Shear-friction truss model for reinforced concrete beams

Transcription

1 Shear-friction truss model for reinforced concrete beams S G Hong, Seoul National University, South Korea T Ha*, Seoul National University, South Korea 26th Conference on OUR WORLD IN CONCRETE & STRUCTURES: August 2001, Singapore Article Online Id: The online version of this article can be found at: This article is brought to you with the support of Singapore Concrete Institute All Rights reserved for CI Premier PTE LTD You are not Allowed to re distribute or re sale the article in any format without written approval of CI Premier PTE LTD Visit Our Website for more information

2 26th Conference on Our World in Concrete & Structures: August 2001, Singapore Shear-friction truss model for reinforced concrete beams S G Hong, Seoul National University, South Korea T Ha*, Seoul National University, South Korea Abstract This paper presents a new model, called the "shear-friction truss model," for slender reinforced concrete beams to derive a clear and simple design equation for their ultimate shear strength. In this model, a portion of the shear strength is provided by shear reinforcement as in the traditional truss model, and the remainder by the shear-friction mechanism. Friction resistance is derived considering both geometrical configuration of the rough crack surface and material properties. The inclined angle of diagonal strut in the traditional truss model is modified to satisfy the state of balanced failure, when both stirrups and longitudinal reinforcement yield simultaneously. The vertical component of friction resistance is added to the modified truss model to form the shear-friction truss model. Test results from published literatures are used to find the effective coefficient of concrete strength in resisting shear on inclined crack surfaces. Keywords: shear-friction, truss model, reinforced concrete beam, aggregate interlock, shear crack 1. Introduction The truss model adopted in current design codes is a good mechanical model to express the behavior of reinforced concrete members after cracking. However, it cannot solely present the ultimate strength of reinforced concrete beams, but must be accompanied by various other equations for concrete contribution, which are usually derived from test results. For example, ACI states that the nominal shear resistance Vn is: (1 ) where V c ' Vs are the shear components carried by concrete and shear reinforcement, respectively [1]. While Vs can be calculated using truss analogy as described later, the prediction of Vc depends only on the empirical equation which lumps the contribution of aggregate interlock, dowel action and uncracked concrete together. For members subject to shear and flexure, the code specifies Vc as: Vc = ijf:bwd (in SI unit) (2) Vc = ikbwd (in SI unit) (2) where b w is the web thickness and d is the effective depth of the member. Compared with the test results on simply supported beams without web reinforcement, Eq.(2) generally underestimates the shear capacity for beams with large longitudinal reinforcement ratios and overestimates the shear strength for small steel percentages [2]. This paper presents a different approach to evaluating shear contribution by concrete in terms of the shear-friction mechanism. It is assumed that rough crack surfaces playa major role in shear 309

3 capacity of cracked concrete and friction resistance is proportional to the strength of concrete, and contact area. Based on these assumptions, a simple friction equation is derived and is added to the truss model with modified crack angle. 2. Review of the conventional truss model The response of uncracked reinforced concrete members to applied loads can be understood by using elastic analysis. However, cracking of the concrete causes stress redistribution and the elastic beam theory cannot be applied. Truss analogy, developed in the early twentieth century, visualizes the load path in the cracked reinforced concrete beam as a statically determinate truss. It is an excellent conceptual model to show the force flows in cracked concrete members. As shown in Figs. 1 and 2, the truss model consists of the several vertical tension members and diagonal compression members running parallel to the inclined shear crack. The stirrups crossing a crack are collected into a vertical tension member, and all the concrete struts meeting at any vertical section playa diagonal compression member. The traditional truss model assumes a 45 constant angle of inclination as a conservative design approach. From the free body diagram of Fig. 1, the shear force can be calculated as: v = A/syjd (3) s where At is the area of vertical reinforcement, fsy is the yield strength of the stirrups, jd is an internal lever arm and s is the spacing of stirrup legs. Although the analysis of reinforced concrete beams using the traditional truss model is clear and simple, this model disregards the shear transferred by the aggregate interlock along the crack suriace, the dowel action of longitudinal reinforcement and the uncracked compression zone [2]. The ACI code uses Eq.(3) for Vs except that jd is replaced with d. I I r ~""'"., "." '1 ~~-_r-,_-r_~-_r~~=-i --~ ~-ICI i I I, V ~f ~fsy: : I / ~~=======r.=... :i/1 Tv +.:::-!... ~... T -... :.y- --!. - -!. - - ;: ~~==~ =... ~... ~...:::::::: v...; ~fsy Fig. 1 Internal forces at an inclined crack Fig. 2 Equivalent truss of a beam Fig. 3 Crack patterns of slender reinforced concrete beam [3] 3. Shear-friction truss model When a slender reinforced concrete beam is loaded, flexural cracks appear and extend vertically up to the neutral axis of the beam. With increased loading, the flexural cracks develop to shear cracks or new inclined shear cracks initiate in the uncracked region. This inclined shear cracks always occur before shear failure and progressively cover the entire depth in the ultimate state (Fig. 3). Additional components other than shear reinforcement (truss action) begin to resist shear loading after inclined shear cracks appear, namely aggregate interlock along the crack surface, the dowel action of longitudinal reinforcement and the uncracked compression zone. Although these components have different mechanisms in resisting shear loading, their contribution can be explained commonly by 310

