Behavior and Analysis of Inverted T-shaped RC Beams under Shear and Torsion.pdf

Engineering Structures 68 (2014) 57–70

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Engineering Structures j o u r n a l h o m e p a g e : : w w w . e l s e v i e r . c o m / l o c a t e / e n g s t r u c t

Behavior and analysis of inverted T-shaped RC beams under shear and torsion A. Deifalla a, , A. Ghobarah b ⇑

a b

BUE, EL-Shourouk City, Postal No.11837, P.O. Box 43, Egypt McMaster University, Hamilton, Ontario, Canada L8S 4L7

a r t i c l e

i n f o

Article history: Received 23 July 2013 Revised 19 February 2014 Accepted 20 February 2014 Available online 27 March 2014 Keywords: T-beams Combined Combined loading Torsion Shear Global behavior Flange stirrup

a b s t r a c t

The 1998 ASCE-ACI ASCE-ACI Committee Committee 445 on shear and torsion identified identified researching researching combined combined shear and torsion torsion as well as giving giving physical significance significance for torsion torsion design as an upcoming upcoming challenge challenge (ASCE-ACI Committee 445 on shear and torsion, 1998). Most of the previous experimental studies were focused on reinforced (RC) beams under flexure, shear or torsion. The behavior of inverted T-shaped beams with both both web and flange flange closed closed stirru stirrups ps are not fully fully explor explored. ed. In this this resear research ch paper, paper, an innova innovativ tive e test test setup setup capable of simulating the behavior of inverted T-shaped beams under combined shear and torsion was develope developed d and implemented. implemented. The behavior behavior of three inverted T-shaped T-shaped beams beams tested under different values for the ratios of the applied torque to the applied shear force is discussed. The value of the torque to shear ratio significantly affects the behavior of the inverted T-shaped beams in terms of cracking pattern; failure failure mode; mode; strut angle of inclination; inclination; cracking cracking and ultimate ultimate torque; torque; post-cracking post-cracking torsional torsional rigidrigidity; cracking and ultimate shear; flange and web stirrup strain. The flange stirrup is more efficient in resisting torsion moment over shear forces. A model capable of predicting the behavior of flanged beams under combined actions was developed and implemented. The model showed good agreement with the experimental results from three different experimental studies.  2014 Elsevier Ltd. All rights reserved.

1. Introduction Reinforc Reinforced ed concrete concrete (RC) inverted inverted T-shape T-shaped d beams beams are being being used as the main main girders girders that support support the the lateral lateral secon secondary dary precast precast beams or slabs which is one of the popular structural systems for many many existing existing bridges and parking parking garages as shown shown in Fig. Fig. 1. The The behavi behavior or of invert inverted ed T-sha T-shape ped d beam beams s is more more compli complicat cated ed than than that of conventio conventional nal either rectangul rectangular ar or T-shaped T-shaped RC beams. beams. Conventional rectangular and T-shaped RC beams fail in flexure, shear, torsion, or a combination of these failure modes. In addition to the conventi conventiona onall modes modes of failure, failure, inverted inverted T-shaped T-shaped beams beams could fail due to other local causes such as hanger failure in the web, web, cantil cantileve everr action action,, or punch punching ing shear shear in the flange flange,, which which was studied by others [2,3] [2,3].. Moreover, inverted T-shaped beams are subjected to significant torsional moments. Thus, these beams must be designed to resist significant torsional moment combined with shear forces. In 1998, the ASCE-ACI Committee 445 on shear and torsion torsion identifie identified d integratin integrating g and designat designating ing a physical physical signifsignificance for the torsion design provisions, as well as reviewing combined bined shear shear and and torsio torsion, n, as an upcom upcoming ing challe challenge nge [1] [1].. Modelin Modeling g of ⇑

Corresponding Corresponding author.

E-mail addresses: [email protected], [email protected], [email protected] [email protected](A. (A. Deifalla). http://dx.doi.org/10.1016/j.engstruct.2014.02.011 0141-0296/ 2014 Elsevier Ltd. All rights reserved.

un-cracked flanged beams is more complex than that of rectangular beams as shown in Fig. in Fig. 2. 2. Conventional Design Codes approach the design of RC beams subjected to combined shear and torsion differen differently, tly, especia especially lly for cases cases that involve involve significa significant nt torsion, torsion, which which was indicated indicated by many researchers researchers [4–6]. [4–6]. Thus, a unified practical practical solution solution is required required for the analysis analysis of these inverted inverted T-shaped beams. Fig. 3 shows shows a typical typical inverted inverted T-shaped T-shaped beam beam loading loading and forces. forces. The test setup simulates simulates the behavior behavior at the inflection point with zero bending moment highlighted in Fig. 3. 3. This segment segment is subjecte subjected d to significa significant nt torsion torsion and shear force, force, while the values of bending moments are relatively insignificant. In addition, it is far from the local effect of the load application mechanism. Although Although diagonal diagonal tension cracks occur in RC beams due to torsion or shear, the behavior of RC beams due to torsion is different from that under shear. In the case of shear forces, the cracks propagate propagate in the same direction on both sides of the beam parallel to the applied shear plane. In case of torsion, the cracks follow a spiral pattern, propagating in opposite directions on the opposite sides of the beam. In addition, the assumptions used in modeling RC beams under shear are different from that used for modeling those under torsion. In the case of shear forces, stresses are assumed to be in the plane of the applied shear and uniform across

