Variable compression panel thickness for concrete members under bending and torsion: Experimental study

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Engineering Structures 279 (2023) 115606

Available online 24 January 2023

Variable compression panel thickness for concrete members under bending and torsion: Experimental study

Joonas Tulonen

^*

, Anssi Laaksonen

Faculty of Built Environment – Concrete and Bridge Structures, Tampere University, Tampere, Finland

A R T I C L E I N F O Keywords:

Reinforced concrete beams Prestressed concrete Bending

Torsion Combined action Space truss model Experiments

Concrete compressive strength

A B S T R A C T

An experimental study was conducted on the behaviour of eight reinforced and prestressed concrete beams subjected to combined action of bending and torsion. The beams were heavily reinforced in the longitudinal direction so that the height of the concrete compression zone under pure bending would be much larger than the concrete diagonal strut thickness under pure torsion and measurements were done so that behaviour of this compressive zone could be observed. The amount of the reinforcement was designed close to the over- reinforcement limits, as would happen if the cross-section dimensions of an actual structure would be mini- mized in the design. The ultimate resistances and beam deformations were compared against the results of a plasticity-based space truss model where the panel on the compressive side of the cross-section had variable thickness in order to account for the resultant force demands of both the bending and torsion. Most heavily reinforced beams failed in a brittle manner due to the ineffectiveness of diagonally cracked concrete to resist the high normal stresses arising from the bending. The proposed model gave better predictions than the classical model where the panel thickness was constant, and the results showed the significance of concrete strength reduction factor contribution in ultimate resistance of concrete members under combined bending and torsion.

1. Introduction

In the design of reinforced concrete beam structures, it is common that the beam reinforcement of any given cross-section is designed to resist multiple resultant forces, most commonly bending moment, shear force and torque. These forces are usually from the same source, like from the self-weight of the structure or from a moving load on a bridge, and therefore, they occur simultaneously. The effect of these combined actions was studied extensively by a number of researchers in the 1960s- 70s when a large amount of experimental research was carried out on reinforced and prestressed concrete beams with different reinforcement arrangements and cross-section shapes. [1–13] Pure torsion was also extensively studied [14–16] during this era and this research effort eventually lead to the formation of space truss models, where the shear stresses induced by torsion are carried by variable angle diagonal concrete struts and the transverse and longitudinal reinforcement at the edge of the cross-section. [17–20] These variable angle space truss models made it to the CEB-FIP model codes and form the basis of the current Eurocode shear and torsion design rules.[21–24] During the 1980s and the 1990s the theories were further developed by research on

the behaviour of reinforced concrete membranes under in-plane shear and normal forces, which yielded new reinforced concrete material models to be included in the calculation of full torsional response of under and over-reinforced concrete members. [25–28] During the later decades, the experimental and theoretical research has focused on advancing deeper understanding of the pure torsion behaviour of concrete members and the numerical modelling of concrete members.

[29–38] Some research on the use of advanced methods in the case combined actions has also been carried out and verified against experimental data collected in databases. [39–43] Plasticity based limit analysis or simplified models of different compression field models for torsion and shear have also gained some research focus lately, where the objective has been to find more design oriented solutions to solving the cross-section ultimate resistance under the combined actions while still including the aspects of advanced concrete material models for shear.

[44–48].

Reinforcement design in practice is mainly performed with plasticity-based space truss models since the more advanced models are not practical for design use. In particular, bending and torsion designs are commonly done separately by a superposition of the reinforcements.

[49] However, this approach overlooks the fact that in the design for

* Corresponding author.

E-mail address: joonas.tulonen@tuni.fi (J. Tulonen).

Contents lists available at ScienceDirect

Engineering Structures

journal homepage: www.elsevier.com/locate/engstruct

https://doi.org/10.1016/j.engstruct.2023.115606

Received 1 July 2022; Received in revised form 2 January 2023; Accepted 8 January 2023

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Nomenclature

Ac area of concrete cross-section, bc•hc

Ae effective area of concrete cross-section enclosed by shear flow path, be•he

Ap total area of prestressing steel in cross- section Ap,n nominal area of prestressing steel strand

As,L,i total area of longitudinal steel reinforcement in layer i As,L,ten/com/side total area of longitudinal steel reinforcement

intension/compression/side -panel

As,L,tot total area of longitudinal steel reinforcement in the whole cross-section

As,T area of single stirrup leg

Fcal-V total load of the test beam calculated with space truss model with variable compression panel thickness Fcal-V,fit total load of the test beam calculated with space truss

model with variable compression panel thickness with concrete strength reduction factor derived from the test results

Fcal-C total load of the test beam calculated with space truss model with constant compression panel thickness Fexp total experimental load

Fexp,max total experimental load at failure DT diameter of stirrup leg

Dten/com/side,L diameter of longitudinal reinforcement steel in the tension/compression/side panel

LVDT Linear Variable Differential Transducer M bending moment in cross-section

Mexp bending moment in the research area during experiment, Fexp/2•0,8 m

Mexp,max bending moment in the research area at failure

MR,y bending resistance of cross- section due to yielding of steel reinforcement

Nbal sum of all longitudinal cross- section forces NL,ten/com/side sum of longitudinal forces in thetension/

compression/side panel Pp total force in prestressing steel

R ratio of the amount of longitudinal reinforcement in the compressive side of the beam to the amount of longitudinal reinforcement in the tension side of the beam

