M the maximum bending moment that can be resisted by the beam (design chart) With the elastic bending strength of the drilled section. Of the five beams with pairs of spans, three had closely spaced spans (1.5 times the center-to-center diameter) and two had wide spans (twice the center-to-center diameter).
Introduction
Use of cellular beams in practice
- Development
The circular openings in cellular beams are more aesthetically pleasing than the hexagonal openings in castellated beams. Eight full-scale cellular beams were tested for destruction and failure loads, failure modes and deflections were recorded.
Previous research
The slenderness of the web post is the ratio of the diameter of the opening to the width of the web post. The fracture was a completely plastic Vierendeel fracture in one of the openings next to the cargo. At a load of 107 kN, the flange in the center of the beam started to buckle.
Figure 2.21 shows a slight deformation of the opening, indicating that Vierendeel failure was imminent. The beam can be analyzed elastically or plastically using the properties of the perforated section. Two assumptions are made to make the structure statically determined. i) The vertical shear force is evenly distributed between the upper and lower tees. (ii) The counter-flection points are located in the center of the tees.
Two assumptions are made viz. i) Shear and bending stresses interact according to the Von Mises yield criterion. Two sections of the beam are checked for shear strength, ie. the center of the opening for vertical cuts and the narrowest part of the web post for horizontal cuts. The theoretical failure load ratios and experimental values are given in Table 3.2:.
For all beams loaded in the middle, Vierendeel failure was predicted in the openings on either side of the load, ie. openings are subjected to combined maximum shear and moment. The vertical axis relates the actual bending m at a section to the ultimate bending moment that the section can develop M. The horizontal axis relates the actual shear force q at a section to the ultimate shear force that the section can develop Q. The ultimate shear strength is that which can resisted if the entire stress capacity of the punched section is used to resist the shear.
Introduction
Test beam details
The ratio of the opening distance to the opening diameter determines the width of the web post and thus the tendency of the beam to fail in web buckling or shear. The spans were chosen to provide a wide variation in the shear-to-moment ratio at the center of the beam while avoiding web buckling or web undercutting.
Fabrication of beams and test rig
- Beams
- Beam preparation and instrumentation
At a load of 151 kN, the beam continued to buckle under a constant load and this load was taken as the failure load. Too few weights were used and at a load of 102 kN the girders were pushed aside due to girder deflection.
Summary
Introduction
Potential modes of failure
- Pure Bending
The stresses caused by the Vierendeel moments are calculated using the section properties of the T-pieces. The primary bending stresses on the critical sections are calculated based on the properties of the perforated profile.
Proposed new methods of analysis
- Plastic analysis
- The Computer Program
- The Spreadsheet
- Design charts
- Finite Element Analysis
On the low moment side of the tee section, the primary block of bending stress is located on the outer threads of the pattern; The available voltage is greater than the requirements and the value of F2 (and when necessary F3) is reduced accordingly. The upper limit for the input force was taken as the smallest of the following three values:
The root-finding technique used here is the same straight-line approach used in the analysis portion of the program. The dominant force in the beam in the first stage of the design map, from mIM = 1 to the horizontal shear limit, is the bending moment. The beam will fail due to horizontal shear of the web when the beam curve touches the horizontal shear failure line.
This is in contrast to the spreadsheet results for beams loaded at the third points where pure bending failure was predicted in the middle third of the beam. The symmetry of the beams was exploited and only one half of the beam was modelled. The initial linear elastic analysis gave a good indication of the load at which the beam was likely to yield.
Summary
This can be explained by the fact that Beam 4B failed to buckle before the plastic fracture was fully developed. Two sets of means and standard deviations were calculated, one including Beam 4B and the other excluding. If the curve is flatter, it indicates a greater spread of values, and even if it is biased towards unity, each individual ratio may not be close to unity.