A. Deifalla, A. Ghobarah/ Engineering Structures 68 (2014) 57–70

58

Nomenclature

Ac Ao As E c f c f y i k l m M x N N vk P c P o 0

q qs qt T V T i t t s t t V i yci

the gross concrete cross section area the area enclosed inside the center of the shear flow loop area of each bar ( j) Young’s modulus of the concrete the compressive strength of concrete the yield stress of the steel panel number number of concrete strips for moment calculations the length of the panel (i) parallel to the shear plane number of steel bars moment around the x-axis the applied axial force on the cross section the shear contribution from each panel (i) the perimeter of the concrete cross section the perimeter of the centerline of the equivalent thin tube the average shear flow of the panel (i) the shear flow due to the shear force (V) the uniform shear flow on the panel due to the torsion the applied torsion moment on the whole cross section the applied shear force on the whole cross section the applied torsion moment on the rectangular subdivision the effective thickness of each element resisting both shear and torsion the thickness of theelement resisting the shear force (V i) the thickness of the element resisting the torsional moment (T i) the applied shear force of each rectangular sub-division (i) distance between the elastic centroid and the centroid of each concrete panel (i)

ysj ysk

t /d /L /t b1 b2

c D Aci

ec e1 e2 e2s 0

e x e y h

qh

rci rsj 0 0

r1 r2 rst r x r y ui W

distance between the elastic centroid and the center of each bar ( j) distance between the elastic centroid and the panel (i) centroid the shear stress the curvature in the direction of angle h the longitudinal curvature the transversal curvature softening coefficient of the concrete stress strain softening coefficient the shear strain of each panel (i) the area of the strip concrete strain at the peak stress the principal average tension concrete strain the average principal compression strain the maximum compression principal strain at the surface of the concrete the average longitudinal strain the average transverse strain the inclination angle of the principal strains the ratio of the transverse steel per unit length of the span to the gross area of the concrete cross section the concrete stress at the centroid of the strip stress in the steel longitudinal reinforcement for each bar ( j) the principal average tension stresses the principal average compression stresses the steel reinforcement stress the average longitudinal stress the average transverse stresses the curvature for each panel the twist rate

(a)

(b)

(c)

Fig. 1. Examples of inverted T-shaped beams under significant torsion.


59

(b)

(a) (c) Fig. 2. R and T-shaped beams under torsion (a) isometric; (b) uncracked and (c) cracked.

the test setup is capable of applying different shear to torsion ratios by varying the ratio between the applied loads. Three inverted T-shaped beams were designed, constructed, and tested while sub jected to various torque to shear ratios. The tested beams represented a scaled concrete inverted T-shaped beam model. The inverted T-shaped beams were tested under torque to shear ratios of 0.5 m, 1.0 m and 0.1 m while being referred to as TB1, TB2, and TB3, respectively. The parameters investigated by the test program were the effect of the torque to shear ratio on the behavior of the RC inverted T-shaped beams subjected to shear, torsion, and an unavoidably small bending moment. In addition, a previous analytical model developed by the authors was extended to predict the full shear and torsional behavior of the inverted T-shaped beams.

2. Research significance and previous work

Fig. 3. Typical inverted T-beam loading and internal forces.

the perpendicular plane to it. In the case of torsion, the diagonal concrete compression strain is assumed to vary linearly across the assumed effective thickness of the walls of the cross-section due to lateral curvature that eventually causes the variation of the stress across the section, both vertically and horizontally [35]. In addition, according to the theory of hollow-tube spacetruss analogy, the effective thickness of the tube varies based on the applied torque, similar to the variation of the effective depth of the beam with the bending moment [35]. In theory, the concrete web and the steel web stirrup carry most of the shear. However, the torsional moment must be distributed between the web and the flange, which can vary based on the dimensions and reinforcements of the section. In this research study, an experimental program was conducted. An innovative test set-up that allows the beams to fail due to combined shear and torsion accompanied by relatively low levels of bending moments, was developed and constructed. In addition,

The 1998 report by the ASCE-ACI Committee 445 on shear and torsion outlined the challenges of reviewing RC beams under combined shear and torsion and integrating and designating a physical significance for current torsion design provisions [1]. Behavior of RC inverted T-shaped beams, despite its frequent use since the 1950s, remained as one of the least investigated until mid-1980s [2,3]. Until that time, no guidance for handling design issues specifically those associated with the inverted-T section was available in design standards. Therefore, engineers have tended to rely on personal judgment and discretion for design of these beams. A careful examination of existing literature has shown the following: (1) very valuable contributions concerning the behavior of RC beams under combined shear and torsion were made by several researchers [29,11,24,17]. However, these studies focused on rectangular beams rather than T-shaped beams with flange stirrups; (2) pioneering works on the behavior of T-shaped beams were conducted by several researchers [36–39,2,34,40,41,3,26,27]; however, they all focused on T-shaped beams under pure shear, pure bending, pure torsion, combined shear and moment, or combined moment and torsion. In addition, many recent investigations were concerned with spandrel L-shaped beams [12,28,42–45]. Kaminski and Pawlak indicated that, despite all the extensive research conducted in the area of beams under combined torsion, not all the questions were answered. In addition, it was pointed out that the behavior of RC beams with a cross section other than rectangular or circular is yet to be explored [45]. Experimental testing remains the most reliable research approach compared to the use of numerical models. The tests provide

