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Neutral axis depth factor. */d

Figure A2.2 — Allowable plastic rotation of reinforced concrete sections

A2.3 Simplified methods (linear members)

(1) When calculating the rotation of plastic hinges by integration of the curvature between the hinges, it will generally be sufficient to use a simplified linear moment-curvature diagram. This diagram may be defined by a straight line from the origin to a point (1/rm, Myk) where Myk is the moment which produces the stress fyk in the reinforcement, calculated on the basis of a cracked section and (1/r)m is the curvature at the moment Myk calculated allowing for tension stiffening (1/r)m may be calculated from the relation:

-sy sym

(1/r)cr (sym/(sy is the curvature calculated on the basis of a cracked section is the yield strain of the reinforcement (= fyk/Es)

is the strain calculated for Bs = fyk = fym allowing for tension stiffening.

(2) The limiting rotation given in Figure A2.2 may be assumed to apply at the moment Myd, corresponding to the attainment of fyd at the hinge considered.

A2.4 Plastic analysis (linear members)

P(1) Methods of analysis involving plastic hinges without any direct check on their rotation capacity may be used provided adequate ductility can be ensured and other factors such as model uncertainty are taken into account.

(3) Normal ductility steel should not be used unless its application can be justified. A2.5 Non linear and plastic approaches for prestressed linear members A2.5.1 Non linear methods

P(1) The internal forces and moments and the resistance shall be calculated taking account of the non-linear behaviour of the prestressing and reinforcing steels and of the concrete.

(2) As behaviour at the ultimate limit state is relatively insensitive to the effects of prestressing, structural analysis may be carried out using Yp = 1.

A2.5.2 Plastic methods

(1) A2.4(1) and (2) apply. The statically indeterminate effects of prestressing may be ignored in the design of sections.

A2.6 Numerical methods of analysis for slabs

(1) In general the methods given in A2.2 may be adopted.

(2) When using non-linear numerical methods (e.g. Finite Element or Finite Difference methods) cracking may be assumed to be either distributed or concentrated within orthotropic elements.

(3) The reinforcement in a slab analysed using numerical methods may be determined using the methods given in A2.8 below.

A2.7 Non-linear analysis of walls and plates loaded in their own plane

(1) Non-linear analysis methods may be used for the ultimate and serviceability limit states using deformation relationships based on material properties appropriate for the limit state considered. The contribution of concrete in tension between cracks should be taken into account.

(2) Prior to analysis it is necessary to make a first estimate of a suitable arrangement and amount of reinforcement. This may be done using the methods given in 2.5.3.6.3.

(3) The results of the analysis may be used to calculate appropriate areas of reinforcement by applying the rules given in A2.8.

A2.8 Reinforcement in slabs

(1) The reinforcement in a slab subject to any general moment field may be assessed using the procedure set out below.

(2) A set of orthogonal axes are chosen and the moments resolved in the directions of these axes to give moments per unit length, mx, my and mxy such that my T mx. Reinforcement is provided in the x and y directions to resist design ultimate moments, mudx, m'udx, mudy and m'udy, mudx and mudy are moments giving tension in the bottom of the slab while m'udx and m'udy give tension in the top of the slab.

(3) The following flow chart is used to establish the values of the design ultimate moments from the values of mx my and mxy.

Design Footing With Moment Eurocode

(4) As an alternative to (3) above, the required design moments may be obtained from Equations A2.4 to A2.7 below:

Y and *' are coefficients which should be chosen so that the equations give values which lie between half and twice the values given by (3) above.

(5) The ability of a section to resist a given combination of moments will be satisfactory if the following conditions are satisfied:

- (mudx - mx) (mudy - my) + mxy2 r 0

(A2.8)

- (m'udx + mx) (m'udy + my) + mxy2 r 0

(A2.9)

mx r mudx

(A2.10)

my r mudy

(A2.11)

mx T - m'udx

(A2.12)

my T - m'udy

(A2.13)

A2.9 Reinforcement in plates (walls)

(1) The reinforcement in an element of a plate subjected to a stress field defined by the stress Bx, By and rxy, referred to an orthogonal co-ordinate system selected so that Bx r By may be calculated using the procedure set out below.

(2) In the flow chart below, ftdx and ftdy are notional design tensile stresses in the material to be reinforced in the x and y directions respectively. Assuming the tensile strength of the concrete to be zero, the reinforcement ratios in the x and y direction are given as:

(negative values should be taken as zero)

(negative values should be taken as zero)

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(3) For walls with reinforcement on both faces, which is properly anchored [e.g. by means of U-stirrups, (see Figure 4.25)], the concrete stress Bc should be limited to Bc r fcd, while at the same time limiting the shear stress to:

when V is determined by Equation (4.21).

Other values of V may be used, if appropriately defined by tests.

In the absence of appropriate test data, the shear stress should be limited to:

iTxyir fcd/fk (A2.14b)

(4) As an alternative to the procedure given in (2) above, the reinforcement may be estimated from Equations A2.15 and A2.16 below:

The coefficient Y should be chosen so that the results from Equations A2.15 and A2.16 lie between one half and twice the values given by (2) above.

The stress in the concrete is given by:

(5) The ability of a section to resist a given combination of stresses will be satisfactory if the following conditions are met:

- (fcd - Bx) (fcd - By) (fcd - By) + Txy2 r 0 (A2.19)

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