# Theoretical Design Strategies, Strengths, Costs, and Environmental Impacts of Triple Composite Beams Utilizing Glass Compressive Reinforcement

^{*}

## Abstract

**:**

## 1. Introduction

## 2. An Explanation of the Potential Economics and Environmental Improvements of Glass Reinforcement

- The theoretical glass strength was set to 1000 MPa, the as-tested glass strength was 217 MPa (based on our research), and the concrete strength was assumed to be 27.6 MPa (4 ksi, an industry-standard concrete strength).

_{ult}is the ultimate strength, which may be substituted with the yield (σ

_{y}) for a given application, cm

_{i}is the cost to mass (essentially the unit price) for a product, and asc

_{i}is the axial strength to cost index.

_{2}generated. These environmental qualities are of increasing importance. Comparisons should be made on a per strength basis, as it allows for a more realistic real environmental impact for each material. The testing of glass produced in preliminary testing has indicated an embodied energy of 92% that of concrete, while the theoretical strength of glass would allow a reduction of 80% of the embodied energy. The advantages regarding CO

_{2}developed indicate that in order to reduce CO

_{2}generated, the glass is required to be recycled or near the theoretical strength. Virgin glass with the reduced weight does pose a disadvantage by producing quadruple the CO

_{2}generated by an equivalent strength component made of concrete.

- The embodied energies for concrete and glass were 1.9 MJ/kg and 12.7 MJ/kg, respectively [27].
- The kilograms of CO
_{2}generated per kilogram of the material was 0.15 for concrete, 4.4 for non-recycled (virgin) glass, and 0.73 for recycled glass [28].

## 3. Stress Equations for Triple Composite Beams Using Euler–Bernoulli Beam Theory and the Transformed Sections Method

#### 3.1. Binderless Beams

_{b}is the distance from the neutral axis to the bottom reinforcement, y

_{t}is the distance from the neutral axis to the top reinforcement, At is the area of the top reinforcement, n

_{t/b}is the ratio of Young’s modulus of the top reinforcement to the binder, d is the distance between the top and bottom reinforcement (i.e., depth), A

_{b}is the area of the bottom reinforcement, n

_{b/b}is Young’s modulus ratio of the bottom reinforcement to the binder, I* is the second moment of area of the transformed section, M is the applied bending moment, σ

_{r,t,x}is the normal stress of the top reinforcement in the x-direction, σ

_{b,t,x}is the normal stress of the top binder in the x-direction, σ

_{r,b,x}is the normal stress of the bottom reinforcement in the x-direction, and σ

_{b,b,x}is the normal stress of the binder at the bottom in the x-direction. While the binder is not considered to contribute to the second moment of area, it can still have maximum stress calculated which will be larger than the maximum stress in analyses that consider the contributions of the binder. The advantage of this arrangement of equations is that the method of designing a beam with a different top and bottom reinforcement can be made significantly easier using this set of equations.

#### 3.2. Compression Only Binder Beams

_{c}is the top cover distance. Note that Equations (6) and (8) also apply in this section, though (7) has been modified to include a top cover (see (14) above) and (9) is neglected. There is no calculation for the binder tension on the bottom half of the beam, as it is assumed that the binder is a ceramic material and fails immediately upon the development of tension (an assumption often used in the concrete analysis, though concrete does have a minimal tensile strength).

#### 3.3. Beams Considering Full Binder

## 4. Finite Element Modeling for Triple Composite Beams

## 5. Establishment of the Flexural Reinforcement Utilization Factor

^{3}, and a strength of 42 MPa, the ratio becomes:

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Quagliarini, E.; Lenci, S. The influence of natural stabilizers and natural fibres on the mechanical properties of ancient Roman adobe bricks. J. Cult. Herit.
**2010**, 11, 309–314. [Google Scholar] [CrossRef] - Kalpakjian, S.; Schmid, S.R. Manufacturing Processes for Engineering Materials, 6th ed.; Pearson: London, UK, 2016. [Google Scholar]
- Mazumdar, S. Composites Manufacturing: Materials, Product, and Process Engineering; CrC Press: Boca Raton, FL, USA, 2001. [Google Scholar]
- Scala, E.P. A brief history of composites in the US—the dream and the success. JOM
**1996**, 48, 45–48. [Google Scholar] [CrossRef] - Gibson, R.F. Principles of Composite Material Mechanics; CRC Press: London, UK, 2016. [Google Scholar]
- Bush, S.; Ashby, M.; Swindells, N.; Bullough, R.; Ellison, G.; Lindblom, Y.; Cahn, R.; Barnes, J. Technology of the 1990s: Advanced Materials and Predictive Design: Discussion. Philos. Trans. R. Soc. A
**1987**, 322, 404–407. [Google Scholar] - Ramani, K.; Moriarty, B. Thermoplastic bonding to metals via injection molding for macro-composite manufacture. Polym. Eng. Sci.
**1998**, 38, 870–877. [Google Scholar] [CrossRef] - Hamilton, R.; Tennyson, S.; Hamilton, W. Analysis by the transformed-section method. In Proceedings of the 2001 American Society for Engineering Education Annual Conference & Exposition, Albuquerque, NM, USA, 24–27 June 2001. [Google Scholar]
- Bos, F.; Louter, C.; Veer, F. Challenging Glass: Conference on Architectural and Structural Applications of Glass, Faculty of Architecture, Delft University of Technology, May 2008; IOS Press: Amsterdam, The Netherlands, 2008. [Google Scholar]
- Northolt, M.G. Compressive strength and glass transition temperature. J. Mater. Sci.
**1981**, 16, 2025–2028. [Google Scholar] [CrossRef] - Chojnacki, J.T.; Chen, W.W. Mechanical Response of Borosilicate and Soda-Lime Glass Under Dynamic Triaxial Compression. JDBM
**2016**, 2, 251–258. [Google Scholar] [CrossRef] [Green Version] - Benayoune, A.; Samad, A.A.; Trikha, D.; Ali, A.A.; Ellinna, S. Flexural behaviour of pre-cast concrete sandwich composite panel–experimental and theoretical investigations. Constr. Build. Mater.
**2008**, 22, 580–592. [Google Scholar] [CrossRef] - Liew, J.R.; Wang, T. Novel steel-concrete-steel sandwich composite plates subject to impact and blast load. Adv. Struct. Eng.
**2011**, 14, 673–687. [Google Scholar] [CrossRef] - Khoddam, S.; Tian, L.; Sapanathan, T.; Hodgson, P.D.; Zarei-Hanzaki, A. Latest developments in modeling and characterization of joining metal based hybrid materials. Adv. Eng. Mater.
**2018**, 20, 1800048. [Google Scholar] [CrossRef] - Louter, P. Adhesively bonded reinforced glass beams. HERON
**2007**, 52, 31. [Google Scholar] - Louter, C.; Cupać, J.; Lebet, J.-P. Exploratory experimental investigations on post-tensioned structural glass beams. J. Facade Des. Eng.
**2014**, 2, 3–18. [Google Scholar] [CrossRef] [Green Version] - Bedon, C.; Louter, C. Finite-element numerical simulation of the bending performance of post-tensioned structural glass beams with adhesively bonded cfrp tendons. Am. J. Eng. Appl. Sci.
**2016**. [Google Scholar] [CrossRef] [Green Version] - Santarsiero, M.; Louter, C.; Nussbaumer, A. Laminated connections for structural glass applications under shear loading at different temperatures and strain rates. Constr. Build. Mater.
**2016**, 128, 214–237. [Google Scholar] [CrossRef] [Green Version] - Le Bourhis, E. Glass: Mechanics and Technology; John Wiley & Sons: Hoboken, NJ, USA, 2014. [Google Scholar]
- Ma, H.; Xu, J.; Ma, E. Mg-based bulk metallic glass composites with plasticity and high strength. Appl. Phys. Lett.
**2003**, 83, 2793–2795. [Google Scholar] [CrossRef] - Bruck, H.; Christman, T.; Rosakis, A.; Johnson, W. Quasi-static constitutive behavior of Zr41. 25Ti13. 75Ni10Cu12. 5Be22. 5 bulk amorphous alloys. Scr. Metall. Mater.
**1994**, 30, 429–434. [Google Scholar] [CrossRef] - Bruck, H.A.; Rosakis, A.J.; Johnson, W.L. The dynamic compressive behavior of beryllium bearing bulk metallic glasses. J Mater. Res.
**1996**, 11, 503–511. [Google Scholar] [CrossRef] [Green Version] - Ashby, M.F. Materials Selection in Mechanical Design; Elsevier Butterworth-Heinemann: Oxford, UK, 2005. [Google Scholar]
- Available online: https://www.alibaba.com/trade/search?fsb=y&IndexArea=product_en&CatId=&SearchText=steel+prestressed+cable (accessed on 21 November 2019).
- Available online: https://www.alibaba.com/trade/search?IndexArea=product_en&CatId=&fsb=y&viewtype=&tab=&SearchText=recycled+glass+cullet (accessed on 21 November 2019).
- Available online: https://www.alibaba.com/products/soda_lime_glass_sheet.html?IndexArea=product_en&sort_type=TRALV (accessed on 21 November 2019).
- Available online: http://www.yourhome.gov.au/materials/embodied-energy (accessed on 22 February 2020).
- Available online: https://winnipeg.ca/finance/findata/matmgt/documents/2012/682-2012/682-2012_Appendix_H-WSTP_South_End_Plant_Process_Selection_Report/Appendix%207.pdf (accessed on 22 February 2020).
- Hibbeler, R.C. Mechanics of Materials; Prentice Hall: Upper New Jersey River, NJ, USA, 2008. [Google Scholar]
- Available online: https://www.engineeringtoolbox.com/young-modulus-d_417.html (accessed on 21 November 2019).
- Available online: http://valleydesign.com/soda-lime-glass.htm (accessed on 21 November 2019).
- ACI. Building Code Requirements for Structural Concrete (ACI 318-08) and Commentary; ACI: Rome, Italy, 2008. [Google Scholar]

