In engineering, Warren girders are trusses consisting of upper and lower members connected by members arranged as a series of isosceles triangles (“Definition of WARREN GIRDER” 2018). They basically offer support to the other beams. They often have an I-beam cross-section with two load-bearing flanges separated by a stabilizing web. (Isami.H,2008). They have a variety of applications including bridges and crane booms while in cantilevered form. Below is a structural model of a girder relevant for this experiment:
Figure 1: A Diagram Showing a Warren Girder with a Central Load
This experiment is aimed at examining and determining the forces in each member of a statically determinate warren girder, an example is above. In simpler terms, the purpose of the experiment is to examine and analyse the forces recorded during the experiment and compare them to their theoretical values determined using the method of joints. Statically determinate structures are those which can only be analysed using static equations, their net force is zero and thus they exist in equilibrium (Cheyenne 2018). In this experiment the forces on each of the members will be determined using the method of joints, which assumes that the sum of vertical forces as well as horizontal forces acting on a joint is equal to zero. The following equations are applied.
E = σ / ɛ Where E=Young’s Modulus (Nm-2)
σ = Stress in the member (Nm-2)
ɛ = Displayed strain
and; σ = F / A Where F = force in the member (N).
A = cross-sectional area of the member (m2).
Rod diameter = 5.98 mm and Esteel = 210 GNm-2
In engineering, especially mechanical and civil engineering, it is crucial to understand the mechanics of trusses and joints. This is usually very important in the building of bridges and other structures. Trusses are considered to be engineering structures that are capable of providing practical and economical solutions to many engineering constructions. They are mostly used in the designing of buildings and bridges which require large spans.
Procedure
In the course of this experiment, the following are the steps that were followed in obtaining the results which enabled the analysis that was done post-test.
1. The first step was to ensure that all the apparatus was properly set up for the experiment in order to avoid possible precision and accuracy errors.
2. Secondly, a preload of a hundred newtons (100 N) was applied and then the load cell was zeroed.
3. The next step was to carefully apply a load of five hundred newtons (500 N) and thereafter the frame was checked for stability and security.
4. The load was then systematically reduced until the digital force display read zero, after which the deflection gauge was carefully zeroed.
5. The next step was the placing of the loads in increments of a hundred newtons (100N) as shown in the tables 1.0 and 2.0 below.
6. During the experiment, all the strain and indicator readings were recorded in the suitable table of results.
Table showing experimental recordings of member strains
Loading condition
(N)
AD
AE
AF
BD
CF
DE
EF
Def
(mm)
0
100
200
300
400
500
Table 1.0.
Readings of true member strains were recorded in the table below
Table showing true member strain recordings.
Loading condition
(N)
AD
AE
AF
BD
CF
DE
EF
Def
(mm)
0
100
200
300
400
500
Table 2.0.
Results
The experiment was conducted with as much accuracy and precision as was possible. Using suitable tables of results, the experiment’s data was collected and analysed after careful observations had been made. Results of experimental recordings were recorded as shown below for both experimental and true strain recordings:
Loading condition
(N)
AD
AE
AF
BD
CF
DE
EF
Def
(mm)
0
89
18
-2
35
43
135
46
0
100
78
7
-13
40
48
146
56
-0.016
200
67
-5
-24
46
53
157
67
-0.034
300
55
-16
-36
52
59
169
78
-0.058
400
44
-27
-47
58
64
180
90
-0.077
500
34
-38
-58
63
69
190
100
-0.094
Table 1.1.
Loading condition
(N)
AD
AE
AF
BD
CF
DE
EF
Def
(mm)
0
0
0
0
0
0
0
0
0
100
-11
-11
-11
5
5
11
10
-0.06
200
-22
-23
-22
11
10
22
21
-0.034
300
-34
-34
-34
17
16
34
32
-0.034
400
-45
-45
-45
23
21
45
44
-0.077
500
-55
-56
-56
28
26
55
54
-0.094
Table 2.1.