4 (a) Frictional forces on the crack surface Fig. 4 Shear-friction mechanism (b) Infinitesimal element on the crack surface the shear transfer across rough crack surfaces by the friction mechanism. As the inclined cracks develop into the uncracked region, the two cracked parts of the beam move relatively to each other as shown in Fig. 4(a). This relative displacement along the rough crack surface is resisted mainly by aggregate interlock, and the shear reinforcement prevents excessive opening of the shear crack to make the interlocking mechanism possible. Both the uncracked compression zone and the dowel action also provide additional confinements to the displacements. Shear failure occurs when crack widths become so large that the friction mechanism no longer controls the relative movement of the two cracked parts. Fig. 4(a) shows the friction forces and two infinitesimal elements involving the crack surface. These two elements are under different states of stress. Since uniaxial tension is applied to the lowerleft element, it cannot transmit friction forces across the crack. On the contrary, the upper-right element is under uniaxial compressive stress which is in equilibrium on the crack surface.. 2 (ja = (jc Sin a 'l"a = (jc sinacosa where (jc is uniaxial stress along the average crack orientation, and (ja' 'l"a stresses normal and tangential to the crack surface, respectively. (4) (S) are the frictional Since shear-friction resistance is the sum of 'l"a in the direction of average crack path, its magnitude depends on the COSa of the deviation angle of the individual contact surface. As shown in Fig. S, crack orientation constantly changes from the average crack path due to the roughness of shear crack. For the relative displacement of the two cracked bodies shown in Fig. 4, parts of the crack surfaces which deviate in the counter-clockwise direction contribute to the shear-friction mechanism. If the crack deviations across the entire crack surface are assumed to cancel out, only half of the crack surface can be used to evaluate friction force, and shear-friction in the direction of average crack path, 'l"f' is obtained by calculating the mean value of the counter-clockwise deviation angle. Various probability density functions, called "contact density functions," have been used to describe the distribution of local deviation angle. These functions p(o) have the following properties (3): a. Symmetry about 0 = O. b. p(o) = 0 outside the region -tr/2::;; 0 ::;; +tr/2 c. [!!p(o)do = 1 2 Li and Maekawa found that the cosine function satisfies all the required properties of the contact density function and suggested taking p(o) = O.Scos(O) in the region -tr/2::;; 0::;; +tr/2. Their final model was reported to produce excellent correlation with specimens having f;::;; SO MPa (3). However, as(-) Fig. S Crack orientations in a shear crack 311

