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 http://www.archive.org/details/cu31924004137539 
 
stress Lines in 
 
 Beams of Unsymmetrical 
 
 Cross Section 
 
 TG 260.887"'" ""'""""'■"'"'» 
 ®!llS'iE.i!;.5,^?™?.?lHn?ym'netricalc 
 
 3 1924 004 137 539 
 
 C. M. BROOMALL 
 
Reprinted from 
 
 Proceedings Dei.aware Codnty Institute ob Science 
 
 Volume VIII, Number 4 
 
 Media, Pa., U. S. A. 
 
 1918 
 
 STRESS LINES IN BEAMS OF UNSYMMETRICAL 
 CROSS SECTION. 
 
 BY C. M. BROOMALL. 
 
 Following up the method already made use of in treating 
 of Stress and Strain in beams of symmetrical cross section,* 
 it is proposed in the present article to extend the same meth- 
 ods to beams of unsymmetrical cross section . As a typical 
 beam of this character, the Channel with web vertical will be 
 first considered. 
 
 As a preliminary to the treatment of the Channel Beam, a 
 brief resume of a few of the facts brought out in the articles 
 mentioned may be useful. These facts may be summarized 
 as follows : — 
 
 1. The horizontal unit shear at any point of the beam 
 is always accompanied by an equal right-angled vertical (or 
 transverse) unit shear. 
 
 2. The total horizontal shear at any point is equal to the 
 difference between the horizontal forces acting on the two 
 
 *" Stress and Strain." Proc. Del. Co. Inst. Sci. Vol. VIII, No. i, 
 May, 1916. 
 
 "Stress and Deformation in the I Beam." Proc. Del. Co. Inst. Sci. 
 Vol. VIII, No. 2, December, 1916. 
 
2 BEOOMALL : 
 
 ends of the elementary "block" of material lying above or 
 beyond the element of length under consideration. 
 
 3. Passing from the middle towards the ends along any 
 level line, the above horizontal forces are all "spilled off" 
 in the shape of horizontal shear, becoming reduced to zero at 
 the ends. The horizontal forces become less and less by reg- 
 ular decrements or subtractions from a previous value. The 
 horizontal shear usually increases towards the ends, but not 
 by additions to a previous value. 
 
 4. The horizontal shear may be regarded as the physical 
 lever-arm of the couples in the upper and lower part of the 
 beam. Without this shear the parts of the beam would not 
 work together as a harmonious " resisting moment." 
 
 5. In the I Beam the compression in the right-hand edge 
 of the upper flange, for instance, must work in conjunction 
 with its most remote comrade, the tension in the left-hand 
 edge of the lower flange. This can only be through the 
 medium of some kind of shear. Hence we must assume the 
 existence of horizontal shear in the flange as well as in the 
 web. The horizontal shear in the flanges, however, has its 
 shearing planes vertically disposed, while in the web the 
 shearing planes are horizontal. Figure i. 
 
 Illlll Jlllll 
 
 rTTnn'>rTTTn 
 
 FIGURE 1 
 
 As a definite problem for the application of these and 
 other principles to beams of unsymmetrical cross section, let 
 
STRESS LINES IN BEAMS. 3 
 
 US take up the determination of the general character of the 
 stresses and deformations in the Channel Beam, set with web 
 vertical and loaded uniformly. For the present let us assume 
 this loading to be in a narrow line along the web, so as to 
 avoid consideration of the bending of the flange by the 
 direct load. 
 
 In order to balance the external forces and reactions, each 
 element of the body will be acted upon by certain forces 
 which for convenience we will resolve vertically, transversely 
 and longitudinally. These are the resultants of the various 
 external forces acting upon the body, and necessary to 
 equilibrium. By resolving the forces acting upon the ele- 
 ments in various directions the actual direction and amount 
 of the true internal stresses acting in the material may be 
 found. In the present article we will consider more particu- 
 larly the direction and characteristics of the lines of maxi- 
 mum stress rather than the absolute value of the stresses. 
 
