Design beam in torsion according to TCVN | Fundamentals for Structural Engineering
Beams subjected to simultaneous torsion and bending are cases that cannot be ignored when designing the structure of most civil works in practice. The reason is that the structure works in space, especially at the positions of edge beams and secondary beams that transmit loads to the main beams, due to horizontal loads. Failure to consider the torsional effect in practice causes torsional cracks on the web of the beam in many practical works. The requirements for calculating the design of reinforced concrete beams subjected to simultaneous bending and torsion are relatively clearly instructed, however, the calculation process is relatively complicated according to TCVN 5574:2012. Therefore, it requires designers to build accurate and convenient calculation tools for practice. The following article presents the process and principles for building a calculation table to check the bearing capacity of beams subjected to simultaneous bending and torsion.
1. Calculation assumptions
When calculating the spatial section (withstanding both bending moment M and torsional moment Mt), the internal forces are determined based on the following assumptions:
Ignore the tensile strength of the concrete
The compression zone of the spatial section is considered to be flat, inclined at an angle θ to the longitudinal axis of the member, the compressive strength of the concrete is taken as $R_b\sin^2\theta$, evenly distributed over the compression zone
The tensile stress in the longitudinal and transverse reinforcements cutting through the tensile zone of the spatial section under consideration is taken as the calculated strength $R_s, R_{sw}$
The stress of the reinforcement in the compression zone is taken as $R_{sc}$ with non-prestressed reinforcement
Control conditions of the problem:
$$M_t\leqslant{0,1}R_sb^2h$$
where b,h are the dimensions of the small and large sides of the rectangular beam section, respectively
The value of Rb for concrete with a strength level higher than B30 is taken as for concrete with a strength level of B30.
2. Calculation procedure
Calculation of spatial cross-section according to strength must be carried out according to the following conditions:
Internal force diagram in the cross-section of reinforced concrete structure subjected to simultaneous bending and torsion when calculating according to durability
The height of the compression zone x is determined from the condition:
ω: characteristic deformation properties of the compressed concrete zone: $\omega=\alpha-0,008R_b$ with α=0.85 for beam material being ordinary heavy concrete.
$\sigma_{sc,u}=$500MPa: limiting stress in the compressed concrete zone
if x<a’ then let $A’_s=0$ to recalculate $x=\frac{R_sA_s}{R_bb}$
The calculation needs to be carried out with 3 diagrams of the location of the compressed zone of the spatial section:
Diagram 1: at the edge compressed by bending of the member;
Diagram 2: at the edge of the member, parallel to the plane of action of the bending moment;
Diagram 3: at the edge stretched by bending of the member
Diagram of the location of the compression zone of the spatial cross-section: a – on the side compressed by bending; b – on the side parallel to the plane of the bending moment; c – on the side stretched by bending
$A_s,A′_s$: cross-sectional area of longitudinal reinforcement located in the tension zone and compression zone corresponding to each calculation diagram
$$\delta=\frac{b}{2h+b}, \lambda=\frac{c}{b}$$
c: projection length of the compression zone limit line onto the longitudinal axis of the structure, the calculation is performed with the most dangerous value c, c is determined by the method of gradual iteration and taken not greater than (2h+b)
The values $\chi, \varphi_q$ characterizing the relationship between the internal forces Mt, M, Q are calculated as follows:
When designing the structure, calculate according to:
Diagram 1: $\chi=\frac{M}{M-t}, \varphi_q=1$
Diagram 2: $\chi=0, \varphi_q=1+\frac{Qh}{2M_t}$
Diagram 3: $\chi=-\frac{M}{M_t}, \varphi_q=1$
When there is no bending moment M: $\chi=0, \varphi_q=1$
Torsional moment Mt, bending moment M and shear force Q are taken at the section perpendicular to the longitudinal axis of the member and passing through the centroid of the compression zone of the spatial section.
The value of the coefficient φw, which characterizes the relationship between transverse and longitudinal reinforcement, is determined by the formula:
