Construction and Building Materials 50 (2014) 736–743

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Construction and Building Materials

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Numerical and experimental evaluation of a double inverted trussedbeam reinforced with steel cable

0950-0618/$ - see front matter � 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.conbuildmat.2013.10.036

⇑ Corresponding author. Tel.: +55 034 3239 4170.E-mail addresses: gesualdo@ufu.br (F.A.R. Gesualdo), thaisalvesc@gmail.com

(T.A. Cunha), romulobrezende@yahoo.com.br (R.B. Rezende).

Francisco Antonio Romero Gesualdo ⇑, Thais Alves Cunha, Romulo Barbosa RezendeFederal University of Uberlandia, School of Civil Engineering, Av. Joao Naves de Avila, 2121, CEP 38400-902 Uberlandia, MG, Brazil

h i g h l i g h t s

� It is an arrangement that generates a system lightweight, easy to handle, stronger and stiffer than a simple beam.� Easy to be mounted and transported.� Indicated to be used in formwork systems.� If compared with simple beam, its resistance and stiffness increase 43% and 97%, respectively.

a r t i c l e i n f o

Article history:Received 12 August 2013Received in revised form 4 October 2013Accepted 19 October 2013Available online 14 November 2013

Keywords:Inverted trussSteel cableDouble sectionReinforced beam

a b s t r a c t

A special type of queen-post truss was evaluated numerically and experimentally. The system consists ofa main beam with a double rectangular cross section, into which are fitted two struts that are connectedto steel cables anchored to transverse steel dowels embedded at the ends of the main beam. The system,which can be used as a support for concrete formwork, for structural rehabilitation and for numerousother applications, is lightweight, easy to handle, and stronger and stiffer than a simple beam. Fromexperimental study, on average, beams are 97% stronger than ones without cable reinforcement, andthe gain in stiffness was 43%. The numerical model was analyzed via the finite element method.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

It should be noted that wood is an important structural mate-rial, particularly due to its high strength-to-weight ratio in com-parison with other materials. It is a natural renewable materialwith adequate mechanical properties, whose compressive strengthparallel to the grain may exceed 60 MPa.

Timber trussed beams can be used as concrete formwork sup-ports because timber beams are easy to transport and adjust to dif-ferent conditions, thus preventing the overloading of shores.

The system called as inverted truss or queen (or king) post trussis little described in the literature. The little information that existsis fairly generic, and considers the structural model with classicalconnections – rigid or hinged, with a lattice arrangement com-posed of members. No reports of a system similar to the one pre-sented here were found in the literature. Therefore, it is a uniquesystem in terms of its assembly.

The use of steel in conjunction with wood has been successfullyused due to appropriated interaction between these materials [1].Herzog et al. [2] present several schemes for distributing membersto form an inverted truss, which is characterized by the existenceof upward thrust at the points where the struts are located, asshown in Fig. 1a and b. In this situation the structure is internallythree-dimensional. Boresi et al. [3] offer general information aboutthe system and present a numerical example of how to calculatethis type of trussed beam. Pfeil [4] and Ritter and Fahert [5] offerthe same information. The latter authors consider V-shaped strutstilted in relation to the vertical position – Fig. 1c – in a planar sys-tem. Rebello [6] offers a generic explanation of how to use this typeof beam, indicating its advantages and efficiency. Pletz and Mello[7] studied another alternative involving V-shaped struts, as illus-trated in Fig. 1d, considering it an easy way to prestress this type ofstructure.

Feio et al. [8] consider this type of beam suitable for structuralrehabilitation, enabling the recovery of existing beams by increas-ing their stiffness and strength by means of easy handling. Thearrangement with wire rope used in this study it is an interestingway to treat this type of problem, once wire rope is flexible andadjustable to curve surface. This topic can be also included as a

Fig. 1. Types of struts described in the literature.

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structural rehabilitation category for solving problems in the samerange as presented by Augelli et al. [9], Borri et al. [10], Pantelideset al. [11] and Branco et al. [12].

A team of researchers associated with the authors has reportedinformation about similar situations. Gesualdo and Lima [13,14]and Gesualdo and Cunha [15] describe different arrangements.

There are also three-dimensional systems for use in bridges, forexample. This type of system consists of a model similar to the onepresented here, but the strut has two inverted V-shaped pieces thatincrease the system’s stability.