60


physical knowledge and information about the behavior of the system studied [53,54]. Moreover, test results are essential in verifying analytical models such as (1) the skew bending theory models based on an inclined plane failure [7–13]; (2) the space hollow tube truss models [14–18,6,19–25]; (3) the finite element and the finite difference numerical models [26–28]; and (4) the empirical models developed by fitting experimental data [29–33]. A milestone point in the analysis of RC beams under combined shear and torsion was the work presented by both Hsu, and Rahal and Collins [34,17]. Hsu presented a unified theory for combined shear and torsion ‘‘Softened Truss Model’’ that was based on: (1) equilibrium equations; (2) compatibility equations; (3) the softened constitutive laws of concrete [34]. Rahal and Collins [17] updated the existing space truss model to include; (1) concrete softening; (2) tension stiffening; (3) improved modeling for the cover spalling; and (4) an equivalent uniform stress distribution block for the concrete strut. Another key point in the history of RC beams under combined actions was the work by Greene and Belarbi [21]. They presented a ‘‘Combined-Action Softened Truss Model,’’ which was based on the ‘‘Softened Truss Model’’ by Hsu and Mo for pure torsion with improvements over existing models [17,16]. More recently, Bernardo and co-workers studied the modeling of RC beams under torsion [19,24,25]. Their work focused on comprehensively examining previous experimental and analytical models to verify and improve existing analytical models. Ultimately, a modified version of the ‘‘Variable Angle Truss-Model’’ by Hsu and co-workers [34,16,22] that is capable of predicting the behavior of the beams for all loading stages was presented. Moreover, they indicated that the next step would be dealing with special beams under combined straining actions. The behavior of RC inverted T-shaped beams is different from RC rectangular beams. The cross-section shape can have a significant effect on the behavior and design, as shown by several researchers [26,27,5,31,46]. In addition, the inverted T-shaped beams with flange stirrups are an asymmetricaly-reinforced section. Moreover, there is no unified approach for the design of RC inverted T-shaped beams under combined loading. The first step in reaching a unified approach is to conduct an experimental program in order to identify the significance of the contribution of various parameters to the behavior.

3. Testing inverted T-shaped beams 3.1. Scale model for the inverted T-beam The concrete dimensions of the tested beams were chosen as half-scaled model for a commonly used precast inverted T-shaped beam [47] or a typical 700 mm girder monolithically cast with a 200 mm slab. Since the study focused on the effect of the torque to shear ratio on the behavior, beams were heavily reinforced in the longitudinal direction to minimize the effect of flexure on the behavior of the tested beams. The stirrups were designed according to the CSA [48]. The concrete dimension and steel reinforcements were kept the same for all tested beams.

3.2. Specimen details All of the test beams had a total depth of 350 mm, a flange thickness of 100 mm, a flange width of 450 mm, and a web width of 150 mm. Fig. 4b shows a typical cross-section of the beam within the test region. The concrete cover was 25 mm for the web and 15 mm for the flange. Fig. 4d shows a typical longitudinal section of the beams and the reinforcements. All transversal and longitudinal reinforcements were ribbed steel bars. The longitudinal reinforcement is 4–20 M (i.e. 4 bars 20 mm diameter) at the bottom

of the web and 2–15 M + 4–10 M (i.e. 2 bars 15 mm and 4 bars 10 mm diameter) in the flange. The transverse reinforcement was determined to be 10M @ 170 mm (i.e. 10 mm stirrup every 170 mm). The clear length of the central region was 1400 mm, as shown in Fig. 4, to ensure that at least one complete spiral crack would occur within the central region. At the two ends of the test region, an end block was created with a rectangular section having a total depth of 350 mm, a width of 450 mm, and a length of 250 mm. These two end blocks were used to apply torsion at one end (active frame) and to restrain the torsion at the other end (reactive frame). To apply the required load and the proper boundary condition far from the test region, the beam was extended at both ends. The extensions were either for applying load (loading arm) or for applying the end restraints (roller arm). The loading arm was 900 mm long while the roller arm was 750 mm as shown in Fig. 6. To ensure that failure would occur within the test region, both arms had additional longitudinal and transverse reinforcement. The shear reinforcement was 10-M @ 70 mm, the bottom reinforcement was 6–20 M, and the top longitudinal reinforcement was 4–10 M + 2–15M. The concrete mix was designed using Type 10 cement, sand, and 10 mm aggregate. The results from the compression testing of standard concrete cylinders are shown in Table 1. The 28-day concrete compressive strength was 25.6 MPa. Compression tests conducted on the same day of the beam testing showed a compressive strength of 35.9 MPa for beams TB3 and TB1, and 33.6 MPa for beam TB2. The longitudinal and transversal steel bars were ribbed high strength steel. The tensile testing of coupons made from the reinforcement bars showed that 10 M bars yielded at 465 MPa, while the 15 M and 20 M bars yielded at 450 MPa. Linear variable differential transformers (LVDTs) were used to measure displacements at different locations of the beam. Ten LVDTs measured the vertical displacements at five sections of the beam—two at each section. The two LVDTs at the tip of the flange of each section were used to calculate the rotation and the average vertical displacement. Strain gauges were used to measure the strain in the longitudinal and transversal reinforcement at different locations, as shown in Fig. 5. Strain in the longitudinal reinforcement was measured at the maximum and at the zero moment section. Strain in the transverse reinforcement was measured at the beginning, middle, and the end of the test region area. Strain gauges were installed at the same location in all the tested beams.

3.3. Test set-up Recently, Talaeitaba and Mostofinejad proposed a test setup using a simple beam with a cantilever in the middle for applying combined shear and torsion [49]. In their test setup, the combined shear and torsion are accompanied by relatively large bending moments. The test setup used in the present research was designed in 2005 with the objective of minimizing the bending moments in the torsion and shear interaction test region. The combined shear and torsion is significant at low bending moment values, including, but not limited to, the following cases: (1) the case of inflection point for a continuous beam; or (2) the case of a section at the support of a simple beam. Fig. 6 shows a schematic of the structural system for the test set-up where three different actuators are used to apply loads to the beam (denoted as L1, L2, and L3), simulating a simple beam with a cantilever at both ends. The middle section of the test region is subjected to combined shear and torsion with zero or near zero bending moment. The two hydraulic actuators L2 and L3 apply the load to the beam through 0.5 m long steel arms to apply the required torque. The hydraulic actuator L1 acts at the center of the cross-section of the beam. The top end condition for actuator L1 is a pin support. The middle region (test region) was


61

Fig. 4. Dimensions and reinforcement details of tested beams.

Fig. 5. Strain gauge location on longitudinal and transversal reinforcement.

subjected to combined shear and torsion while the torque to shear ratio was kept fixed throughout the test by controlling the three different applied loads. After installing the T-beams in the test setup and attaching the instruments to the data acquisition system, the beam was loaded with low-level load combinations within the elastic range of concrete. Measurements from this test were verified to ensure that all the instruments were correctly installed and functioning properly. The load values L1, L2, and L3 that give the desired shear and torsion combination were calculated from simple structural analysis. The loads were applied in small steps of 2 kN in order to exercise better control over the loading values and achieve the required torque to shear force ratio. After each

load step, the beam was inspected for cracks and any possible signs of failure. During the tests, it was possible to maintain good control over the torque to shear ratio all the way to near failure of the beam. Fig. 7 shows a photo for the test setup with a specimen in place. Four load cells (L1, L2, L3, and L4) were used in the test set-up. Three of the load cells (L1, L2 and L3) were used to measure the actual applied loads at Points A, D and E on the beam. The fourth load cell (L4) was used at point F to measure the reaction at the support of the beam. Fig. 8 shows the boundary conditions at points F, D, E and A. Due to the complexity of the test set-up, the assumptions made concerning the beam boundary conditions were verified. This was done by comparing the measured values


62

TB3) were tested under torque to shear ratios of 0.5 m, 1.0 m and 0.1 m, respectively. The torque to shear ratio was chosen to cover a wide range of practical shear torsion interactions. In addition, Table 2 shows the ratio of the applied torque to shear ratio to the ultimate torque to shear ratio, which was chosen to vary from 1 to 10. Based on this range, the applied torque to shear ratios were chosen to be either 0.1 m, 0.5 m, or 1.0 m.

4. Experimental results 4.1. Cracking pattern and failure mode

Fig. 6. Schematic structural system and interal forces for the tested beams.

Table 1

Concrete strength at different dates. Batch I

Batch II

Date

f c (Mpa)

7 days 28 days TB1

17.7 25.6 35.9

0

F

Date

f c (Mpa)

28 days TB3 TB2

25.6 35.9 33.6

0

H

E

C

B

G A

D

The concrete cracking pattern for beams TB1, TB2, and TB3 are shown is Figs. 11–13, respectively. In addition, the failure modes are listed in Table 3. For beamTB1( T /V = 0.5 m), the onset ofcracking was observed at the bottom of the web at a total load value of 56 kN. Afterwards, more diagonal cracks were initiated within both the web and the flange, which were spiral and uniformly distributed, as shown in Fig. 11(a–d). Before failure, significant concrete cover spalling from the flange (as shown in Fig. 11b and c) and additional longitudinal cracks in the flexure compression zone side were observed, as shown in Fig. 11a and b. These additional longitudinal cracks are due to the diagonal compression stress from the shear and torsion, and that from the flexure. The major diagonal cracks were formed at an average angle of inclination with the longitudinal axis of the beam ( h) value of 51. Beam TB1 failed due to stirrup yielding before concrete compression at a load value of 162 kN. For beam TB2 (T /V = 1.0 m), the onset of cracking occurred at an applied load of 33 kN. The cracks propagated in a helical form around the beam in a similar manner to those of beam TB1 (as shown in Fig. 12a–d), where concrete cover spalling from both the flange and the web was observed. However, on average, the major cracks formed at an average (h) value of55, which is steeper than beam TB1. Beam TB2 failed due to stirrup yielding before concrete compression at a load value of 75 kN. In comparing Fig. 11b and Fig. 12b, it is clear that beam TB2 exhibited significant web spalling with respect to beam TB1. For beam TB3 (T /V = 0.1 m), the onset of cracking was observed at a load value of 130 kN. Significant diagonal cracks were observed in the web compared to that in the flange, as shown in Fig. 13a–d. The cracking pattern varied along the test region and between both sides of beam TB3. For the web side, where the shear stresses due to the torsion and shear were added together, the average (h) for the cracks was 30 , which is lower than that of the other web side, where shear stresses due to the torsion and shear will subtract. Beam TB3 failed due to diagonal concrete compression before stirrup yielding at a load value of 342 kN. Comparing Fig. 13(a–d) with Fig. 11(a–d) and Fig. 12(a–d), the angle of inclination of the cracks of beam TB3 was lower than those of either beam TB1 or TB2. The spacing between the cracks of beam TB3 was smaller than that of either beam TB1 or TB2. The cracking patterns of beams TB1, TB2, and TB3 were significantly influenced by the torque to shear ratio.

