Indian Journal of Engineering & Materials SciencesVo\. 5, June 1998, pp. 111-116

Strain energy ofRC beams subjected to a moving load

Terezia Niimbergerova", Martin Krizma«, Lubomir Bolhab & Sabah Shawkat''.

"Institute of Construction and Architecture, Slovak Academy of Sciences, Dubravska 9, 842 20 Bratislava,Slovak Republic

bSlovak Technical University, Department of Civil Engineering, Starohorska 2,8\3 68 Bratislava, Slovak Republic

Received 2 May 1997; accepted 28 April 1998

Evaluation of strain energy from the tests of reinforced concrete beams subjected to a movingload is presented in this paper. The effect of a moving load on the deformations was investigated on 3reinforced and 4 partially prestressed concrete beams. The moving load was simulated by activating anddeactivating- a set of hydraulic jacks placed along the beam resulting in a force similar to the moving load.The arrangement of measurements enabled the separation of the effect of bending moments from that ofshear forces. The run of a loading force along the beam enabled the evaluation of the elastic and inelasticpart of the strain energy. The results show that the effect of the shear force, on the irreversible part of thedeflections, is higher than that due to the bending moment.

Concrete structures, as a rule, have been designedon the basis of strength criteria, applied both tomaterials and structural elements. In the design ofstructures in accordance with the limit statesmethod, the deformation characteristics have alsobeen introduced, which may characterise theoccurrence of a certain limit state. For developinga computational model, the whole failure process,which is generally expressed by the completestress-strain diagram, has to be known 1-3. Thedeveloped computational models describing thebehaviour of material use the amount of strainenergy that can be absorbed by a specimen as a

. further characteristic of the material.The values of the failure load and ultimate

deformations, as well as those of crack width andcrack orientation are usually obtained by testingreinforced-concrete elements, while the evaluationof the strain energy in structural members is' stillnot common. Nevertheless, the strain energycharacterizes the state of a structure as efficientlyas (or even more efficiently than) the deformationor the stresses. The occurrence of a limit state canbe interpreted as the total dissipation of a certainpart of the strain energy. The soundness of such aninterpretation results from the strain energy being ascalar quantity, which expresses the reserve of a

structure in a given state of stress with respect tothe ultimate limit state or to the limit state ofserviceability. The reliability condition can also beexpressed by means of strain energy". However,the general application of this approach ishampered by the scanty data available so far.Consequently, the necessary data should beobtained by carrying out further tests.

In this paper the evaluation of the strain-energyfrom the tests of RC beams subjected to a movingload is presented. Moving loads are very oftenapplied to important structures (e.g. bridges, rails,beams in movable structures like cranes andgantries). However, the first tests under movingloads' did not show any significant effect of amoving load on the ultimate capacity of the beams,but truly speaking the experimentalinvestigations on beam behaviour are far lessexhaustive than it should have been. Since 1984,many tests on reinforced and partially prestressedconcrete beams6

,7 have been carried out at theInstitute of Construction and Architecture inBratislava. The tests were focused on the effects ofa moving load on both deformations and cracking.The difference between the crack pattern due to astationary load and that due to a moving load wasfound to be significant. Moreover, compared to the

112INDIAN 1. ENG. MATER. SCI., JUNE 1998

Experimental ProcedureThe tests on RC beams, aimed at the evaluation .

of the' strain energy, were a part of an extensiveexperimental programme, which included thetesting of 3 reinforced concrete and 4 partially-prestressed concrete beams. The beams differedfrom each other only in the reinforcement.The cross-section of the beams is shown inFig. 1 and the elevations in Fig. 2. The beams werecast of concrete with Portland cement content of470 kg m-3, fine to coarse aggregate ratio of 1:1.4,and water to cement ratio w/c = 0.4. The averagemechanical properties of concrete are given inTable 1. The beams were reinforced in the tension

720 720 720 720zone with 11 high-bond deformed bars (cj1 = 16 .r-==....,r-="--,t--..:.=..---,f'--'-=-"--l'-----'=--4'---="---+=

mm). The average properties of high-bond barswere: yield stress/sy = 454 MPa, tensile strength z,= 649 MPa, modulus of elasticity E, =213 GPa.The stirrups were made of high-bond wire (cj1 = 8mm, spacing 180 mm). The mechanical propertiesof steel used for stirrups were similar to those of16 mm reinforcing bars (e.g. the yield stress /sy =

'485 MPa).

