Traditionally, given the configuration layout of a low or medium height building structure, as it has been decided by architectural, esthetic or functional norms, the practicing engineer starts the structural design by estimating the design (base) shear, *V*
_{
d
}, required by code provisions. This horizontal force, which is specified as a fraction of the total dead (and portions of live) load *W*, in relation to a first period dependant coefficient *β*, as follows:

$$ V_{d} = \beta W $$

(2a)

can also be seen as a first estimate of the yield force of an equivalent inelastic SDOF system. The characteristics of this system (mass, frequency, yield displacement) are derived by the following considerations:

As stated in the previous section, the equivalent SDOF system is constructed by assuming first that the real building (Fig. 1a) is subjected to an increasing lateral loading vector, proportional to MΦ (i.e.: V = αMΦ), where M is the mass matrix and Φ is the assumed mode (vector) of deformation (among the various deflection profiles shown in the mentioned figure, Φ is selected to represent a shape of deformation reflecting an advanced inelastic stage). This inelastic static analysis is ended when story drift limits (critical for non-structural members) or strains capacities (critical for structural members) are reached. The base shear—top deflection curve, *V* − *Δ*, obtained from this analysis (Fig. 1c) may be approximated by a bilinear curve (shown in the same figure by the dotted line, where the peak load is denoted with *V*
_{
do
} and the corresponding top displacement with *Δ*
_{
y
}) and then it is transformed into the capacity curve, *A*-*u* (Fig. 1d) by using the following formulation:

$$ A = V/M_{e}^{*} \quad {\text{and}} \quad u = \Delta /\varGamma \varPhi_{r} $$

(2b)

where *M*
^{*}_{
e
}
, *u* represent the effective (modal) mass and displacement respectively of the SDOF system shown in Fig. 1b. The first of these quantities is given as

$$ M_{e}^{ * } = ({\varvec{\Phi}}^{\rm T} {\mathbf{M1}})^{2} /{\varvec{\Phi}}^{\rm T} {\mathbf{M\varPhi }} $$

(2c)

and, in the second of Eq. (2b), *Φ*
_{
r
} is the value of the assumed vector *Φ* at the top (roof) of the structure and, *Γ* is the (modal) participation factor equal to

$$ \varGamma = {\varvec{\Phi}}^{\rm T} {\mathbf{M1}}/{\varvec{\Phi}}^{\rm T} {\mathbf{M\varPhi }} $$

(2d)

where 1 is the unit vector. The diagram shown in Fig. 1d may be interpreted as the normalized force -displacement relationship of the elasto-plastic SDOF system shown in Fig. 1b, which yields when it is pushed by a static force equal to *V*
_{
do
}. This force is, in general, higher than the design shear, *V*
_{
d
}, and constitutes the over-strength of the structure. There are many possible sources for this reserve strength: effects of gravity loads, order in which the various plastic hinges are formed, redistribution of internal forces, etc. (Humar and Rahgozar 1996). With a proper strength assignment through the structure, as it is described further below, the yield force, *V*
_{
do
}, may be close to *V*
_{
d
}, but in any case the horizontal acceleration causing yield of the SDOF system will be equal to

$$ A_{y} = V_{do} /M_{e}^{ * } $$

(2e)

The yield acceleration*, A*
_{
y
}, and the corresponding yield deformation, *u*
_{
y
}, are also shown in Fig. 1d, together with the slope of the initial elastic branch, *ω*
^{2}_{
e
}
, which represents the square value of the effective frequency.