5 since the crack profiles are less tortuous for higher strength, a different function is needed for high strength concrete. This paper uses Li and Maekawa's contact density function to find the average angle of local crack deviation, and depends on the effective coefficient to explain the effect of concrete strength. gacosada aavg = = 32.7 gcosada (6) '!"f = CT c sin32.7 cos = 0.206CT c (7) According to Eq.(7), shear-friction capacity of a rough surface is only about 20 percent of the uniaxial compressive stress present in the direction of average crack angle. This uniaxial compression is in the same direction as the diagonal compression strut in the traditional truss model but is resisted by the cracked concrete element. Based on the above arguments, the following assumptions are made to describe the overall response of reinforced concrete beams to shear loading. a. The shear-friction mechanism is responsible for the additional shear capacity to the truss model for slender reinforced concrete beams with inclined shear cracks. b. The dowel action of longitudinal reinforcement and the uncracked compression zone contribute to the prevention of excessive opening of the inclined shear cracks, and are included in the shear-friction mechanism. c. Frictional resistance is uniformly distributed along the effective contact area of the crack surface and depends on the strength of the concrete. The first assumption can be expressed as: V=~+v, (8) where ~, V, are the shear components carried by truss action and friction mechanism, respectively. According to the second assumption, V f comprises the resistance by aggregate interlock, dowel action and the uncracked compression region. When combining Eq.(7) with the concept of effective contact area (Ae ), the magnitude of friction can be calculated as: F = '!"faa = 0.206CT bw (0.5d) (9) c sin(} = v,f; ~wd sm(} where CT c is replaced by v,f;, the effective strength of concrete in resisting shear on an inclined crack surface. Eq.(9) is different from the usual friction equation, which has been adopted by many codes and previous researches and is given by: F=J1N (10) where J1 is the coefficient of friction and N is the normal compressive force on the contact surface. Because this normal force on the contact surface is indispensable for the friction mechanism to be available, there have been various attempts to determine the source of normal forces. These forces include shear-resisting forces of shear reinforcement and tensile forces of longitudinal reinforcement. However, the shear-friction mechanism on the inclined shear cracks of reinforced concrete members is different from the usual friction theory in that it involves force transfer across the crack surface, and thus depends more on the concrete strength than the normal forces on the contact surface. The truss component ~ is calculated using the same approach as that of the traditional truss model except that the inclined angle of the diagonal strut, i.e. the crack angle, is determined for the case of balanced failure of the stirrup, crack surface and longitudinal reinforcement. For the free body diagram of Fig. 6, equilibrium requirements for the left free body yield T' = 0.5~ cot(} + F cos (). Summing the moments of forces applied to the right free body about the point at the upper right end gives: A/y d - T' d +FCOS(}.~ -(~ + FSin(})( a- ~ cot(}) = 0 (11 ) 312

6 T' elf d T'- ~==========='--'T -± Fig. 6 Free body diagram of cracked reinforced concrete Because this quadratic equation for colo involves an unknown variable, v" an iteration process is needed to calculate both (J and v,. The crack angle has a maximum value if only the longitudinal reinforcement yields and has a minimum value when only stirrups crossing the inclined crack yield. Therefore, the crack angle gets larger as the amount of stirrup increases. When (J is determined, ~ is given as: v. = A/syd (12) t stan(j Considering that V, Inclined crack angle (J is the vertical component of the friction force F, Eq.(8) can be rewritten as: V = A/sy jd v,f;b w d (13) stan(j and effective coefficient v, are obtained by the following procedure: 1. Assume a linear relationship of f; and v,: v, = A. f; + B 2. Calculate (J and V using Eq.(11) and Eq.(13). 3. Find new coefficients A, B using linear regression of f; and v,. 4. Repeat steps 2 and 3 until satisfactory convergence is reached. 4. Effective coefficient of concrete strength Many researches on the shear-friction mechanism were carried out by direct shear tests or precracked pushoff tests. Test results show that shear-friction resistance increases with the concrete strength and the confinement effect. However, since the general configuration of the specimens is quite different from that of a beam under transverse loading, most of the test results are not directly applicable in evaluating the shear capacity of a beam. The shear transfer mechanism of the well known shear-friction specimens is developed only by the aggregate interlock along a precracked straight surface confined by tension ties. It does not include the effect of dowel action or uncracked compressive region. Table 1. Comparison with test results by Mphonde [4, 5] Specimen No. b d aid f' Av s fy Vexp (Jpre Vpre Vexp/Vpre c B B B B B B B B B B B B Note: The units are mm for length, mm 2 for area, kn for force and MPa for stress 313

7 ~ 118" or 3/16" or 118" + 3/16" stirrups at 3.5" spacing. / I I I ~rmlli l' f 3 No. :\" cover on A. 84 in 90 in Fig. 7 Specimen detail [4] Table 1 and Fig. 7 give the specimen details and test results of twelve simply supported reinforced concrete beams loaded at midspan [4]. These beams have an identical sectional property and span ratio, but differ in concrete strength, which ranges from 22 MPa to 83 MPa, and the amount and yield strength of stirrups. The effect of concrete strength and shear reinforcement on the shear-friction mechanism is shown in Fig. 8(a) and (b). Shear-friction components of the measured shear strength are obtained by subtracting truss components from the test results. As shown in the figure, shearfriction resistance of a beam increases for higher concrete strength, but shows only marginal increase as the amount of shear reinforcement increases. This relationship between concrete strength and friction component confirms the third assumption, and the effective coefficient of concrete strength in resisting shear-friction is developed by following the steps explained in the previous section as shown in Fig. 9. > v, = f; (f; in MPa) (14) 120 V -V (kn) exp s V-V (kn) exp s f; (kn) A/y (kn) o o (a) (b) Fig. 8 Effect of (a) concrete strength and (b) amount of shear reinforcement on shear-friction V f f' c Fig. 9 Effective coefficient of concrete strength in resisting shear-friction 314