STM space truss model T torque in cross-section

Texp torque in the research area during experiment, Fexp/2•esup

TR,exp torque in the research area at failure

TR,y torque resistance of cross-section due to yielding of steel reinforcement

bc concrete cross-section width

be effective width of concrete cross- section for torsion, bc - te

d the effective height of the cross- section in bending dLVDT distance from the concrete surface to the LVDT strain

gauge measurement plane

esup eccentricity of the support from the beam central axis fc cylinder compressive strength of concrete

fp0.01 characteristic value of prestressing steel strength, 1% proof strength

fsy,T stirrup steel yield strength

fsy,com/ten/side,L longitudinal steel reinforcement yield strength in tension/compression/side panel

hc height of the concrete cross- section

he effective height of concrete cross- section for torsion, hc - te

nL plane stress of 2D element in L- direction nT plane stress of 2D element in T- direction q shear flow at the edge of 2D element

ss,L spacing of the reinforcement steel of 2D element in L- direction

ss,T spacing of the reinforcement steel of 2D element in T- direction or stirrup spacing

te effective wall thickness of panels in space truss model, Ac/ uc

te,com effective wall thickness of compressive panel in space truss model

te,com,fit effective wall thickness of compressive panel in space truss model where strength reduction value for concrete has been fitted to experimental data

uc perimeter of the concrete cross- section

ue effective perimeter of the shear flow path in the space truss model

uef effective perimeter of the shear flow path in the space truss model

yp distance of the of prestressing steel from the bottom of the beam

ys,L,i distance of the of the longitudinal steel reinforcement steel in layer i from the bottom of the beam

ze internal lever arm of the cross- section for bending moment α angle of strain gauge from the beam longitudinal axis ε_1,com principal tensile strain of the compressive concrete panel ε_1,com,fit principal tensile strain of the compressive concrete panel in

space truss model where strength reduction value for concrete has been fitted to experimental data

ε_cor correction value for measured surface strains due to strain gauge distance from the surface

Φ twist of the beam around the longitudinal axis κ bending curvature of the beam in the longitudinal

direction

θten/com/side direction of diagonal compressive stress of tension/

compression/side panel

θ_thrust direction of the diagonal compressive failure stress of compression panel determined visually from the failure surface

ρ_T relative amount of stirrups, AsT•uc/(Ac•ss,T)

ρ_L relative amount of longitudinal steel, (As,L,tot +Ap,tot)/Ac

σ_c concrete stress

σc,ten/com/side concrete stress in the tension/compression/side panel σ_c,y diagonal concrete stress at the point of reinforcement

yielding

σ_s,L stress of steel reinforcement of 2D element in L-direction σs,L,ten/com/side steel stress of longitudinal steel reinforcement in the

tension/compression/side panel

σ_s,T stress of steel reinforcement of 2D element in L-direction σs,T,ten/com/side steel stress of stirrup steel reinforcement in the

tension/compression/side panel

σ_p0 initial stress in the prestressing steel after the lock-in losses σp stress in the prestressing steel

ν_c compressive strength reduction factor for theconcrete under shear stresses

ν_c,fit strength reduction factor for concrete determined so that the beam capacity calculated with variable compression panel space truss model equals the experimental maximum load

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bending of solid cross-section the parameter that is being solved is the height of the concrete compression zone, which might be a substantial portion of the whole cross-section height if the cross-section is heavily reinforced, and that in design for torsion the diagonal compression struts have thickness based on the twisting properties of the cross-section under pure torsion and that thickness usually has rather small value when compared to the cross-section dimensions. [30] An example of where this can have a large effect is a continuous post-tensioned T- shaped bridge beam at the pier if the beam cross-section remains the same from the sagging region to the hogging region. In this common case, the beam cross-section at the pier can be heavily reinforced in the longitudinal direction, even over-reinforced, since there is no deck slab on the compression side and the amount of prestressing steel is the same as in the sagging region.

A total of 300 tests on solid rectangular beams with purely combined bending and torsion have been found from the literature. Almost all of the tests have been made during 1950–1970, 267 of the specimens had rather small cross-sections (minimum side length less than 200 mm) and only 19 were prestressed. Many of the specimens were rather lightly reinforced for bending and only 83 of them had mechanical reinforcement ratio over 0.25, of which had median torque-to-bending ratio of 0.36. [1–5,7,8,11,13,14,17,20] To investigate prestressed concrete structures with high reinforcement ratios and low torque-to-bending ratios this study examined a plasticity-based space truss model with a variable compression panel thickness which can be used to calculate the ultimate resistance of reinforced and prestressed concrete cross-sections under torsion and bending. In the presented model, the concrete compression failure happens in the diagonal direction on the

Fig. 1.Test setup, end support and the location of supports, loads and deformation measurements (LVDTs) underneath the beam.

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compressive side after the transverse and longitudinal steel in the tensile side has yielded. This way it was possible to combine aspects from the truss model for bending with the space truss model for torsion into a single model, which previously have mostly been done with computa- tionally heavy strain compatibility based calculations. [39] The model was verified with eccentric four-point beam bending experiments, where the amount of unsymmetric longitudinal reinforcement was large and the torque-to-bending ratio was relatively low (<0.5). The chosen torque-to-bending ratio corresponded approximately to the state of stress encountered at the piers of straight multiple-beam post-tensioned beam bridges with spans less than 50 m.

2. Experimental program 2.1. Test specimen details

Eight concrete beams with differing reinforcement arrangements and torque-to-bending -ratios were tested to failure with eccentric four- point-bending experiments. Composite construction with tubular steel beams was used in the beam ends to ensure that the failure and the change of bending and torsional stiffness during the loading would occur in the middle of the beams where the bending moment and torque moment have constant values. Fig. 1 shows the test setup. The eccentricity and amount of torque in relation to the bending moment was adjusted by moving the end bearings relative to the main beam central axis.