Fig. 7. The test setup with a specimen in place.

of the reaction at point F (L4) to the theoretically predicted reaction at the same location (R1) using a linear structural analysis, assuming actual hinges at R2 and R3, and an actual roller at R1, as shown in Fig. 9.

3.4. Torque to shear ratio Fig. 10 shows the applied torque versus the applied shear for the tested inverted T-shaped beams. The beams (TB1, TB2, and

4.2. Torsional behavior Fig. 14 shows the relationship between the applied torque and the angle of twist for the tested beams. Before cracking, the behavior was similar for all of the tested inverted T-shaped beams, with a pre-cracking torsional rigidity value of approximately 2110 kN m2. The value of the cracking strength was taken as the minimum of either the strength at which the torsion behavior deviated from the initial linear behavior or the strength at which cracks were observed during the testing of the beam. The recorded values of


63

Fig. 8. Details of the test setup; (a) roller support at point F, (b) actuator used to apply load at points D and E and c) actuator used to apply load at point A.

120 160 ) N 140 k ( 1 R 120 , e c r 100 o f n o 80 i t c a e r 60 d e t a 40 l u c l a 20 C 0

TB1

TB2

TB3 (0.1 m)

100

TB3

) N k ( 80 2 Q , e 60 c r o f r a 40 e h S

TB1 (0.5 m) TB2 (1.0 m)

20 0 0

20

40

60

80

100

120

Measured reaction force, R1 (kN) Fig. 9. Physical verification of the test setup.

140

160

0

5

10

15

Torque, T (kN. m) Fig. 10. The applied shear and torsion.

20

25


64 Table 2

Selected torque to shear ratios.

B

C

(V - T)

(T /T ult )/(V /V ult )

(T /V ) (m)

1 5 10

0.1 0.5 1.0

(V - T)

(a) (V - T)

(V - T)

(V + T)

(V + T)

C

B

B

C

(b)

(a) (V + T)

(V + T)

B

C

C

B

(c)

(b) B

C

C

B

(c)

(d) Fig. 12. Cracking pattern for TB2 (1.0); (a) south, (b) north, (c) bottom and (d) top.

B

C

B (V - T)

C

(d)

(V - T)

Fig. 11. Crack pattern for TB1 (0.5); (a) south, (b) north, (c) bottom and (d) top.

the cracking torque and corresponding twist for all the tested beams are shown in Table 3. After cracking, the behavior of beams TB1 and TB2 was similar because they were subjected to high torque to shear ratios. However, the behavior of beam TB3 was different compared to beams TB1 and TB2. In examining Fig. 15, it can be seen that the average post-cracking torsional rigidity of beam TB3 was higher than that of either beam TB1 or TB2 that is due to wider cracks associated with the high torsion to shear ratio for beams TB1 and TB2. This is commonly observed after steel yielding, which is the case for both TB1 and TB2. The value of the ultimate strength was taken as the maximum strength observed during the testing of the beam. Table 3 shows the ultimate torque and the corresponding twist for all the tested beams. As shown in Table 3, the shear– torsion interaction affected the value of the ultimate torque.

4.3. Shear behavior The shear behavior of the tested beams was affected by the torque to shear ratio. Fig. 13 shows the relationship between the applied shear force and the maximum strain in the transverse steel reinforcement. The stirrup strain increased substantially with the increase in the torque to shear ratio. The applied shear force at

(a) C

(V + T)

(V + T) B

(b) C

B

(c) B

C

(d) Fig. 13. Crack pattern for TB3 (0.1); (a) south, (b) north, (c) bottom and (d) top.


65

Table 3

Summary of the experimental results. Beam

T /V (m)

Cracking torque (kNm)

Twist at cracking (deg/m)

Cracking shear force (kN)

Ultimate torque (kNm)

Twist at ultimate (deg/m)

Ultimate shear (kN)

Observed failure mode

TB1

0.5

11.6

0.25

17

23

2.82

46

TB2

1.00

11

0.33

11

22.7

3.16

21.4

TB3

0.1

4

0.13

42

10.8

0.5

105

Stirrup yield before concrete crushing Stirrup yield before concrete crushing Concrete diagonal crushing

the onset of cracking for all tested beams is shown in Table 3. The value of the ultimate shear strength for all tested beams is also shown in Table 3. It is clear in the table that the shear–torsion interaction affected the ultimate and cracking shear force.