The purpose of this work was to evaluate the aforementionedtype of beam based on a numerical and experimental analysis.The analysis involved the connections, joints and details of anchor-age including the positioning and the elastic and geometric charac-teristics of all the components of the system.

2. Experimental description

Lightweight, strong, stiff systems that are easy to assembly are essential forapplications as concrete formwork supports. Moreover, the structural repair andrehabilitation of buildings requires simple procedures for adding new elementsto existing beam systems. Therefore, a system was developed consisting of ahorizontal double cross section connected to two vertical elements – struts –with steel cables anchored at the ends to transverse steel dowels, as illustratedin Fig. 2. This type of system is known as an inverted truss or inverted queenpost truss. However, the system devised here differs from the conventional sys-tem described generally in the literature, since it includes a continuous cableanchored to transverse steel dowel pins located close to each end of the beam.The ends of the cable are connected by a stretcher that makes it continuous,thus transforming it into a double cable with back and forth movement (seeFig. 3).

Fig. 2 shows the position of the cable anchored to the transverse struts. Thecable, which distributes the stress, passes through a standard stainless steel thimbleand the radius of curvature of the cable is compatible with its diameter. The size ofthe thimble is larger than the diameter of the cable in order to accommodate thecable properly, since its bidirectional location causes a differential in its vertical po-sition, Fig. 3d.

Fig. 4 shows details of the cable in contact with the strut. A transverse pin wasused to better distribute the stress caused by the cable in contact with the strut.

To analyze the gain in stiffness, a direct comparison can be made of the beams,since the experiment allows the same beam to be tested under two conditions, i.e.,with and without the cable system.

However, an analysis of rupture loads requires a comparison of pairs of beamsthat are considered acceptable. Therefore, three simple beams without cables weretested in bending to compare their ultimate failure loads.

Fig. 2. Overall view of the beam under testing.

All the beams were tested with an unsupported span of 1800 mm, using232.5 mm long struts, as illustrated in Fig. 5. The main beam was made of a Brazil-ian wood species known popularly as Roxinho (Peltogyne cf. subsessilis), which isreadily available in the Brazilian market in the form of sawn timber. All the beamscame from the same lot and had the same moisture content (10.4%), compressivestrength parallel to the fibers (79.6 MPa) and density (877 kg/m3) determinedaccording to Brazilian standard ABNT NBR 7190:1997. These values are generalfor all the beams, but the main analysis involved a direct comparison of pairs ofthese beams.

Fig. 5 illustrates the loading scheme. Loads were applied on the beam at thepoints where the struts were located, i.e., at ‘/3 from the supports.

The vertical displacements were measured with two inductive displacementtransducers positioned laterally at the midpoint of the beam, Fig. 6. The average va-lue of two measurements was considered representative of the vertical displace-ment. The transducers were then fixed to the top of the beam to enable readingsof displacements up to failure. To ensure the lateral stability of the system, a lateralbracing system was employed to prevent transverse movements, as illustrated inFig. 6.

3. Experimental results

3.1. Procedures

The beams were tested in different configurations, as describedin Table 1. The initial tests of the beams tested as inverted trusseswere conducted without the cable (Case B), with loads below theelastic limit (Case A). This enabled a direct comparison of the mod-ulus of elasticity of the simple beam and the cable-reinforcedbeam. To compare the ultimate load, three pairs of these beams– V1, V4 and V5 – were mounted as simple supported beams (CaseA), which were cut from the same piece of timber and loaded up tofailure. Each pair of beams, one a simple beam and the other an in-verted truss, were loaded up to failure.

The diagrams shown here refer to ‘‘simple beams’’, i.e., beamswithout reinforcement cables (Case A), and ‘‘inverted trusses’’(Case C), i.e., the complete system. The inverted trusses were alsotested in two steps. The first test, to accommodate the cable andadjustments, involved loading to a level below the elastic limit.In the second test, the beam was unloaded and the cable stretchedagain to regain the deformation produced by the natural accom-modation of the cable, after which the beam was loaded up tofailure.

The cables used to reinforce the beams consisted of six wires of6.4 mm diameter, each wire composed of 19 strands, according toinformation provided in a technical manual for steel cables. Fig. 7ashows a thimble deformed and undeformed. In order to preventadditional local deformation the thimbles were reinforced, basedon previous experimental observations. To prevent crushing, thethimble’s radius of curvature, a metallic element was welded toits internal surface, thus reducing the bending of the thimble, asillustrated in Fig. 7b.