The test area of the beams had a square cross-section of 232x235 mm and total span of the beams was 2.9 m. The beams were reinforced with longitudinal steel bars and evenly spaced closed stirrups. Four of the beams were of ordinary reinforced concrete construction and the other four were centrally prestressed with two steel strands in smooth plastic ducts filled with cement grout. Due to the smoothness of the ducts, the bond between strands and concrete may have not been perfect, but the location of the prestressing steel in the centre of cross-section means it should not have a large effect, since they were close to the neutral axis for bending.

Reinforcement details and cross-section dimensions of the beams in the test area are presented in Fig. 2 and in Table 1. The beams were named in a following way: P#-0XX-NN, where # stands for the beam number, 0XX stands for the torque-to-bending ratio of the beam (e.g.

050 =50%) and NN stands for the type of reinforcement ratio of the stirrups and the longitudinal steel respectively (H =high, L =low). An additional letter “p” was added to the end of the name if the beam was

prestressed.

2.2. Material properties

All beams were cast at the same time with a single batch of self- compacting concrete (SCC) with a design strength class of C30/37.

During the casting the beam was upside down, so that the top of the beam, which was the main area of interest, was at the bottom for a good compaction and smooth surface. The beams were tested at different ages which caused some variation in the concrete strength properties between the beams. The tested concrete strengths are given in Table 1. The concrete strength was tested from D100x300 cylinders cast from the same batch of concrete as the beams and were cured in a similar con- ditions. The average density of the concrete cylinders was 2186 kg/m³.

Reinforcement yield strengths were determined from manufacturers quality assurance documents and some steel samples were also tested in the laboratory. Average yield strength for the diameter 16 mm bars was 588 MPa and 538 MPa for the diameter 8 mm bars. The average ultimate strengths were 691 MPa and 642 MPa respectively. The bars with 8 mm diameter were cold formed (de-coiled) so they lacked a distinct yielding plateau. The prestressing steel strands of nominal diameter of 15.7 mm were of type St1600/1860 and were not tested since the stresses were not expected to increase to yielding levels during the experiments due to a central placement of the prestressing strands in the cross-section. The target stress for the prestressing was 1000 MPa which would translate to 300 kN of total prestressing force and 5.7 MPa of mean concrete compressive stress in the test area when excluding the two 50 mm plastic ducts for strands from the cross-section. Both strands were stressed simultaneously from one end and the stressing force was estimated from the hydraulic pressure of the prestressing jack, which had been cali- brated against a reference load cell, and from the elongation of the strands which was estimated from the movement of the hydraulic jack.

In both ends of the beams, the prestressing strands were anchored to Paul Type 34 0.62^′′ Grip Anchors with wedges. A steel plate with thickness of 20 mm was used under the anchors to distribute local concrete stresses to larger area. Results from a separate test to determine the lock-in losses for the chosen anchor type were also used in the estimation of the prestress.

2.3. Testing procedure and instrumentation

Test specimens were placed on pinned end supports which were made of spherical steel ball bearings that allowed rotations in any

Fig. 2. Arrangement of reinforcement bars and cross-section dimensions in the test area.

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direction and provided a small contact area with the test specimen so that the torsion arm length, e_sup, would not change too much with large rotations. Both supports were on rollers so that one end allowed movement in the longitudinal direction of the beam and the other one allowed movement in the transverse direction.

The loading was applied by two hydraulic jacks, with maximum a capacity of 250 kN each, which were placed at a distance of 0.8 m from the beam end supports. The hydraulic jacks were supported by an external steel frame. The forces from jack end plates were concentrated with smaller steel pieces and rollers. With this test setup, and assuming an even loading from both hydraulic jacks, the bending moment and torque in the test area were:

Mexp=0.5Fexp•0.8m Texp=0.5Fexp•esup

where.

esup is the horizontal distance from the contact point of the spherical ball bearing to the centre of the main beam cross-section, shown in Fig. 1. The distances were 0.125 m, 0.25 m and 0.375 m for beams with torque/bending ratios of 0.16, 0.31 and 0.47, respectively.

Fexp is the total loading of both hydraulic jacks.

Experiments for each beam were started with smaller load cycles.

After the initial cycles the load was linearly increased until failure was exceeded. The loading was force controlled and the force was increased in both jacks at the same rate.

The loading force was measured from both hydraulic jacks with electronic force transducers. Vertical deformations of the beams were measured from five different locations with Linear Variable Differential Transformers (LVDT) as shown in Fig. 1. Two LVDTs were placed at each end of the test area on both sides of the beam, which allowed the measurement of the beam deflection from bending at the beam centreline and the angle of twist around beam longitudinal axis due to torque.

Beam deflection at the centreline in these locations was interpolated from the measurements, since the distance of the LVDTs from the centreline was known. One LVDT was placed in the middle of the beam to measure the maximum beam deflection. The average twist and average curvature of the test area can be derived from the LVDT results. The measurement area was larger than the test area so the average twist and curvature were affected by the thicker concrete parts of the beams. The thicker cross-sections had 7% more height and 14% more cross-section area than the cross-section in the test area and made up 30% of the test measurement area length. It can be estimated that the measured average curvature and twist would appear approximately 5% lower, compared to the situation if the cross-section would be the same for the whole measurement length. This effect should be considered if the results are compared with other tests or calculations, where the beam stiffness is studied.

LVDTs were also used for the measurement of the average concrete surface strains on both of the side faces and on the top face of the beam.