The procedure described above presumes an estimate of *V*
_{
d
} and more importantly a distribution of strength through the building to assess bending moment capacities and flexural rigidities of the various members. However, as strength and stiffness are interrelated, the designer has a considerable choice to allocate strengths in a rather arbitrary way, say according to his experience, with the only restriction being that the limits (on deflections, crack widths, etc.) imposed by the code in the serviceability limit state, where member rigidities are based on lightly cracked sections (under bending moments well below the yield values), should be satisfied. With these considerations, it may be decided, prior to any calculations, just by engineering judgment, what proportion of the design shear *V*
_{
d
} (say *V*
_{
df
} = *λV*
_{
d
}) is to be resisted by the frame sub-system, and the rest of it, *V*
_{
dw
} = (1−*λ*)*V*
_{
d
}, by the wall sub-system. Typical values of *λ* vary between 0.3 and 0.4, as modern codes (e.g. EC8 2004) define the wall-equivalent dual system as that where the shear resistance of walls exceeds 50% of the total resistance of the building. It is reminded here, that elastic analyses have demonstrated (Paulay and Priestley 1992) that the wall shear in the upper stories is opposite in sense to the external load shear and, as a result, the frame shear exceeds the external shear in these stories. The overall frame shear profile presents little variation from the base to the top of the structure and this means that the allocated frame shear, *V*
_{
df
} = *λV*
_{
d
}, may be considered constant over the height of the frame subsystem (Garcia et al. 2010). Further than that, it should be noticed that as the wall sub-system is composed by purely flexural members, their flexural strength lies mainly on their capacity to undertake the overturning moment *V*
_{
dw
}
*H*
_{
e
}, where *H*
_{
e
} represents the effective (modal) height of the equivalent SDOF system. This height may be determined as the effective modal height using the mode vector *Φ* (e.g. Priestley et al. 2007; Priestley 2000), i.e.:

$$ H_{e} = {\varvec{\Phi}}^{\rm T} {\mathbf{Mh}}/{\varvec{\Phi}}^{\rm T} {\mathbf{M1}} $$

(2f)

where h is the vector of the heights of the floor masses from the level of excitement. However, a further simplification can be made taking into account that cantilever (building) systems analyzed by the approximate method of the continuous medium have shown that the first mode effective height varies from 0.726*H* for purely flexural systems, to 0.636*H* for purely shear-type systems (Chopra 2008; Clough and Penzien 1993). It is therefore appropriate, in common types of wall-frame buildings, to assume that *H*
_{
e
} may be taken, with reasonable accuracy, equal to 2/3 of the total height.

### 2.1 Assigning Strength and Rigidity to the Frame Sub-system

Let’s assume that the shear force sustained by the particular *f*-frame is equal to *V*
_{
f
}, where Σ*V*
_{
f
} = *V*
_{
df
}, and that the i-column resists a shear force equal to *V*
_{
i
} (Σ*V*
_{
i
} = *V*
_{
f
}). Prior to an assignment of strength in the frame members it is worth demonstrating the relation among deflections, rigidities and strength. Envisaging the beam-column sub-assemblage of Fig. 2a, with half story heights above and below the joint and half beam lengths on either side of it, and the bending moment diagrams on each member, the elastic inter-story drift of a frame with story heights equal to *h*, when the joint centre is restrained against rotation, is equal to (Fig. 2b):

$$ \theta_{c} = \frac{{V_{i} h}}{{6\left( {{{EI_{co} } \mathord{\left/ {\vphantom {{EI_{co} } {h + {{EI_{cu} } \mathord{\left/ {\vphantom {{EI_{cu} } h}} \right. \kern-0pt} h}}}} \right. \kern-0pt} {h + {{EI_{cu} } \mathord{\left/ {\vphantom {{EI_{cu} } h}} \right. \kern-0pt} h}}}} \right)}} $$

(3a)

In the expression above, *EI*
_{
co
}
*, EI*
_{
cu
} are the rigidities of the column sections, above and below the joint under consideration, which are still unknown. In the case that *I*
_{
co
} = *I*
_{
cu
} = *I*
_{
c
} and taking into account the diaphragmatic action of the floor slabs (that is, taking *θ*
_{
c
} to be the same for all columns), the equation above takes the form

$$ \theta_{c} = \frac{{V_{i} h}}{{{{12EI_{c} } \mathord{\left/ {\vphantom {{12EI_{c} } h}} \right. \kern-0pt} h}}} = \frac{{\varSigma (V_{i} h)}}{{12\varSigma {{(EI_{c} } \mathord{\left/ {\vphantom {{(EI_{c} } {h)}}} \right. \kern-0pt} {h)}}}} = \frac{{V_{f} h}}{{12\varSigma {{(EI_{c} } \mathord{\left/ {\vphantom {{(EI_{c} } {h)}}} \right. \kern-0pt} {h)}}}} $$

(3b)

