The applied stresses in the ℓ-t coordinate system,   σ l ,   σ t and τ l t are in equilibrium with the average internal stresses in the rebar (f _l and f _t ) and in the concrete (   σ 2 c ,   σ 1 c and τ 21 c ). Based on 2D stress transformation, the following equations were developed. σ l = σ 2 c cos 2 ⁡ α 2 + σ 1 c sin 2 α 2 + τ 21 c 2 ⁡ sin ⁡ α 2 ⁡ cos ⁡ α 2 + ρ l f l (1a) σ t = σ 2 c sin 2 α 2 + σ 1 c cos 2 α 2 − τ 21 c 2 ⁡ sin ⁡ α 2 ⁡ cos ⁡ α 2 + ρ t f t (1b) τ l t = − σ 2 c + σ 1 c sin ⁡ α 2 ⁡ cos ⁡ α 2 + τ 21 c cos 2 α 2 − sin 2 α 2 (1c) Here, σ 2 c ,   σ 1 c and τ 21 c are the normal and shear stresses in concrete in the two to one coordinate system, respectively. ρ l and ρ t are the reinforcement ratios in the ℓ- and t-directions, respectively.

The strains in the ℓ- t coordinate system ( ε l ,   ε t and γ l t ) are expressed in terms of the strains in the two to one coordinate system ( ε 2 ,   ε 1 and γ 21 ) based on 2D strain transformation. ε l = ε 2 cos 2 α 2 + ε 1 sin 2 α 2 + γ 21 2 2 ⁡ sin α 2 cos ⁡ α 2 (2a) ε t = ε 2 sin 2 α 2 + ε 1 cos 2 α 2 − γ 21 2 2 ⁡ sin ⁡ α 2 ⁡ cos ⁡ α 2 (2b) γ l t 2 = − ε 2 + ε 1 sin ⁡ α 2 cos ⁡ α 2 + γ 21 2 cos 2 α 2 − sin 2 α 2 (2c)

It is to be noted that γ 21 generates due to the shear-extension coupling in an anisotropic element. This leads to an increase in γ _lt .

Based on extensive tests of panels under biaxial tension-compression, the following relationships were developed.

1) Concrete under compression (Belarbi (1991); Belarbi and Hsu (1995))

For   ε 2 u / ζ ε 0 ≤ 1 , σ 2 c = ζ f c / 2 ε 2 u ζ ε 0 − ε 2 u ζ ε 0 2 (3a)

For   ε 2 u / ζ ε 0 >   1 , σ 2 c = ζ f c / 1 −   ε 2 u / ζ ε 0 − 1 4 / ζ − 1 2 (3b) Here, ε 2 u and ε 0 are the uniaxial component of compressive strain, and compressive strain corresponding to peak stress in a concrete cylinder, respectively. The symbol f c / . represents the compressive strength of concrete cylinder. The compressive strength of concrete in the panel is represented as ζ f c / . The softening of concrete under compression due to orthogonal tensile strain is quantified by the coefficient ζ, which is defined as follows (Zhang and Hsu, 1998). ζ = 0.9 1 + ε 1 u η / (4) η / is taken as η or reciprocal of η whichever is less than 1.0. η is the ratio of the capacities of the rebar along the transverse and longitudinal directions ( ρ t f y t / ρ l f y l ). This is a measure of asymmetry in the reinforcement grid.

2) Concrete under tension (Belarbi and Hsu, 1994)

For   ε 1 u ≤ ε c r , σ 1 c = E c ε 1 u (5a)

For ε 1 u > ε c r , σ 1 c = f c r ε c r ε 1 u 0.4 (5b) Here, f c r , ε c r , ε _1u and E c are the cracking stress, cracking strain, uniaxial component of tensile strain and the elastic modulus of concrete in uniaxial tension, respectively. For the post-cracking analysis, only Eq. 5B is required.

