38limit state of collapse : flexure

38.1 Assumptions

Design for the limit state of collapse in flexure shall be based on the assumptions given below:

• a) Plane sections normal to the axis remain plane after bending.
• b) The maximum strain in concrete at the outermost compression fibre is taken as 0.0035 in bending.
• c) The relationship between the compressive stress distribution in concrete and the strain in concrete may be assumed to be rectangle, trapezoid, parabola or any other shape which results in prediction of strength in substantial agreement with the results of test. An acceptable stress-strain curve is given in Fig. 21. For design purposes, the compressive strength of concrete in the structure shall be assumed to be 0.67 times the characteristic strength. The partial safety factor $\gamma_{\mathrm{m}}=1.5$ shall be applied in addition to this.

NOTE — For the stress-strain curve in Fig. 21 the design stress block parameters are as follows (see Fig. 22):

Area of stress block $=0.36 f_{c} \cdot x_{o}$

Depth of centre of compressive force = 0.42 x

from the extreme fibre in compression where

$\begin{aligned} f_{\mathrm{cu}} & =\text { characteristic compressive strength of concrete, and } \\ x_{0} & =\text { depth of neutral axis. }\end{aligned}$

• d) The tensile strength of the concrete is ignored.

e) The stresses in the reinforcement are derived

• from representative stress-strain curve for the type of steel used. Typical curves are given in Fig. 23. For design purposes the partial safety factor $\gamma_{\mathrm{m}}$, equal to 1.15 shall be applied.
• f) The maximum strain in the tension reinforce- ment in the section at failure shall not be less than:

$\frac{f_{y}}{1.15 E_{s}}+0.0002$

where

Figure description:

Stress-Strain Relationship for Concrete

Stress-Strain Curve for Concrete

Legend

• fck (Characteristic strength — color: black; symbol: Solid line
• 0.67 fck — color: black; symbol: Solid line
• 0.67 fck / γm (Design strength — color: black; symbol: Solid line

Annotations

• PARABOLIC CURVE (text_label - position: Points to the curve between strain 0 and 0.002)
• 0.002 (vertical_line - position: x = 0.002)

FIG. 21 STRESS-STRAIN CURVE FOR CONCRETE

Figure description:

Image Content: Stress Block Parameters (Fig. 22

• $f_y$: Characteristic strength of steel
• $E_s$: Modulus of elasticity of steel
• $X_u$: Depth of the neutral axis
• $0.42 X_u$: Distance from the extreme compression fiber to the centroid of the compressive force
• $0.36 f_{ck} X_u$: Total compressive force in the concrete stress block (where $f_{ck}$ is characteristic compressive strength of concrete
• Diagram: Illustrates the parabolic-rectangular stress distribution in a concrete section under compression.

Figure description:

Stress-Strain Relationship for Steel Reinforcement

Representative Stress-Strain Curve for Steel

Legend

• Characteristic curve (fy — color: black; symbol: Solid line (upper
• Design curve (fy/1.15 — color: black; symbol: Solid line (lower

Annotations

• Es = 200000 N/mm2 (text_label - position: bottom right, above strain axis)
• Offset lines (dashed_line - position: Parallel to initial slope, intersecting x-axis at labeled offset values)
• fy/1.15 (text_label - position: y = 0.87 fy)