**Figure 1.**The types of composites: (

**a**) Fiber/particle reinforcement distributed throughout the interior of a component. (

**b**) Laminate composites, which alternate different materials between layers. (

**c**) Sandwich composites. (

**d**) Rod composites; these are representative of reinforced concrete.

**Figure 2.**A comparison of the cross-section of different composite materials when transformed using the transformed sections method with Euler–Bernoulli beam theory. (

**a**) The distributed particles sum to a increase in width in the transformed (right-side) section. (

**b**) With sandwich composites, the stiffer material is placed at the extreme top and bottom, allowing the contributed area to increase the width at the extremities. (

**c**) Rod composites have top and bottom cover included to allow for reinforcement protection.

**Figure 3.**As material moves from being a homogenized material to an increasingly complex composite, several changes are made. While the strength becomes more specialized so too does the cost of construction per additional compositing item added. This, however, is offset by decreasing material costs per strength. The relative increases in specialized strength for beams greatly outweighs the disadvantage of the increased labor costs. An additional example is fiber-reinforced concrete; while it does have superior general strength, it cannot outperform steel-reinforced concrete for flexural requirements due to the advantages of locating the reinforcement at the extremities of a beam section.

**Figure 4.**A typical post-tensioned glass composite is shown. These are constructed so that the view through the glass is minimally obstructed while providing increased load capacity to the glass beam through the post-tensioning of the steel rods.

**Figure 5.**(

**a**) The beams considered in this study, which feature a purposefully wide section to limit alternate failure modes. (

**b**) An alternate design featuring rod-type reinforcement at the top and bottom. (

**c**) Another alternative, but a sheet of glass is placed as top reinforcement.

**Figure 6.**A summary of the binder models considered. (

**a**) The general arrangement of the beams considered. (

**b**) The binderless model, where the reinforcement is converted to binder-material; due to the differential Young’s moduli between the materials. (

**c**) The compression-only binder model, which is commonly used for the concrete design. (

**d**) In cases where the binder still contributes significantly to the beam’s flexural stiffness, the full binder model may need to be used. Note the dark gray regions at the top (c and d) and bottom (d), which denotes cover regions. These small components contribute little strength, with the true intention being protection for the reinforcement.

**Figure 7.**The dimensions utilized for the FEM of the triple composite beams. (

**a**) Side view showing the length of the beam. (

**b**) The configuration used for rod-type reinforced models. (

**c**) The configuration used for double-plate reinforced beams. Note that the reinforcement spacing was maintained between cross-sections (b and c) to provide comparable stresses between the two cross-sections.

**Figure 9.**A section cut near the center of the beam for the double-plate configuration with glass reinforcement in the top plate and steel in the bottom. Probe values are shown in MPa.