3) Concrete under shear (Zhu et al., 2001)

The shear stress and strain in concrete are related through the normal stresses and strains so as to use the previous constitutive relationships and avoid an empirical shear modulus.

4) Rebar under tension (Belarbi and Hsu, 1994)

The following expressions are in generic notations which are applicable for the bars along the ℓ- and t-directions.

For ε s u ≤ ε n ∗ , f s = E s ε s u (7a)

For ε s u > ε n ∗ , f s = 0.91 − 2 B f y + 0.02 + 0.25 B E s ε s u = f n ∗ + E p ε s u (7b) Here,   f s and ε s u are the stress and strain in the bars, respectively. ε n ∗ = 0.91 − 2 B ε y approximates the apparent yield strain. ε y and f y are the yield strain and the yield stress of a bare bar coupon, respectively. f n ∗ is the apparent yield stress of the rebar embedded in concrete. B = 1 / ρ f c r / f y 1.5 is a measure of tensile strength of concrete with respect to the yield stress of rebar. Eqs. 7A,7B are termed as uniaxial relationships, as they were developed by testing panels under uniaxial tension.

As mentioned before, in the SMM, the Poisson’s effect is considered through an orthotropic formulation of 2D strains in the two to one coordinate system. The uniaxial components of the strains ( ε 1 u and ε 2 u ) are related to the total strains ( ε 1 and ε 2 ) in terms of apparent Poisson’s ratios (Hsu/Zhu ratios) ( ν 12 and ν 21 ) as follows (Sengupta and Belarbi (2001); Bavukkatt (2008)). ε 2 ε 1 = 1 − ν 21 − ν 12 1 ε 2 u ε 1 u (8a)

Considering ν 21 = 0 (based on tests it was found that the effect of tension on compressive strain is negligible), the strains are expressed as shown in Eqs. 8B, 8C. The uniaxial strains ε 1 u and ε 2 u are denoted as ε ¯ 1 and ε ¯ 2 in the reference. ε 2 = ε 2 u (8b) ε 1 = ε 1 u − ν 12 ε 2 u (8c)

The Hsu/Zhu ratio ν 12 is defined as follows (Eqs. 9A, 9B).

For ε s f ≤ ε y , ν 12 = 0.2 + 850 ε s f (9a)

For ε s f > ε y , ν 12 = 1.9 (9b) Here, ε s f is the average tensile strain of bars along the ℓ- and t-directions that yield first.

The above equations are solved simultaneously to develop the shear stress versus strain behaviour of a membrane element.

A biaxial panel testing facility is available at the Structural Engineering Laboratory of Indian Institute of Technology Madras to test RC panels under in-plane loading. Originally the facility was used to test prestressed panels under biaxial tension (Achyutha et al., 2000). This was subsequently reconfigured to conduct biaxial tension-compression tests (Sengupta et al., 2005). The facility consists of a horizontal self-equilibrating system made of frames and beams, supported on a fiber reinforced concrete floor. Loading of capacity 2000 kN can be applied in each horizontal direction. A schematic sketch of the setup is shown in Figure 4A. The components are:

1) Two stiff built-up beams and high strength tie rods of 32 mm diameter, for transferring tension. The beams are placed on heavy duty sliding bearings.

The two sets of jacks are operated separately by two pumps, and two pairs of distribution blocks. The oil pressure from each pump is controlled by a hand operated lever. Each distribution block maintains approximately equal pressure in the four jacks connected to it. A view of the test setup is shown in Figure 4B.

To investigate the effect of the chosen parameters, panel specimens were tested under sequential tension–compression. Although, increasing shear corresponds to proportional increase of tension and compression, a sequential tension–compression load path was selected to segregate the effects of tension (in terms of R ( f t ) and compression (in terms S σ 2 c on the shear strain   γ 21 . Initially, tension was applied along the 1-direction up to a predetermined level based on R f t . It was maintained constant during the subsequent compression phase. The compression was applied along the 2-direction up to the crushing of concrete S σ 2 c = 1.0 .

