RC Beam Flexure Design to TCVN 5574:2018

What this check does

For a beam under bending, TCVN 5574:2018 (clause 8.1.2) checks that the design bending moment $M_{Ed}$ does not exceed the section bending capacity $M_{Rd}$:

$$M_{Ed} \le M_{Rd}$$

Unlike Eurocode 2, TCVN 5574 tabulates the design strengths directly - the concrete design strength $R_b$ and the steel design strengths $R_s$ (tension) and $R_{sc}$ (compression) already contain the material safety factors. You look them up in a table and use them as-is; there is no separate $\gamma_c$ / $\gamma_s$ division at the point of use.

The rectangular stress block

The method assumes a rectangular compression stress block: the concrete carries its full design strength $R_b$ uniformly over a compression depth $x$ (measured from the compression face), and the tension steel $A_s$ carries $R_s$.

For a singly reinforced section, horizontal (axial) equilibrium in pure bending gives the compression-zone depth:

$$R_b\, b\, x = R_s A_s \quad\Rightarrow\quad x = \dfrac{R_s A_s}{R_b\, b}$$

where:

• $R_b$ - concrete design compressive strength (MPa), from the grade table

• $b$ - section width (mm)

• $R_s$ - steel design tensile strength (MPa)

• $A_s$ - area of tension reinforcement (mm^2)

• $x$ - depth of the rectangular compression zone (mm)

The moment capacity is then taken about the tension steel:

$$M_{Rd} = R_b\, b\, x \left(h_0 - \dfrac{x}{2}\right)$$

with the effective depth $h_0 = h - a$, where $h$ is the overall depth and $a$ the distance from the tension face to the centroid of the tension steel.

The ductility limit - relative boundary height

A section is safe in bending only if the tension steel yields before the concrete crushes (a ductile failure). TCVN 5574 controls this with the relative compression-zone height $\xi = x/h_0$ and the relative boundary height $\xi_R$ (clause 8.1.2.2.3):

$$\xi_R = \dfrac{0.8}{1 + \dfrac{\varepsilon_{s,el}}{\varepsilon_{b2}}}, \qquad \varepsilon_{s,el} = \dfrac{R_s}{E_s}, \qquad \varepsilon_{b2} = 0.0035$$

where:

• $\xi_R$ - relative boundary height (the value of $\xi$ at which steel yielding and concrete crushing occur simultaneously)

• $\varepsilon_{s,el}$ - the elastic yield strain of the steel

• $E_s$ - steel modulus of elasticity (MPa)

• $\varepsilon_{b2} = 0.0035$ - the ultimate concrete compressive strain (heavy concrete)

The rule is:

• If $\xi \le \xi_R$ the section is under-reinforced - the tension steel yields first, the failure is ductile, and the capacity above is valid.

• If $\xi > \xi_R$ the section is over-reinforced - the concrete crushes first (brittle). The usable capacity is capped at the boundary depth $x_{lim} = \xi_R\, h_0$, and you should add compression steel or enlarge the section rather than rely on the extra tension steel.

Doubly reinforced sections

When compression steel $A_s'$ (area, at distance $a'$ from the compression face) is present and active, it enters both equilibrium and the moment:

$$x = \dfrac{R_s A_s - R_{sc} A_s'}{R_b\, b}$$

$$M_{Rd} = R_b\, b\, x \left(h_0 - \dfrac{x}{2}\right) + R_{sc} A_s'\,(h_0 - a')$$

where $R_{sc}$ is the steel design compressive strength and $a'$ the cover to the compression steel. If the resulting $x$ is very small ($x < 2a'$) the compression steel is not effective in compression and the section is treated as singly reinforced.

Design strengths (TCVN 5574:2018 tables)

Representative ULS design strengths used below:

Worked example

Section: $b = 300$ mm, $h = 600$ mm, $a = 50$ mm, concrete B25, tension steel CB400-V, $A_s = 1473$ mm^2 (about 3 x phi 25), singly reinforced. Design moment $M_{Ed} = 250$ kN.m.

Design strengths: $R_b = 14.5$ MPa, $R_s = 350$ MPa, $E_s = 200000$ MPa.

Effective depth:

$$h_0 = h - a = 600 - 50 = 550\ \text{mm}$$

Relative boundary height:

$$\varepsilon_{s,el} = \dfrac{R_s}{E_s} = \dfrac{350}{200000} = 0.00175$$

$$\xi_R = \dfrac{0.8}{1 + 0.00175/0.0035} = \dfrac{0.8}{1.5} = 0.533$$

Compression-zone depth:

$$x = \dfrac{R_s A_s}{R_b\, b} = \dfrac{350 \times 1473}{14.5 \times 300} = 118.5\ \text{mm}$$

$$\xi = \dfrac{x}{h_0} = \dfrac{118.5}{550} = 0.215 \le \xi_R = 0.533 \quad\checkmark\ \text{(ductile)}$$

Moment capacity:

$$M_{Rd} = R_b\, b\, x\left(h_0 - \dfrac{x}{2}\right) = 14.5 \times 300 \times 118.5 \times \left(550 - \dfrac{118.5}{2}\right) \times 10^{-6} \approx 253\ \text{kN.m}$$

Result: $M_{Ed} = 250 \le M_{Rd} \approx 253$ kN.m - the section passes in flexure, with the steel yielding first (ductile). The utilisation is close to 1.0, so there is little spare capacity.

Key points

• TCVN 5574 uses tabulated design strengths - no separate material-factor division at use.

• Bending capacity comes from the rectangular stress block: find $x$ from equilibrium, take the moment about the tension steel.

• Always check $\xi \le \xi_R$: if not, the section is over-reinforced (brittle) and the capacity is capped - add compression steel or resize.

• Doubly reinforced sections add the $R_{sc} A_s'(h_0 - a')$ term once the compression steel is effective ($x \ge 2a'$).