For each strain measurement, two M10 bolts were attached to the concrete surface with strong adhesive and the LVDT and a metal-strip were fixed to these bolts. Average concrete surface strains were measured in multiple directions as is shown in Fig. 3 so that the principal strains and their angle can be determined from the results for the top surface of the beam. The crossed strain gauges of the side faces were used to measure the shear strain. The most significant source for error in this type of strain measurement with the LVDTs is that the measurement was not done directly on the surface, but some distance from it. Therefore, in addition to the plane strains of the concrete surface, the strain measurements also include the rotations of the surface due to plane curvature. In the top surface of the beam, the strains from the LVDTs can be transformed to surface strains by assuming that the measured average beam curvature and twist of the test area corresponds to the curvature and twist of the surface. To transform LVDT strain measurement to surface strain, the value to be removed from the actual measurement is:

Table 1 Material properties and reinforcement ratios. Beam name fc [MPa] Stirrups Longitudinal steel, tension (bottom) Longitudinal steel, compression (top) Longitudinal steel, sides Prestressing steel ρT [%] ρL [%] ss,T [mm] DT [mm] fsy,T [MPa] No. Dten,L [mm] fsy,ten,L [MPa] No. Dcom,L [mm] fsy,com,L [MPa] No. Dside,L [mm] fsy,side,L [MPa] No. Ap,n [mm2] σp0 [MPa] fp0.01 [MPa] P1-031-LL 37.4 130 8 538 5 8 538 4 8 538 3 +3 8 538 0 – – – 0.66 1.4 P2-016-LL 37.4 130 8 538 5 8 538 4 8 538 3 +3 8 538 0 – – – 0.66 1.4 P3-031-LH 42.4 130 8 538 5 16 588 4 8 538 3 +3 8 538 0 – – – 0.66 2.9 P4-016-LH 42.4 130 8 538 5 16 588 4 8 538 3 +3 8 538 0 – – – 0.66 2.9 P5-031- LHp 44.5 130 8 538 5 16 588 4 8 538 3 +3 8 538 2 150 985 1600 0.66 3.4 P6-047- LHp 44.5 130 8 538 5 16 588 4 8 538 3 +3 8 538 2 150 1021 1600 0.66 3.4 P7-031- HHp 44.5 65 8 538 5 16 588 4 8 538 3 +3 8 538 2 150 1035 1600 1.3 3.4 P8-047- HHp 44.5 65 8 538 5 16 588 4 8 538 3 +3 8 538 2 150 1019 1600 1.3 3.4 Notes: ss =spacing, D =diameter, fsy =yield strength, Ap =prestressing steel strand nominal area, σp0 =prestressing steel stress after the lock in lossess, fp0.01 =0.01 limit strength, ρT =(AsTuc)/(Acs),ρL =(As,L,tot +Ap,tot)/Ac, uc =0.934 m, Ac =0.0545 m2

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εcor=(

ϕsin(2α) +κcos²(α))

dLVDT (1)

Where ϕ is the beam twist, κ is the beam curvature, α is the angle between the strain gauge direction and the beam longitudinal axis and dLVDT is the distance of the measurement plane from the concrete surface.

3. Space truss models for the analysis of experimental results 3.1. Shear flow idealization in space truss models

In this section, two plasticity-based models are presented for the calculation of ultimate resistance in combined bending and torque. Both models are based on space truss analogy, where normal and shear forces of the cross-section are carried by panels that are located at the edges of the cross-section, as is shown in Fig. 4. The first model is the ordinary truss model where the concrete panel thickness is assumed to be constant and is presented as the base for the second model which is modified so that, under combined torsion and bending, the effective concrete panel on the compressive side of the beam has a variable thickness value similar to a pure bending case. If the beam reinforcement ratio for bending is high, the effective height of compression area in the ultimate limit state is also higher and the internal moment lever arm is lower. It is assumed in the modified model that this reduction in lever arm also applies for the torsional lever arm.

3.2. Plasticity-based model with constant panel thickness te

The calculation of plastic torque resistance of the concrete section can be done by integrating the plastic shear flow resistance of the cross- section walls, or panels, that contain the hoop reinforcement and the longitudinal reinforcement, multiplied with the distance to the centre of twist, over the perimeter that is enclosed by the panel centre lines. [30]

The panels in the calculation represent the location of shear flow of solid concrete cross-section that is cracked. The equilibrium equations for section of 2D -panel (in Fig. 5) with thickness t are:

nL=As,L

ss,L

σs,L− qcot(θ) (2)

nT=As,T

ss,T

σs,T− qtan(θ) (3)

σct= − q(tan(θ) +cot(θ) ) (4)

σctcos²(θ) = − qcot(θ) (5)

Assuming an even distribution of longitudinal reinforcement in the perimeter of the cross-section and yielding of all of the reinforcement in both directions, the fully plastic torque resistance of the cross-section is:

Fig. 3. Arrangement of the LVDTs used for measurement of the average surface strains.

Fig. 4.Cross-section idealization for models and shear flow (q) distribution due to torque.

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TR,y=2Ae

̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅

fsy,LAs,L,tot

ue

fsy,TAs,T

ssT

√

(6) Where Ae and ue are the area and the perimeter enclosed by the shear panel centre lines, fsy is the yield strength of reinforcement, As,L,tot is the total area of longitudinal reinforcement, As,T is the area of single stirrup leg and ss,T is the stirrup spacing.