All the panel specimens were of dimensions 800 mm × 800 mm × 100 mm. The horizontal dimensions were fixed based on the requirement that a minimum of three to four cracks form along the direction of tension within the test region. The thickness of 100 mm was selected such that the capacity of a panel with normal strength concrete, when tested under uniaxial compression, was less than the capacity of the testing facility i.e., 2000 kN. The reinforcement was provided in two layers and details are shown in Figure 5.

The following features were added to avoid premature failures.

1) Stitching reinforcement was provided along the tension edges of the panel to avoid premature cracking of the edges.

Load cells of capacity 500 kN were used to measure the tension load applied by the hydraulic jacks. As there was no gap to place load cells on the compression side, a hydraulic jack connected in series to the compression jacks was placed in a separate reaction standalone frame outside the panel tester, to measure the compression load.

Deformations were measured using linear variable differential transducers (LVDTs). LVDTs were fixed only on the top face of the panel. As the bottom face was inaccessible, no LVDT was placed below the panel. The average strains were calculated from the measured deformations. Arrangement of the LVDTs is shown in Figure 6. LVDTs one and two were used to record deformation along the compression direction (ε₂). LVDTs three and four recorded the deformation along tension direction (ε₁). LVDTs five to eight recorded the deformations along the diagonals to quantify the average shear strain (γ₂₁).

Figure 7 shows a typical cracked specimen at different stages of the loading during the test. Panel P45-4-2 A is chosen for demonstration. It can be observed from the figure that the cracks which started to form perpendicular to the direction of tension rotate gradually to become parallel to the λ-direction with increasing load. This can be attributed to the higher stiffness in the λ-direction due to the presence of higher amount of reinforcement along the λ-axes.

Shear strain γ 21 can be computed from three measured strains from a rosette, using 2D strain transformation equations. Since four strains were measured along 1-, 2-, A- and B- directions, as shown in Figure 8A, first 2D Mohr’s compatibility condition was checked with the measured strains. Figure 8B shows the Mohr’s circle for measured strains. The compatibility condition ε 1 + ε 2 = ε A + ε B for P45-4 series panels is demonstrated in Figure 8C. It can be noted from the plot that the measured strains are consistent and satisfy the criteria during the initial phase of loading. However, the equality slightly deviates with increase in load. This can be attributed to the substantial cracking which occurred due to the application of tension close to the yield load. Thus, a shear strain value calculated from three measured strains may not be consistent. This is demonstrated in Figure 9. If the compatibility is maintained, all the strains would have fallen on the points marked as “Expected values”. As the experimental values do not coincide with the expected values, a unique Mohr’s circle cannot be drawn considering all the four points. A best fit Mohr’s circle can be drawn by calculating the root mean square (RMS) value of γ 21 given by the following equation. γ 21 2 = ε 0 − ε A 2 + ε 0 − ε B 2 2 (13a) Here, ε 0 is the average location of the centre of the circle and is given by the following expression. ε 0 = ε 1 + ε 2 + ε A + ε B 4 (13b)

Based on the method of separation of variables, γ 21 was modelled as a function of the identified parameters as shown in Eq. 7. Although, sequential tests were conducted, the effects of the two parameters causing the non-linear variation of the response, R(f _t ) and S ( σ 2 c ), act simultaneously under proportional loading (Figure 12). Therefore, γ 21 was expressed as the product of two functions F 1 S σ 2 c and F 2 R f t .

The other two parameters H and α 2 affect the magnitude of γ 21 . The maximum value of γ 21 in panels with difference in amounts of reinforcement is modelled by F 3 H . The maximum value of γ 21 in panels with rebar grid inclined at angle other than 45° is modelled by F 4 α 2 . Since a membrane element can have both cases of asymmetry simultaneously, additive functions were selected. γ 21 = F 1 S σ 2 c F 2 R f t F 3 H + F 4 α 2 (14)