Stresses in Column with Biaxial Eccentric Loading

When vertical load on the column is not coincide with center of gravity of column cross section and does not act on either axis (X and Y axis), then the column is called biaxially eccentric loaded column.

Load Acting Eccentric to Both Axes

If the axial load P is placed at a point (ex, ey) eccentric to both x-axis and y-axis as shown in Figure, then the system can be assumed to consist of

(i) a direct compressive force P acting at the centroid,

(ii) a couple(Bending Moment) P x ey, about the x-axis and

(iii) a couple P x ex, about the y-axis.

The moment of P about these axes are respectively.

Mx = P.ey and My= P.ex

As seen for the case of load acting eccentric to one axis, the stress at any point can be written as

σ = σa + σbx + σby

σ = P/A + Mx/Zx + My/Zy

The maximum or minimum fiber stress will occur at the corner point A, B, C or D in the given figure.

If the load is in first quadrant ie. ex and ey are +ve, then the stresses will be calculated as below

σ = σa + σbx + σby

σA = σa + σbx - σby

σB = σa - σbx - σby

σC = σa - σbx + σby

σD = σa + σbx + σby

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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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