Problems on Unsymmetrical bending of Beams

Assignment :

1. the stresses and deflection for the mid section of the I beam by unsymmetrical method Also identify the position of the neutral axis

2. A 240 mm × 120 mm steel beam of I-section is simply supported over a span of 6m and carries two equal concentrated loads at points 2 m from each end. The properties of the section are Ixx = 6012.32 × 104mm4, Iyy = 452.48 × 104 mm4. a) Determine the magnitude of the loads when the plane of the loads is vertical through YY. The permissible stress is 150 N/mm2 in compression and tension. b) Determine the degree of inclination of the plane of these loads to the vertical principal plane YY that will result in 20 percent greater bending stress than permitted under (A)

3. A T-Section of dimensions 150 wide x 200 mm deep, with 10 mm thickness of flange and web, is used as simply supported a beam on a span of 6 m. Find the maximum value of ‘w’ in kN/m, the permissible stress in the material is 120 MPa. The plane of loading is inclined at an angle of 400 to the vertical plane.

4. A beam having an I section 5m in length carrying a uniformly distributed load of 15 kN/m and having the section properties listed below. Calculate maximum bending stresses induced in the member when the trace of load plane is inclined at 180 to the principal axis YY. Calculate the maximum deflection in the beam. IXX = 13158 cm4 , IYY = 631.9 cm4 , ZXX = 751.9 cm3 ZYY =76.6 cm3, h = 350mm, b = 165mm.

5. A beam of rectangular section 100mm wide and 180mm deep is subjected to a bending moment of 12kN.m The trace of the plane of loading is inclined at 450 to the y-y axis of the section. Locate the natural axis of the section and calculate the maximum bending stress induced is the section.

6. A rectangular section of dimensions 120 x 200 mm is used as a beam on a 3 m span, If the beam is loaded by a concentrated load (P) at the centre at 30o to the vertical (Y-Y axis). Find the maximum value of the load ‘P’ in kN, if the maximum bending stress is not to exceed 12 MPa

Short Answer Questions:

1.Differentiate between symmetrical and unsymmetrical bending
2 What is principal moment of inertia?
3 Explain graphical method for locating principal axes.
4 What are the conditions that should be satisfied for a beam to bend without twisting?
5 State the assumptions made in analyzing a beam for unsymmetrical bending.
6 Determine the principal moments of Inertia for an angle section 225x175x15 mm.

Comments

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PORTAL METHOD and CANTILEVER METHOD

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Shearing Stresses Distribution in Circular Section

Show that the shearing stress developed at the neutral axis of a beam with circular cross section is τ max = (4/3)(F/πr2). Assume that the shearing stress is uniformly distributed across the neutral axis. Solution : Let us consider the circular section of a beam as displayed in following figure. We have assumed one layer EF at a distance y1 from the neutral axis of the circular section of the beam Shear stress at a section will be given by following formula as mentioned here Where, F = Shear force (N) τ = Shear stress (N/mm2) A = Area of section, where shear stress is to be determined (mm2) ȳ = Distance of C.G of the area, where shear stress is to be determined, from neutral axis of the beam section (m) Q = A. ȳ = Moment of the whole shaded area about the neutral axis I = Moment of inertia of the given section about the neutral axis (mm4) For circular cross-section, Moment of inertia, I = ПR4/4 b = Width of the given section where shear stress is to be determined. Let us consider on

Prof.Kodali Srinivas

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