Because of the beam flexure, the joint rotation adds an inter-story drift (Fig. 2b) equal to

$$ \theta_{b} = \frac{{V_{i} h}}{{6\left( {{{EI_{1} } \mathord{\left/ {\vphantom {{EI_{1} } {l_{1} + {{EI_{2} } \mathord{\left/ {\vphantom {{EI_{2} } {l_{2} }}} \right. \kern-0pt} {l_{2} }}}}} \right. \kern-0pt} {l_{1} + {{EI_{2} } \mathord{\left/ {\vphantom {{EI_{2} } {l_{2} }}} \right. \kern-0pt} {l_{2} }}}}} \right)}} $$

(3c)

where *EI*
_{
1
}
*, EI*
_{
2
} and *l*
_{
1
}
*, l*
_{
2
} are the rigidities and lengths of the beams shown in Fig. 2a. Assuming that *θ*
_{
b
} is the same for all joints, and expressing the beam rigidity with the general term *EI*
_{
b
} and its length as *l*
_{
b
}, the equation above gives:

$$ \theta_{b} = \frac{{\varSigma (V_{i} h)}}{{12\varSigma {{(EI_{i} } \mathord{\left/ {\vphantom {{(EI_{i} } {l_{i} )}}} \right. \kern-0pt} {l_{i} )}}}} = \frac{{V_{f} h}}{{12\varSigma {{(EI_{b} } \mathord{\left/ {\vphantom {{(EI_{b} } {l_{b} )}}} \right. \kern-0pt} {l_{b} )}}}} $$

(3d)

The total elastic drift, for an elastic frame, therefore is

$$ \theta_{f} = \theta_{b} + \theta_{c} = \frac{{V_{f} h}}{12}\left[ {\frac{1}{{\varSigma {{(EI_{b} } \mathord{\left/ {\vphantom {{(EI_{b} } {l_{b} )}}} \right. \kern-0pt} {l_{b} )}}}} + \frac{1}{{\varSigma {{(EI_{c} } \mathord{\left/ {\vphantom {{(EI_{c} } {h)}}} \right. \kern-0pt} {h)}}}}} \right] $$

(4)

For buildings designed according to the strong column - weak beam concept, the yield drift corresponds to the condition that the beam elements yield at their ends. Under the assumption that the frame shear *V*
_{
f
} is constant all over the height of the frame, and assuming further that equal yield moments are formed at the ends of all beams at any story, that is (with reference to Fig. 2a) assuming that

$$ M_{b1} = \, M_{b2} = \ldots .M_{by} $$

(5)

then, the sum of beam yielding moments, at the ends of all beams in any story, will be equal to

$$ \varSigma M_{by} = V_{f} h $$

(6a)

For a frame with *N* columns, the first part of Eq. (6a), is equal to 2(*N*−1)*M*
_{
by
}, and therefore the equation above specifies the beam yield moments as

$$ M_{by} = \frac{{V_{f} h}}{2(N - 1)} $$

(6b)

Note here that in practice positive and negative bending moment capacities are not necessarily equal in concrete sections, depending mainly on the magnitude of the gravity loads and also on the tensile reinforcement into the effective slab width affecting the hogging (negative) moment capacity. However, using moment redistribution rules, it is quite possible to end up with positive and negative moments close to each other. Further than that, for ductility reasons, the code suggestions are to provide compression reinforcement exceeding half of the tensile reinforcement, bringing closer the two capacities and this is particularly notable in advanced post elastic stages, due to the deep compression zone for negative moments and strain hardening for positive (sagging) moments (Priestley 1996). In any case, the moments of Eq. (6b) may be taken as the mean values of the two bending capacities and can be used to determine the flexural rigidities of beams and columns in Eq. (4), in combination with Eq. (1a) and (1c). That is: for any beam whose moment–curvature diagram is idealized by the dash bilinear curve of Fig. 3a, its effective rigidity is equal to

$$ (EI)_{be} = \frac{{M_{by} }}{{\varPhi_{by} }} = \frac{{M_{by} }}{{k_{b} \varepsilon_{y} }}d_{b} $$

(7a)

which means that, for computational purposes, the beam effective second moment of area can be determined as

$$ I_{be} = \frac{{(EI)_{be} }}{{E_{c} I_{bg} }}I_{bg} = a_{be} I_{bg} $$

(7b)

where *E*
_{
c
} is the concrete modulus of elasticity and *I*
_{
bg
} is the second moment of area of the gross concrete section about the centroidal axis ignoring the reinforcement. Similarly, from the equilibrium of moments around the joint of Fig. 2a, and taking into account that column yield is prevented, the average column moment (the mean value of the moments above and below the joint in the case that the points of contraflexure in the two stories are not at the mid height) will be equal to