Another option is to use solid steel thimbles, Fig. 7c. Althoughthis is not a conventional product available in the market, itis undoubtedly a more rigid element. However, it is not

(a) Cable anchoring system − bottom view(b) Cable anchoring system − top

view

(c) Left side view (d) Right side view

Fig. 3. Cable position and anchoring.

Fig. 4. Details of the tip of the strut in contact with the cable.

Fig. 5. Loading scheme and position of struts.

lateral verticalbracing

main beam: doublecross section

Fig. 6. Displacement transducer fixed to the top of the beam.

738 F.A.R. Gesualdo et al. / Construction and Building Materials 50 (2014) 736–743

recommended for this purpose because the internal surface of thehole is flat, while the stretcher end is curved and must be accom-modated inside this hole, which thus provides an unsuitable con-tact surface.

3.2. Beam V1

This beam geometry was analyzed based on two specimens,which were cut from the same timber sample. The first specimen,called a ‘‘simple beam,’’ was loaded up to failure to determine itsreal properties without struts and cables, based on which the gainin strength and rigidity can then be calculated. The second beam,which was assembled as an inverted truss, was tested in three dif-ferent configurations. First, as a simple beam, it was loaded belowits elastic limit to determine its modulus of elasticity. This test wasdubbed the ‘‘ITB modulus of elasticity’’ test. The beam was thenassembled as an inverted truss with struts and cable and loadedto its elastic limit to accommodate the cable. This was called ‘‘Test1’’. Lastly, after stretching the cable again to eliminate residualdeformation, the beam was loaded up to failure, which involved‘‘Test 2 – rupture.’’ Fig. 8 illustrates all the results for beam V1.

As can be seen in Fig. 9, beam V1 reached failure in tension per-pendicular to the grain. This failure was produced by downwardstress on the transverse pin to which the cable was attached closeto the end supports. Table 2, where all results of the tests of the fivebeams tested are summarized (showed at the end of this section),describes the numerical results of beam V1.

3.3. Beam V2

Beam V2 was tested in two steps. The first test consisted ofapplying a load below the elastic limit to determine the beam’smodulus of elasticity without the presence of a cable. The cable

Table 1Test scheme for beams.

Simple beam (case A) Beam without cable (case B) Inverted trussed beam (case C)

(b) Reinforced thimble

(a) Comparison of deformed and undeformed standard thimble

(c) Cast solid thimble

Fig. 7. Condition of the thimbles of the steel cable.

Fig. 8. Beam V1.

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was then attached and the beam was loaded up to failure. The‘‘Modulus of elasticity’’ and ‘‘Test 2 – rupture’’ curves in Fig. 10 de-pict the results of these tests. The failure mode was rupture due tostress on the fibers and pronounced curvature. No simple beamwas tested up to rupture for purposes of comparison with thisbeam. Fig. 10 also shows an unload and reload zone between 2kN and 12 kN in the ‘‘Test 2 – rupture’’ curve. As can be seen, thefinal result was the same with both loading and unloading.Although all the tests were carried out using this procedure, someof them are not shown in the respective diagram to allow for bettervisualization. No information is available for ‘‘Test 1’’, whose pur-pose was simply to check the general behavior of the beam. Thebeam was found to regain the same curve that it had formed inthe first step. Table 2 lists all the numerical results for beam V2.

3.4. Beam V3

Beam V3 was tested similarly to beam V2. An initial test wasconducted to determine the beam’s modulus of elasticity withoutcable. Fig. 11 shows the diagrams obtained in these tests. The fail-ure mode was rupture due to stress on the lower fibers at midpointof the span. Additional numerical details are given in Table 2.

3.5. Beam V4

During the ‘‘Test 2’’, upon reaching a force of 13.26 kN there wasan abrupt increase in deformation, resulting in a visible failure inthe proximity of the left strut. Nevertheless, it was possible to con-tinue loading up to 23.05 kN, whereupon the beam failed. Thisaccommodation, which appears in the ‘‘Test 2 – rupture’’ curve, de-notes a reduction in the beam’s bearing capacity. Rupture occurredthrough severe loss of stability, despite the lateral restraining sys-tem used in the experiment. Fig. 12 presents the force–displace-ment curve obtained for this beam.

