Usually the systems are idealized as lumped-mass systems. In the first part, we will find the structural response as a function of time when the system is subjected to ground acceleration This is known as the response history analysis (RHA) procedure.
Earthquake analysis of linear systems
Usually the systems are idealized as lumped-mass systems. In the first part, we will find the structural response as a function of time when the system is subjected to ground acceleration This is known as the response history analysis (RHA) procedure. In the second part, we can compute peak response of a structure during an earthquake directly from earthquake response or design spectrum without the need for response history analysis. The values given by response spectrum analysis (RSA) are fairly accurate.
Let us assume that earthquake-induced motion u˙˙g ( t ) is identical to all support points. The equation of motion can be written as
Γn is called modal participation factor, implying that it is a measure of the degree to which the nth mode participates in the response. Γ is not independent of how the mode is normalized nor a measure of modal contribution to response quantity.
Displacement in physical coordinates may be obtained as
The equivalent static force is the product of two quantities:
• nth mode contribution;
• pseudo-acceleration response of the nth mode SDF system.
The nth modal contribution to any response quantity R(t) may be determined by static analysis of structures subjected to external force fn(t). If Rnst is the static value
Rnst may be
positive or negative and is independent of how the mode is normalized.
1 Total response
The response contributions of some of the higher modes may under appropriate circumstances be determined by simple static analysis instead of dynamic analysis. For short periods Tn ≤ 1/33
If the period range included is the natural periods from Nd+1
to N, then,
2 Interpretation of modal analysis
At first the dynamic properties natural frequencies and mode shapes of the structure are computed and the force distribution vector mi is expanded into modal components. The rest of the analysis procedure is shown schematically in Table 18.8.
3 Analysis of response to base rotation
The modal analysis procedure is applicable after slight modification when the excitation is base rotation. Consider the cantilever frame shown in Fig. 18.19.
Multi-storey buildings with symmetrical plan
The equation of motion for this structure is (see Fig. 18.20)
The two storey shear frame shown in Fig. 18.21 is excited by a horizontal ground motion u˙˙g ( t ). Determine
(a) modal expansion of effective earthquake forces;
(b) the floor displacement response of Dn(t);
(c) the storey shear in terms of An(t);
(d) the first floor and base overturning moments in terms of An(t).
= D2 ( t ) which is same as [ k ] –1( F2 )ω n22 D2 ( t ) –.207
Combining we get
u1(t) = 0.854 D1(t) + 0.146 D2 (t) u2(t) = 1.207 D1(t) – 0.207 D2 (t)
(c) Storey shear can be determined as follows (see Fig. 18.23). Substituting this we get storey shear as
Vbst = 1.4565 m A1 ( t ) + 0.0425 m A2 ( t )
V1 (t ) = 0.6035 m A1 (t ) – 0.1035 m A2 (t )
(d) Static analysis of the structure for external floor stress FN gives static responses M bst , M1st for the overturning moments Mb, and M1 at the base and first floor respectively.
Mbst ( t ) = 2.062 mh A1 ( t ) – 0.062 mh A2 ( t )
M1(t) = 0.604 mh A1(t) – 0.104 mh A2(t)
Figure 18.24 shows a two storey frame with flexural rigidity EI for beams and columns (span of the beam = 2h). Determine the dynamic response of the structure to horizontal ground motion u˙˙g (t ) and express
(a) floor displacement and joint rotations in terms of Dn (t); the bending moments in a first storey column and in the second floor
(b) beam in terms of An(t).
18.12.1 Modal responses
The relative lateral displacement Uin(t) is written as
Ujn(t) = ΓnϕijDn(t)
The storey drift is
Djn(t) = Ujn(t) – Uj–1,n(t)
= Γn(ϕjn – ϕj–1, n)Dn(t)
The equivalence static force for the nth mode fn(t) is given by
fn(t) = FnAn(t)
fjn(t) = FjnAn(t)
where fjn(t) is lateral force at any jth floor level.
Rn(t) due to nth mode is given by
Rn ( t ) = Rnst An ( t )
The modal static response Rn(st) is determined by static analysis of building due to the external force Fn. The modal static responses are presented in Table 18.9, where
2 Total response
Combining the response contribution of the entire mode gives the earthquake response of the multi-storey building
The steps of analysis are given below:
1. Define ground acceleration u˙˙g ( t ) numerically at every time step ∆t.
2. Define structural properties:
(a) determine mass and stiffness matrix,
(b) estimate modal damping ratio.
3. Determine natural frequencies (Tn = 2π/ωn) and natural modes of vibration.
4. Determine modal components RN of the effective earthquake force distribution.
5. Compute the response contribution of nth mode by following steps:
(a) Perform static analysis of building subjected to Fn forces.
(b) Determine pseudo-acceleration response An(t) for the nth mode of SDOF system.
(c) Determine An(t).
6. Combine modal contributions Rn(t) to determine the total response. As already seen, only lower fewer modes contribute significantly to the response. Hence steps 3, 4 and 5 need to be implemented only for these modes.