Deflection check (beam and slab: EC2)
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
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))
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))
where
| 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' |
|
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 |
|
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 |
|
|
|
|
|
|
|
|
| f 2 |
|
|
|
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 |
For design in accordance with EC2 Recommendations the NDP value of f3 is given by3;
f3 = 1.5
For design in accordance with UK NA, Irish NA, Malaysian NA and Singapore NA the NDP value of f3 is given by4;
f3 = 1.5
For design in accordance with UK NA, EC2 Recommendations, Irish NA, Malaysian NA and Singapore 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 | 1.3 |
| 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 |
1 BS 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 BS EN 1992-1-1:2004 Section 5.3.2.2(1)
3 BS 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 other others refer to Table NA.5