1.4 Design example – strut-tie model representing combined mechanism

1.4.1 Problem statement

In this section, the deep beam is designed by the strut-tie model representing the combined load transfer mechanism shown in Fig. 1-1(c). As illustrated in the previous sections, the dimensions and the information on the bearing plates and loads are determined in the same way.

1.4.2 Construction of strut-tie model

The ‘Beginning Mode’ is switched to the ‘Modeling Mode’ to construct a strut-tie model for the deep beam. As the shear span-to-effective depth ratio of the deep beam is 1.27, the indeterminate strut-tie model that represents a combined load transfer mechanism is selected automatically from the template for deep beam. In the model, the top and bottom horizontal elements are placed 127 mm and 102 mm away from the top and bottom surfaces of the deep beam, respectively.

Fig. 1-10 Constructed strut-tie model

1.4.3 Analysis of strut-tie model

The cross-sectional forces of struts and ties are determined by carrying out the finite element analysis of the constructed strut-tie model. As the model is an indeterminate truss structure, an iterative procedure is employed to find the converged axial rigidities and cross-sectional forces of struts and ties.

Fig. 1-11 Strut and tie forces

1.4.4 Strength verification and required rebars

1.4.4.1 Strength under bearing plates

The strength conditions under bearing plates are examined by the method illustrated in Section 1.2.4.1.

1.4.4.2 Required area of rebars

The required areas of main reinforcing bars are determined by the following equation. The requirement on the reinforcing bars is examined in the ‘Design Review’ as shown below.

A_s,req = F_u / φ f_y                                                        (1-10)

where,
φ = strength reduction factor of steel tie (= 0.75)
F_u = cross-sectional force of steel tie
f_y = yield strength of steel (= 414 MPa)

The spacing of shear reinforcing bars is determined by the following equation.

φF_n = (φ A_v f_y w_eff,tie ) / s_v ≥ F_u                                           (1-11)

where,
φ = strength reduction factor of steel tie (= 0.75)
A_v = area of vertical shear reinforcement
f_y = yield strength of steel (= 414 MPa)
w_eff,tie = effective width of vertical tie (The half of the shear span is a default. A user can define the width in ‘Assign-Outer Element’)
F_u = cross-sectional force of steel tie

1.4.4.3 Strength verification of struts

The strength conditions of concrete struts are verified by comparing the required widths with provided (available) widths of concrete struts, as shown below.

The strength conditions of concrete struts can also be verified visually as shown below.

Fig. 1-12 Required/proposed area of concrete strut

1.4.4.4 Strength verification of nodal zones

The strength condition of a nodal zone is verified by comparing the required width with the provided width of the nodal zone boundary, as shown below.

1.4.5 Minimum rebars for crack control

Since the effective strength coefficient 0.75 was taken for the two inclined struts of the strut-tie model, the ACI 318M-14 requirement for minimum reinforcing bars for crack control must be satisfied.

∑ (A_si / b_ss_i) sin γ_i ≥ 0.003                                                    (1-12)

where A_si is the total area of distributed reinforcement at spacing s_i in the i-th direction of reinforcement crossing a strut at an angle α_i to the axis of a strut, and b_s is the width of the strut.