25 TB1 (0.5 m) TB2 (1.0 m)

20 ) m . N 15 k ( T , e u 10 q r o T

4.4. The shear–torsion interaction

TB3 (0.1 m)

5

0 0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

Angle of twist (deg/m) Fig. 14. Torsional behavior.

Fig. 16a shows a plot for the observed absolute values of the cracking and the ultimate shear forces versus the torsion moment. We can see that the relationship is not linear and there is a clear curvature in the interaction. Huang et al. proposed a circular dimensionless relationship for the torque–shear interaction based on the theory of plasticity [30]. In an attempt to quantify the shear–torsion interaction, the shear forces and torsion moments were normalized and compared with the interaction relationship proposed by Huang et al. and are shown in Fig. 16b [30]. We can see that the experimentally observed values agreed fairly well with the relationship, with an error less than ±10%.

4.5. Transverse steel strain 120 TB3 (0.1 m)

100

) N k ( 80 2 Q , e 60 c r o f r 40 a e h S 20

d l e i y l e e t s e s r e v s n a r T

TB1 (0.5 m) TB2 (1.0 m)

0 0.00

0.50

1.00

1.50

2.00

2.50

Stirrup strain at the mid of the test zone (1000 microstrain) Fig. 15. Shear behavior.

(a)

3.00

Fig. 17 shows the transverse steel strain for the flange and web stirrups versus the total load. In case of beam TB2 (high torque to shear ratio), the strain measured in the flange was similar to the strain measured in the web. However, in the case of beam TB3 (low torque to shear ratio), the strain in the web was larger than the strain in the flange. The flange was more effective in cases of higher torque to shear ratios. Fig. 18 shows the relationship between the flange stirrup strain at both the top and bottom branch versus the total load. The strain gauges were installed as shown in Fig. 5, with the exception of beam TB1, where the bottom strain gauge was installed in the middle of the bottom branch within the overlapping zone of the flange stirrup. The strain of beam TB1 (under low torque to shear ratio) was significantly lower than that of TB3, which agrees well with the assumption that the flange stirrup primarily carries forces from torsion.

(b)

Fig. 16. Ultimat and cracking experimentaly observed shear–torsion interaction (a) absolute and (b) normalized.


66

rectangular RC beams up to failure using a displacement control solution scheme rather than a force control solution scheme; (2) include the FRP material modeling; (3) model external bonded reinforcements with different arrangements; and (4) improved the concrete constitutive modeling [17,23]. All of these models focused on rectangular beams under combined torsion, although structural members subjected to torsion may be of different configurations, such as rectangular beams, T-shaped beams, L-shaped beams, and box beams. In this study, the model by Deifalla and Ghobarah was adapted and further extended to predict the behavior of cross-sections with different shapes subjected to torsion, shear, and bending moments [23]. In the development of the proposed model, the following assumptions were made:

450 400 N k , ) 3 L + 2 L + 1 L ( d a o l l a t o T

350 300 250 200 150 100 50 0 0

0.5

1

1.5

2

2.5

Strain (1000 microstrain) Fig. 17. Stirrup strain versus total applied load (L1 + L2 + L3).

N k , ) 3 L + 2 L + 1 L ( d a o l l a t o T

400 350 300 250 200 150 100 50 0

-1

-0.5

0

0.5

1

1.5

2

2.5

3

Strain (1000 microstrain) Fig. 18. Flange stirrup strain versus total applied load (L1 + L2 + L3).

5. Analytical model Several models were developed for predicting thebehaviorof RC beams subjected to combined straining actions. Numerous contributionsby manyresearchers attempting to improvethe space truss model by Rausch were found in existing literature [50,15– 18,51,6,19,21,20,22–25] . Deifalla and Ghobarah [23] adapted the model by Rahal and Collins [17] to: (1) predict the behavior of

(a)

(1) The longitudinal strain follows the Bernoulli–Navier hypothesis, which indicates that plane section before bending will remain plane after bending. (2) Mohr Circle can be used to evaluate the strain, curvature, and stress status at any point in the plane. (3) The direction of the principal stresses at any point in the plane is coincident with the direction of the principal strain evaluated at the same point in the plane [35]. (4) The torsional behavior is dominated by Saint–Venant’s torsion, which indicates that the torsion will be resisted by shear flow in the perimeter of the cross section [34,17,23]. (5) The effective thickness of the diagonal concrete struts is function of the external loading [33,35] which is similar to beams subjected to bending where the effective depth is function of the bending moment. (6) The equivalent hollow tube is being divided into four panels; each panel is subjected to uniform bi-axial stresses [33,17,18,23,21] . (7) The diagonal compressive strain distribution within the concrete diagonal struts is assumed to be linear and consequently the diagonal compressive stress is assumed to be non-uniform [33,35,20,23]. (8) The torsion stresses and the uniform shear stresses are being replaced by one equivalent uniform stress block as shown in Fig. 19 [17].