The diagonal concrete compressive stress at the point of yielding can be checked with equation:

σc,y=1 te

(fsy,LAs,L,tot

ue

+fsy,TAs,T

ssT

) (7)

Too high a concrete stress will cause the structure to fail in a brittle manner before yielding of the reinforcement. The limit value for the amount of reinforcement in which the concrete crushes before yielding of both longitudinal and transverse reinforcement can be solved by checking with Eq. (7) that σ_c,y≤νfc, where ν is the strength reduction factor for concrete under shear stress. Different reduction factors have been suggested by many researchers, but essentially the accurate value cannot be solved without solving the cross-section strains. Similarly, the amount of reinforcement in the case that neither reinforcement nor reinforcement in only one direction yields before the concrete crushes, cannot be solved without solving the strains of the cross-section. In a torsion design done with Eurocode 2 the strength reduction factor can be taken as:

ν=0.6(1− fc/250) (8)

where fc is concrete compressive cylinder strength in MPa. For the beams presented in this paper, the factor would be approximately 0.5.

[23].

Prestressing steel can be taken into account in equations (6) and (7) by adding prestressing steel force σ_pAp to the longitudinal force of the reinforcement, fsy,LAs,L,tot. Ap is area of prestressing steel in the cross- section and σ_pis the stress of prestressing steel, which can be taken as the prestress value σ_p0or as the yield strength of the steel, fp0.01, depending on how much additional elongation is expected in the prestressing steel.

In the case of combined bending (M) and torque (T), let us first consider the longitudinal resultant forces of different panels - tension

panel, side panel and compression panel - in the case of that the longitudinal steel in tension panel is yielding in tension:

NL,ten=fsy,LAs,L,ten− qbecot(θten) (9)

NL,side= − qhecot(θside) (10)

NL,com= (

fsy,LAs,L,ten− 2M ze

)

− qbecot(θcom) (11)

Assuming that the stirrups are also yielding and that their force is in an equilibrium with the transverse component of diagonal concrete stress (n_Tequals zero) in each panel, then the equation (3) yields:

tan(θten) =tan(θside) =tan(θcom) =2Ae

T fsy,TAs,T

ssT (12)

The cross-section equilibrium demands that the sum of all the longitudinal forces must be zero:

Nbal=NL,ten+2NL,side+NL,com=2fsy,LAs,L,ten− 2M ze

− T² 4A²_e

ssT(2be+2he) fsy,TAs,T

=0 (13) Which yields the interaction equation of bending and torque resistances:

M zefsy,LAs,L,ten

+ T²uessT

8Ae2fsy,TAs,Tfsy,LAs,L,ten

=1 (14)

The equation transforms into the dimensionless form, when taking into account equation (6) and assuming that the pure bending resistance MR,y =MR,y,ten =zefsy,LAs,L,ten and that in the calculation of pure torque resistance TR,y (Eq. (6)), As,L,tot =2As,L,ten:

M MR,y

+T²

T_R,y² =1 (15)

Prestressing steel can be taken into account as stated before as longitudinal reinforcement, if also effective internal lever arm ze is changed to respect the true location of the tension force resultant. Diagonal concrete stresses can be checked by using the diagonal angle from Eq.

(12) in Eq. (4).

Equation (15) is equivalent to the design method where the torque resistance and bending resistance are calculated separately, so that the evenly distributed longitudinal reinforcement required by the torsion is subtracted from the cross-section when calculating the bending resistance.

The interaction formula needs to be refined for the case where there is an uneven distribution of the longitudinal reinforcement, so that the compression side has less longitudinal reinforcement than the tension side, so that when the torque-to-bending ratio is high, the longitudinal tensile force induced by the torque may be higher than the compressive force induced by bending, thus, causing the reinforcement in the compressive side of the beam to yield in tension before the reinforcement in the tension side exceeds its yield stress, in which case the interaction formula becomes:

− M zefsy,LAs,L,com

+ T²uefssT

8Ae2fsy,TAs,Tfsy,LAs,L,com

=1 (16)

For a rectangular cross-section where R = As,L,com/As,L,ten, the dimensionless interaction curve may be written as:

M MR,y

+ T²

T²_R,y=1 if M>(0.5− 0.5R)MR,y (17)

− M MR,y

+ T²

T²_R,y=R if M<(0.5− 0.5R)MR,y (18) MR,y and TR,y are calculated with the same assumptions as in Eq. (15).

Fig. 5.Force equilibrium of 2D -panel element of reinforced concrete.

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3.3. Plasticity-based model with variable compressive panel thickness te

Similar equations as (17) and (18) have been presented by others authors before, since it is, essentially, the space truss model developed in the 1970 s. [49,50] However, the equations (6–18) do not take into account the reduced bending lever arm of cross-sections with a high amount of tensile reinforcement where the area of compressed concrete must be larger for the cross-section to be in equilibrium. In fact, it is assumed in the previous equations that the compression component of bending is solely carried by the longitudinal reinforcement located in the compressed panel since the diagonal concrete stress is a function of the torque and the amount of transverse reinforcement.

In a similar fashion to solving the height of the compressive concrete block in a bending problem, the thickness of a compressive panel in combined bending and torsion can be solved if it is assumed that the concrete compressive stress eventually reaches failure stress, the effective thickness of the other panels remains unchanged and the longitudinal reinforcement in the compression panel is neglected. The longitudinal equilibrium (Eq. (2)) of the compressive panel without the longitudinal reinforcement yields the angle of compression diagonal stress:

θcom=atan (qbe

NL

)

=atan (Tzebe

2AeM )

=atan

(T

2M )

(19) The internal lever arm for bending is:

ze=d− 0.5te,com (20)

In case the tensile reinforcement is located at the centreline of the panel in tension, the effective height of the cross-section is:

d=hc− 0.5te,ten (21)

When inserting equation (19) into equation (4), te,com becomes solvable from equation:

fcte,com=T²+4M²

4AeM = T²/M+4M 4be

(d− te,com

)

te,com=d−

̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅

d²+ 1 fcbe

(T² 2M+2M

√ )

(22) If the tensile reinforcement is not concentrated in the centre of the tensile panel, eg. when there is centrally placed prestressing steel or the sides of the beam have longitudinal steel reinforcement, the internal lever arm ze for bending is decoupled from the torsional lever arm Ae in equation (19) and the equation for the cross-section effective height becomes:

d=hc− Apσpyp+∑

As,L,ifs,L,iys,L,i

Apσp+∑

As,L,ifs,L,i (23)

where y denotes the distance of the reinforcement i or prestressing steel from the edge of the cross-section. This will lead to a much more complex solution for the compressive panel thickness, te,com, since it will require solving a quartic equation.

