Worked Examples .1 Worked Example 5.1

Calculate the factored bending moment at the centre of a simply supported floor slab, for normal temperature design and for fire design. Use these bending moments to calculate the load ratio for fire design.

The slab has a span of 3.5 m. The characteristic dead load is 2.4 kN/m² and the characteristic live load is 2.0 kN/m². Make all calculations for a strip of slab 1 m wide. Refer to Figure 5.8.

Calculate load combinations:

Characteristic dead load G_k 2 40. kN/m² Characteristic live load Q_k 2 00. kN/m² Factored load for cold design 1 35 3 24

1 2 1 5 5 88

2 2

. .

. . .

G

G Q

k

k k

kN/m

kN/m governs Factored load for fire design G_k 0 4. Q_k 3 20. kN/m²

Convert to uniformly distributed loads on a strip 1 m wide:

Factored load for cold design w_c 5 88 1 5 88. . kN/m Factored load for fire design w_f 3 20 1 3 20. . kN/m Calculate bending moments in centre of 3.5 m span:

Design bending moment for cold design M_cold^* w L_c ²/8 5 88 3 5 8 9 00. . ²/ . kNm Design bending moment for fire design M^*_fire w L_f ²/8 3 20 3 5 8 4 90. . ²/ . kNm Calculate the required flexural design capacity of the slab, using a strength reduction factor Φ = 0.8, and the load ratio for fire design, assuming that the slab is provided with exactly the required strength.

Flexural design capacity R_cold M_cold^* / 9 00 0 8 11 25. / . . kNm Load ratio for fire design r_load M^*_fire/R_cold 4 90 11 25 0 44. / . .

This shows that the slab would not be expected to fail in a fire until its strength drops to 44%

of its strength at normal temperatures.

5.7.2 Worked Example 5.2

Recalculate the bending moments from Worked Example 5.1 for a slab which is continuous over several supports. Assuming that the slab has equal positive and negative flexural capacity, calculate the load ratio considering redistribution to equal positive and negative moments.

Refer to Figure 5.13.

Calculate the design bending moments for cold conditions:

Design bending moment at support (negative moment)

M_cold^* w L_c ²/12 5 88 3 5 12 6 00. . ²/ . kNm

the  negative and positive moments is 3 27 1 63 4 90. . . kNm, the same as M* in Worked _fire Example 5.1.

Consider the effect of a fire which causes the flexural capacity R_cold to drop at the same rate for both positive and negative bending. If there is no redistribution of moments, the slab would fail at the supports when the flexural capacity drops to 3.27 kNm. The load ratio is r_load 3 27 7 50 0 44. / . . . However, with redistribution, the bending moment diagram can take  any position provided that the sum of positive and negative moments remains at w L_f ²/8 4 90. kNm.

If positive and negative flexural strengths are equal, the optimum location of the bending moment diagram is the shape shown by the dotted curve, with positive and negative moments both M^*_{fire red,} 4 90 2 2 45. / . kNm (which could also have been obtained from Equation 5.26).

This gives a revised load ratio for fire design of:

r_load M^*_{fire red,} R_cold

. . .

/

2 45 7 50 0 33 /

Final failure of the slab will not be expected until its strength in fire conditions has reduced to 33% of its strength at normal temperatures.

5.7.3 Worked Example 5.3

Reconsider the slab from Worked Example 5.2, assuming that the slab can have different positive and negative flexural capacities. Calculate the revised load ratio assuming that the flexural capacity at mid‐span (positive moment) drops to zero during fire exposure. Refer to Figure 5.14.

Before moment redistribution, the bending moments shown by the solid curve (M_cold^* ) and the dashed curve (M^*_fire) are the same as in Worked Example 5.2. The maximum flexural capacities are shown by the two lines marked R_cold for unequal positive and negative values.

With moment redistribution, the bending moment diagram M^*_fire can again take any position, provided that it remains the same shape. The dotted line marked M^*_{fire red,} in Figure 5.14 shows the bending moment diagram lifted to give a maximum value at the supports and a zero value at mid‐span.

In this case the slab could survive a fire even though the positive flexural capacity at mid‐

span drops to zero during the fire. The slab must be able to resist a negative bending moment at the supports of M^*_{fire red,} 4 90 kNm, which corresponds to:.

Load ratio for fire design / / r_load M^*_{fire red,} R_cold

. . .

4 90 7 50 0 655

Failure of the slab will not be expected until its cantilever strength in fire conditions has reduced to 65% of its strength at normal temperatures and the mid‐span strength has dropped to zero.

5.7.4 Worked Example 5.4

Consider the continuous beam shown in Figure 5.17. The span AB is 6 m and BC is 2 m. The uniformly distributed load during the fire conditions is w = 22 kN/m.

(a) Calculate the minimum flexural capacity of the positive and negative plastic hinges.

(b) Calculate the required flexural capacity of the negative plastic hinge if the strength of the positive plastic hinge decreases to 30 kNm under fire exposure.

(a) Span BC

By the principle of the conservation of energy, the magnitude of the external virtual work is equal to the internal virtual work.

external work internal work w L_BC /2 M_p

Substituting L_BC

wL²_BC /2 M_p Substituting w = 22 kN/m and L_BC = 2 m gives:

M_p 22 2

2 44

2

kNm (1) Span AB

Assume that the positive hinge occurs in the centre of span AB.

external work internal work w L_AB /2 w L_BC /2 M_p 2M_p

M_p 44 kNm M_p 55 kNm (b) If M_p⁺ is reduced to 30 kNm, substituting into (2) gives:

M_p 94 kNm

Structural Design for Fire Safety, Second Edition. Andrew H. Buchanan and Anthony K. Abu.

© 2017 John Wiley & Sons, Ltd. Published 2017 by John Wiley & Sons, Ltd.

Steel Structures

This chapter provides the information needed for calculating the performance of steel buildings exposed to fires. Simple methods are described for designing individual steel members to resist fire exposure, including calculations of elevated temperatures, methods of fire protection, and information on thermal and mechanical properties of steel at elevated temperatures. Fire behaviour of large steel buildings is also discussed.

This chapter draws information from Eurocode 3 Part 1.2 (CEN, 2005b), which with the other structural Eurocodes, summarizes the results of a large international cooperative programme over recent decades.