Reinforcement Calculator for Beam, Slab or footing by limit State method

Beam Footing or Slab Design by Limit State Method

Note: The above design is applicable to M35 grade only. For grades above M35, various coefficients change as shown in the following Design Procedure.

Shear Design Flow Diagram

Procedure for Design Checks

For a given cross-section, the ultimate axial load carrying capacity can be checked for uniform strain of $\epsilon_{c2}$ if the applied force is pure compressive and for $0.9\,\epsilon_s$ if the applied axial force is pure tensile. The ultimate axial force carrying capacity of the section for a given bending moment shall always be between these two limits. The position of the neutral axis $x$ for which the ultimate force carrying capacity matches the factored axial force is obtained by solving:

$f(x) = P_{us}(x) + P_{uc}(x) + N = 0$

where $N$ is the factored axial force, and $P_{us}(x)$ and $P_{uc}(x)$ are the contributions from steel and concrete respectively:

$P_{us}(x) = \sum \sigma_s(x) A_s$

$P_{uc}(x) = \int \sigma_c(x) \, dA_c$

Direct Solution for Rectangular Sections

The magnitude of compressive force $C_u$ and its location are determined from the concrete stress block in compression. Ultimate failure occurs when the extreme compressed fibre reaches limiting strain $\epsilon_{cu}$. Two cases are considered:

1) Neutral axis within the section

2) Neutral axis outside the section

1) Neutral axis within the section

a) Parabolic-Rectangular Stress Block

When the neutral axis lies within the section, the strain at the extreme compressed fibre is $\epsilon_{uc2}$ and the corresponding design stress is $f_{cd}$. For a rectangular section of width $b$ and neutral axis depth $x$, the resultant compressive force is:

$C_u = \beta_1 f_{cd} b x$

The position of $C_u$ from the extreme compressed edge is $\beta_2 x$.

Table C8.1 – Values of β₁ and β₂ (Parabolic-Rectangular Block)

Concrete Grade (fck)	β₁	β₂
≤ M40	0.80	0.42
M45	0.79	0.42
M50	0.78	0.42
M55	0.77	0.42
≥ M60	0.76	0.42

Alternative Detailed Values of β₁ and β₂ for Higher Grades

Parameter	fck (N/mm²)
	≤ 60	70	75	90	100	115
β₁	0.8095	0.74194	0.69496	0.63719	0.59936	0.58333
β₂	0.416	0.39191	0.37723	0.36201	0.35294	0.35294

b) Simplified Rectangular Stress Block

A simpler rectangular stress block may be used instead of the parabolic-rectangular block. The factors are defined as:

$\beta_1 = \lambda \eta$

$\beta_2 = \frac{\lambda}{2}$

Table C8.2 – Values of λ and η (Rectangular Stress Block)

Concrete Grade	λ	η
≤ M55	0.80	1.00
M60	0.79	0.99
M65	0.78	0.98
M70	0.77	0.97
M75	0.76	0.96
M80	0.75	0.95
≥ M85	0.74	0.94

Direct Values of β₁ and β₂ for Rectangular Stress Block

Parameter	fck (N/mm²)
	≤ 60	70	75	90	100	115
β₁	0.80	0.76781	0.73625	0.675	0.61625	0.56
β₂	0.40	0.39375	0.3875	0.375	0.3625	0.35

2) Neutral axis outside the section

a) Parabolic-Rectangular Stress Block

When the neutral axis lies outside the section (i.e., the entire concrete section is in compression), the strain at the less compressed edge will be less than the ultimate strain while the strain at the highly compressed edge remains at $\epsilon_{uc2}$.

In this case, the compressive stress distribution consists of:

• The full parabolic-rectangular stress block over the depth corresponding to the ultimate strain $\epsilon_{uc2}$, and
• An additional parabolic portion over the remaining depth where the strain varies linearly from $\epsilon_{uc2}$ to the strain at the less compressed edge.

The resultant compressive force $C_u$ and its location are calculated by superposing the contributions from both parts of the stress block. The factors $\beta_1$ and $\beta_2$ used previously for the case when the neutral axis is within the section are no longer directly applicable. Detailed integration or simplified expressions provided in relevant design codes/handbooks should be used for accurate evaluation.

The adoption of the assumptions in Clause 8.2.1 of the Code leads to a range of possible strain diagrams at the ultimate limit state under different loading conditions. When the neutral axis lies outside the section, numerous intermediate conditions arise between two extreme cases of strain distribution:

• One with uniform compressive strain $\epsilon_{c2}$ over the entire section (pure compression with no bending).
• The other with strain varying from $\epsilon_{cu2}$ at the extreme compressed edge to zero at the neutral axis (as in the case when the neutral axis is at the edge of the section).

b) Rectangular Stress Block

For the case where the neutral axis is outside the section, a simplified rectangular stress block can also be adopted with appropriate modification factors.

Table C8.4 – Values of λ, η and k for Rectangular Stress Block
(Neutral Axis Outside the Section)

Concrete Grade (fck)	λ	η	k
≤ M55	0.80	1.00	1.00
M60	0.79	0.99	0.99
M65	0.78	0.98	0.98
M70	0.77	0.97	0.97
M75	0.76	0.96	0.96
M80	0.75	0.95	0.95
≥ M85	0.74	0.94	0.94

Note: The factor k accounts for the reduction in the effective stress block when the entire section is under varying compression. The resultant compressive force and its location are determined using these modified parameters in conjunction with the geometry of the strain diagram.