**Figure 10.**A section cut near the center of the beam for the double-rod configuration with glass reinforcement in the top plate and steel in the bottom. Probe values are shown in MPa.

**Table 1.**The cost of materials required to resist 100 Mn applied to a variable cross-sectional area (i.e., axially loaded) and at a unit length of 1 m. The sections are divided by the loading types for the given materials so as not to directly compare compression-only to other material types.

Material | Density (kg/m^{3}) | Mat. Str. (MPa) | Length (m) | Load (Mn) | Area Req. (m^{2}) | Vol. Req. (m^{3}) | Weight Req. (kg) | Est. Cost Per Kg | Cost Each |
---|---|---|---|---|---|---|---|---|---|

Compression Only Materials | |||||||||

Concrete, 4000 psi or 27.6 MPa | 2324 | 27.6 | 1 | 100 | 3.62 | 3.62 | 8420 | $0.03 | $219 |

Glass, Max. Theoretical, Virgin Glass | 2520 | 1000 | 1 | 100 | 0.1 | 0.1 | 252 | $0.17 | $42.8 |

Glass, Max. Theoretical, Recycled Glass | 2520 | 1000 | 1 | 100 | 0.1 | 0.1 | 252 | $0.18 | $45.4 |

Glass, As Tested Thus Far, Virgin Glass | 2520 | 217 | 1 | 100 | 0.461 | 0.461 | 1160 | $0.17 | $197 |

Glass, As Tested Thus Far, Recycled Glass | 2520 | 217 | 1 | 100 | 0.461 | 0.461 | 1160 | $0.18 | $209 |

Compression and Tension | |||||||||

HDPE, Typical Strength, Recycled and Virgin | 950 | 22 | 1 | 100 | 4.55 | 4.545 | 4320 | $0.25 | $1079 |

Steel, Mild A36 | 7800 | 400 | 1 | 100 | 0.403 | 0.403 | 3145 | $0.45 | $878 |

Tension Only | |||||||||

Steel, Prestressed Cable | 7800 | 1770 | 1 | 100 | 0.0565 | 0.0565 | 441 | $0.45 | $198 |

**Table 2.**The required weight from Table 1, for the compression only materials, is used to compare the transportation and mining requirements for each material. This is done to show that glass, having a higher strength to mass, greatly outperforms concrete. This would help to reduce the amount of greenhouse gas emissions developed during the transportation of materials.

Material | Compressive Strength (MPa) | Weight Req. (kg) | Weight of Materials Transported to Site as % of Final Weight | Total Weight of Transported Materials (kg) | Transported Material Weight as % Weight of Concrete |
---|---|---|---|---|---|

Concrete, Typical Strength | 27.60 | 8420.29 | 145% | 12,209.42 | 100.00% |

Glass, Max. Theoretical, Virgin Glass | 1000.00 | 252.00 | 100% | 252.00 | 2.06% |

Glass, Max. Theoretical, Recycled Glass | 1000.00 | 252.00 | 100% | 252.00 | 2.06% |

Glass, As Tested Strength, Virgin Glass | 217.00 | 1161.29 | 100% | 1161.29 | 9.51% |

Glass, As Tested Strength, Recycled Glass | 217.00 | 1161.29 | 100% | 1161.29 | 9.51% |

**Table 3.**The embodied energy (an estimate of the energy to produce a material, which correlates with fossil fuel usage during the manufacturing of the material) and CO

_{2}generated for concrete and glasses are shown. Glass outperforms concrete in certain situations.

Material | Compressive Strength (MPa) | Weight Req. (kg) | Embodied Energy (MJ/kg) | Req. Energy (MJ) | % Req. Energy of Concrete | CO_{2} Generated (kg CO_{2}/kg) | CO_{2} Generated (kg) | % CO_{2} Generated Compared to Concrete |
---|---|---|---|---|---|---|---|---|

Concrete, Typical Strength | 27.60 | 8420.29 | 1.90 | 15,998.55 | 100.00% | 0.15 | 1263.04 | 100.00% |

Glass, Max. Theoretical, Virgin Glass | 1000.00 | 252.00 | 12.70 | 3200.40 | 20.00% | 4.40 | 1108.80 | 87.79% |

Glass, Max. Theoretical, Recycled Glass | 1000.00 | 252.00 | 12.70 | 3200.40 | 20.00% | 0.73 | 183.96 | 14.56% |

Glass, As Tested Strength, Virgin Glass | 217.00 | 1161.29 | 12.70 | 14748.39 | 92.19% | 4.40 | 5109.68 | 404.55% |

Glass, As Tested Strength, Recycled Glass | 217.00 | 1161.29 | 12.70 | 14748.39 | 92.19% | 0.73 | 847.74 | 67.12% |

**Table 4.**The FEM stresses are compared against all-binder and no binder assumptions. The percent error for both against the FEM results is also presented.

