 1-4 EXAMPLE 1-5

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1 EXAMPLE 1- 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 50 mm. wl V 15.75kN V max MPa A 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 5 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. V P bh I mm 1 1 VQ V h ( y ) Ib I V h ( y ) ( I (10000 y ) y ) 1

2 EXAMPLE 1- 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. N and is simply supported with span length L= 0 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. MPa. (b) the allowable bending stress in the plastic is 8 MPa. (a). w 10N / m 0 bh 0 0 I 7500mm MPa allow P wl P V 1. Q mm VQ ( P / 1.) 000 Ib ( P / 1.) P N 000 (b) allow 8MPa PL wl M max 0.08P My (0.08P 0.18) 15 I 7500 (0.08P 0.18) P 18 8.N 80 15

3 EXAMPLE 1-7 A wood pole of solid circular cross section is subjected to a horizontal force P= 50 lb. The length of the pole is L= ft, and the allowable stresses in the wood are 1900 psi in bending and 10 psi in shear. Determine the minimum required diameter of the pole based upon the allowable shear stress. r I b r r 1 Q r r V r VQ V Ib r A r 1V 1V max d d d min.5in allow. 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 MPa. bh I mm Q mm VallowQ allow Ib allowib Vallow 7kN Q 1.810

4 EXAMPLE 1-9 A box beam of wood is constructed of two (0 mm x 50 mm) boards and two 0 mm x 5 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=100 N, what is the maximum allowable shear force V max?

5 EXAMPLE 1-10 Two wood box beams (beams A and B) have the same outside dimensions (00 mm x 0 mm) and the same thickness (t=0 mm) throughout, as shown in the figure. Both beams are formed by nailing, with each nail having an allowable shear load of 50 N. The beams are designed for a shear force V =. 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? (c) BEAM B IS MORE EFFICIENT because the shear flow on the contact surfaces is smaller and therefore fewer nails are needed. 5

6 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=000n/mm on the welded joint. M=P

7 EXAMPLE 1-1 A wood box beam shown in the figure is constructed of two boards, each 180x0mm in cross section, that serve as flanges and two plywood webs, each 15mm thick. The total height of the beam is 80mm. 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=5mm 7

8 EXAMPLE 1-1 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 1mm diameter bolts spaced 00mm apart. b) Find the maximum shearing stress in the glued joint. 8

9 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 8 kn/m. Determine the normal stress σc and shear stress τc at point C, which is located 5 mm below the top of the beam and 00 mm from the right support. Show these stresses on a sketch of a stress element at point C. H.W. 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. MPa. 9

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