Table C8.4 – Values of λ, η and k for Rectangular Stress Block
(when Neutral Axis is Outside the Section)

fck (N/mm²)	λ	η	k
	≤ 60	0.80	1.00	0.75
70	0.7875	0.975	0.73016
75	0.775	0.95	0.70968
90	0.75	0.90	0.66667
100	0.725	0.85	0.62069
115	0.70	0.80	0.57143

Table C8.3 – Values of β₃ and β₄

Table C8.3 – Values of β₃ and β₄ for Parabolic-Rectangular Stress Block
(when Neutral Axis is Outside the Section)

x/h	fck = 60 N/mm²		fck = 70 N/mm²		fck = 75 N/mm²		fck = 90 N/mm²		fck = 100 N/mm²		fck = 115 N/mm²
	β₃	β₄	β₃	β₄	β₃	β₄	β₃	β₄	β₃	β₄	β₃	β₄
	1	0.80952	0.41597	0.74194	0.39191	0.69496	0.37723	0.63719	0.36201	0.59936	0.35482	0.58333	0.35294
1.2	0.89549	0.45832	0.83288	0.43765	0.78714	0.42436	0.72968	0.41022	0.69249	0.40355	0.6772	0.40186
1.4	0.93409	0.4748	0.88197	0.45841	0.84129	0.44724	0.78831	0.43492	0.75381	0.42907	0.73986	0.42761
1.6	0.95468	0.48304	0.91168	0.4699	0.87615	0.46046	0.82826	0.44975	0.79679	0.44461	0.78422	0.44335
1.8	0.96693	0.48779	0.93113	0.47702	0.90007	0.46895	0.85695	0.45954	0.82834	0.45499	0.81702	0.45389
2	0.97481	0.49077	0.9446	0.48178	0.9173	0.47478	0.87838	0.46644	0.85234	0.46237	0.84211	0.4614
2.5	0.9855	0.49475	0.96464	0.48861	0.9442	0.48347	0.91348	0.47705	0.89255	0.47385	0.88448	0.47311
5	0.99702	0.49893	0.9906	0.49705	0.98285	0.49512	0.96937	0.49234	0.95972	0.49089	0.95622	0.49057

Table C8.5 – Values of β₃ and β₄ for Rectangular Stress Block
(when Neutral Axis is Outside the Section)

x/h	fck = 60 N/mm²		fck = 70 N/mm²		fck = 75 N/mm²		fck = 90 N/mm²		fck = 100 N/mm²		fck = 115 N/mm²
	β₃	β₄	β₃	β₄	β₃	β₄	β₃	β₄	β₃	β₄	β₃	β₄
1	0.80000	0.40000	0.76781	0.39375	0.73625	0.38750	0.67500	0.37500	0.61625	0.36250	0.56000	0.35000
1.2	0.88000	0.44000	0.85478	0.43750	0.82500	0.43125	0.76000	0.42000	0.69500	0.40625	0.63200	0.39250
1.4	0.91429	0.45714	0.89048	0.45625	0.86321	0.45089	0.80000	0.44000	0.73571	0.42589	0.67143	0.41429
1.6	0.93750	0.46875	0.91563	0.46875	0.89063	0.46563	0.82813	0.45313	0.76563	0.44063	0.70000	0.43125
1.8	0.95556	0.47778	0.93542	0.47813	0.91250	0.47656	0.85000	0.46250	0.78889	0.45156	0.72222	0.44444
2	0.97000	0.48500	0.95100	0.48563	0.93000	0.48500	0.86750	0.47000	0.80750	0.46063	0.74000	0.45500
2.5	0.98400	0.49200	0.96880	0.49375	0.95200	0.49500	0.89200	0.48000	0.83200	0.47250	0.76800	0.47000
5	0.99600	0.49800	0.99040	0.49875	0.98400	0.50000	0.93600	0.49000	0.88800	0.48500	0.84000	0.48500

The above tables are taken from IRC guidelines.

Estimation of Required Steel Area (A_st)

Example: Ultimate moment = 150 kNm, effective cover = 50 mm, concrete grade M45, steel Fe (assume Fe415 or Fe500 as per context), provided depth = 250 mm. Determine the required steel assuming 20 mm diameter bars.

Step 1: Check if provided depth is adequate

Required D ≈ 142.2 + 10 (half bar dia) + 50 (cover) = 202.2 mm < 250 mm.
Hence, the provided depth is adequate.

Effective depth d_eff = 250 – 50 – 10 = 190 mm.

Step 2: Determine neutral axis position

x/d = 0.290 < 0.617 (limiting value for Fe415/M45).
Hence, steel yields (under-reinforced section).

Note: An increase of 47 mm in depth reduces x/d from 0.617 (balanced) to 0.290.

Step 3: Calculate required steel area

If the balanced section lever arm is assumed instead (incorrect for this case), the calculated A_st would be higher:

This results in approximately 18% over-estimation of steel.
The neutral axis depth changes from 117.23 mm to 55.1 mm, and x/d from 0.617 to 0.290.

Conclusion: When the provided depth is greater than required, it is better to accurately estimate the neutral axis position rather than assuming a balanced section, to avoid over-estimation of reinforcement.