3.6. Beam V5

Beam V5 was tested following the same procedure as the otherbeams, i.e., a test to determine the modulus of elasticity, a test foradjustment of the cable, and finally loading up to failure. Upon fail-ure there was a significant loss of lateral stability. The results arerepresented in the diagram shown in Fig. 13.

3.7. Summary of experimental results

The beams are identified by number and type, e.g., V2-C: beamV2 tested under condition C, as described in Table 1. As indicated inTable 2, this beam was tested under almost the same condition as asimple supported beam. Type A beam is an additional beam cutfrom the same timber and loaded up to failure in order to provideinformation about the maximum force that can be applied to thesimple beam and to compare it with the inverted truss. This situa-tion applies only to beams V1, V4 and V5. Beams V2 and V3 arecompared with the mean value obtained from these tests. All thebeams came from the same lot of timber.

Table 2 summarizes the results of the tests of the five beams.The modulus of elasticity (E) of the inverted trussed beam de-

scribed in Table 2 was inferred from equivalent beams, i.e., the in-verted trussed beam was considered a simple beam to calculatethe displacements.

Based on Table 2 it is possible to correlate some of the valuesbased on mean results. Considering the simple beams V1, V4 andV5 representative of all the tested beams, their average ruptureforce was equal to 15.69 kN, while that of the five inverted trussed

(a) Rupture of the beam at the cable docking pin

(b) Rupture of the beam at the cabledocking pin

(c) Failure – overall view, left side

Fig. 9. Failure mode of beam V1 – longitudinal crack.

Table 2Summarized results of the inverted truss with double rectangular section.

Beams b1 (mm) h1 (mm) b2 (mm) h2 (mm) Inertia (mm4 � 104) E (MPa) Frup (kN)

V1-C 20.98 93.74 20.82 93.57 286.1347 29244.16 37.75V1-B 23586.18V1-A 21.05 90.19 21.40 89.85 258.0667 14031.85 13.89

V2-C 21.15 94.36 20.52 94.30 291.4962 31045.66 45.91V2-B 19543.61

V3-C 18.78 94.07 18.99 95.02 266.0661 30819.66 20.90V3-B 21198.89

V4-C 18.81 93.14 18.71 93.96 255.9769 22659.51 23.05V4-B 15643.32V4-A 20.80 93.28 20.85 93.74 283.7891 16846.94 17.60

V5-C 18.86 92.82 18.72 93.13 251.6697 25349.29 27.43V5-B 17582.26V5-A 20.51 92.79 20.60 93.27 275.8291 13284.64 15.59

Fig. 10. Beam V2.

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beams was 31.01 kN. This represents a gain of 97.6%, i.e., a two-foldincrease in strength. On the other hand, in terms of the rigidity ofthe system, a comparison of the modulus of elasticity of the in-verted trussed beams tested with and without cables led to theconclusion that the use of cables increased the system’s stiffnessby about 43.5%, i.e., the percent gain in stiffness was almost halfthat of strength. The letters A, B and C refer to Table 1 for indicatingthe type of specimen tested.

4. Numerical analysis

The tested beams were analyzed numerically using ANSYS�

software [16]. To generate more accurate results, this softwareconsiders real conditions, i.e., it uses contact elements, solid ele-ments, support conditions, and fixation of the cable to the beam,etc. A second computer program was developed to generate a codein APDL language used in the ANSYS� software. This code was

Fig. 11. Beam V3.

Fig. 12. Beam V4.

Fig. 13. Beam V5.

(a) Lateral view of the support

(b) Internal view

internal bracing

support

support

internal bracing

Fig. 14. Model and mesh division by FEM – 1/4 of the complete model.

Fig. 15. Schematic diagram of the boundary condition used in the numerical modelfor a coarse mesh.

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inserted and read by ANSYS�. The code optimizes the data inputprocess, facilitating the user’s work since it requires only the inputof elastic and geometric data. The software then generates themodel automatically. This software was used for four types of ele-ments: SOLID45, LINK10, TARGE170 and CONTA174. The mesh

division produces solid hexahedral elements for the entire model(see Fig. 14). The contact surface of the support between the steelplate and the wooden beam was also considered. At the bottompart of the strut, a contact surface is considered between the pairssteel bolt/cable and steel bolt/strut. The transfer of stresses be-tween strut and main beam was also considered as compressioncontact.

