Two pinned arch experiment.

Student Name: xxxx

Title: xxx

Abstract

An arch is a structural member that transmits forces and stresses purely by compression [ CITATION Cli11 l 1033 ]. By the resolving the external forces it is subjected to into compressive stresses, and in turn eliminating tensile stresses, arches have found common application in structures requiring long spans, most notably bridges. This process of resolving forces to compressive stresses is referred to as arch action[ CITATION Vai04 l 1033 ].

When a load is applied at the top of the arch, it pushes outward at the base, referred to as thrust. As the load at the top increases, the arch tends to flatten out, resulting in more thrust at the bases. At this moment, all the fibres in a true arch are under compression. In order to ensure the arch does not flatten completely, or collapse, there is need for restraints. This can be done with the help of internal ties joining the ends of the arch or external bracings at the supports, usually in the form of abutments [ CITATION Amb12 l 1033 ].

There are various types of arches that have been used for a long time in the construction industry. The
fixed arch is constructed using reinforced concrete for use in bridges and tunnels with small spans due to the buildup of internal stresses temperature differences. This makes the fixed arch statically indeterminate. The two- hinged arch has pinned connections at the base, allowing the structure to rotate freely to compensate for the internal thermal stresses. This allows two- hinged arch to support longer spans. The free movement of the structure could lead to development of internal stresses which makes the two- hinged arch statically indeterminate [ CITATION Rey08 l 1033 ]. The three- hinged arch has an additional hinge at its mid-span, which allows for movement in two directions to compensate for any internal stresses. This makes the arch statically determinate, able to support medium spanning structures, such as roofs for large buildings [CITATION Lue16 l 1033 ]. This report focusses on the load distribution of a two- hinged arch subjected to a 500g load at incremental distances from the left support.

Key words: arch, statically indeterminate, load distribution

Table of Contents

iAbstract

iiList of Tables

iiList of Equations

iiList of Figures

1Introduction 1.

1Material and Methodology 2.

12.1 Material

22.2 Methodology

3Results and Discussion 3.

33.1 Reactions

63.2 Influence value

11Conclusion 4.

11References 5.

List of Tables

3Table 3‑1: Comparison between Experimental and Theoretical Values for Horizontal Reaction

3Table 3‑2: Theoretical vertical reaction with changing load position

6Table 3‑3: Relationship between load position and influence value

List of Equations

2Equation 2‑1: Theoretical Relationship between Position of Load and Reaction

6Equation 3‑1: Influence value equation

List of Figures

1Figure 2‑1: TecQuipment(R) two hinged apparatus

4Figure 3‑2: Relationship between Load Position and Horizontal Reaction

5Figure 3‑3: Relationship between Load Position and Vertical Reaction at Support B

7Figure 3‑4: Influence value for the load at different positions

  1. Introduction

In the experiment, a fixed load is applied at various positions across the arch and the effect this load has on the horizontal reaction investigated. In the case of a two- hinged arch, there are four unknown reactions but only three equations of static equilibrium, giving the structure an indeterminacy of one. This requires the use of the flexibility method. However, this experiment uses the secant formula which is a simplified formula which gives relatively good results for parabolic arch ribs. This gives the theoretical values for the reactions, which are compared with the experimental values collected during the experiment. Any discrepancies in the results obtained are noted and discussed.

  1. Material and Methodology

2.1 Material

TecQuipment® two- hinged arch apparatus shown below is used to conduct the experiment. The apparatus allows for the recording of the thrust the arch places on its ends, which corresponds to the reaction forces at the supports. A 500g load is moved at intervals of 50 mm from the left end support.

Two pinned arch experiment.

Figure 2‑1: TecQuipment(R) two hinged apparatus

2.2 Methodology

When an arch is subjected to a load, it develops internal stresses which are transferred to its supports, generating reactions. The main aim of this experiment is to determine the relationship between the position of the load and the horizontal reactions developed at the support. The dial on the apparatus gives the experimental values for the reaction.

The theoretical values for the horizontal reaction are calculated using the simplified formula below, derived from the secant assumption;

Equation 2‑1: Theoretical Relationship between Position of Load and Reaction

Two pinned arch experiment. 1

Where Two pinned arch experiment. 2 is the horizontal reaction at B (N)

W is the load (N)

L is the span of the arch (m)

x is the distance from the left hand side (m)

r is the rise of the arch (m)

The theoretical vertical reaction at support B is also calculated.

