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<p><strong>a) </strong>A reinforced concrete beam has a rectangular section of breadth b = 225 mm and effective depth d = 450 mm. It has to carry a design moment of 300.0 kNm. Show that compression reinforcement will be required. &nbsp;</p>
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<p><em>f<sub>cu</sub> </em>= 30 N/mm<sup>2</sup>, <em>f<sub>y</sub> </em>= 460 N/mm<sup>2</sup>.</p>
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<p><strong>b) </strong>Calculate the number of 16 mm diameter steel bars that will be required as compression reinforcements. The distance of the center of compression reinforcement from the compression concrete surface may be taken as 40 mm.</p>
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<p><strong>c) </strong>Calculate the number of 25 mm diameter steel bars that will be required as tensile reinforcements.</p>
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<p><strong>Answer:</strong></p>
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<figure class="wp-block-image size-large"><img src="https://civilengineeringlectures.com/wp-content/uploads/2021/01/28-1.png" alt="Doubly Reinforced Beams " class="wp-image-1397"/></figure>
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<p><strong>Given&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; b = 225 mm,&nbsp;&nbsp; &nbsp;&nbsp; &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; d = 450 mm,&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; d<sup>’</sup> = 40 mm,</strong></p>
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<p><strong>M = 300.0 kNm, &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;<em>f<sub>cu</sub></em>= 30 N/mm<sup>2</sup></strong>, &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <strong><em>f<sub>y</sub></em>=460 N/mm<sup>2</sup></strong></p>
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<p><strong>a)                     </strong>M         =          300.0 kNm      =          300 x 10<sup>6</sup> Nmm</p>
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<figure class="wp-block-image size-large"><img src="https://civilengineeringlectures.com/wp-content/uploads/2021/01/28-2-1024x178.png" alt="Doubly Reinforced Beams" class="wp-image-1398"/></figure>
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<p>                                    <strong><u>Therefore, compression reinforcement will be required.</u></strong></p>
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<figure class="wp-block-image size-large"><img src="https://civilengineeringlectures.com/wp-content/uploads/2021/01/28-3-1024x187.png" alt="Doubly Reinforced Beams" class="wp-image-1399"/></figure>
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<figure class="wp-block-image size-large"><img src="https://civilengineeringlectures.com/wp-content/uploads/2021/01/28-4.png" alt="" class="wp-image-1400"/></figure>
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<p>The compression steel will yield and the stress is equal to 0.95<em>f<sub>y</sub></em>.</p>
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<figure class="wp-block-image size-large"><img src="https://civilengineeringlectures.com/wp-content/uploads/2021/01/28-5.png" alt="" class="wp-image-1401"/></figure>
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<figure class="wp-block-image size-large"><img src="https://civilengineeringlectures.com/wp-content/uploads/2021/01/28-6-1024x198.png" alt="" class="wp-image-1402"/></figure>
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<figure class="wp-block-image size-large"><img src="https://civilengineeringlectures.com/wp-content/uploads/2021/01/28-7.png" alt="" class="wp-image-1403"/></figure>
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<figure class="wp-block-image size-large"><img src="https://civilengineeringlectures.com/wp-content/uploads/2021/01/28-8.png" alt="" class="wp-image-1404"/></figure>
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<figure class="wp-block-image size-large"><img src="https://civilengineeringlectures.com/wp-content/uploads/2021/01/28-9-1024x214.png" alt="" class="wp-image-1405"/></figure>
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<p>*        The force in the compression reinforcement is dependent on the stress and consequently the strain ζ<em><sub>cs</sub></em>. In order to ensure that this is greater than or equal to the yield stress, the ratio of (<em>d</em>′<em>/x</em>) is limited to 0.37 as shown below.</p>
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<figure class="wp-block-image size-large is-resized"><img src="https://civilengineeringlectures.com/wp-content/uploads/2020/12/28-10-1.png" alt="" class="wp-image-1408" width="-24" height="214"/></figure>
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<figure class="wp-block-image size-large"><img src="https://civilengineeringlectures.com/wp-content/uploads/2021/01/28-10-2.png" alt="" class="wp-image-1409"/></figure>
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<figure class="wp-block-image size-large"><img src="https://civilengineeringlectures.com/wp-content/uploads/2021/01/28-11.png" alt="" class="wp-image-1407"/></figure>
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<p><em>This value is given as 0.37 in the code to accommodate redistribution of moments in excess of 10%.</em></p>
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