 As the Channel stands on edge subjected to its load, we 
 know it is acted upon by vertical shear and bending moment. 
 Further, we know that if a vertical unit shearing stress exists 
 at any point, an equal and opposite longitudinal unit shear- 
 ing stress must exist also, as required by the principles of 
 statics. This opposing longitudinal unit shear is produced 
 by or results from the decrements in the value of the horizon- 
 tal direct stresses already mentioned. Hence, each element 
 of the body must be acted upon by two equal right-angled 
 shearing unit stress. As a consequence of the bending 
 moment we know also that each element must in addition be 
 acted upon by direct stress, tension or compression. 
 
 As far as the web is concerned these are the same forces as 
 are met with in a beam of rectangular cross section. 
 
 In the flanges we meet a rather different array of forces, 
 similar to those already mentioned in connection with the 
 I Beam. A longitudinal shear, with vertically arranged 
 shearing planes must exist in order to transfer the decrements 
 of longitudinal stresses to the web and thence by ordinary 
 
4 BROOMALL : 
 
 horizontal shear to the neutral axis. Along with this longi- 
 tudinal shear in the flange there must also exist its right- 
 angled companion shear likewise with vertical shearing planes. 
 
 As regards the variation in value of the stresses in a given 
 cross section of the Channel, it is evident that in the web the 
 direct stress varies as the distance from neutral surface and 
 the horizontal and vertical shear vary inversely as the square 
 of the distance from the neutral surface. It is also evident 
 that in the flanges the direct stresses are independent of the 
 distance from web and are constant, while the longitudinal 
 and cross shears vary inversely as the distance from the web. 
 
 Knowing the resultant forces acting upon the elements of 
 the body, it is easy to predict the characteristics of the 
 actual lines of maximum internal stress. In Figure 2 are 
 indicated the forces above enumerated which the elements 
 must resist, and the nature of the resulting lines of maximum 
 stress. The formulas for tracing these lines may be found in 
 any advanced work on the mechanics of materials, and are 
 
 s 
 
 Direct Stress : cot 26^ (i) 
 
 2 v 
 
 s . ^ 
 
 Shear : tan 2 4> = — (2) 
 
 2 v 
 
 Where 6 = angle of direct stress with longitudinal direction 
 
 <ji = angle of shearing plane with longitudinal direction 
 
 S = direct unit stress, tension positive, compression 
 
 negative 
 V = unit longitudinal shear 
 
 The figure shows respectively top view, elevation and 
 bottom view of the beam. The various forces are represented 
 by arrows or by the letters T for tension, C for compression 
 and S for shear. 
 
 So far the stress lines as determined are based upon stat- 
 ical principles. When the Channel deflects under its loads 
 these lines of stress will vary to a small extent. Under 
 
STRESS LINES IN BEAMS. 
 
 
 
 C 
 
 m 
 
6 BROOMALL : 
 
 deflection the upper portion of the beam expands in all direc- 
 tions, while the lower portion contracts. This means that a 
 transverse unit tension in all directions must exist in the 
 upper portion of the beam and a transverse unit compression 
 in the lower portion. These forces may be resolved vertically 
 and horizontally and their effect considered. The vertical 
 components in the web and the horizontal forces in the 
 flanges are those which will act to modify the lines of maxi- 
 mum stress as already found. The actual effect, of course, 
 is small, and if the deformation were known these forces 
 might be added to those of Figure 2 before making the reso- 
 lution. 
 
 Another effect of the expansion and contraction just men- 
 tioned is to raise the centre of gravity of the section and con- 
 sequently the neutral axis. This alters the amount of the 
 horizontal forces, producing another modification in the lines 
 of maximum stresses. These modifications in the stress lines 
 are naturally small, and for all practical purposes may be 
 neglected. 
 
 lyCt us now consider the Channel used as a beam with web 
 horizontal and flanges vertical, for example, turned upward. 
 The position of the neutral plane will usually be in the lower 
 part of the flanges. The lines of maximum stress in the 
 flanges will be calculated in the usual way, and will have the 
 general characteristics shown in Figure 3. The figure also 
 suggests the forces acting on the elements of the body and the 
 manner in which they vary in value from top to bottom. 
 
 The lines of direct stress cut the upper edges of the flanges 
 at 90° and 0°, and they cross the neutral axis at 45°. They 
 do not, however, intercept the upper surface of the web at 90° 
 and 0°, since the horizontal shear does not fall to zero. The 
 lines as sketched are only meant to indicate the general char- 
 acteristics of the curves. 
 
