{"id":14409,"date":"2025-01-28T13:49:04","date_gmt":"2025-01-28T13:49:04","guid":{"rendered":"https:\/\/support.buildsoft.eu\/?post_type=ht_kb&p=14409"},"modified":"2025-06-23T21:56:19","modified_gmt":"2025-06-23T21:56:19","slug":"dc-ec-11-longitudonal-reinforcement-in-beam-under-pure-bending-doubly-reinforced-uls","status":"publish","type":"ht_kb","link":"https:\/\/support.buildsoft.eu\/fr\/knowledge-base\/dc-ec-11-longitudonal-reinforcement-in-beam-under-pure-bending-doubly-reinforced-uls\/","title":{"rendered":"DC EC 11: longitudinal reinforcement in beam under pure bending (doubly reinforced – ULS)"},"content":{"rendered":"

Description<\/h2>\n

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\n

\n\n\n\n\n\n\n\n\n
Geometry<\/th>\nCross-section:<\/td>\n\"b=200mm\"
\n\"h=400mm\"<\/td>\n<\/tr>\n
Material:<\/td>\nC25\/30<\/td>\n<\/tr>\n
Concrete cover:<\/td>\n\"c=40mm\"<\/td>\n<\/tr>\n
Load<\/th>\nSelf-weight:
\nDead loads:<\/td>\n		neglected
\n\"P=30kN\" at end of span<\/td>\n<\/tr>\n
Internal forces<\/th>\n<\/td>\n\"M_{Ed.ULS}=162\text{kNm}\"<\/td>\n<\/tr>\n
Standard<\/th>\n<\/td>\nEN 1992-1-1 [- -]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n

Independent reference results<\/h2>\n
<\/span>Open handcalculations<\/div>
\n

This model will require both upper and lower reinforcement. If you’re into the formula’s for designing reinforced concrete, you know that there’s only two equilibrium equations [Van Hooymissen.L, \u00a75.3.4.3] while there are 5 unknows (\"b\", \"d\", \"\xi\", \"A_{s1}\" and \"A_{s2}\").
\nIn Diamonds, you impose \"b\" and \"d\" by defining the cross-section and concrete cover. That leaves 3 unknows \"\xi\", \"A_{s1}\" and \"A_{s2}\" with 2 equations…. Diamonds will solve this system by looking for a solution so that \"A_{s1} + A_{s2}\" is as small as possible (= the most economic reinforcement).<\/p>\n

However that’s not a condition we can easily impose when we only have the tables from [Minne, P. Gewapend beton: numeri<\/em>] at our disposal. As a work around we’re going to assume that the compressive reinforcement \"A_{s2}\" calculated by Diamonds, is also given. Then we can calculate \"\xi\" and \"A_{s1}\".<\/p>\n

Required reinforcement:<\/p>\n

  <\/span>   <\/span>\"\[d=h-c=400mm-40mm=360mm\]\"<\/p>\n

  <\/span>   <\/span>\"\[\delta_1=h-c=\frac{d_1}{d}=\frac{40mm}{360mm}=0.11\]\"<\/p>\n

  <\/span>   <\/span>\"\[\mu _d=\frac{M_{Ed.ULS}}{b\cdot d^2\cdot f_{cd}}=\frac{162\text{kNm}}{200mm\cdot (360mm)^2\cdot 16.7\text{MPa}}=0.375\]\"<\/p>\n

  <\/span>   <\/span>\"\[\omega _2=\frac{A_{s2}\cdot f_{yd}}{b \cdot d \cdot f_{cd}}=0.052\]\"<\/p>\n

Table 5.7b from Gewapend beton: numeri<\/em> (click here to open<\/a> the Table) shows \"\omega _1\" and \"\xi\" as a function of \"\mu _d\" and \"A_{s2}\". We find:<\/p>\n

  <\/span>   <\/span>\"\[\omega _1=0.487\]\"<\/p>\n

  <\/span>   <\/span>\"\[A_{s1}=\omega _1 \cdot b\cdot d\cdot \frac{f_{cd}}{f_{yd}}=0.487 \cdot 200mm \cdot 360mm \cdot \frac{16.7\text{MPa}}{434.8\text{MPa}}=1344mm^2\]\"<\/p>\n

Minimum reinforcement:<\/p>\n

(9.1N) <\/span>   <\/span>\"\begin{align*}<\/p>\n

Diamonds calculates the minimum reinforcement with \"h\" instead of \"d\" because for more complex cross-section shapes, \"d\" is not unambiguously defined. Therefor \"h\" is a safe alternative.<\/p>\n<\/div><\/div><\/div>\n

Diamonds results and comparison<\/h2>\n

\"\"<\/p>\n

Longitudinal reinforcement calculated by Diamonds (EN 1992-1-1 [- -])<\/strong><\/p>\n

\n\n\n\n\n\n
Results<\/th>\nIndependent reference<\/th>\nDiamonds<\/th>\nDifference<\/th>\n<\/tr>\n
Compressive longitudinal reinforcement \"A_{s2}\"<\/td>\n144 mm\u00b2<\/td>\n144 mm\u00b2<\/td>\n0,00%<\/td>\n<\/tr>\n
Tensile longitudinal reinforcement \"A_{s1}\"<\/td>\n1344 mm\u00b2<\/td>\n1304 mm\u00b2<\/td>\n-2,98%1<\/a><\/sup><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n1. Although the difference here is certainly acceptable, it is still larger than expected. But that’s because we’re using tables. If you solve the equilibrium equations analytically, you will obtain higher accuracy that matches the Diamonds results even better.<\/sup><\/p>\n

References<\/h2>\n