Continuous Systems, Infinite Degrees of Beams in Flexure Freedom - PowerPoint PPT Presentation

Continuous Systems, Infinite Degrees of Freedom Giacomo Boffi Continous Systems Continuous Systems, Infinite Degrees of Beams in Flexure Freedom Giacomo Boffi Dipartimento di Ingegneria Strutturale, Politecnico di Milano June 7, 2011

Outline Continuous Systems, Infinite Degrees of Freedom Giacomo Boffi Continous Systems Continous Systems Beams in Flexure Beams in Flexure Equation of motion Earthquake Loading Free Vibrations Eigenpairs of a Uniform Beam Simply Supported Beam Cantilever Beam Mode Orthogonality Forced Response Earthquake Response Example

Intro Continuous Systems, Infinite Degrees of Freedom Giacomo Boffi Continous Systems Discrete models Beams in Flexure Until now we have described or approximated the structural behaviour using the dynamical degrees of freedom , either directly contructing a model with lumped masses or using the FEM to derive a stiffness matrix and a consistent mass matrix or using the FEM stiffness with a lumped mass matrix reducing the degrees of freedom with the procedure of static condensation. Multistory buildings are ecellent examples of structures for which a few dynamical degrees of freedom can describe the dynamical response, using only 3 deegres of freedom for each storey under the assumption of fully rigid floors.

Intro Continuous Systems, Infinite Degrees of Freedom Giacomo Boffi Continous Systems Beams in Flexure Continuous models For some type of structures (e.g., bridges, chimneys) a lumped mass model is not the first option. While a FE model is however appropriate, there is no apparent way of lumping the structural masses in a way that is obviously correct, and a great number of degrees of freeedom must be retained in the dynamic analysis. An alternative to detailed FE models is deriving the equation of motion for the continuous systems in terms of partial derivatives differential equation.

Continuous Systems Continuous Systems, Infinite Degrees of Freedom Giacomo Boffi Continous Systems The equation of motion can be written in terms of partial Beams in Flexure derivatives for many different types of continuous systems, e.g., ◮ taught strings, ◮ axially loaded rods, ◮ beams in flexure, ◮ plates and shells, ◮ 3D solids. Today we will focus our interest on beams in flexure.

Continuous Systems Continuous Systems, Infinite Degrees of Freedom Giacomo Boffi Continous Systems The equation of motion can be written in terms of partial Beams in Flexure derivatives for many different types of continuous systems, e.g., ◮ taught strings, ◮ axially loaded rods, ◮ beams in flexure, ◮ plates and shells, ◮ 3D solids. Today we will focus our interest on beams in flexure.

Continuous Systems Continuous Systems, Infinite Degrees of Freedom Giacomo Boffi Continous Systems The equation of motion can be written in terms of partial Beams in Flexure derivatives for many different types of continuous systems, e.g., ◮ taught strings, ◮ axially loaded rods, ◮ beams in flexure, ◮ plates and shells, ◮ 3D solids. Today we will focus our interest on beams in flexure.

EoM for an undamped beam Continuous Systems, Infinite Degrees of Freedom x m ( x ) , EJ ( x ) Giacomo Boffi At the left, a straight beam with characteristic depending on position x : m = m ( x ) and Continous L EJ = EJ ( x ) ; with the signs conventions for Systems displacements, accelerations, forces and Beams in Flexure p ( x , t ) bending moments reported left, the equation of Equation of motion vertical equilibrium for an infinitesimal slice of Earthquake Loading beam is Free Vibrations Eigenpairs of a u ( x , t ) Uniform Beam ∂x d x ) + m d x∂ 2 u V − ( V + ∂V Simply Supported ∂t 2 − p ( x , t ) d x = 0. ∂x d x Beam Cantilever Beam p d x V + ∂V Mode Orthogonality Rearranging and simplifying d x , Forced Response Earthquake ∂x d x ∂x = m∂ 2 u Response ∂V ∂t 2 − p ( x , t ) . Example M + ∂M M V The rotational equilibrium, neglecting d f I rotational inertia and simplifying d x is d x ∂M ∂x = V . d f I = m d x ∂ 2 u ∂t 2

EoM for an undamped beam Continuous Systems, Infinite Degrees of Freedom x m ( x ) , EJ ( x ) Giacomo Boffi At the left, a straight beam with characteristic depending on position x : m = m ( x ) and Continous L EJ = EJ ( x ) ; with the signs conventions for Systems displacements, accelerations, forces and Beams in Flexure p ( x , t ) bending moments reported left, the equation of Equation of motion vertical equilibrium for an infinitesimal slice of Earthquake Loading beam is Free Vibrations Eigenpairs of a u ( x , t ) Uniform Beam ∂x d x ) + m d x∂ 2 u V − ( V + ∂V Simply Supported ∂t 2 − p ( x , t ) d x = 0. ∂x d x Beam Cantilever Beam p d x V + ∂V Mode Orthogonality Rearranging and simplifying d x , Forced Response Earthquake ∂x d x ∂x = m∂ 2 u Response ∂V ∂t 2 − p ( x , t ) . Example M + ∂M M V The rotational equilibrium, neglecting d f I rotational inertia and simplifying d x is d x ∂M ∂x = V . d f I = m d x ∂ 2 u ∂t 2