The overall length of the beam was 4.35 m andits span was 3.6 m. The moving load was simulatedby means of 5 hydraulic jacks located as shown inFig. 2 (locations A - E). The experimental set-up is

beams under a stationary cyclic load", a movingload causes an increase of the deflections becauseof the interaction between the shear force and thebending moment. This phenomenon also affectsthe values of the strain energy.

In the case of a reinforced beam, a certainamount of the strain energy is dissipated in crackformation and propagation, and in otherirreversible deformations. The value of this energycan be determined from the tests under cyclicloads"!'. However, the motion of the loading forcealong the beam leads to the loading and unloadingof the sections, and in the end makes it possiblealso to determine the amount of the dissipatedenergy.

The tests have been devised in order toinvestigate the above mentioned phenomena. Apeculiar method for measuring the deformationshas been used, with the objective of separating thebending-induced deformations from the shear-induced deformations.

Table I-Average values of principle properties

Beam Age Cube Prism Modulus Tensiledays strength strength of strength

MPa MPa elasticity MPaGPa

14 53.26 47.38 39.05 2.37PZ03 28 58.99 47.80 42.55 2.50

65 60.87 52.35 41.76 2.5314 42.65 39.03 39.12 2.12

PZ02 28 56.07 48.68 40.69 2.4350 58.33 53.45 43.72 2.4814 52.59 45.87 36.06 2.36

PZPI6 28 60.30 46.47 38.84 2.5243 63.48 51.32 41.69 2.5914 45.48 40.22 37.33 2.19

PZPI9 28 51.67 48.29 39.66 2.34

2 ~ Vl0 8-0N

8-~~=+~~r- 5 QlV 16

.1 2_l_0__ ....,

Fig. I--Cross-section of the beams.

360 seo 3eo 360seoo4leO

Fig. 2-Side-view of the beams: A, B, C, D, E - location ofhydraulic jacks; I, 2, 3, 4, 5 - dial gauges for measurement ofdeflections; a - network of bases for mechanical measurementof deformations.

NURNBERGEROV A et al.: STRAIN ENERGY OF RC BEAMS SUBJECTED TO A MOVING LOAD 113

shown in Fig. 3. The loading started at the locationA. The loading force in the jack A was graduallyincreased until the value of the first load level (100kN) was reached. After all planned measurementswere performed, the loading force at the jack Awas gradually decreased simultaneously with thegradual increase of the loading force in the jack B.In this way, the loading force in two adjacent jackswas successively increased and decreased so thatthe resultant force was a "moving force" with aconstant value. Twelve loading levels varying from100 kN to 340 kN with step-by-step increments of20 kN were applied. At each load level, one "run"of the load from end to end and backward wasperformed. The readings of the gauges, crackwidth, and crack pattern were registered when theloading force had reached its full value in each ofthe locations A-E. After each complete "run" ofthe load the total unloading was done.

Strains of continuously linked-up bases at thecompressed and tensioned edges, as well as at thecrossing diagonals were registered. The metalplugs embedded into the surface layer of the beamserved to make these measurements. The basesshown in Fig. 2 form a "truss" consisting of"struts" and "ties" enabling the calculation not onlyof the strains but also of the deflections, by using amethod based on Williot-Mohr translocationpolygons. The reliability and checking of testresults were increased because of the possibility tocompare the deflections computed from theelongations measured on truss with those measureddirectly by the dial gauges (1-5 in Fig. 2). Thismethod offered the possibility of separating thedeflections due to shear from those due to bending.The network of measurement bases (Fig. 2) wasdecomposed into two truss systems. Theelongations were then evaluated by using theabove-mentioned method of translocationpolygons. More detailed information about theloading procedure and the evaluation of thedisplacements has been reported earlier'".

