Dynamic behavior of buildings with non-uniform stiffness along their height assessed through coupled flexural and shear beams

Abstract

A novel model for assessing building behavior has been developed by coupling a Bernoulli beam with a quartic stiffness variation and a shear beam with a parabolic stiffness variation, trends that are expected in buildings designed for earthquake actions. Then the partial differential equation of motion governing the behavior of the model has been solved, obtaining analytic expressions (Closed form solutions) for mode shapes in terms of Legendre functions. These closed form solutions were validated with finite element model analyses and effects of non-uniformity of stiffness were assessed in a generalized manner. It was found that period lengthening is mild for the first mode, but for higher modes can be far more noticeable if shear stiffness at beam top is <20 % of its base value. Mode shapes also change notoriously for reductions beyond the same limit, potentially inducing large floor acceleration demands at unexpected locations. Also it was found that drift demands can be noticeably enhanced even if shear stiffness at top is 75 % of the base value, in what would be considered uniform buildings. This model has several applications for assessing the response or large stocks of buildings, calibrate complex models, assess damage on building contents, establishing in short time damage scenarios for large cities, and could be helpful for education, as emphasis is brought back on fundamental concepts.

Appendices

Appendix 1: Derivation of a solution for the differential modal equation that governs the behavior of a flexural with four order stiffness variation along height coupled with a shear beam with parabolic stiffness variation along height

Consider the following second order differential equations:

$$\left( {1 - x^{2} } \right)\frac{{d^{2} u}}{{dx^{2} }} + 2x\frac{du}{dx} + \nu \left( {\nu + 1} \right)u = 0$$

(37)

$$\left( {1 - x^{2} } \right)\frac{{d^{2} u}}{{dx^{2} }} + 2x\frac{du}{dx} - \left( {\tau^{2} + 1/4} \right)u = 0$$

(38)

A solution of Eq. 37 is:

$$P\left( {\nu ,x} \right) + Q\left( {\nu ,x} \right)$$

(39)

while a solution of Eq. 38 is:

$$A_{3} M\left( {\tau ,x} \right) + A_{4} M\left( {\tau , - x} \right)$$

(40)

If Differential operator 37 is applied in Eq. 38, or differential operator 38 is applied in Eq. 1, either way, following is obtained:

$$\left[ {1 - x^{2} } \right]^{2} \frac{{d^{4} \phi }}{{dx^{4} }} + \left[ {8x^{3} - 8x} \right]\frac{{d^{3} \phi }}{{dx^{3} }} + \left[ {12x^{2} - 4 - \alpha_{0}^{2} \left\{ {1 - x^{2} } \right\}} \right]\frac{{d^{2} \phi }}{{dx^{2} }} + 2\alpha_{0}^{2} x\frac{d\phi }{dx} - \left( \omega \right)^{2} \frac{{\rho H^{2} }}{{EI_{0} }}\phi \left( x \right) = 0$$

(41)

where

$$\omega^{2} \frac{{\rho H^{2} }}{{EI_{0} }} = \nu \left( {\nu + 1} \right)\left( {\tau^{2} + \frac{1}{4}} \right)$$

(42)

$$\alpha_{0}^{2} + 2 = \nu \left( {\nu + 1} \right) - \left( {\tau^{2} + \frac{1}{4}} \right)$$

(43)

In the first case where operator 37 is applied to 38, is evident that the solution of differential Eq. 38 is also a solution of 41. In the latter case where operator 38 is applied to 37 the solution differential Eq. 37 is shown to be also a solution of 41. It must be emphasized that both solutions are linearly independent as the zero order terms in Eqs. 37 and 38 are distinct. Therefore a general solution of Eq. 15 can be formulated:

$$\varphi \left( x \right) = A_{1} P\left( {\nu ,x} \right) + A_{2} Q\left( {\nu ,x} \right) + A_{3} M\left( {\tau ,x} \right) + A_{4} M\left( {\tau , - x} \right)$$

(44)

However, the case considered so far involves zero shear and flexural stiffness at top. This is unrealistic, and therefore of no practical interest. Differential equation of a beam with non-zero stiffness at top is:

$$\begin{aligned} &\left[ {1 - \left( {1 - \delta } \right)x^{2} } \right]^{2} \frac{{d^{4} \phi }}{{dx^{4} }} + {\kern 1pt} \left[ {8\left( {1 - \delta } \right)^{2} x^{3} - 8\left( {1 - \delta } \right)x} \right]\frac{{d^{3} \phi }}{{dx^{3} }} \\ &\quad + \left[ {12\left( {1 - \delta } \right)^{2} x^{2} - 4\left( {1 - \delta } \right) - \alpha^{2} \left\{ {1 - \left( {1 - \delta } \right)x^{2} } \right\}} \right]\frac{{d^{2} \phi }}{{dx^{2} }} \\ & \quad + {\kern 1pt} 2\alpha_{0}^{2} \left( {1 - \delta } \right)x\frac{d\phi }{dx} - \omega^{2} \frac{{\rho H^{2} }}{{EI_{0} }}\phi \left( x \right) = 0 \\ \end{aligned}$$

