Comparing the first failure of the cross-sections of the parabolic and

One of the reasons for searching for an optimized shape of the arch is to establish greater resistance. In other words, a favourable arch structure is one that carries a specified design load case or cases for the minimum weight of the construction material. Concerning other applications of the momentless arch, the first failure of the section of the momentless arch is evaluated with the first failure of the section of the parabolic arch. The first failure of arch cross-section is emerged under ultimate load beyond the initial yield.

For the first step of finding the failure of these arches, the OpenSees program (http://OpenSees.berkeley.edu/) was used. OpenSees is an interactive Tool Command Language (Tcl) software framework in which the commands can be changed at any time and performed at the MS-DOS/Unix prompt. This program works in parallel with MATLAB as a post-processing tool generated by means of Tcl

scripting language. Each Tcl command is bound with a C++ procedure. Therefore, this finite element analysis is executed to simulate the response of the arch structure to the applied loading. Both parabolic and momentless arches are modelled using straight elements, which are defined as displacement-based beam- column elements. Since this programme performs as displacement control, the displacement increased by 1 mm at the crown of the arch. As a result, the required structural responses are achieved at each displacement increment using

OpenSees. The first input to the OpenSees program is the nodes that define the arch curve. Parabolic and momentless arch shapes are established by the positions of x-y nodes, as described in subsection 5.6.5. The other input is the material where, arches are assumed to be reinforced concrete. The area of the reinforcement for arch cross-section is assumed to be 193396 mm2 that gives

comparison reason having the same section across arch length. However, the momentless arch doesn’t require any reinforcement. Having set the pin as arch constraints in the program, the third input is the nodal masses. The nodal masses are produced by the summation of nodal weight caused by SW and UDL at each individual node. The final step is the introduction of the desirable outputs that involve reactions, displacements, all forces, and bending moments. Furthermore, a final vertical displacement of 10000 mm at the crown of the parabolic and momentless arches is chosen in the OpenSees programme. The MATLAB file for this analysis can be run for a targeted vertical displacement at the arch crown. Finally, the reactions, forces, and bending moments at the nodes are obtained at each displacement for parabolic and momentless arches.

The second step of finding the first failure of the arch cross-section is accomplished by using an interaction curve between the axial forces and bending moments. Hence, the obtained axial forces and bending moments from OpenSees are plotted. This diagram is plotted for each displacement. To do so, a programme was written in Excel as “Visual Basic” that only requires a vertical displacement at the arch

crown as input and gives a diagram of the axial force and bending moment for that displacement. The interaction curve of the arch cross-section can be plotted based on finding some points on the curve such as, the points relevant to the squash load, decompression, balance, and pure moment (EN1992-1-1: 2004). These points are calculated using dimension, reinforcement, and material properties of the cross- section for each arch. The generated interaction curve of each arch is then plotted on the same diagram of axial force and bending moments. To have a safe cross- section, the diagram of axial forces and bending moments of the arch must be under the interaction curve of the arch cross-section. Therefore, the displacement at which the diagram of axial forces and bending moments from OpenSees crosses the interaction curve of the cross-section is considered as the first failure of the

cross-section. The parabolic arch shows the first failure of the cross-section when the vertical displacement at the arch crown is 3950 mm; see Figure 5.14. Meanwhile, from Figure 5.15, the first failure of the cross-section of the momentless arch is seen for a vertical displacement of 5200 mm at the arch crown.

Figure 5.14. Comparison of the interaction curve of the parabolic arch cross-section and diagram of the axial force and bending moment results from OpenSees when the vertical displacement at the arch crown is 3950 mm

Figure 5.15. Comparison of the interaction curve of the momentless arch cross- section and diagram of axial force and bending moment results from OpenSees

The green, purple, and blue straight lines in Figures 5.14 and 5.15 connect the minimum eccentricity, decompression, and nominal balance points to the origin. The minimum eccentricity point shows the maximum moment strength at the maximum axial compression load permitted by Eurocode. The decompression point shows compression and moment strength at zero strain in the tension side reinforcement. The nominal balance point presents compression and moment strength at 50% strain in the tension side reinforcement. The interaction curve crosses the axial force axis at the squash load point that gives the axial compression at zero moment. The interaction curve also crosses the bending moment axis at pure moment point when the axial force is zero.

Although the interaction curve of the parabolic arch is similar to that of the momentless arch, the axial force and bending moment diagrams from OpenSees

are completely different for these two arches. According to the OpenSees results, the total load that causes first failure of the parabolic arch is 211992 kN. First failure of the momentless arch is initiated at 221722 kN. Comparing the first failures of the parabolic and momentless arches, the momentless arch could carry 5% higher load before reaching the first cross-sectional failure.

5.7.

Concluding Remarks

An analytical form-finding procedure that can optimize shape is used to find the shape of a momentless two-pin arch. The shape of the momentless form has been investigated using equilibrium equations from Brew (2013) for a pin-ended arch with constant cross-section. The geometry of the momentless arch is then obtained using the equation of zero shear force having axial force only for any L:h ratio. A functional relationship is found that represents the x and y coordinates individually for the three categories of loading, that is, UDL:SW < 1, UDL:SW = 1, and UDL:SW

solved by iterative method using the boundary condition. The parabolic arch function is used to calculate the initial value in the case of UDL:SW > 1. Meanwhile, the initial value in the iterative procedure of finding the x and y coordinates is chosen from the equation of a catenary arch when UDL:SW < 1. Consequently, the geometry of the optimal arch is given as a function of loading for any L:h ratio and any ratio of UDL:SW via the analytical form-finding technique. However, the effect of arch shape is more tangible for arches of L:h≤5.

The application of the momentless arch was shown by comparing a momentless arch and a parabolic one in a theoretical case study. This study was carried out for

UDL:SW > 1, when the parabolic arch is the best known arch shape. The momentless arch was compared with the parabolic arch, both made of concrete which is relatively weak in tension subjected to the same loading for L:h=3.

Case study shows that the maximum vertical displacement of momentless arch to be almost half of the maximum vertical displacement of the parabolic arch for a similar load case. To have the same maximum displacement, the thickness of the momentless arch could be reduced to half of its initial value. This leads to mass reduction in the case of momentless arch down to about half on its initial mass. From case study, momentless arch could also carry 5% greater load before reaching the first cross-sectional failure compared to the parabolic form under the same conditions. The momentless arch shows this first failure for 24% higher vertical displacement at the arch crown compared to the parabolic one.

Overall, the comparison confirmed the superiority of using the momentless arch in practice which demonstrates the significant effect of shape on structural response.