EC2 concrete shrinkage

Shrinkage.txt

/**Target mean cylinder strength at 28 days, in MPa.
Ref: EC2 §Table 3.1
*/
get_f_cm = LAMBDA(f_ck_28days, f_ck_28days + 8);


/**Notional size of member. Used for determining creep factor. 
Ref: EC2 §3.1.4(5)
*/
get_h_0 = LAMBDA(cross_sectional_area_mm2, perimeter_exposed_to_drying_mm,
    LET(
        A_c, cross_sectional_area_mm2,
        u, perimeter_exposed_to_drying_mm,
        2 * A_c / u
    )
);


/**Autogenous shrinkage strain (at time t in days), in με. 
Ref: EC2 §3.1.4(6)(3.11)
*/
get_autogenous_shrinkage_strain_time_ε_ca_t = LAMBDA(f_ck28days, time_in_days,
    LET(
        t, time_in_days,
        ε_ca_∞, get_autogenous_shrinkage_strain_final_ε_ca_∞(f_ck28days),
        β_as_t, 1 - EXP(-0.2 * SQRT(t)),
        β_as_t * ε_ca_∞
    )
);


/**Final autogenous shrinkage strain, in με. 
Ref: EC2 §3.1.4(6)(3.12)
*/
get_autogenous_shrinkage_strain_final_ε_ca_∞ = LAMBDA(f_ck28days,
    2.5 * (f_ck28days - 10)
);


/**Autogenous shrinkage strain at time t in days (με). Binder types = {"portland cement", "silica fume", "fly ash", "ggbs"}
percent_binder should be given if binder_type is given. percent binder is given as a percent of total binder weight. 
Ref: CIRIA C766 A4.6. 

Args:
    binder_type: "portland cement", "ggbs", "silica fume, "pulverised fuel ash", "fly ash", "fuel ash", "fa", "pfa" (default: "portland cement").
    percent_binder: (default: 0)
*/
get_autogenous_shrinkage_strain_CIRIA_C766 = LAMBDA(time_in_days, water_binder_ratio, [binder_type], [percent_binder],
    LET(
        t, time_in_days,
        w_b, water_binder_ratio,
        _binder, IF(ISOMITTED(binder_type), "portland cement", LOWER(binder_type)),
        _percent_binder, IF(ISOMITTED(percent_binder), 0, percent_binder),
        ε_ca_CEM_1, 240 * t^0.15 - 650 * (w_b),
        // The mod_factor is a tuple of: {percent change to ε_ca_CEM_1, for every percent of binder}
        mod_factor, IFS(
            _binder = "portland cement", {1, 0},
            _binder = "silica fume", {0.1, 1}, //i.e increase by 10% for every 1% of silica fume
            OR(_binder = "pulverised fly ash", _binder = "pulverised fuel ash", _binder = "fly ash", _binder = "fuel ash", _binder = "fa",  _binder = "pfa"),
                {-0.01, 1}, //i.e reduce by 1% for every 1% of fly ash
            _binder = "ggbs", {0.08, 0.1} //i.e increase by 8% for every 10% of ggbs
        ),
        IFNA(ε_ca_CEM_1 + (ε_ca_CEM_1 * PRODUCT(mod_factor) * _percent_binder),"Invalid binder")
    )
);


/**Basic drying shrinkage strain, in με. 
Ref: EC2 §3.1.4(6)(3.9) & Annex B.2 (B.11)(B.12). 

Args:
    cement_class: "R", "N", "S" (default: "R").
*/
get_basic_drying_shrinkage_strain_ε_cd_0∞ = LAMBDA(f_ck28days, relative_humidity_percent, [cement_class],
    LET(
        f_ck, f_ck28days,
        _cement_class, IF(ISOMITTED(cement_class), "R", UPPER(cement_class)),
        f_cm, get_f_cm(f_ck),
        f_cmo, 10,
        α_ds1, IFS(
            _cement_class = "S", 3,
            _cement_class = "N", 4,
            _cement_class = "R", 6
        ),
        α_ds2, IFS(
            _cement_class = "S", 0.13,
            _cement_class = "N", 0.12,
            _cement_class = "R", 0.11
        ),
        RH, relative_humidity_percent,
        RH_0, 100,
        β_RH, 1.55 * (1 - (RH / RH_0)^3),
        0.85 * β_RH * ((220 + 110 * α_ds1) * EXP(-α_ds2 * f_cm / f_cmo)) 
    )
);


/**Basic drying shrinkage strain (at time t in days), in με. 
Ref: EC2 §3.1.4(6)(3.9). 

Args:
    cement_class: "R", "N", "S" (default: "R").
*/
get_drying_shrinkage_strain_ε_cd_t = LAMBDA(
    f_ck28days,
    relative_humidity_percent,
    cross_sectional_area_mm2,
    perimeter_exposed_to_drying_mm,
    time_in_days_at_consideration,
    time_in_days_of_beginning_of_drying,
    [cement_class],
    LET(
        A_c, cross_sectional_area_mm2,
        u, perimeter_exposed_to_drying_mm,
        RH, relative_humidity_percent,
        h_0, get_h_0(A_c, u),
        t, time_in_days_at_consideration,
        t_s, time_in_days_of_beginning_of_drying,
        _cement_class, IF(ISOMITTED(cement_class), "R", UPPER(cement_class)),
        ε_cd_∞, get_final_drying_shrinkage_strain_ε_cd_∞(f_ck28days, RH, A_c, u, _cement_class),
        β_ds_t, (t - t_s) / ((t - t_s) + 0.04 * SQRT(h_0^3)),
        MAX(β_ds_t * ε_cd_∞,)
    )
);


/**Final drying shrinkage strain, in με. 
Ref: EC2 §3.1.4(6)(3.8)(3.9) & Table 3.3. 

Args:
    cement_class: "R", "N", "S" (default: "R").
*/
get_final_drying_shrinkage_strain_ε_cd_∞ = LAMBDA(
    f_ck28days,
    relative_humidity_percent,
    cross_sectional_area_mm2,
    perimeter_exposed_to_drying_mm,
    [cement_class],
    LET(
        _cement_class, IF(ISOMITTED(cement_class), "R", UPPER(cement_class)),
        ε_cd_0, get_basic_drying_shrinkage_strain_ε_cd_0∞(f_ck28days, relative_humidity_percent, _cement_class),
        A_c, cross_sectional_area_mm2,
        u, perimeter_exposed_to_drying_mm,
        h_0, get_h_0(A_c, u),
        // Linearly interpolating between values of h_0 for k_h from Table 3.3
        k_h, IFS(
            h_0 <= 100, 1,
            h_0 >= 500, 0.7,
            h_0 <= 200, FORECAST(h_0, {1, 0.85}, {100, 200}),
            h_0 <= 300, FORECAST(h_0, {0.85, 0.75}, {200, 300}),
            h_0 <= 500, FORECAST(h_0, {0.75, 0.7}, {300, 500})
        ),
        k_h * ε_cd_0
    )
);


/**Total free shrinkage strain, in με. 
Ref: EC2 §3.1.4(3.8)
*/
get_total_shrinkage_strain_ε_cs = LAMBDA(drying_shrinkage_strain_ε_cd, autogenous_shrinkage_strain_ε_ca, 
    drying_shrinkage_strain_ε_cd + autogenous_shrinkage_strain_ε_ca
);