Equation of motion, 2 Continuous Systems, Infinite Degrees of Freedom Deriving with respect to x both members of the rotational Giacomo Boffi equilibrium equation, it is Continous ∂x = ∂ 2 M ∂V Systems ∂x 2 Beams in Flexure Equation of motion Substituting in the equation of vertical equilibrium and Earthquake Loading rearranging Free Vibrations Eigenpairs of a m ( x ) ∂ 2 u ∂t 2 − ∂ 2 M Uniform Beam ∂x 2 = p ( x , t ) Simply Supported Beam Cantilever Beam Mode Using the moment-curvature relationship, Orthogonality Forced Response Earthquake M = − EJ∂ 2 u Response Example ∂x 2 and substituting in the equation above we have the equation of dynamic equilibrium m ( x ) ∂ 2 u ∂t 2 + ∂ 2 EJ ( x ) ∂ 2 u � � = p ( x , t ) . ∂x 2 ∂x 2

Equation of motion, 2 Continuous Systems, Infinite Degrees of Freedom Deriving with respect to x both members of the rotational Giacomo Boffi equilibrium equation, it is Continous ∂x = ∂ 2 M ∂V Systems ∂x 2 Beams in Flexure Equation of motion Substituting in the equation of vertical equilibrium and Earthquake Loading rearranging Free Vibrations Eigenpairs of a m ( x ) ∂ 2 u ∂t 2 − ∂ 2 M Uniform Beam ∂x 2 = p ( x , t ) Simply Supported Beam Cantilever Beam Mode Using the moment-curvature relationship, Orthogonality Forced Response Earthquake M = − EJ∂ 2 u Response Example ∂x 2 and substituting in the equation above we have the equation of dynamic equilibrium m ( x ) ∂ 2 u ∂t 2 + ∂ 2 EJ ( x ) ∂ 2 u � � = p ( x , t ) . ∂x 2 ∂x 2

Effective Earthquake Loading Continuous Systems, Infinite Degrees of Freedom Giacomo Boffi If our continuous structure is subjected to earthquake excitation, we will write, as usual, u tot = u ( x , t ) + u g ( t ) and, Continous Systems consequently, Beams in Flexure u tot = ¨ u ( x , t ) + ¨ u g ( t ) ¨ Equation of motion Earthquake Loading and, using the usual considerations, Free Vibrations Eigenpairs of a Uniform Beam Simply Supported p eff ( x , t ) = − m ( x ) ¨ u g ( t ) . Beam Cantilever Beam Mode Orthogonality In p eff we have a separation of variables: in the case of Forced Response Earthquake earthquake excitation all the considerations we have done on Response Example expressing the response in terms of static modal responses and pseudo/acceleration response will be applicable. Only a word of caution, in every case we must consider the component of earthquake acceleration parallel to the transverse motion of the beam.

Effective Earthquake Loading Continuous Systems, Infinite Degrees of Freedom Giacomo Boffi If our continuous structure is subjected to earthquake excitation, we will write, as usual, u tot = u ( x , t ) + u g ( t ) and, Continous Systems consequently, Beams in Flexure u tot = ¨ u ( x , t ) + ¨ u g ( t ) ¨ Equation of motion Earthquake Loading and, using the usual considerations, Free Vibrations Eigenpairs of a Uniform Beam Simply Supported p eff ( x , t ) = − m ( x ) ¨ u g ( t ) . Beam Cantilever Beam Mode Orthogonality In p eff we have a separation of variables: in the case of Forced Response Earthquake earthquake excitation all the considerations we have done on Response Example expressing the response in terms of static modal responses and pseudo/acceleration response will be applicable. Only a word of caution, in every case we must consider the component of earthquake acceleration parallel to the transverse motion of the beam.

Effective Earthquake Loading Continuous Systems, Infinite Degrees of Freedom Giacomo Boffi If our continuous structure is subjected to earthquake excitation, we will write, as usual, u tot = u ( x , t ) + u g ( t ) and, Continous Systems consequently, Beams in Flexure u tot = ¨ u ( x , t ) + ¨ u g ( t ) ¨ Equation of motion Earthquake Loading and, using the usual considerations, Free Vibrations Eigenpairs of a Uniform Beam Simply Supported p eff ( x , t ) = − m ( x ) ¨ u g ( t ) . Beam Cantilever Beam Mode Orthogonality In p eff we have a separation of variables: in the case of Forced Response Earthquake earthquake excitation all the considerations we have done on Response Example expressing the response in terms of static modal responses and pseudo/acceleration response will be applicable. Only a word of caution, in every case we must consider the component of earthquake acceleration parallel to the transverse motion of the beam.