5.1. Modeling T-shaped beams This section describes the capability of predicting the behavior of the flanged beams. The flanged cross-section is divided into several rectangular sub-divisions. Each rectangular sub-division

(b)

(c)

Fig. 19. Compression Stress distribution within the concrete strut (a) actual stress distribution, (b) equivalent stress distribution, and (c) equivalent uniform stress distribution.


(a)

(b)

67

(c)

Fig. 20. Rectangular divisions (a) Solution I, (b) Solution II, (c) Solution III.

where T i is the torsion carried by each rectangular sub-division (i) at the same angle of twist and n is the total number of rectangular subdivisions. The applied shear force ( V ) is calculated as follows: Input section details and applied

n

V

internal actions.

¼

Set T and V as 0.1 of the applied internal actions.

X

ð2Þ

V i

i 1

¼

where V i is the shear carried by each rectangular sub-division ( i) at the same angle of twist. The stirrup strain e is calculated such that:

Arbitrary assume tt

n

e¼

Calculate Ao and po, Eqs. (19-20)

X

ei

ð3Þ

i 1

¼

where ei is the stirrup strain for each rectangular sub-division (i) at the same angle of twist.

Calculate average shear stress, Eqs. (4-7)

5.2. Modeling rectangular sub-divisions Calculate average stresses and strains for each panel, Eqs. (8-14 and 23-29)

For predicting the full behavior of each rectangular sub-division, the model proposedby Deifalla and Ghobarah is implemented [23]. The model is briefly listed in Eqs. (4)–(29); however, details regarding the development of the adapted model is to be found in both Deifalla [6] and Deifalla and Ghobarah [6,23].

“

Use the Panel “subroutine shown in fig (20) to calculate longitudinal stresses and strain for the whole section, Eqs. (21-22).

¼ 2T A

qt

No Check tt , Eqs. (15-17) yes Calculate

Check T and V applied actions

i

ð4Þ

o

Input average shear stress and

using Eq. (18-20)

longitudinal strain

No

Assume the diagonal strain Increase the T and V by 0.1 applied actions.

Assume the angle of inclination

yes End

Solve the wall element

Fig. 21. Flow chart for the main program.

NO Check the angle

is analyzed independently while subjected to the applied combined shear and torsion. For example, the T-shaped beam is divided into rectangular sub-divisions as shown in Fig. 20. After modeling each rectangular section, the principle of superposition is applied to obtain the strength and the deformations of the complete T-shaped beam, while assuming that the angle of twist for the T-shaped beam and the sub-divisions are the same. The applied torque (T ) on the whole cross section is calculated such that: n

T

¼

X

T i

i 1

¼

ð 1Þ

Yes NO Check the diagonal stress

Yes Return to main program Fig. 22. Flow chart for the ‘‘Panel’’ subroutine [23].


68

¼ V l q  t þ q  t q¼ i

qs

t

t

s

s

t

q t

m¼

ð5Þ

r2 ¼ b1 f c 0

ð6Þ

b1

ð7Þ

5.3. Mohr circle for the average concrete strains of each panel 2

c¼

e y ¼

ðe þ e Þ tanðhÞ x

2

c

ð Þe e ¼ e þ e þ e 1

2 tan h x

2

2

tan

r2 r2

r2

h

r x r1

s

1

tan h

tan h

ðÞ

r1

L

ð27Þ

0:33 f c 0

1

ð28Þ

þ p 500e

ffiffi ffi ffi ffi ffi ffi 1

ð29Þ

A flow chart for the solution technique is being shown in Figs. 21 and 22. A force driven solution technique is being used limiting the model predictions to the ultimate strength.

5.7. Model validation

ð16Þ

i 2

ð26Þ

rs ¼ E s es 6 f y

ð14Þ ð15Þ

2s

t

1: 0

Three rectangular RC beams (N1, N2, and N3) were found in the literature. The beams were tested under combined significant

15

ð13Þ

¼ eu

¼ / sin ðhÞ þ /

500e1

1:0

b2

Experimental [52]

ui ¼ w sin2hi

/d

ð25Þ

ð9Þ

ð12Þ

r y ¼ qh rst þ r1  s tan h

t ti

1

ffiffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffiffi ffi ffi ffiþffi ffi ffi ffi ffi ffi ffi ffi ffi ¼ p

r1 ¼

ð11Þ

   ð Þ¼    ¼ ð Þþ  2

ð24Þ

¼ p 1:0 0þ:9400e

ð8Þ

5.4. Equilibrium and compatibility conditions for each panel

r x ¼ r2  r y  r1

e2 P1 b2 e0c

r1 ¼ E c e1

ð10Þ

y

if

cos2 h

ð Þ þ w sinð2hÞ

Anaylitical

) m 10 . N k ( T , e u q r o 5 T

ð17Þ

5.5. Panel assemblage

0 0

5

10

150

¼ c

i 1 li i

ð18Þ

2 Ao

4

¼ A 

10

150

5

10

15

li

c

i 1

¼

(a)

(b)

(c)

Fig. 23. Torque versus angle of twist for (a) N1; (b) N2 and (c) N3.

t t i

X X ¼  X þX

Ao

5

Angle of twist (deg/m)

