On the Stress Distribution in a Deep Beam having Two Circular Holes subjected toUniform Bending Moment

Abstract

In connection with the problem of stress distribution in a deep beam with one hole(1)it is of interest to study also the more practical case where two circular holes are perforated for the purpose of the weight saving, etc. In this paper, the problem is considered as that of a deep beam having two circular holes, the depth of which is great enough compared with the size of holes. The solution is given for bipolar co-ordinates, for which the co-ordinate curves are co-axial circles.Since the problem is considered as that of plane stress, the stresses can be expressed in terms of a single functionX, say stress function, as given in (3) in this particular coordinates.(2) We assume this functionXto be (X1+X0) and suppose thatX1gives no stresses at great distance from two holes and the stresses in terms ofX0only satisfy the conditions at that point, though each function satisfies independently the linear partial differential equation of the same type as given in (4) or (5) in our particular co-ordinates.(2) Two solutions are obtained, (1) for the case of a deep beam containing two circular holes of equal radii on each side of the neutral axis and (2) for the case of a deep beam containing two circular holes on the neutral axis.The results of mathematical investigation may be summarised as follow;For the case (I). In this case, thex-axis is the neutral axis and they-axis is the vertical axis, on which the centers of two holes lie. These two circular holes lie at equal distance on either side of the neutral axis. The stress function which satisfies the conditions above mentioned is given by (8). The stresses in beam are given in (9), (10), (11) and (12). The constants in (12) which are determined by the conditions on the periphery of holes (13), namely, (14) are given in (15). The stress distribution round each hole is as shown in Fig. 3 by plotting the values of stresses on the normals to the periphery of hole. The stress round each hole has two maximum values at the symmetrical points A and B on the vertical axis through centers of holes. These two maximum values are some multiples of the stresses at the corresponding points in a non-perforated beam. These multiples are 2.85 atAand 3.22 atBfor P/D= 3.75, Pbeing the center distance and Dthe dia. cf two holes. For larger values of P/D, these multiples are nearly constant as shown in Fig. 1. The stress distribution on the neutral axis and on the vertical axis through centers of holes are given in Fig. 4 by plotting the values of stresses on the normals to the neutral and vertical axis respectively.For the case (2). In this case, the y-axis is the neutral axis of beam, on which two equal circular holes lie. The stress function for this case is given in (17) and the stresses in beam are given by (18), (19), (20) and (21). The constants in (21) are determined by the conditions round each circular hole (22), namely, (23) and are given in (24). The stress distribution round each circular hole in this case is not symmetry about the vertical axis through its center as shown in Fig. 5, since the existence of the another Hole on the same neutral axis has great influence upon the stress distribution in the neighbourhood of the other hole. But this effect due to the another hole will be decreased as the center distance of two holes is increased and, for larger value of P/D=10, the stress round of each hole may be approximately given by that for the case of a deep beam having one circular hole as shown in Fig. 2. The stress for the latter case is given byββ=MR/I (sin θ-sin 3θ) M=uniform bending moment.R=radius of circular hole.