When the compressive panel thickness is solved, the longitudinal force of the compressive panel in equation (13) can now be replaced with values from equation (5):

NL,com=fcbete,comcos²(θcom) (24)

Essentially, the equations (9) and (10) remain the same, but the torsional lever arm A_eand side panel length h_eare now also functions of te,com, which itself is a function of the bending moment and torque, and the longitudinal equilibrium function for the cross section becomes:

Nbal(T,M) =fcbete,com(T,M)cos²(θcom(T,M) ) +As,L,tenfsy,L

− T²(be+2he(T,M) )ssT

4(Ae(T,M) )²fsy,TAs,T

(25) Solving the roots of the function gives the interaction curve for bending and torque. Due to the non-linear nature of function, numerical methods should be used. Again, if the tensile reinforcement or prestressing steel is not located in the panel in tension, the internal lever arm for bending should be changed accordingly.

The assumption in the equation (25) is that the transverse steel in the compression panel is not yielding. The stress of transverse steel in the compression panel can be calculated from the transverse equilibrium:

σsT= T² 4Ae(T,M)M

ssT

AsT (26)

The angle and the magnitude of the diagonal concrete stress in the bottom and side panels can be checked with Eqs. (12) and (4).

The reduction in the concrete compressive strength at the compressive panel due to shear stress can be taken into account by replacing the fc in equations (24) and (25) with ν_cfc, where ν_cis the strength reduction factor. Again, the magnitude of strength reduction factor depends on the principal strains of the compressive panel, which can be estimated in the ultimate limit state by calculating transverse strains from the transverse equilibrium in equation (26) and by assuming the value of ultimate compressive strain of concrete is ε_cu=-0.002 and that the direction of principal strains coincidence with the direction of compression diagonal stress calculated with equation (19). Now Mohr’s circle of strains can be used to determine principal tensile strain of the compressive panel at the failure:

ε1,com=εT+ (εT− ε2)tan²θ=σsT

Es

+ (σsT

Es

− εcu

)(Tzebe

2AeM )₂

(27) It can be seen that the principal strain value is affected by the effective cross-section area for torsion Ae, since it is component of both transverse strains and principal strain angle. A_eis a function of t_e,com which is a function of the strength reduction factor and therefore also the principal tensile strain, so using this equation would require an iterative calculation.

In this experimental study, the strength reduction factor of the compressive panel in the beam failure was evaluated from the experimental data by assuming that all of the longitudinal and stirrup reinforcement in the tension side of the beam reached the yield stress in the experiment. The factor can then be solved from equation (25) since M = Mexp and T = Texp. Schematic bending moment – torque resistance interaction curves calculated with model with constant compressive panel thickness and with model with variable compressive panel thickness using different concrete strength reduction factors are shown in Fig. 6.

3.4. Balanced amounts of reinforcement for pure torsion and bending The amount of reinforcement in the concrete panel can be said to be in balance if the reinforcement in one direction is yielding and the concrete crushes simultaneously with the yielding of the reinforcement in the other direction. Fig. 7 shows the curves for balanced reinforcement spacings in two directions when the panel thickness is approximately 60 mm according to Fig. 5. Values have been calculated with

“full” concrete strength (40 MPa) and “reduced” strength (20 MPa) and with and without the prestressing force of 300 kN, which in the calculation was distributed evenly to the perimeter uef. These curves do not show if either direction yields before the other. Reinforcement spacings of the top panels of the test beams are also shown, and it can be seen that for a pure torsion case, the beams LL, LH and LHp were clearly under- reinforced. HHp beams can be a little over-reinforced if the concrete strength reduction in shear and the prestressing force are both taken into

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account.

For a pure bending case, the cross-section is balanced if the tensile reinforcement achieves the yield strain with concrete achieving its failure strain (-0.0035) simultaneously. Here the amount of tensile reinforcement in the balanced state is solved from equation:

Pp+fs,LAs,L,ten=fcbc0.8 (

1− fs,L/Es

fs,L/Es+0.0035 )

d (28)

where Pp is the prestressing force, which is seen as constant and an increase of the stress arising from the deformations is neglected. Fig. 7 shows the balanced amount of steel for different steel yield strengths and prestressing forces. It can be seen that that LL and LH beams were well in the under-reinforced side, but LHp and HHp beams were close to the balanced state. It can be stated that all of the beams were under- reinforced or close to the balanced reinforcement for pure bending and pure torsion.

Fig. 6. Bending moment – torque interaction curves with different calculation assumptions.

Fig. 7. Over-reinforcement limits for transverse and longitudinal reinforcement of top panel in shear and beam tensile reinforcement in bending.

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4. Experimental and analytical results

The maximum loads achieved in the experiments and the values calculated with the plasticity-based models with both constant and variable compression panel thickness are shown in Table 2. The experimental findings are presented and discussed in the following chapter.

4.1. Comparison of ultimate resistances

The experimental ultimate resistance of the beams was substantially lower than the calculated ultimate plastic resistances for all beams, except for the beam P2-016-LL in which bending clearly dominated the failure mode. The model with constant panel thickness te grossly over- estimated the resistances of most beams and the model with variable compression panel thickness, where concrete strength reduction factor was one, was more accurate.

The results show that both the higher torque-to-bending ratio and the amount of tensile reinforcement correlates inversely with the accuracy of the model, which is logical in a sense that the model did not directly take in to account the strength reduction of concrete due to shear stress from the torque. If the strength reduction factor ν_cis solved so that the calculated plastic resistance is equal to the experimental result, it can be seen that beams with a torque-to-bending ratio of 0.31 and light torsional reinforcement had a strength reduction factor of approximately 0.5 and the one beam with the same torque-to-bending ratio but heavy torsional reinforcement required only a factor of 0.8. For the beams with the highest torque-to-bending ratio of 0.47, it was not possible to determine fitted ν_c-value because the compressive panel thickness would have become so large that it would have required the whole cross-section which goes against the logic behind the space truss model. It may be that in these beams the fully plastic nature of the applied models was not valid at all and that the steel failed to yield. It can also be seen in the Table 2 that the calculated concrete stress at the side of the beams with high torque-to-bending ratio was high (>0.5fc, see Eq. (8)), which would indicate that the concrete failure could have happened at the beam side before or at the same time as the failure at the compressive side.

It can also be seen from the Table 2, that the calculated principal strain ε_1,comof the compressive panel was almost equal to the ε_1,com,fit_, even though the thickness of the compression panel increased consid- erably when strength reduction value was decreased. With this obser- vation it is possible to work out relationship between the load effects and the principal strain ε_1,comthat does not require iterative calculation of te, com, as was stated with Eq. (27). If for the determination of the tensile principal strain, the thickness of compressive panel is held constant, te, com =te, combining Eqs. (26) and (27), yields a closed form estimation of the ε_1,com:

ε1,com,est= zebe

16A⁴_e T² M³

(

4A²_eM² ssT

EsAsT

+T²z²_eb²_e ssT

EsAsT

− 4zebeA²_eMεcu

)

(29) Fig. 8 shows a diagram of fitted strength reduction factors against the calculated principal tensile strains of the compressive panel and also a regression model of the data. The beams where fitted strength could not be determined, were omitted from the diagram. The regression model shows that the strength reduction factor drops to low values with quite low principal strain values compared to models proposed by others, where factor for principal strain is in the range of 200–400 [26,49].

This regression model was used with Eqs. (29) and (25) to recalculate the plastic resistances of the tested beams and the results are shown in Table 3. θcom, ε_1,com, θten/side and σc,ten/side were omitted from the table since they changed only little from the initial model with concrete strength reduction value of one.

The match between experimental resistances and calculated plastic resistances is clearly improved. If the beams with high side face concrete

diagonal stresses, P6-047-LHp and P8-047-HHp, are not included the Table 2 Results of experiments and models. Beam Experimental values at ultimate load STM with constant panel thickness STM with variable compression panel thickness Fitted model t=tσ=fσ=fσ=fσ=fσ=υfυ=1 ε=ε=-0.002 F/F=1 e,come s,Ts,T s,Ls,L s,T,ten/sides,T s,L,ten/sides,L c,comcc c2,comcuexp,maxcal-V FMTFFexp,maxexp,max exp,max exp,max cal-C 1) 1) F[kNm][kNm][kN] [kN] cal-C

θ [◦] σc [MPa] Fcal-V [kN] Fexp,max Fcal-V te,com [mm] θcom [◦] σs,T,com [MPa] ε1,com [x10−3] θten/side [◦] σc,ten/side [MPa] υc,fit te,com,fit [mm] ε1,com,fit [x10−3] P1-031-LL 28.5 8.9 71.2 85.1 0.837 53.1 −6.6 83.8 0.850 35 7.4 50 0.287 54.6 −6.3 0.432 85 0.275 P2-016-LL 39.2 6.1 98.0 94.8 1.034 66.7 −5.0 95.5 1.026 41 3.8 15 0.086 67.0 −4.9 1.192 34 0.082 P3-031-LH 58.1 18.2 145.2 197.7 0.735 29.9 −16.9 174.7 0.831 56 9.3 138 0.761 32.6 −14.5 0.473 124 0.806 P4-016-LH 84.1 13.1 210.2 253 0.831 41.0 −9.7 223.4 0.941 74 4.8 50 0.264 42.6 −9.2 0.772 98 0.254 P5-031-LHp 61.6 19.2 153.9 232.4 0.662 26.0 −21.8 196.5 0.783 72 8.1 142 0.763 28.5 −18.4 0.524 148 0.750 P6-047-LHp 40.1 18.8 100.2 182.5 0.549 22.3 −29.1 156.7 0.639 55 12.3 249 1.399 25.2 −23.2 - 2) >180 – P7-031-HHp 82.8 25.9 207.1 270.7 0.765 40.0 −20.3 224.1 0.924 86 8.1 84 0.469 42.4 −18.5 0.798 110 0.470 P8-047-HHp 52.3 24.5 130.8 224.1 0.584 33.7 −27.2 189.3 0.691 71 12.2 156 0.912 36.7 −23.5 - 2) >180 – μ 0.750 μ 0.836 σ 0.156 σ 0.130 1) Values do not include the effects of self-weight (approx. 1 kNm for bending) 2) Calculation of the value would require too large compression panel thickness

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mean value for Fexp,max/Fcal-V,fit would be 0.982 with standard deviation of 0.075, which can be seen as an excellent match.

