Tutorial on Shear Stresses in Beams PDF

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Shear Stresses in Beams...

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Mechanics of Structures 2A Example Sheet 4 - Shear stresses in beams

1. A simply supported beam AB of span length of 3.5 m carries a concentrated load W at a distance of 1.5 m from support B as shown in Fig.1. The beam has a rectangular cross-section of depth 250 mm and width 150 mm. If the maximum allowable shear stress is 3.5 N/mm2, calculate the allowable value of the load W.

W A

B 1.5 m 3.5 m

Fig. 1

2. A cast iron beam is made from the stocky, unsymmetrical I-section shown in Fig.2. It carries a vertical shear force of 140 kN. Calculate (a) the maximum vertical shear stress. (b) the vertical shear stresses in the web at the top and bottom web/flange intersections (c) the horizontal shear stresses 42.5 mm from the edge of the top flange and 80 mm from the edge of the bottom flange. 125 mm 50 mm

250 mm 40 mm

50 mm 200 mm

Fig. 2

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3. The 2nd moment of area of the inverted T-section shown in Fig.3 is 2.354×106 mm4. (a) Derive and sketch the vertical shear stress distribution on the section for a vertical shear force of 70 kN, indicating key values. (b) Calculate the allowable applied shear force if the maximum shear stress is not to exceed 75 N/mm2. 4. A vertical shear force of 50 kN acts on the section shown in Fig.4 (a) Determine and sketch the shear stress distribution, indicating key values. (b) Estimate the proportion of the applied shear force which is carried by the section above A-A. 50 mm

10 mm 100 mm

100 mm

A

A 100 mm

10 mm

100 mm

100 mm

Fig. 3

Fig.4

5. A symmetrical I-beam has an overall depth of 350 mm and a breadth of 150 mm. The flanges are 17.5 mm thick, the web is 10 mm thick and the 2nd moment of area is 171.3×106mm4. The beam carries a shear force V and a bending moment M. Determine (a) the percentage of the shear force carried by the web (b) the percentage of the bending moment carried by the flanges. (c) the ratio of the maximum shear stress in the web to the average shear stress in the web, assuming the web carries all of V. 6. A simply supported beam of length L and rectangular cross-section of depth d and width b carries a uniformly distributed load q per unit length over its entire length. Derive an expression for the ratio of the maximum shear stress to the maximum bending stress in the beam.

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Solutions 1. 153.2 kN 2. (a) τ max = 18.4 N/mm2 at centroid (b) top of web: τ = 15.0 N/mm2, bottom of web: τ = 17.0 N/mm2 (c) top flange: τ = 4.1 N/mm2 bottom flange: τ = 5.5 N/mm2 3. (a) parabolic distributions through web and flange τ = 0 at top and bottom, τmax = 89.3 N/mm2 at centroid, τ = 81.8 N/mm2 & 8.2 N/mm2 above & below the web/flange junction resp. (b) 58.8 kN 4. (a) parabolic distributions through top and bottom parts τ = 0 at top and bottom, τ = 3.79 N/mm2 at centroid, τ = 7.28 N/mm2 & 3.64 N/mm2 above & below AA respectively. (b) 45% 5. (a) 95.4%

(b) 84.8%

(c) τmax/ τave = 1.032

6. τ max σ max = d L

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