The distributions in Figure 3.61 show that the SCI method is very conservative, while the plastic quadrilateral spreadsheet method and the FEA method are both unconservative. The distributions for the SCI, spreadsheet and graph method are more "peaked" when the ratios for Beam 4B are excluded. It can be clearly seen from Figure 3.62 that the plastic four-part design chart method is the most accurate.
The distribution for the chart is the most "peaked" and is centered closest to unity.
Introduction
Existing methods of analysis
- British Steel Construction Institute Method
- Simplified British Steel Construction Institute method
Virtual work is the basis for the SeI deflection method. The deflection at a point in the beam is found by applying a unit load at that point. This method takes into account five components of deflection: iii) Axial force in the tee. iv) Shear in the tee (vertical shear). v) Shear in the web post (horizontal shear). The original SeI method of adding deflections assumed that the forces acting on the free body and causing the deflections were the forces to the left of the free body (Figure 4.
A new method of summing deflections was therefore developed where the forces causing deflections were assumed to be those at the middle of the section (Figure. These ratios were calculated for each of the three positions at which deflections were measured. The average of the three ratios for each beam has been found and these are given in the chart below.
The deflection results for Test Beams 2A and 3B are shown in Figures 4.7 and 4.8 respectively, and the deflection ratios for all test beams are shown in Figure 4.9.
Proposed new methods
- Vierendeel Method
- Finite Element Analysis
This includes the tee that spans half the opening and a quarter of the adjacent track post, which forms a cantilevered section that can be approximated as a stepped tee (Figure 4.10). The total Vierendeel deflection was the sum of the Vierendeel deflections at all openings between the support and the position where the deflection was calculated. The total deflection was the sum of the primary deflection deflection and the Vierendeel deflection.
The results for test beams 2A and 3B are shown in Figures 4.11 and 4.12, respectively; and bending ratios for all test members are shown in Figure 4.13. Figure 4.13 shows that this is true for all beams except beam 1A. One output of a finite element analysis program is the displacements at each nodal point.
This is seen to be the case for all the beams except beams 1A and 3A in Figure 4.17.
Summary
Deflections calculated with this method in beams with a small shear-to-moment ratio are larger than the experimental ones and in beams with a large shear-to-moment ratio are smaller than the experimental ones, although the average of the deflection ratios close to unity are . This can be seen from the scatterplot by the lack of clustering of deflection ratios around unity. The deflection ratios of the Four-part method are clustered around unity for the lower shear-to-moment ratios.
As with the other methods, the deflection ratios decrease with increasing shear-moment ratios, but the agreement between the theoretical and experimental values is good. This scatterplot shows a clustering of deflection ratios around unity, as well as a decrease in deflection ratios with an increase in shear-moment ratios. All these observations can be summarized by plotting normal distributions for the deflection ratios.
All deflection ratios were multiplied by the reciprocal of their mean, and the normal distributions were reslotted, as in Figure 4.23.
Introduction
The design charts predicted the Vierendeel error for all test beams, which was correct for beams 1, 2, and 3 for the midspan and third point load cases. Beam 4A failed due to pure bending, and it can be seen from the design diagrams in Section 3.4.1.3 that the beam line lies closer to the moment axis than the beam line for the other test beams. Design charts are quick and easy to use and can be used with any load case, unlike a spreadsheet.
This method would be impractical for use in the design office as it is very time consuming. The computer program and the spreadsheet are designed to handle simply supported beams with point loads at midspan or at the third points. The design charts, which are also based on the plastic Four-Part method, can be used for any combination of internal shear and bending moment.
This enables them to be used for beams with different end conditions, continuous beams or beams with loads other than mid-span or third point loads.
Deflections
Summary
The plastic Four-Part design chart method has been found to be the most accurate and practical method for calculating failures and predicting failure modes. The Four Part Deflection method has been found to be the most accurate and practical method to calculate deflections.
Future Work
Journal of the Structural Division, Proceedings of the American Society of Civil Engineers, Januarie 1968, pp 1-17.
