In the design of reinforced concrete beams, if the design ultimate moment is greater than the ultimate moment of resistance i.e. MEd > MRd, then compression reinforcement is required. Provided that d

_{2}/x ≤ 0.38 (i.e. compression steel has yielded) where d

_{2}is the depth of the

compression steel from the compression face and x = (d − z)/0.4.

The area of compression reinforcement, A

_{s2}, is given by:

Area of compression reinforcement A

_{S2}= (M

_{Ed}- M

_{Rd}) / (0.87f

_{yk}(d - d

_{2})) ------------- (1)

Area of tension reinforcement A

_{s1}= M

_{Rd}/ (0.87f

_{yk}z) + A

_{S2}----------- (2)

Where z = d[0.5+ √(0.25 - 0.882K')]

where

K’ = 0.167

**DESIGN EXAMPLE**

To show how this is done, let us consider the flexural design of support A of the beam loaded as shown below. The depth of the beam is 600mm, and the width is 400mm.

f

_{ck }= 35 N/mm

^{2 }; f

_{yk }= 460 N/mm

^{2}; concrete cover = 40mm

**Top reinforcement (Hogging moment)**

**Support A**

Effective depth (d) = 600 - 40 - 16 - 10 = 534 mm

M

_{Ed}= 761.24 KNm

But since the flange is in tension, we use the beam width to calculate the value of k (this applies to all support hogging moments)

k = M

_{Ed}/ (f

_{ck}b

_{w}d

^{2}) = (761.24 × 10

^{6}) / (35 × 400 × 534

^{2}) = 0.1906

Since k < 0.167, compression is required

Area of compression reinforcement A

_{S2}= (M

_{Ed}- M

_{Rd}) / (0.87f

_{yk}(d - d

_{2}))

M

_{R}d = 0.167f

_{ck}bd

^{2}= (0.167 × 35 × 400 × 534

^{2}) × 10

^{-6}= 666.69 KNm

d

_{2}= 40 + 16 + 10 = 66 mm

A

_{S2}= ((761.24 – 666.69) × 10

^{6}) / (0.87 × 460 × (534 – 66)) = 504.22 mm2

Provide 3X16 Bottom (A

_{sprov}= 603 mm

^{2})

Area of tension reinforcement A

_{s1}= M

_{Rd}/ (0.87f

_{yk}z) + A

_{S2}

Where z = d[0.5+ √(0.25 - 0.882K')]

K’ = 0.167

z = d[0.5+ √((0.25 - 0.882(0.167))] = 0.82d

A

_{s1}= M

_{Rd}/ (0.87f

_{yk}z) + A

_{S2}= ( 666.69 × 10

^{6}) / (0.87 × 460 × 0.82 × 534) + 504.22 mm

^{2}= 4308.66 mm

^{2}

Provide 6X32mm TOP (A

_{Sprov}= 4825mm

^{2})

*READ ALSO IN THIS BLOG*

Comparison of shear design procedure according to Eurocode 2 and BS 8110-1:1997

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