Service-limit state

The Swiss standard, Basics of Structural Design SIA 260 [52] recommend verifying different situations for the service-limit state. Three different load cases (near-permanent, frequent and occasional) with different consequences (reversible or irreversible) are given. Only the

‘occasional’ load case leads to irreversible consequences, and constructional measures are recommended rather than upgrading the stiffness. Furthermore, three limit states are given:

functionality, comfort and appearance. Functionality is again divided into three sub-categories: fittings with brittle behaviour, fittings with ductile behaviour and use and opera-tion. The long-term effects have only to be considered for three limit states: fittings with brittle behaviour, fittings with ductile behaviour and appearance. For the two limit states considering the fittings only, the deformation taking place after the installation of the relevant equipment or non-loadbearing component has to be considered. In the presented example it is assumed that all dead loads are installed prior to installing the fittings with ductile behaviour.

Identifying the deformation prior and due to intervention (equations (43) and (51)) allows the calculation of the deformation due to service loads at time zero. The elastic-bending defor-mation due to service loads can be calculated using well-known equations [36], and the bending stiffness of the composite section (EIy,co equation (38)). According to Heuer [42], the shear stiffness of the composite section (GA_co) can be calculated using equation (76) below.

The knowledge of the composite shear stiffness allows the determination of the shear defor-mation at time zero using well-known [36] equations.

= ^,

( ) ( )

( ) ( )

(76)

Where:

GAco = shear stiffness of the composite section is regarded as constant over the beam length EIy,co = bending stiffness of the composite section is regarded as constant over the beam length G = shear modulus of the relevant material

MOE = modulus of elasticity of the relevant material b = width of the section

zt = distance between the neutral axis and the top of the composite section

zb = distance between the neutral axis of the composite section and the bottom of the timber section z = position over the beam height

The deformation at time zero is the sum of the deformation due to prestressing at time = 0, the elastic deformations due to service loads and the creep deformations prior to interven-tion. The deformation due to shear is less than 3% of the deformation due to bending, and therefore, not significant for building purposes if one is considering the variation in the elastic properties of timber.

For long-term deformation, the different creep behaviour of the material have to be consid-ered. As mentioned above in the example, timber only is prone to creep which results in a load redistribution and increasing deformation due to long-term effects. Furthermore, accord-ing to Navi and Heger [81], it is assumed that the creep deformation prior to intervention stayed within its linear elastic limit. In order to determine the deformation, the load history has to be considered (Table 4). Therefore, the creep relevant loads are only the additional near-permanent loads and the additional dead loads. The creep deformation of the loads prior to intervention is the permanent deformation at the time of reinforcing, and can be measured on-site. In the presented example, the permanent deformation is assumed to be equal to 60% of the elastic deformation of the permanent and near-permanent loads prior to reinforcing, and calculated using the bending stiffness of the timber section. The long-term deflection due to bending loads of a reinforced beam can be assumed using equation (77).

, ( ) = ^, 2 _, + _,

EIy,co = bending stiffness of the composite section at time = 0 EIy,co, = bending stiffness of the composite section at time = EIy,ti = bending stiffness of the timber section

sp = span of the beam

q_np,a = creep relevant loads after intervention q_np,pi = creep relevant loads prior to intervention

= creep factor

P = prestress force at time = a = eccentricity of the prestress force x = position over the beam length

The deflection due to shear forces can be determined in a similar way to that for bending moments. The shear stiffness of the composite section at time = can be estimated using equation (78).

GA_co, = shear stiffness of the composite section at time = is regarded as constant over the beam length EI_y,co, = bending stiffness of the composite section at time =

G = shear modulus of the relevant material (for timber G is divided by 1+ )

MOE = modulus of elasticity of the relevant material (for timber MOE is divided by 1+ ) b = breadth of the section

zt, = distance between the neutral axis and the top of the composite section at time =

z_b,ti, = distance between the neutral axis of the composite section and the bottom of the timber section at

time =

z = position over the beam height

The deformation due to shear is less than 4% of the deformation due to bending. This is not significant for building purposes if one is considering the variation in the elastic properties of timber.

In the example presented, the service-limit state for appearance is not fulfilled. However, the estimation of the deformation is only 8% higher than the limit, and the deformation could be accepted for an historic building. The service-limit state for use and operation is reached without any discussions; and the same is true for a case where the timber beam has not been reinforced. The service limit for fittings with ductile behaviour is not reached by 6%, and in cases where there are such fittings, construction measures need to be taken to avoid damage. The service-limit state for fittings with brittle behaviour is not possible to reach, and therefore construction measures must be taken. The comparison of the reinforced section with the timber beam only shows a large improvement due to strengthening using the cam-ber method (Figure 63).

Figure 63: Service-limit state of the reinforced beam (solid lines) and the timber beam only (dashed lines). The dotted lines represent the limit given in the SIA 260 [52].

The green lines show the limit state for use and operation (uo), the blue lines for near-permanent loads, regarding appearance (np) and the red lines for fit-tings with ductile behaviour.