ResultsIt should be noted that the crack pattern of a

beam subjected to a moving load differs from thecrack pattern induced by a stationary load (Fig. 4).In Fig. 4 the crack patterns of a beam SUbjected toa mid-span stationary load (top) and to a movingload (bottom) are sketched. In the central part of

Fig. 3-Expcrimcntal set-up.

Fig. 4--Crack pattern in a stationary-loaded beam (upper part)and in a beam subjected to a moving load (lower part):concrete crushing is marked with dots.

the beam subjected to a moving load the cracks atright angles with different inclinations wereformed as a consequence of the change of theprincipal stresses. It is obvious that the differentcrack patterns influenced the overall deformationswhich is not considered in standards so far.

The method adopted for measuring thedeformations enabled to calculate not only thework of the external forces but also the work of theinternal forces (bending moment and shear). Thevalue of the work of the bending moment Wbm at agiven load level and at a given location of theloading force was determined numerically byevaluating at first the energy of a cross-section. Inaccordance with the formulae of buildingsmechanics the energy of the cross-section j whenthe force is applied in C at the load level k wasobtained from the formula

1 5

w} ="2 ~(Mj'B'k + Mj,C,k) x ~(1 / rj), ... (1)

114 INDIAN 1. ENG. MATER. sci., JUNE 1998

where .J(lIr) is the increment of the curvature due(i)to the variation of the location of the loadingforce, (ii) to the increment of the loading level;M.J.B,k (M.J,c,J is the bending moment in the cross-sectionj when the loading force is applied in B (C)and at the loading level k.

The bending-related energy of the whole beamat the load level k was obtained by the numericalintegration of the values ~ along the length of thebeam according to the formula

111 n

Whm = 2~(Wj_1 + Wj), ... (2)

where .Ji is the base length (the same along thewhole span of the beam), n is the number of thecross-sections considered including beam ends,~.I (~) is the energy of the cross-sectionj-l (j).

The work of the shear forces was calculatedfrom the vertical displacements (deflections)caused by shear forces. The value of the work ofthe shear forces at any given location of theloading force at the load level k was calculatedaccording to the formula

1 n

W (! = "2 ~ Qj x 11Sj , ..• (3}

where Qj is the shear force at the cross-section j,Llsj is the increment of the vertical displacementbetween two adjacent sections.

The value of the work of the external forces atthe load level k and location A of the loading forcewas determined from the formula

1Wk.A=Wk-I,O+2"xFkXI1WA, ... (4)

where Wk.I,O is the dissipated energy at the loadlevel k-l , F, is the external force at the load level k,L1WA is the increment of the deflection at the loadpoint.

The work of the external forces at the otherlocations of the loading force was calculatedaccording to the formula

L 1Wk,L = LWk,l + F, x 2" x (/1 WI. + /1 WI.-I) ... (5)

I=A

where Wk,l is the work at the i-th location of theloading force, F, is the external force at the loadlevel k, L1Wl. is the increment of the deflection inthe load point considered.

As an example, the increment of the strainwork as a function of both: the location of the

oA It c ED

Fig. 5--Distribution of the total work of the external forcesalong the beam; 1 - the work at the run of the loading forcefrom the left end of the beam to the right one; 2 - the work atloading force running backward.

5

...,

..•• 3~.

0~~~~-------+------4-~o 5 10 15

f. mm

Fig. 6--Variation of the work of external forces with themidspan deflection.

loading force and the direction of its motion alongthe beam (forward-curve I) or Backward-curve2) is plotted in Fig. 5. In Fig. 6 the total work ofthe external forces ensuing from the abovementioned loading procedure and reflecting themotion of the loading force along the beam isgiven. The evaluation of the strain work made itpossible to plot the value of the strain work versusthe midspan deflection (Fig. 7), or in other wordsthe work of the external forces - the envelope(curve I), as well as the work of the bendingmoments (curve 2) and the work of the shear forces(curve 3). The values of the midspan deflections inFigs 7-10 represent the deflections measured at thelocation C of the loading force. The amount ofwork done by the bending moment and by theshear force, as a part of the total strain work isplotted in Fig. 8, Figs 7 and 8 show that the amountof the bending-related work is a decreasingfunction of the deflection, while the amount of the