  1. Results and Discussion

3.1 Reactions

The experimental results obtained for horizontal reaction at support B and the theoretical results obtained by calculation are as shown below;

Table 3‑1: Comparison between Experimental and Theoretical Values for Horizontal Reaction

Distance from Left

Fraction of Span

Experimental Horizontal Reaction

Theoretical Horizontal Reaction

Variation

The table below gives the theoretical values of the vertical reaction at the right support with increasing distance of the load from the left support;

Table 3‑2: Theoretical vertical reaction with changing load position

Distance from Left

Fraction of Span

Theoretical Vertical Reaction

The results above show that there is strong correlation between the results obtained theoretically and those collected experimentally, with the greatest variation being just above 7%. This small variation can be brought about by various factors, such as;

  • Imperfections in the geometry of the experimental apparatus could result in the internal stresses not being distributed as is expected

  • Lack of uniformity in the composition of the arch rib, which means the deflection is not uniform, resulting in uneven load distribution

  • The secant formula is an approximation, and as such, does not produce perfect results for each load position- support reaction

The horizontal reaction at support B varies, depending on the position of the load on the arch beam. The graph below shows the load position- reaction relationship for both the experimental and theoretical results;

Two pinned arch experiment. 3

Figure 3‑1: Relationship between Load Position and Horizontal Reaction

The relationship between the load position and the vertical reaction at the right support is as below;

Two pinned arch experiment. 4

Figure 3‑2: Relationship between Load Position and Vertical Reaction at Support B

The reaction at the right support increases as the distance from the left support is increased, with the maximum reaction occurring when the load is at the mid- span of the arch. The reaction then reduces as the load nears the right support. When the load is close to the left support, most of the load is transmitted to the support as gravitational load, meaning that the vertical reaction at the left support is largest when the load is at this point. Concurrently, the components of horizontal reaction and vertical reaction at the right support are very small. As the load moves away from the support, the vertical reaction offers no support and the horizontal component of the reaction increases as the load is increasingly being carried by compression forces.

3.2 Influence value

In order to determine the position where the load will cause the largest horizontal reaction at the right support, an influence value is calculated using the formula;

Equation 3‑1: Influence value equation

Two pinned arch experiment. 5

The table below gives the experimental and theoretical influence value for the horizontal reaction for each load position;

Table 3‑3: Relationship between load position and influence value

Distance from Left

Fraction of Span

Experimental Horizontal Reaction Influence value

Theoretical Horizontal Reaction Influence value

Variation

There is a strong correlation between the experimental values collected and the calculated theoretical values, with very little variation. The results show that the load at 250mm has the greatest influence on the horizontal reaction, at 0.97.

Two pinned arch experiment. 6

Figure 3‑3: Influence value for the load at different positions

Inferences

The following can be inferred from the results of the experiment conducted;

  1. The maximum horizontal reaction at the right support occurs when the load is at the mid- span of the arch (250mm from the left support)

  2. The secant formula used is suitable as the deviation in the theoretical horizontal forces calculated and the experimental values recorded

  3. The horizontal influence line value using a unit load being loaded at different points on the arch are as below;

Table 3‑4: Influence value for unit load on span

Unit Load

Distance from Left

Fraction of Span

Horizontal Reaction

Influence value

Table 3‑5: Relationship between unit load position and influence value

Two pinned arch experiment. 7

  1. Find the horizontal reaction at A for a load of 4.9N (500g) at the crown (Ha = Hb with the load at the crown)?

Two pinned arch experiment. 8

Two pinned arch experiment. 9Two pinned arch experiment. 10

  1. The bending moment for the load on the arch span is as follows;

Table 3‑6: Bending Moment

Distance from Left

Fraction of Span

Horizontal Reaction

Bending Moment

Two pinned arch experiment. 11

Figure 3‑4: Bending Moment Diagram

  1. Bending moment diagram to scale for beam loaded with 4.9 N at the centre

Two pinned arch experiment. 12

Figure 3‑5: Bending Moment Diagram for Central Load on beam

  1. Conclusion

The experiment has proven that there is a good correlation between the experimentally measured and theoretically calculated values. The horizontal reaction at the right support increases as the load nears the mid- span and decreases afterwards. The load was found to have the greatest influence value on the horizontal force when at the mid- span.

  1. References

Ambrose, J. (2012). Building Structures. Hoboken, New Jersey: John Wiley & Sons, Inc.

Clive, D., & Williams, H. (2011). Stress and Displacement Estimates for Arches. Journal of Structural Engineering, 49-58.

Luebkeman, C. (1998). Architectonics: The Science of Architecture. Retrieved from Support and Connection Types: http://web.mit.edu/4.441/1_lectures/1_lecture13/1_lecture13.html

Reynolds, C. E. (2008). Reynold’s Reinforced Concrete Designer’s Handbook. New York, NY: Psychology Press.

Vaidyanathan, R., & Perumal, P. (2004). Structural Analysis. Vol 2: Laxmi Publications.