Dynamics of Structures - Lectures - Introduction

University of Architecture, Civil Engineering and Geodesy 1, Hristo Smirnenski blvd., 1046 Sofia, Bulgaria http://www.uacg.bg/

Faculty of STRUCTURAL ENGINEERING Degree Program: Structural Engineering (Строителство на сгради и съоръжения)

Dynamics of Structures (Course Id: 039.00, ECTS: 4.0, form of assessment: exam)

Alexander Taushanov Department "Structural Mechanics"

LECTURES The course includes many topics in the theory of structural dynamics and application of this theory to earthquake analysis, response and design of structures. The necessary background of the course includes: • Static analysis of structures, including statically indeterminate structures and matrix formulation of analysis procedures • Basic structural design • Dynamics of rigid body • Ordinary differential equations • Linear algebra • Partial differential equations Х. ВЪРБАНОВ УСТОЙЧИВОСТ И ДИНАМИКА НА ЕЛАСТИЧНИТЕ СИСТЕМИ

Most of the course is covered by the book “Dynamics of Structures” – author professor Anil Chopra from the University of California at Berkeley.

ТЕХНИКА

The lectures are unfinished. This is not the final version and therefore can’t be use for sufficient training for the exam.

LECTURE № 1. INTRODUCTION. SDOF SYSTEMS A static load is one which does not vary. A dynamic load is one which changes with time. If it changes slowly, the structure's response may be determined with static analysis, but if it varies quickly (relative to the structure's ability to respond), the response must be determined with a dynamic analysis.

p

p(t)

inertial forces Static loading and dynamic loading

Structural dynamics is a subset of structural analysis which covers the behavior of structures subjected to dynamic loading. Dynamic loads include wind, earthquake, people, traffic, wave, blast and impact. Any structure can be a subject to dynamic loading. Structural dynamics was developed historically along two distinct paths. Vectorial dynamics, based on Newton’s Laws •

First Law. A particle (body) remains in its state of rest, or of uniform rectilinear motion, unless compelled by force to change that state.

•

Second Law. The acceleration of a given element is proportional to the force, applied to it and acts in the direction of that force.

•

Third Law. To every action there is an equal and opposite reaction.

Virtual work dynamics, based on the principle of conservation of energy •

Bernoulli’s Principle of static equilibrium. A mechanical system is in a state of static equilibrium if the virtual work done by all real forces and moments is zero for every virtual displacement consistent with the constraints.

•

d’Alembert’s Principle of dynamic equilibrium. A mechanical system is in a state of dynamic equilibrium at any instant if the virtual work done at that instant by all real and inertial forces and moments is zero for every virtual displacement consistent with the constraints.

Both concepts lead to the same equations of motion.

We begin the study in structural dynamics with simple structures, which can be idealized as a concentrate (lumped) mass m, supported by a massless structure with stiffness k (in the lateral direction). For now let assume that the lateral motion of these structures (displacement u) is small in the sense that deformation of the supporting structure is within their elastic limit.

m ½k

m

u(t)

½k

u(t) k

.. ug(t)

.. ug(t)

Some single degree of freedom systems (SDOF)

•

We will consider the motion of structures as a function of time. We will use overdot ( ) to denote •

the differentiation with respect of time. In this train of thought the velocity of the mass is u , the ••

acceleration of the mass is u . It is shown on the figure also the ground motion, presented with ii

acceleration ug , which is the basic case of earthquake, considered by civil engineers. Stiffness is the value of reaction in the spring due to a given displacement (or rotation) of 1 unit.

force (or moment)

spring constant ku [kN/m] spring constant kv [kN/m]

k – spring constant 1

displacement (or rotation)

Some review from Strength of materials.

spring constant kθ [kNm/rad]

Mass is the value of lumped mass in a point of structure. Every object contributes mass to the structure from the mass density of its material. Mass values are in engineering design mass is in

force×time2 units. In structural length

kN×sec2 kN×sec2 which is equal to ton ( 1 =1 ton ). m m

Mass moments of inertia The rotational mass moments of inertia are in force×length×time2 units. In structural engineering design mass moment of inertia is in kN×m×sec2 , i.e. 1 kN×m×sec2 =1 ton×m2 . Degrees of Freedom. In mechanics, degrees of freedom (DOF) are the set of independent displacements and/or rotations that specify completely the displaced or deformed position and orientation of the body or system. This is a fundamental concept relating to systems of moving bodies in mechanical engineering, aeronautical engineering, robotics, structural engineering, etc.