Discussion of Results
From the data collected, analysis of various parameters can be obtained. Suitable line graphs can also be drawn from the data.
In calculating the true strain of each member (ɛT), we use the formula;
ɛT = ln (1+ ɛ) where ɛ is the displayed strain.
Recall Stress, σ = F / A, also, E = σ / ɛ; ɛ = σ / E = F / A * E
Cross-sectional area of member, A = πD2/4 = π (5.98 x 10-3) = 2.80 x 10-5m2.
0 N; ɛ = 0 / (2.80 x 10-5m2 x 210 x 109) = 0 ɛT =ln (1+ 0) = 0
100 N; ɛ = 100 / (2.80 x 10-5m2 x 210 x 109) = 1.7007 x 10-5; ɛT = 1.7007 x 10-5
200 N; ɛ = 200 / (2.80 x 10-5m2 x 210 x 109) = 3.4014 x 10-5; ɛT = 3.4013 x 10-5
300 N; ɛ = 300 / (2.80 x 10-5m2
x 210 x 109) = 5.1020 x 10-5; ɛT
= 5.1019 x 10-5
400 N; ɛ = 400 / (2.80 x 10-5m2
x 210 x 109) = 6.8027 x 10-5; ɛT
= 6.8025 x 10-5
500 N; ɛ = 500 / (2.80 x 10-5m2
x 210 x 109) = 8.5034 x 10-5; ɛT
= 8.5030 x 10-5
In order for one to be able to calculate the equivalent experimental member forces at an applied load of 500 N, we use the values from the experimental recordings table (Table 1.1.);
Recall the formula for Young’s Modulus; E = σ / ɛ; Thus, σ = E. ɛ……………(i).
Also; σ=F/A; Thus, F= σA……………………………………………(ii).
Combining equations (i) and (ii) and substituting σ;
F=E.ɛ. A…………………………………………………………. (iii).
We will now use F from equation (iii) to calculate the experimental member forces at 500N applied load.
AD; F= 210 x 109
x (34 x 10-6) x 2.80 x 10-5 = 199.92N
AE; F=210 x 109
x ( -38 x 10-6) x 2.80 x 10-5= -223.44N
AF; F=210 x 109 x ( -58 x 10-6) x 2.80 x 10-5= -341.04N
BD; F=210 x 109 x (63 x 10-6) x 2.80 x 10-5= 370.44N
CF; F=210 x 109 x (69 x 10-6) x 2.80 x 10-5= 405.72N
DE; F=210 x 109 x (190 x 10-6) x 2.80 x 10-5= 1117.2N
EF; F=210 x 109 x (100 x 10-6) x 2.80 x 10-5= 588N
In calculating the theoretical member forces at an applied load of 500 N, we use values of true strain from the true strain recordings table (Table 2.1);
AD; F=210 x 109 x ( -55 x 10-6) x 2.80 x 10-5= -323.4N
AE; F=210 x 109 x ( -56 x 10-6) x 2.80 x 10-5= -329.28N
AF; F=210 x 109
x ( -56 x 10-6) x 2.80 x 10-5= -329.28N
BD; F=210 x 109
x (28 x 10-6) x 2.80 x 10-5= 164.64N
CF; F=210 x 109 x (26 x 10-6) x 2.80 x 10-5= 152.88N
DE; F=210 x 109 x (55 x 10-6) x 2.80 x 10-5= 323.4N
EF; F=210 x 109
x (54 x 10-6) x 2.80 x 10-5= 317.52N
As with most experiments, the experimental values often differ from the theoretical values. The error is calculated by subtracting the theoretical value from the experimental value.
Therefore, Percentage Error,
PE = (Experimental value – Theoretical value) / (Experimental value) * 100 %
The percentage error, for instance for member EF under a load of 500 N;
PE = (100 – 54) / (100) x 100% = 46%
The high percentage error in the experiment may have resulted from overloading the truss members beyond or up to the yield point, or measurement drift of the electronic digital force display as the experiment proceeded.