As columns remain into the elastic stage, the above *M*
_{
c
} column moments are fractions of their yield values and if the corresponding curvatures are defined as *Φ*
_{
c
}, the slope *M*
_{
c
}/*Φ*
_{
c
} is higher than the column flexural rigidity, defined as *M*
_{
cy
}/*Φ*
_{
cy
}. This is because the bilinear shape of column moment–curvature relationship, shown in Fig. 3b by the dotted line, is an approximate shape, based mainly on the bending moment capacity of the column. Neglecting the effect of the axial load on the column bending capacity and magnifying its value by a factor of 1.25 to ensure that column yielding is prevented (i.e., *M*
_{
cy
} = 1.25*M*
_{
by
}) and assuming further that *Φ’*
_{
c
} = 0.8*Φ*
_{
cy
} (Fig. 3b) the column stiffness can be estimated by the following expression

$$ (EI)_{ce} = \frac{{M_{c} }}{{\varPhi_{c} }} = \frac{{M_{cy} }}{{\varPhi^{\prime}_{c} }} \approx 1.5\frac{{M_{by} }}{{\varPhi_{cy} }} = 1.5\frac{{M_{by} }}{{k_{c} \varepsilon_{y} }}d_{c} $$

(9a)

As for the case of beams, for computational purposes, the column effective second moment of area may be taken as

$$ I_{ce} = \frac{{(EI)_{ce} }}{{E_{c} I_{cg} }}I_{cg} = a_{ce} I_{cg} $$

(9b)

Evidently, in exterior beam-column joints the column moment will be equal to half of that of Eq. (8) and therefore the stiffness of edge columns will be half of that given by Eq. (9a). Note here that induced axial compression loads in column sections affect their flexural strengths (typical bending moment-axial load (M–N) interaction diagrams indicate this relationship). When the axial load is below the ‘balance point’ of the mentioned diagrams, the column bending moment capacity assessed by Eq. (8) is underestimating the true column capacity, but this assessment does not really affect the yield drift of the frame as explained further below.

Buildings designed according to the capacity design concept (strong column-weak beam model) require further the yield moment at the ground column bases. An estimate of these values can be made by assuming that the point of contraflexure in the first story columns is at a height 0.6*h*. Therefore, if at the top of these columns, the moments developed to maintain equilibrium (around the interior joint of Fig. 2a) are given by Eq. (8), the yield moments at their bases, when a magnifying factor of 1.25 has been taken into account, may be estimated as

$$ M_{cy} = \left( { 6/ 5} \right) 1. 2 5M_{by} $$

(10)

For edge columns, the base yield moments will be half of those assessed by the equation above.

### 2.2 Assigning Strength and Rigidity to the Wall Sub-system

Wall elements should be designed to resist an overturning moment equal to *V*
_{
dw
}
*H*
_{
e
}. Again the designer has the choice to allocate different fractions of this bending moment to the various wall elements: from the classical method, in proportion to the traditionally defined (elastic) stiffness, to the methodology of having the same longitudinal steel ratio in all walls, as proposed by Paulay (1998).

Let’s assume that *M*
_{
w
} is the bending moment capacity of the particular *w*-Wall, where

$$ \varSigma M_{w} = V_{dw} H_{e} . $$

(11)

Its effective flexural rigidity, in combination with Eq. (1b), will be equal to

$$ (EI)_{we} = \frac{{M_{w} }}{{\varPhi_{wy} }} = \frac{{M_{w} }}{{k_{w} \varepsilon_{y} }}d_{w} $$

(12a)

and the corresponding wall effective second moment of area

$$ I_{we} = \frac{{(EI)_{we} }}{{E_{c} I_{wg} }}I_{wg} = a_{we} I_{wg} $$

(12b)

Note that axial compression loads sustained by wall sections affect their flexural strengths and, when the mean compression stress reflects a low fraction of the concrete compression strength (as suggested by many building codes), the moment capacity may be higher. As a result the effective second moment of area determined by Eq. (12b) represents a conservative estimate of this property at the base of the wall. In practice however, the reinforcement content is gradually reduced at higher levels, resulting in lower bending capacities and therefore in reduced flexural rigidities. Therefore, for a preliminary structural design, it is considered satisfactory to assess the wall effective second moment of area by means of Eq. (12b).