4

w

P ¼

ð19Þ

2

25

4

P o

P c

ð20Þ

t t i

i 1

¼

k

m

r0ci D Aci

i 1

j 1

¼

¼

k

¼

X

ð21Þ

N k v

k 1

¼

m

Xð i 1

4

r0sj Asj ¼ N þ

r0ci D Aci Þ yci þ

4

Xð j 1

¼

r0sj Asj Þ ysj ¼ M x þ

X

N k ysk

k 1

¼

v

ð22Þ

20 ) m . N 15 k ( T , e 10 u q r o T 5

Experimental Solution I Solution II Solution III

0

5.6. Material modeling

0

1

2

3 0.0 0.8 1.6 2.4 3.2 0.00

0.25

Angle of twist (deg/m)

"  # 

r2 ¼ b1 f c 0

2

e2 e0c

e2 e0c

2

if

e2 61 b2 e0c

ð23Þ

(a)

(b)

(c)

Fig. 24. Torque versus angle of twist for (a) TB1; (b) TB2 and (c) TB3.

0.50


close agreement with the experimental results. However, only up to the ultimate strength as the model employs a force driven solution technique. From the current study, three T-shaped beams (TB1, TB2, and TB3) tested under torque to shear ratio values of 0.5, 1.0, and 0.1 m. Each T-shaped beam was divided into two rectangular sub-divisions using each of the three proposed solutions, as shown in Fig. 20(a–c). The comparison between the behavior (i.e., torque versus twist and shear force versus stirrup strain) predicted by the model and the experimentally observed behavior is shown in Fig. 24(a–c) and Fig. 25(a–c). The figures show that the model prediction agrees well with the experimental results. Two L-shaped beams were found in the literature tested under combined torsion [45]. Each L-shaped beam was divided into two rectangular sub-divisions using each of the three proposed solutions, as shown in Fig. 20(a–c). The comparison between the torsional behavior predicted by the model and the experimentally observed behavior is shown in Fig. 26. The figure shows that the model predictions agree well with the experimental results. Table 4 shows the experimentally observed ultimate torque and the corresponding angle of twist versus the analytically calculated ones using the three solutions shown in Fig. 20. From the table, we can see that any of the three solutions showed good compliance with the experimentally observed results for flanged beams. However, solution II predictions were more consistent compared with those of solutions I and III for beams under combined torsion. This might be because solution II follows the stirrups configuration.

120 Experimental

100 ) N 80 k (

Solution I Solution II Solution III

2 Q , e 60 c r o f r a e h 40 S

20

0 0.00

0.60

1.20 0.00

1.00

2.00

3.00 0.00

0.50

1.00

Transversal steel strain (1000 micro-strain)

(a)

(b)

69

(c)

Fig. 25. Shear force versus transversal steel strain for (a) TB1; (b) TB2 and (c) TB3.

6. Conclusions 1. An innovative test setup capable of simulating the behavior of T-shaped beams under combined shear and torsion was designed and implemented. 2. The behavior of the tested inverted T-shaped beams was affected by the value of the torque to shear ratio. Decreasing the applied torque to the applied shear force ratio resulted in the following: (1) a significant reduction for the spacing between diagonal cracks, the strut angle of inclination, cracking and ultimate torque, flange and web stirrup strain; (2) a significant increase for the failure and cracking load, post-cracking torsional rigidity, cracking and ultimate shear; and (3) the stirrups efficiency was reduced, thus, beams failed due to concrete diagonal failure rather than stirrups yield. 3. The proposed analytical model showed remarkable agreement with the experimental results for the behavior of flanged beams under combined actions.

Fig. 26. Torque versus angle of twist for L-shaped beams [45].

torsion [52]. Beams had the same cross-section dimensions, but the stirrups spacing were different. The model was used to predict the torsional behavior of three rectangular RC beams up to ultimate torsion. The comparison between the model predictions and the experimental results for the tested RC beams are shown in Fig. 23(a–c). The predicted behaviors were found to be in

Table 4

Strength and deformation predicted using the proposed model with solutions I, II, and III versus measured. Beam

TB1 TB2 TB3 BK-Ta BK-TVM-1a

Experimentally observed ultimate

Predicted by the model

Torque

Angle of twist

Torque

(kN m)

(/m)

I

II

III

I

II

III

I

23 22.7 10.8 16.8 18.6

2.82 3.16 0.5 1.9 2.5

25.4 21.5 9.3 16.2 16.2

23 21.3 11.8 16.8 16.8

20.3 21.2 8.6 17.3 17.3

2.8 3.2 0.49 1.91 1.91

2.6 3.2 0.49 1.9 1.9

2.82 3.16 0.49 1.9 1.9

0.91 1.06 1.17 1.04 1.15

1.00 1.07 0.98 1.00 1.10

1.07 10% ±0.1

1.03 1.10 5.0% 10% ±0.05 ±0.1

Average Coefficient of variation 95% Confidence interval a

Ref. [45].

Experimental/predicted Angle of twist

Torque

Angle of twist II

III 1.14 1.07 1.26 0.97 1.08

I

II

III

1.00 1.00 1.01 0.99 1.31

1.07 1.00 1.01 0.98 1.29

1.00 1.00 1.01 0.98 1.29

1.06 13% ±0.14

1.07 12% ±0.13

1.06 12% ±0.13

70


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Behavior and Analysis of Inverted T-shaped RC Beams under Shear and Torsion.pdf

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