The experimental findings are quite different from the true values of force as are the values of strain. This may have been caused by the tensile forces applied on the girder steel beams, which resulted in the necking of some parts on the individual beams and thus changing the cross-sectional areas of the beams. Change in the cross sections caused significant variations in the two stresses and consequently the member forces. This is also a reason for the significant error when comparing the experimental and true strain readings.
For most trusses, when the central members are subjected to a gravitational load, the members that are likely to shrink are in compression while those that may stretch are in tension( www.quora.com). The diagonal members of the Warren Girder exchange compression and tension forces while the center members are similar.
Suppose AD is selected as one of the compression members (struct members) and CF selected as a member under tension (tie member), a graph of load against strain for the two members can be drawn. Considering Table 2.1;
The graph above is a linear graph, which shows a linear relationship between an applied load and the corresponding strain on a member. For tie members, the graph has a negative gradient due to the effects of the tension forces which cause necking at some sections on the member and consequently a decrease in cross-sectional area.
A relationship between load applied to a member and the joint deflection can be demonstrated using the graph shown below. Taking readings from Table 2.1, a graph showing the variation of load with joint deflection was plotted and analysed.
Careful analysis of the graph resulted in the realization of a linear curve. It was evident that there existed a linear relationship between the load applied and the joint deflection for the Warren Girder. The negative deflection was as a result of the action of both compressional and tensional forces acting on the members of the truss upon application of increasing loads.
Conclusion
Generally, trusses are considered to be engineering structures that are capable of providing practical and economical solutions to many engineering constructions. They are mostly used in the designing of buildings and bridges which require large spans. An example of a truss structure is the Warren girder truss, the individual members are assumed to be connected in such a way as to permit rotation, and thus carry an axial force either in compression or in tension (Krenk, S and Hogsberg, J,2013).
Besides measurement drift in the electronic instruments and the effect of tension forces ion the steel beams, the nature of material of the beams may also be a contributing factor to the errors experienced. In addition to that, the lab environment may also have influenced the member’s reaction. It is therefore recommended that beams with pure steel be used instead since the experiment results depended on the material under study.
From the experiment, there are positive and negative forces with tensile and compressional conditions at all members. Structural failure may arise from the load effect exceeding the stable conditions of the structure. It was discovered that the best method of keeping the structure stable was to avoid overloading the it. This would result in the structure holding its ability and thus it will be safe and stable. Application of this knowledge is useful in designing bridges and other structures that require a wide span.
Bibliography
Chen, X, Wu S, Zhou J (2014). “Compressive Strength of Concrete Cores with Different Lengths.”
Journals Materials and Civil. Engineering, 26(7), DOI 04014027.
“Definition of WARREN GIRDER”. 2018. Merriam-Webster.Com https://www.merriam- webster.com/dictionary/Warren%20girder.
Hirol, I (2008). Plate-Girder construction. BibilioBazaar. ISBN 978-0-554-88802-6.
Krenk, S; Hogsberg, J (2013, XI p.680) Statics and Mechanics of Structures.
Mani, Ermand. 2015. “Forces In a Warren Girder – Laboratory Report”. Academia.Edu. https://www.academia.edu/12248465/Forces_in_a_Warren_Girder_-_Laboratory_Report.
Seshu, A (2009). “Structural Steel Design: Plate Girders (class notes)”
Song.W, Ma. Z, Vadivelu J, Burdette, E (2004). Transfer Length and Splitting Force Calculation for Pretension Concrete Girders with High-Capacity Stands. Journal of Bridge Engineering. 19(7), DOI 04014026. “What is the difference between true stress strain and engineering stress strain diagram.?” Retrieved from; https://www.quora.com/What-is- the-difference-between-true-stress-strain-and-engineering-stress-strain-diagram
Forces in a Warren Girder-Laboratory Report. Emerald Mania-Academia.edu. Retrieved from; https://www.academia.edu/Forces_in_a_Warren_Girder_Laboratory_Report/