(45)

Following change of variable is proposed:

$$r = \sqrt {1 - \delta } x$$

(46)

By substituting Eq. 46 with its derivatives on Eq. 45, following is obtained:

$$\left[ {1 - r^{2} } \right]^{2} \frac{{d^{4} \phi }}{{dr^{4} }} + 8r\left[ {r^{2} - 1} \right]\frac{{d^{3} \phi }}{{dr^{3} }} + \left[ {12r^{2} - 4 - \frac{{\alpha^{2} }}{1 - \delta }\left\{ {1 - r^{2} } \right\}} \right]\frac{{d^{2} \phi }}{{dr^{2} }} + 2\frac{{\alpha^{2} }}{1 - \delta }r\frac{d\phi }{dr} - \frac{{\left( {\omega \tau } \right)^{2} }}{{\left( {1 - \delta } \right)^{2} }}\frac{{\rho H^{2} }}{{EI_{0} }}\phi \left( r \right) = 0$$

(47)

General solution of Eq. 47 is:

$$\phi \left( r \right) = A_{1} P\left( {\nu ,r} \right) + A_{2} Q\left( {\nu ,r} \right) + A_{3} M\left( {\tau ,r} \right) + A_{4} M\left( {\tau , - r} \right)$$

(48)

If Eq. 46 is replaced in Eq. 48 the general mode shape Eq. 17 will be obtained. A comparison of Eqs. 45 and 47 show that α and ω have been scaled in the following way:

$$\omega_{e} = \frac{\omega }{{\left( {1 - \delta } \right)}}$$

(49)

$$\alpha_{e} = \alpha \frac{{\sqrt {1 - \delta } }}{1 - \delta }$$

(50)

Appendix 2: Computation of Legendre functions

First and Second degree Legendre functions with real order are solutions of Eq. 37. When their order is an integer value, Legrendre polynomials are obtained; as they are orthogonal are widely used in mathematics and physics, being few applications: Gauss Quadrature; Solutions to rime-dependent Schrodinger Equation; fluid dynamics, particularly compressible gasses and vibration in bodies and membranes (Olver et al. 2010).

Solutions to Eq. 38 define Conical or Mehler functions, a particular kind of Legendre functions with a complex degree; still for any real argument, a real outcome is obtained. Particularly, Mehler functions are helpful for solving problems of vibration and heat flow considering toroidal coordinates, which arise in the design of Tokamaks, for example (Olver et al. 2010). It is observed that Legendre functions with real order have an oscillatory nature, crossing the abscissa (zero ordinate) at uneven intervals between 0 and 1; whilst Mehler functions are monotonically increasing or decreasing between 0 and 1. Legendre functions are natively implemented in Mathematica^© (Wolfram Research Inc 2012). For this study the authors implemented them in Matlab^© (The Mathworks Inc 2012) following algorithms developed by Zhang and Jian-Ming (1996). Higher order derivatives can be computed following recursive relationships (NIST 2012):

$$P_{v}^{\prime } \left( x \right) = \frac{v}{{1 - x^{2} }}\left[ {P_{v - 1} \left( x \right) - xP_{v} \left( x \right)} \right]$$

(51)

or

$$P_{v}^{\prime } \left( x \right) = \frac{v + 1}{{1 - x^{2} }}\left[ {xP_{v} \left( x \right) - P_{v + 1} \left( x \right)} \right]$$

(52)

(52) and (51) applies also to the second type Legendre and Mehler functions. Through chain rule differentiation, successive derivatives can be taken directly:

$$P_{v}^{\prime \prime } \left( x \right) = \frac{2vx}{{\left( {1 - x^{2} } \right)^{2} }}P_{v - 1} \left( x \right) - \frac{{v\left[ {\left( {1 - v} \right)x^{2} + v + 1} \right]}}{{\left( {1 - x^{2} } \right)^{2} }}P_{v} \left( x \right)$$

(53)

$$\begin{aligned} P_{v}^{\prime \prime \prime } \left( x \right) = & {\kern 1pt} \frac{{2v\left( {v - 1} \right)x}}{{\left( {1 - x^{2} } \right)^{3} }}P_{v - 2} \left( x \right) + \left[ {\frac{{\left( {8 - 2v} \right)vx^{2} }}{{\left( {1 - x^{2} } \right)^{3} }} + \frac{{\left( {2 - 2v} \right)v}}{{\left( {1 - x^{2} } \right)^{2} }} - \frac{{v\left( {v - 1} \right)\left[ {\left( {2 - v} \right)x^{2} + v} \right]}}{{\left( {1 - x^{2} } \right)^{3} }}} \right]P_{v - 1} \left( x \right) \\ & - x\left[ {\frac{{\left( {2 - 2v} \right)v}}{{\left( {1 - x^{2} } \right)^{2} }} + \frac{{\left( {4 - v} \right)v\left[ {\left( {1 - v} \right)x^{2} + v + 1} \right]}}{{\left( {1 - x^{2} } \right)^{3} }}} \right]P_{v} \left( x \right) \\ \end{aligned}$$

(54)