8 (a) Normal strength concrete Fig. 10 Crack profiles (b) High strength concrete The effective coefficient is found to decrease linearly as the concrete strength increases, which is in agreement with other experimental results showing that the shear-friction mechanism is less effective in high strength concrete than in normal strength concrete [3, 7]. The theoretical basis for this phenomenon is strongly related to the properties of the constituents of concrete. As stated earlier, the shear-friction mechanism involves force transfer across crack surfaces. For normal strength concrete, these crack surfaces are formed along the interface of the aggregate and the cement matrix due to the higher stiffness of the aggregate than the cement matrix (Fig. 10(a». Failure of the interface to transmit shearing force results from loss of contact when crushing of the cement matrix occurs. In the case of high strength concrete, cracks usually penetrate into individual aggregate particles to form much smoother profiles as shown in Fig. 10(b). The test results are compared with the predicted values by the shear-friction truss model (Eq.(13» and the ACI code (Eq.(1» in Fig. 11. While the predicted values according to the ACI code are very conservative compared with the measured values, those of the shear-friction truss model are in accordance with them for all ranges of concrete strength. The mean value and standard deviation of the measured Ipredicted values are and 0.091, respectively. Fig. 11 also shows that Eq.(13) underestimates the capacity of beams with higher stirrup ratio compared with beams with low stirrup ratio. This tendency can be explained if the equation for the effective coefficient (Eq.(14» is improved to reflect the marginal increase of shear-friction resistance for higher shear reinforcement amount as shown in Fig. 8. Shear-friction contribution is about 53 percent of overall shear capacity and has higher values for members with higher concrete strength and lower shear reinforcement as shown in Fig. 12. This again proves the strong correlation of the concrete strength with the friction resistance. v.,xp -1.6 Vpred $ 0 1 o AC Shear-friction prediction 1. 1 ~ l I! I! 1 I., I I * Specimen No. Fig. 11 Comparison with test results V,/V pred (%) Specimen No. Fig. 12 Shear-friction contribution 5. Conclusion 1. Existing truss models are generally conservative and incomplete in predicting the ultimate shear strength of reinforced concrete beams. 2. The shear-friction truss model is developed to accommodate the contribution by the dowel action of longitudinal reinforcement, aggregate interlock, and uncracked compression zone. A shear-friction component is added to the traditional truss model which is modified to satisfy 315

9 the balanced failure of stirrups and longitudinal reinforcement. Shear-friction resistance is proportional to concrete strength and contact area. The effective coefficient is introduced to describe the influence of concrete strength on the shear-friction mechanism. The shearfriction model is well in agreement with test results. 3. Shear-friction resistance assumes about 50 percent of total shear strength, and is more effective in the case of normal strength concrete than high strength concrete. Acknowledgements This work was supported by the Brain Korea 21 Project. Opinions, findings, and conclusions in this paper are those of the authors and do not necessarily represent those of the sponsors. References: [1] ACI Committee 318, "Bui/ding Code Requirements for Reinforced Concrete (ACI }", American Concrete Institute, [2] MacGregor, James G., "Reinforced Concrete, Mechanics and Design, 3rd Edition", Prentice Hall, [3] Ali, Mohamed A., White, Richard N., "Enhanced Contact Model for Shear Friction of Normal and High-Strength Concrete", ACI Structural Journal, American Concrete Institute, 1999, [4] Chen, Simon A., "A Shear-Friction Truss Model for Reinforced Concrete Beams Subjected to Shear", Ph.D. Thesis, University of Alberta, [5] Mphonde, Andrew G., "Use of Stirrup Effectiveness in Shear Design of Concrete Beams", ACI structural Journal, American Concrete Institute, 1989, [6] Peng, Liying, "Shear strength of Beams by Shear-Friction", Master Thesis, University of Calgary, [7] Walraven, Joost, Frenay, Jerome, and Pruijssers, Arjan, "Influence of Concrete Strength and Load History on the Shear Friction Capacity of Concrete Members", PCI Journal, Prestressed Concrete Institute, 1987,