 The lines of shear in the flanges cut the upper edge 
 of flange at 45", and the neutral axis at 90° and 0°. 
 They do not cut the upper surface of the web at 45°, because 
 
STRESS LINES IN BEAMS. 
 
 o 
 
 73 
 
 m 
 
8 
 
 BEOOMALL : 
 
 the horizontal shear, as before, does not reduce to zero. 
 As regards the lines of stress in the web, another method 
 of reasoning must be used. In the first place, we will 
 encounter longitudinal shear with vertical shearing planes, as 
 only in this way can the two flanges be tied together to work 
 in harmony with the web. The planes of shearing for the 
 whole section are more or less as indicated in Figure 4. 
 
 " iNiir iiiMiM I MiiimiiJiiiiiii] mirl 
 
 FIGURE 4 
 
 The distribution of forces acting on the elements of the 
 body in the four quadrants of the web and their variation in 
 value are indicated in Figure 5. 
 
 The resolution of these forces will give lines of maximum 
 internal stress for the plane of the web after the manner 
 shown to the left in the figure. 
 
 The lines of direct stress cut the middle line of the beam 
 
 at o" and 90°. They cut the edges of the web at angles dif- 
 
 s 
 fering from 45° a certain amount, since the ratio — — does 
 
 2 V 
 
 not reduce to zero. 
 
 The lines of shear cut the middle line at 45° and approach 
 0° and 90° at the edges of the web. 
 
 As mentioned, the lines of stress cut the boundaries in cer- 
 tain cases at angles differing more or less from limiting 
 values. These angles may be found by substitution of the 
 proper values in Formulas (i) and (2). They will depend, 
 of course, upon the relative values of the unit direct stress 
 and unit horizontal shear. 
 
 If the flange and web are of same thickness, the shear 
 and direct stress, at A and B, Figure 4, have nearly the same 
 
STRESS LINES IN BEAMS. 
 
 Q 
 
 C 
 
 01 
 
lO BROOMALL : 
 
 values. The maximum stress lines in the web would, there- 
 fore, cut the junction of web and flange at nearly the same 
 angles as the corresponding lines in the flange. As commonly 
 the web is thinner than the flange, it results that the unit 
 longitudinal shear on plane B is greater than on plane A, 
 which means a different angle of cutting. A consideration 
 of Formulas (i) and (2) will show whether the lines in the 
 web will make a greater or less angle with the longitudinal 
 direction than the corresponding lines in the flanges. 
 
 In case the Channel is used the other side up, that is, 
 flanges directed downwards, the same line of reasoning may 
 be used and lines corresponding to those already shown may 
 be easily sketched. We will leave it to the fertile imagination 
 of the reader to picture them. 
 
 Another common unsymmetrical cross section to examine 
 is that of the Angle Beam. We will consider it in one posi- 
 tion only, for example, horizontal leg on top, and vertical leg 
 directed downwards, referring to same as flange and web 
 respectively. Since the Angle is to all intents nothing more 
 than part of a Channel, the nature of its stresses may be 
 easily indicated by reference to the previous figures. In the 
 Angle Beam the shearing planes will be similar to those in the 
 web and upper half flange of the I Beam or in the web and 
 upper flange of Channel Beam on edge. The maximum 
 longitudinal shear will occur at the neutral axis, which of 
 course passes through the centre of gravity of the section. 
 
 The lines of maximum stress in the web of the Angle will 
 resemble those of the flanges of the Channel Beam laid flat. 
 Figure 3 inverted and with stresses appropriately reversed will 
 indicate them very well. 
 
 In the flange, looked at from above, the lines can be eas- 
 ily shown to have the characteristics of the corresponding 
 lines in the upper flange of the Channel Beam on edge, 
 Figure 2, upper sketch. 
 
 In beams of thin cross section with the parts vertically 
 and horizontally disposed, it is not difficult, as we have seen, 
 
STRESS LINES IN BEAMS. 
 
 to determine the characteristics of the lines of maximum 
 internal stress. In the foregoing the underlying principles 
 have been sufficiently developed to clearly exemplify the 
 method of treatment. In the case of thick cross sections, 
 other than the rectangular, the matter is by no means so 
 simple. In a subsequent article it is expected to treat of cir- 
 cular, triangular and other cross sections. 
 
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