Summary of Calculations | Calculation Method | Percent Error | ||||
---|---|---|---|---|---|---|

System | Measurement | FEM | No Binder | All-Binder | No Binder | All-Binder |

SHS-R | σ_{s,t,avg} | −33.8 | −33.1 | −30.3 | −2% | −10% |

σ_{h,t,max} | −0.20 | −0.13 | −0.15 | −37% | −27% | |

σ_{s,b,avg} | 33.8 | 33.1 | 30.3 | −2% | −10% | |

σ_{h,b,max} | 0.19 | 0.13 | 0.15 | −34% | −23% | |

SHS-DP | σ_{s,t,avg} | −35.1 | −33.1 | −31.4 | −6% | −10% |

σ_{h,t,max} | −0.20 | −0.13 | −0.13 | −37% | −37% | |

σ_{s,b,avg} | 35.1 | 33.1 | 31.4 | −6% | −10% | |

σ_{h,b,max} | 0.21 | 0.13 | 0.13 | −39% | −39% | |

GHS-R | σ_{s,t,avg} | −25.5 | −33.1 | −23.4 | 30% | −8% |

σ_{h,t,max} | −0.40 | −0.38 | −0.31 | −5% | −24% | |

σ_{s,b,avg} | 35.3 | 33.1 | 30.4 | −6% | −14% | |

σ_{h,b,max} | 0.20 | 0.13 | 0.16 | −34% | −20% | |

GHS-DP | σ_{s,t,avg} | −28.2 | −33.1 | −26.7 | 18% | −5% |

σ_{h,t,max} | −0.40 | −0.38 | −0.31 | −4% | −23% | |

σ_{s,b,avg} | 37.5 | 33.1 | 33.6 | −12% | −10% | |

σ_{h,b,max} | 0.16 | 0.13 | 0.13 | −15% | −13% |

**Table 5.**A select subset of Table 4 in which the differences in reinforcement stress can be better comprehended for comparison purposes.

Summary of Calculations | Percent Error | ||
---|---|---|---|

System | Measurement | No Binder | All-Binder |

SHS-R | σ_{s,t,avg} | −2.09% | −10.35% |

σ_{s,b,avg} | −2.09% | −10.35% | |

SHS-DP | σ_{s,t,avg} | −5.59% | −10.37% |

σ_{s,b,avg} | −5.59% | −10.37% | |

GHS-R | σ_{g,t,avg} | 29.84% | −8.39% |

σ_{s,b,avg} | −6.34% | −14.02% | |

GHS-DP | σ_{g,t,avg} | 17.55% | −5.07% |

σ_{s,b,avg} | −11.78% | −10.34% | |

Average | 1.74% | −9.91% | |

Standard Deviation | 14% | 2% |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Cotter, J.; Guldiken, R.
Theoretical Design Strategies, Strengths, Costs, and Environmental Impacts of Triple Composite Beams Utilizing Glass Compressive Reinforcement. *J. Compos. Sci.* **2020**, *4*, 22.
https://doi.org/10.3390/jcs4010022

**AMA Style**

Cotter J, Guldiken R.
Theoretical Design Strategies, Strengths, Costs, and Environmental Impacts of Triple Composite Beams Utilizing Glass Compressive Reinforcement. *Journal of Composites Science*. 2020; 4(1):22.
https://doi.org/10.3390/jcs4010022

**Chicago/Turabian Style**

Cotter, John, and Rasim Guldiken.
2020. "Theoretical Design Strategies, Strengths, Costs, and Environmental Impacts of Triple Composite Beams Utilizing Glass Compressive Reinforcement" *Journal of Composites Science* 4, no. 1: 22.
https://doi.org/10.3390/jcs4010022