\

NURNBERGEROV A et al.: STRAIN ENERGY OF RC BEAMS SUBJECfED TO A MOVING LOAD 115

6 6

5 5

4 4

...•...3 ~ 3

~ ~.2 2

O~--~~r-------~----~~o 5 15

t. mm f. mm10

O+---~~r-------~----~~o 5 10 15

Fig. 7-Re1ationship between the work of load and the Fig. 9--Elastic and the dissipated energies; I - the total workmidspan deflection; 1 - the total work of the external forces of the external forces (the envelope); 2 - elastic energy; 3 -(the envelope); 2 - work of bending moments; 3 - work of dissipated energy.shear forces.

1.00 -r----r----,-----,,,

0.75

1:1 0.50+----+----t-----H

0.25

O+-------+-------t--------~o 5 10 15

f. mm

Fig. 8-Shares of the works of the bending moments (curve 1)and shear forces (curve 2) versus midspan deflection.

shear-related work is an increasing function. Forinstance in Fig. 8 the amount of the shear-relatedwork at the beginning of the loading process was0.25 and it increased to 0.44. It can be noticed thatthe amounts of the work done by the internalforces is a linear function of the deflections. Somescattering of the results at the beginning may beascribed to cracking.

During the motion of the loading force along abeam a gradual unloading of the cross-sections tothe left of the load point takes place and therefore apart of the stored energy is released. Furthermore,after a full cycle (with the load back to the left endof the beam) the beam was completely unloaded.In this way the calculation of the dissipated energyat each load level was carried out. In Fig. 9 theelastic energy (curve 2) and the dissipated energy(curve 3) are plotted versus the midspan deflection.Fig. 9 shows that the relations between the

2+-----~---~~-~

O+-~~~~-----+------~~o 5 10 15

f. mm

Fig.1O-Elastic and dissipated strain energies due to thebending moment and shear force; 1 - work of bendingmoments; 2 - work of shear forces; dashed lines - elasticenergy; dotted lines - dissipated energy.

midspan deflection and the dissipated energy andthe elastic energy, can be described via ascendingcurves. However, after reaching the yield limit ofthe reinforcement, the dissipated energy grew morequickly than the elastic energy. It can be seen inFig. 10 that the dissipated energies related tobending and shear respectively are approximatelythe same, although the relative amount of theshear-related work varied from 0.25 to 0.44(Fig. 8), which implies that the contribution of theshear force on the irreversible deformation issomewhat higher than the contribution of thebending moment.

ConclusionA method for the evaluation of the strain

\

116 INDIAN 1. ENG. MATER. scr., JUNE 1998

energy RC beam subjected to a moving load wasdeveloped. The procedure presented here makes itpossible to evaluate separately the bending-relatedwork and the shear-related work. The values of theelastic and dissipated energies showed that theshear forces have a relatively greater influence onthe deflections (either elastic or inelastic) than thebending moments.

AcknowledgmentThe authors are grateful to the Slovak Grant

Agency for science VEGA (Grant No 2/1260/97 &Grant No 2/4086/98) for partial support of thiswork.

NomenclatureA,B,C,D,E denomination of the location of the loading

forcemodulus of elasticity of concretemodulus of elasticity of steeltensile strength of concreteprism strength of concretetensile strength of steelyield stress of steelloading force at the load level kindex of the given cross-sectionindex of the given load levelbending moment in the cross-section} withthe load applied in B at the load level kbending moment in the cross-section} withthe load applied in C at the load level knumber of cross-sections (including both endsections) considered for numerical integrationshear force in the cross-section}stirrup spacingbending-related work for a given load leveland location of loading forcebending-related work

EcE;,h,J;"/s,f.yr,}kJ~,B,k

n

= work of the external forces with the loadapplied inAdissipated energy at the load level k-Iwork of the external forces with the loadapplied in an arbitrary locationwork of the external forces with the loadapplied in locations other than Ashear-related worklength ofthe measuring baseincrement of the curvature due to the changein the location ofthe loading force and theload levelincrement of the vertical displacement(deflection) between the two adjacent cross-sections (j-I,})increment of the deflection in the crosssection Aincrement of the deflection at the given loadpoint

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