4.1. Definitions of the model

The inverted truss system analyzed here is a 1800 mm longbeam having two struts of 232.5 mm located at a distance of600 mm from the support, dividing the beam’s span into threeequal parts. The dimensions, inertia and modulus of elasticity ofthe beams showed only minor variations since they all came fromthe same lot of timber. The material’s properties were definedaccording to the Brazilian Technical Standards (ABNT). The timberlot had a representative ultimate strength of 796.1 MPa and amoisture content of 11.3%.

Fig. 16. Experimental vs. numerical analysis.

742 F.A.R. Gesualdo et al. / Construction and Building Materials 50 (2014) 736–743

The model of both types of beams – simple beam and invertedtrussed beam – was analyzed numerically with concentrated forcesapplied at the struts, based on the experimental analysis. Thenumerical analysis considered the modulus of elasticity of casesB and C. Due to the double symmetry, one quarter of the completemodel was used to represent the whole structure. A suitableboundary condition was used at the edges to represent well thereal cases and their symmetry. Fig. 15 shows the schematic dia-gram of the boundary condition used in the numerical model fora coarse mesh. One support is used for each node of the mesh.

4.2. Results

The results of the numerical analysis are represented in the dia-grams in Fig. 16, together with the response obtained in the test 2of the experimental analyses. The modulus of elasticity of the woo-den beam was determined experimentally for the beam withoutthe cable. The steel cable was modeled with a value of 8000 MPa,according to the manufacturer’s specifications.

It was found, numerically, that the effect of geometric nonlin-earity was practically null; hence, it was not considered in the

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calculations. Experimentally, it was found that the simple beamsloaded up to failure presented an almost linear load–displacementrelationship. There, no effect of nonlinearity was considered in thenumerical calculations.

The diagrams in Fig. 16 indicate that the numerical and exper-imental results were very similar, particularly in the elastic range.This applies to beam V1, for which the numerical and experimentalcurves showed a very similar initial portion, after which the valuesbecame discrepant. The numerical and experimental values ofbeams V2 and V3 were in good agreement. Beam V4 showed thehighest discrepancy, since the model overestimated the real condi-tion of the beam by 15%.

Although some points show significant differences between theexperimental and numerical results, the model used for thenumerical analysis is a good representation of reality. Undoubt-edly, some effects of deformation need to be better adjusted. Thesedeformations may be attributed to the conditions of the testedmaterial or even to the positioning and anchoring of the parts.

5. Conclusions

The type of structure examined here uses accessories – thimblesand turnbuckles – with curved shapes which concentrate stressesand allow for localized strains. This affects the elasticity of the steelcable by producing additional deformations. These aspects must bewell controlled and taken into account in the calculation for a thor-ough evaluation of the system. The experiment revealed that theweakness of the system is its stiffness. While the strength showeda considerable increase of almost 97.6%, the increase in stiffnesswas only 43%.

Steel cables require pre-stretching to ensure accommodation ofthe wires. Doing so avoids additional stretching �0.50% to 0.75% ofthe cable length –, which interferes in the cable’s modulus of elas-ticity. This procedure can be controlled by using turnbuckles toregulate the force in the cable, thus introducing a force prior toerecting the structure. The force can be estimated from the struc-tural deflection in the opposite direction. This involves tighteningthe turnbuckles and releasing them before introducing an expectedforce into the system. During the preparation of the beam it is veryimportant to ensure that the end portions of the turnbuckle (hookand eye or eye and eye) are located as distant from each other aspossible to allow for their subsequent adjustment.

These beams show a significant tendency for loss of lateral sta-bility due to the thickness of the individual elements. It is not al-ways easy to physically simulate how to prevent the occurrenceof this phenomenon in the laboratory. However, this phenomenonaffects the results. On the other hand, from the standpoint of thepractical application of these beams as elements for concrete form-works systems, where a lateral restraint is provided by the form-work along the entire top part of the beam, produced by theplywood in direct contact with the top part of the beam under

compression. In other cases the designer must treat this point witha special attention to avoid this problem.

It is clear that further studies are required to improve the afore-mentioned aspects. In this system, the most significant effect wasfound to be its lateral stability. Hence, this point must be addressedeffectively to increase the system’s efficiency.

Acknowledgement

The authors gratefully acknowledge the Brazilian researchfunding agency FAPEMIG (Minas Gerais State Research Founda-tion) for its financial support of this work.

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