Note that "Ec" is not a constant property due to the non-liner stress-strain response of concrete so you'll need to clarify which Ec you are interested in. Tangent at small strains? Secant at service levels? etc. It is also not constant and the poissons ratio of r/c will not be constant due to cracking.

Most code-based shear models for R/C do not directly take into account Ec. However, some mechanical models account for the general shape of the concrete stress-strain response. For example, the Modified Compression Field Theory (Vecchio and Collins, ACI 1986) assumes a parabolic stress-strain response for concrete in uniaxial compression and code models derived from this (e.g. CSA A23.3, AASHTO LRFD) assume a strain at peak stress for the parabolic shape. Also, under the MCFT formulation, the shape of the parabola, and hence Ec, will change based on the amount of cracking from diagonal tension. Look for papers co-authored by Collins, Vecchio, Bentz, Mitchell, etc if you want to learn more about the underlying mechanics of these models.

http://library.jsce.or.jp/jsce/open/00670/No41/CLI-41-0113.pdf

Dear Adel,

thanks for the very interesting paper. However, the effect of Ec is not clearly shown in the paper. The consideration of Ec in relation to the characteristic length lch is not good enough to express the influence of Ec on shear strength because the shear equation is just empirically formulated in form of lch.

Best regards

Ngoc Linh Tran

The relation between modulus of elasticity E and modulus of rigidity G proves that.

This relation is: G=E/2(1+v) , in which v: Poissonْ s ratio.

Best regards.

Dear Abbas,

it is not really. The shear modulus shows only the stiffness in shear, regarding displacement, but is not related to failure as well as failure load.

Best regards,

Ngoc Linh Tran

Poisson's ratio (v) as well as modulus of elasticity (E) are important factors in determining the shear modulus (G) and clearly the larger role related to Poisson's ratio (v)

Note that "Ec" is not a constant property due to the non-liner stress-strain response of concrete so you'll need to clarify which Ec you are interested in. Tangent at small strains? Secant at service levels? etc. It is also not constant and the poissons ratio of r/c will not be constant due to cracking.

Most code-based shear models for R/C do not directly take into account Ec. However, some mechanical models account for the general shape of the concrete stress-strain response. For example, the Modified Compression Field Theory (Vecchio and Collins, ACI 1986) assumes a parabolic stress-strain response for concrete in uniaxial compression and code models derived from this (e.g. CSA A23.3, AASHTO LRFD) assume a strain at peak stress for the parabolic shape. Also, under the MCFT formulation, the shape of the parabola, and hence Ec, will change based on the amount of cracking from diagonal tension. Look for papers co-authored by Collins, Vecchio, Bentz, Mitchell, etc if you want to learn more about the underlying mechanics of these models.

Dear Mr. Lubell,

Thank you for the answer.

As you commented, Ec is not directly taken into account in most code-based shear models. Let's take the shear model in Model Code 2010 as an example, if we assume the effect shear depth z = 0.9d, the calculated shear strength is independent from Ec.

It is noted that the form fc^0.5 is often used to estimate the shear strength of concrete (related to aggregate interlock), the concrete tensile strength (mean value approx. 0.56*fc^0.5) and also the tangent modulus of elasticity of concrete (4700*fc^0.5). Therefore, the influence of Ec on shear strength is not clearly indentified if we use assumptions or estimations regarding to concrete properties in the calculation.

Come back to the question, actually I would like to know whether there is an experimental evidence of the influence of Ec on shear strength, or a series of shear strength tests, in which different aggregate types are studied. (Ec also depends on aggregate types).

Best regards,

Ngoc Linh Tran

For different aggregate types, see the research by Sherwood, Bentz and Collins published in ACI Structural Journal

The modulus of elasticity of concrete influences the shear strength of a RC or PC members. Recently developed mechanical models consider that the shear transferred by the uncracked compression zone has a relevant role on the shear strength of beams. The uncracked compression zone depends the neutral axis depth, which depends on the amount of longitudinal reinforcement and on the modular ratio, n=Es/Ec. Therefore, there is a clear dependence between Ec and the shear strength.

Please, look at the following references (among many others) for more information:

Marí, A., Bairán, J., Cladera, A., Oller, E., & Ribas, C. (2015). Shear-flexural strength mechanical model for the design and assessment of reinforced concrete beams. Structure and Infrastructure Engineering, 11(11), 1399-1419.

Cladera, A., Marí, A., Bairán, J. M., Ribas, C., Oller, E., & Duarte, N. (2016). The compression chord capacity model for the shear design and assessment of reinforced and prestressed concrete beams. Structural Concrete, 17(6), 1017-1032.

Cladera, A., Marí, A., Bairán, J. M., Oller, E., & Ribas, C. (2017). One-Way Shear Design Method Based on a Multi-Action Model. Concrete International, 39(9), 40-46.

To summarise, the higher Ec, the lower modular ratio n, the lower the neutral axis depth, and the lower shear strength. However, the concrete tensile strength (and the compression concrete strength) have a stronger influence, and in the other direction. For that reason, the higher concrete strength, the higher shear strength (in spite of the higher Ec).

Best,

Toni

Dear Mr. Cladera,

Thank you very much for your response.

I agree with you that the shear transferred by the uncracked compression zone depends on the modular ratio n = Es/Ec. If the shear strength in the compression zone is dominant (especially for large members), which mainly depends on the depth of the compression zone, the size effect may be not strong as expressed in some existing empirical formulae. Up to now, the maximum depth of concrete members without stirrup is 4 m (Toronto test 2015), and it is ONLY one. Unfortunately, the information of concrete properties is very limited, e.g. missing the tensile strength, fracture energy, elastic modulus of concrete. This is due to the fact that for this test the researchers assumed the shear capacity mostly depends on shear strength of crack concrete (aggregate interlock). Therefore, only the compression strength is required (for shear strength is a square root of fc).

For larger members, the typical way is to do the investigation with numerical models. In my opinion, the use of a numerical model always goes with experience, in which some empirical values can not be avoided. Therefore, the results from numerical analyses should be very carefully considered.

If fracture energy is also taken into account in shear strength calculation, the characteristic length lch = Gf*Ec/fct^2 shows that the Ec has a positive influence on shear strength (in the tension zone). Therefore, total influence of Ec on shear strength (in both compression zone and tension zone) is small, as you also identified.

Best regards,

Ngoc Linh Tran

Dear Mr. Tran,

thanks for you comment.

In any case, I think that the size effect is even important in the models based in the compression chord transfer action. I don't think the size effect is related to a reduction of aggregate interlock due to larger cracks, but it's also due to the size effect in the uncracked compression zone.

@Giovanni Muciaccia recently published the paper "Compressive failure and size effect in plain concrete cylindrical specimens" with other coworkers (Construction and Building Materials, 2017) in which they study the size effect just in the compressive failure. It's very intersting.

Best,

Toni Cladera

Dear Mr. Cladera,

thank you for the interesting comment and also the suggestion of the paper of Giovanni Muciaccia.

In any model, which are based on Rankine's criterion of failure, the shear strength is always determined by the strength of concrete (fc and fct). This means the size effect should be considered in the input parameters (fc and fct). Therefore, if the input parameters are used as measured by standard specimens (kept constant independent from member size, easy for use), an additional factor for size effect should be formulated in shear models.

Best regards,

Ngoc Linh Tran

In this paper, the influence of Ec on the shear capacity is evaluated through an example. Enjoy reading!

Article A mechanical model for the shear capacity of slender reinfor...

Best regards,

Ngoc Linh Tran