Deflection check (concrete beam and slab: EC2)
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The deflection of reinforced concrete beams is not directly calculated and the serviceability of the beam is measured by comparing the calculated limiting span/effective depth ratio L/d to the maximum allowable values as given by;1 |
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| IF ρ | ≤ | ρ0 |
| (L/d)max | = |
Kss*f1*f2*(11+1.5*(fck) 1/2*(ρ0/ρ)+3.2*(fck)1/2*(( ρ0/ρ)-1)3/2)*(500*Ast,prov/(fyk*A st,reqd)) |
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| IF ρ | > |
ρ0 |
| (L/d)max | = |
Kss*f1*f2*(11+1.5*(fck) 1/2*(ρ0/(ρ-ρ'))+(1/12)*(fck)1/2*( ρ'/ρ0)1/2)*(500*Ast,prov/(fyk*A st,reqd)) |
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where |
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| ρ | = | the designed tension reinforcement ratio at mid-span (or at support for cantilevers) required to resist bending |
| = | A st.reqd /(b w *d) for rectangular beams | |
| = | A st,reqd /(b eff *d) for flanged beams | |
| ρ' |
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the designed compression reinforcement ratio at mid-span (or at support for cantilevers) required to resist bending |
| = | A s2,reqd /(b w *d) for rectangular beams | |
| = | A s2,reqd /(b eff *d) for flanged beams | |
| A st,reqd |
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the designed area of tension reinforcement at mid-span (or at support for cantilevers) required to resist bending |
| A st,prov | = | MIN(the area of tension reinforcement provided at mid-span (or at support for cantilevers), f 3 *A st,reqd ) |
| A s2,reqd | = | the designed area of compression reinforcement at mid-span (or at support for cantilevers) required to resist bending |
| f 1 | = |
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| = |
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| = |
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| f 2 | = |
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| = | 7/L eff IF L eff > 7 m with L eff in metre units | |
| L eff | = | the length of the beam between the centre of its supports2 |
| f 3 | = | an NDP factor as given below |
| K ss | = | the structural system factor which is an NDP and is given below |
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For design in accordance with UK NA, as per Table NA.5 Note 6 the absolute value of span/depth may not exceed 40K. |
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For design in accordance with EC2 Recommendations, Finnish NA, Norwegian NA and Swedish NA, the NDP value of f3 is given by3; |
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| f3 | = | 1.5 |
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For design in accordance with UK NA, Irish NA, Malaysian NA and Singapore NA the NDP value of f3 is given by4; |
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| f3 | = | 1.5 |
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For design in accordance with UK NA, EC2 Recommendations, Irish NA, Malaysian NA and Singapore NA, Norwegian NA and Swedish NA the NDP value of Kss is given by the following table: |
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| Span Detail | LH End Fixity | RH End Fixity | Kss |
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| LH End Span | Fixed | Fixed | 1.3 |
| Fixed | Pinned | 1.0 | |
| Pinned | Fixed | 1.3 | |
| Pinned | Pinned | 1.0 | |
| Internal Span | Fixed | Fixed | 1.5 |
| Fixed | Pinned | 1.3 | |
| Pinned | Fixed | 1.3 | |
| Pinned | Pinned | 1.0 | |
| RH End Span | Fixed | Fixed | 1.3 |
| Fixed | Pinned | 1.3 | |
| Pinned | Fixed | 1.0 | |
| Pinned | Pinned | 1.0 | |
| Cantilever | 0.4 |
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For design in accordance with Finnish NA the NDP value of Kss is given by the following table; |
| Span Detail | LH End Fixity | RH End Fixity | Kss |
|---|---|---|---|
| LH End Span | Fixed | Fixed | 1.0 |
| Fixed | Pinned | 0.8 | |
| Pinned | Fixed | 1.0 | |
| Pinned | Pinned | 0.8 | |
| Internal Span | Fixed | Fixed | 1.2 |
| Fixed | Pinned | 1.0 | |
| Pinned | Fixed | 1.0 | |
| Pinned | Pinned | 0.8 | |
| RH End Span | Fixed | Fixed | 1.0 |
| Fixed | Pinned | 1.0 | |
| Pinned | Fixed | 0.8 | |
| Pinned | Pinned | 0.8 | |
| Cantilever | 0.3 |
Footnotes
1 EN 1992-1-1:2004 Section 7.4.2
2 This definition of
effective length will return conservative results when the width of
the support is greater than the depth of the beam - see EN
1992-1-1:2004 Section 5.3.2.2(1)
3 EN 1992-1-1:2004 is silent on the recommended value to
use therefore adopt 1.5 since if f3 is greater than
1.5 no benefit arises.
4 For Irish
NA refer to Table NA.3 and for others refer to Table NA.5