Fig. 9 shows the results of the different models in the form of bending moment-torque -interaction curves with the experimental results. From the figure it can be seen that the difference between interaction curves calculated with model with constant panel thickness and with the model with variable compressive panel thickness is dependent on the amount of reinforcement. The higher the amount of reinforcement, ordinary or prestressed, the bigger the difference.

4.2. Crack patterns and failure modes

The observed crack patterns and the angles of the cracks of the tensile, side and compressive faces of the beams are shown in Fig. 10.

The cracks at the ultimate stage were observed from photographs that were taken after the experiments. The cracks on the side face were also observed from photographs taken during the loading. All beams eventually failed with concrete fracturing and spalling on the compressive (top) face of the beams. It was not possible to determine crack angles in the top face due to excessive spalling after the ultimate load, so a vector was fitted by hand on the failure area to approximate the direction of the compressive diagonal stresses. During the experiments, compressive face cracks were almost invisible until very close to the failure, though it was not possible to observe them from very close distance due to safety concerns. Fig. 11 shows the final two seconds of the beam P7-031-HHp from a video, which shows that the failure happened quickly when cracks in the top face visibly opened in the surface.

Comparison of the crack patterns of the beams P1-031-LL and P3- 031-LH shows that the increase of tensile reinforcement from 251 mm²to 1005 mm²significantly lowered the crack angles in all faces.

Introducing the prestress force of 300 kN in the beam P5-031-LHp lowered the angles a few degrees more. Doubling the amount of stirrups in the beam P7-031-HHp increased the crack angles. Changes of the crack angles were more pronounced in the side face of the beams than in the bottom face. Increasing the torque-to-bending -ratio caused the top face thrust angle to increase, but generally slightly lowered the crack angle for the side and bottom faces, except in the LHp -series. Change, or rotation, of the crack angles on the side faces during the experiments was evident in all beams.

When the crack angles observed in the beams is compared to the crack angle calculated with the plasticity-based model with the variable compression panel thickness presented in the Table 2, it can be seen that, generally, the widest observed crack on the side of the beam in the ultimate stage corresponded very well with the calculated values in all beams except in the beam P2-016-LL. In the beam P2-016-LL it can be assumed that the lower ultimate load due to a lower amount of longitudinal tensile reinforcement combined with low torque-to-bending -ratio also meant fairly low shear stresses, and since the torque remained very low, just the above cracking limit, the diagonal cracks did not be fully develop when failure occurred. The values of crack angles observed in the tension (bottom) face of the beams were, generally, higher than the calculated values. These observations would indicate that the steel stress in the stirrups of the bottom faces were closer to the yield strength than in the side face, which is contrary to the assumptions made for fully plastic models where all the stirrups in both the tension and side faces of the beams were assumed to be yielding. If the angle of the widest observed crack at the ultimate load is used as the angle θ in the Eq. (3), the stirrup steel stress can be estimated for the side and the bottom panels. The estimated stresses are shown for each beam in the Table 4. In the estimation, the value of effective cross-section area for torsion A_e, was assumed to be same as in the model with constant panel thickness. The table shows that in the side panels the stirrup steel stresses were estimated to be under the yield strength of 538 MPa, but considering the large variation of the observed angles in the Fig. 10 and the vagueness in the method of the crack angle determination, the stress value should be seen as highly approximate. Generally, it can be stated that the beams that had lowest stresses and therefore were furthest from the fully plastic state, were also beams where the experimental results and the calculation model results deviated the most.

Apart from P2-016-LL, the accuracy of the model with variable compression panel thickness, with ν_c=1, was best for beams P4-016-LH and P7-031-HHp, where the calculated side face and top face crack angles were also closest to the observed values. Table 2 also shows that the calculated compression (top) face thrust angles were, generally, quite a lot lower than the observed values shown in Fig. 10. Since the calculated values are based on the direction of principal compressive force from the bending and torsion, which directly depends on the eccentricity of the end support, the deviation of the failure direction from this direction would indicate that the stress state of the compressive panel is more complex than just compressive stress in diagonal direction and tensile stress in the transverse direction.

4.3. Experimental load-deformation curves

The values of the mean beam curvature in the vertical direction determined from the deflection measurements in Fig. 12 show that the mean bending moment–curvature relationship for the beams P1-031-LL, P2-016-LL and P4-016-LH had a distinct yielding plateau close to failure, but for other beams the relationship did show signs of brittle failure. The values of the average beam twist in Fig. 13 show that the torsion stiffness degraded more gradually when the load was increased. Even though some of the mean moment–curvature relationships lacked the distinct gradual stiffness degradation near the ultimate load, the absolute values Fig. 8.Relationship of the calculated principal tensile strain and the strength

reduction factor of the concrete on the compression side fitted to match experimental results.

Table 3

Results from model with variable compression panel thickness using strength reduction factor model derived from the test results.

Beam STM with constant panel thickness

σs,T,ten/side=fs,T σs,L,ten/side=fs,L σ_c,com=υ_cfc υ_c=1/(1+3200ε₁)^0.5ε_2,

com=ε_cu=-0.002

Fcal-V,fit

[kN]

Fexp,max

Fcal-V,fit

υ_c _t_e,com

[mm]

P1-031-LL 80.8 0.882 0.751 48

P2-016-LL 94.1 1.042 0.900 45

P3-031-LH 157.5 0.922 0.592 97

P4-016-LH 209.9 1.002 0.781 96

P5-031-LHp 167.8 0.917 0.608 125

P6-047-LHp 131.2 0.764 0.489 123

P7-031-HHp 195.4 1.060 0.701 127

P8-047-HHp 158.4 0.826 0.576 131

μ _0.927

σ _0.104