16. Nonlinear Dynamic Analysis of MDOF Structures• Basic Concept– During earthquake response, physical properties of a structure vary. – Accuracy of linear analysis becomes uncertain. –Nonlinear dynamic analysis must then be performed to obtain realistic solutionNonlinear dynamic analysis must then be performed to obtain realistic solution. – Few examples requiring nonlinear analysis:• If elastic limit of material is reached, causing yielding of certain members, global stiffness of structure modified during dynamic response. • For slender structure, axial loads in columns can induce important second order (P-Delta) effects causing a reduction of the structure’s stiffness during dynamic response.• If the content of a reservoir is evacuated during the vibrations, then its mass will vary.– In nonlinear analysis, coupled equations of motion are integrated directly. Approach called a“timestep”(or piecewise) integration procedureCIE 619 Chapter 4 – Seismic Analysis163–Approach called a time-step (or piece-wise) integration procedure. • Complete response of the system divided into short time increments (time-steps).• System’s response obtained at each time-step by assuming that the system is linear based on the calculated properties at the beginning of the time-step. • At the end of each time-step, the system’s properties are modified according to the levels of stress and strains in the members. • Nonlinear response estimated by a succession of linear analyses with variable dynamic properties.6. Nonlinear Dynamic Analysis of MDOF Structures• Applications– For very tall and/or highly irregular important structures li l d l d dnonlinear structural model needed– Cyclic behavior of structural elements deemed to respond in the inelastic range of the material needs to be included– Realistic representation of limit states– Ground motion input represented by an ensemble of acceleration time-historiesCIE 619 Chapter 4 – Seismic Analysis164• Scaled historical ground acceleration time-histories• Synthetic records– Usually performed at the end of the design process for verification purposes26. Nonlinear Dynamic Analysis of MDOF Structures• Incremental Equations of Motion– Consider multi-storey nonlinear structure shown. – Assume a single concentrated mass at CIE 619 Chapter 4 – Seismic Analysis165each floor, mi, only one horizontal DDOF per floor, xi, and a base acceleration, xs(t).¨¨6. Nonlinear Dynamic Analysis of MDOF Structures• Incremental Equations of MotionThe equation of motion for mass i at time t is : where Fsi(t) = total nonlinear restoring force applied on mass i at time t FDi(t) = total nonlinear damping force applied on mass i at time t (t)x (t),x (t),xiii = relative displacement, relative velocity and relative acceleration of mass i (t)x m - = (t)F + (t)F + (t)x msiDisiii (4.334)CIE 619 Chapter 4 – Seismic Analysis166ass(t)xs = base acceleration36. Nonlinear Dynamic Analysis of MDOF Structures• Incremental Equations of MotionThe restoring force, Fsi(t), and the damping force, FDi(t), represent the influence of all the elements of the system on massiand are written as :the elements of the system on mass iand are written as : (t)f = (t)F (t)f = (t)F DijN1=jDisijN1=jsi CIE 619 Chapter 4 – Seismic Analysis167where s i j(t) = nonlinear force applied to mass i at time t to induce a displacement of mass j, xj(t), at time t (all other masses have a zero displacement) D i j(t) = nonlinear force applied to mass i at time t to induce a velocity in mass j, .xj(t), at time t (all other masses have a zero velocity ) 6. Nonlinear Dynamic Analysis of MDOF Structures• Incremental Equations of Motion– Assume that nonlinear properties of the system and characteristics of the base accelerogram are known CIE 619 Chapter 4 – Seismic Analysis16846. Nonlinear Dynamic Analysis of MDOF Structures• Incremental Equations of MotionThe equations of motion for mass i are written for one time increment later t+ Δt. t)+(tx m - = t)+(tF + t)+(tF + t)+(tx m siDisiii Equation 4.334 is subtracted from equation 4.335. ())( (t)F - t)+(tF + (t)F - t)+(tF + (t)x - t)+(tx mDiDisisiiii (4.335)CIE 619 Chapter 4 – Seismic Analysis169(t)x-t)+(txm-=ssi or (t)x m - = (t)F + (t)F + (t)x msiDisiii 6. Nonlinear Dynamic Analysis of MDOF Structures• Incremental Equations of MotionThese equations can be combined for all N masses of the system and they can be written in a matrix formatwritten in a matrix format. (t)x )(r ][m - = (t)F + (t)F + (t)x s1xNNxND1xNs1xN1xNNx][mN where x(t)k(t)=1xNNxN1x(t)F sN(4.338)CIE 619 Chapter 4 – Seismic Analysis170(t)x - t)(t+x = (t)x (t)x c(t) = (t)F ()()sss1xNNxND1xN1xNNxN1x()sN (4.339)56. Nonlinear Dynamic Analysis of MDOF Structures• Incremental Equations of MotionThe elements of matrices [k(t)]and[c(t)] are calculated for each time-step using theThe elements of matrices [k(t)] and [c(t)] are calculated for each timestep using the secants shown earlier. This procedure requires iteration at each step because, to calculate the secants, it is necessary to know the displacement and velocity vectors atthe end of each step. To reduce the amount of time required for the calculation, the tangent can be used at the beginning of the integration step to solve for the elements ofthese matrices.  xdüd (t)cjdijtijCIE 619 Chapter 4 – Seismic Analysis171 xdüd (t)kjsijtijt6. Nonlinear Dynamic Analysis of MDOF Structures• Incremental Equations of MotionEquation 4.339 is substituted in equation 4.338 to obtain the incremental equations of(t)x (r)[m] - = x(t) k(t) + (t)x c(t) + (t)x [m] sqq qmotion. The assumption that the mass matrix is constant, proves to be arbitrary and is thereforeunnecessary. However, this assumption is reasonable when a civil engineering structurei bj t d t th k Th i l t h i t l ti(4.341)CIE 619 Chapter 4 – Seismic Analysis172is subjected to an earthquake. There are many numerical techniques to solve equation4.341. In this