Analysis and Algorith Developers and sponsors assue no responsibility for the use of MIDAS Faily Progra (MIDAS/GeoX, MIDAS/Civil, MIDAS/FX+, MIDAS/Abutent, MIDAS/Pier, MIDAS/Deck, MIDAS/GS, MIDAS/Gen, MIDAS/ADS, MIDAS/SDS, MIDAS/Set ; hereinafter referred to as MIDAS package ) or for the accuracy or validity of any results obtain fro the MIDAS package. Developers and sponsors shall not be liable for loss of profit, loss of business, or financial loss which ay be caused directly or indirectly by the MIDAS package, when used for any purpose or use, due to any defect or deficiency therein.
Developers and distributors assue no responsibility for the use of MIDAS Faily Progra (idas Civil, idas FEA, idas FX+, idas Gen, idas Drawing, idas SDS, idas GS, SoilWorks, idas NFX ; hereinafter referred to as MIDAS package ) or for the accuracy or validity of any results obtained fro the MIDAS package. Developers and distributors shall not be liable for loss of profit, loss of business, or financial loss which ay be caused directly or indirectly by the MIDAS package, when used for any purpose or use, due to any defect or deficiency therein. Accordingly, the user is encouraged to fully understand the bases of the progra and becoe failiar with the users anuals. he user shall also independently verify the results produced by the progra.
CONENS. Structural Eleents. Overview -----------------------------------------------------------------------------------------------. russ Eleent ---------------------------------------------------------------------------------------- 9.3 Bea Eleent --------------------------------------------------------------------------------------- 4.4 Shell Eleent ---------------------------------------------------------------------------------------- 9.5 Plane Stress Eleent ----------------------------------------------------------------------------- 55.6 Plane Strain Eleent ------------------------------------------------------------------------------ 66.7 Axisyetric Eleent ----------------------------------------------------------------------------- 73.8 Solid Eleent ---------------------------------------------------------------------------------------- 84.9 Spring -------------------------------------------------------------------------------------------------- 98.0 Rigid Link ------------------------------------------------------------------------------------------ 0. Reinforceent Eleent. Introduction ----------------------------------------------------------------------------------------- 07. Reinforceent ypes ---------------------------------------------------------------------------- 08.3 Linear Analysis ------------------------------------------------------------------------------------ 4.4 Reinforceent in Plane Strain Eleent ---------------------------------------------------- 0.5 Reinforceent in Axisyetric Eleent --------------------------------------------------- 5.6 Reinforceent in Plane Stress Eleent ---------------------------------------------------- 30.7 Reinforceent in Solid Eleent -------------------------------------------------------------- 33.8 Reinforceent in Plate Eleent -------------------------------------------------------------- 39.9 Prestress of Reinforceent -------------------------------------------------------------------- 44 i
CONENS 3. Interface Eleent 3. Introduction ----------------------------------------------------------------------------------------- 49 3. Coordinate Syste and Relative Displaceent ------------------------------------------ 5 3.3 Point Interface Eleent ------------------------------------------------------------------------- 53 3.4 Line Interface Eleent -------------------------------------------------------------------------- 55 3.5 Surface Interface Eleent---------------------------------------------------------------------- 58 3.6 Finite Eleent Forulation -------------------------------------------------------------------- 63 3.7 Interface Eleent Output ----------------------------------------------------------------------- 64 4. Linear Analysis 4. Linear Static ---------------------------------------------------------------------------------------- 65 4. Linear Static with Nonlinear Eleents ------------------------------------------------------ 66 5. Modal Analysis 5. Introduction ----------------------------------------------------------------------------------------- 68 5. Lanczos Iteration Method----------------------------------------------------------------------- 73 5.3 Subspace Iteration Method -------------------------------------------------------------------- 75 5.4 Optional Paraeters ----------------------------------------------------------------------------- 76 6. ie History Analysis 6. ie History Analysis ---------------------------------------------------------------------------- 80 6. Modal Superposition Method ------------------------------------------------------------------ 8 6.3 Direct Integration Method----------------------------------------------------------------------- 83 6.4 Optional Paraeters ----------------------------------------------------------------------------- 87 6.5 Cautionary Notes --------------------------------------------------------------------------------- 89 ii
CONENS 7. Response Spectru 7. Response Spectru Analysis ----------------------------------------------------------------- 90 7. Spectru Function ------------------------------------------------------------------------------- 96 8. Linear Buckling Analysis 8. Introduction ----------------------------------------------------------------------------------------- 98 8. Geoetric Stiffness ------------------------------------------------------------------------------ 0 8.3 Critical Load Factor Extraction ---------------------------------------------------------------- 09 8.4 Optial Paraeter ------------------------------------------------------------------------------- 0 9. Load and Boundary 9. Constraint Condition of Degrees of Freedo --------------------------------------------- 9. Skewed (Inclined) Support Condition ------------------------------------------------------- 4 9.3 Constraint Equation ------------------------------------------------------------------------------ 5 9.4 Nodal Load ----------------------------------------------------------------------------------------- 8 9.5 Eleent Pressure Load ------------------------------------------------------------------------- 9 9.6 Body Force ----------------------------------------------------------------------------------------- 9.7 Prescribed Displaceent ----------------------------------------------------------------------- 0. Construction Stage Analysis 0. Introduction --------------------------------------------------------------------------------------- 3 0. Coposition of Construction Stages------------------------------------------------------- 5 0.3 ie Dependent Material Properties ------------------------------------------------------ 9 iii
CONENS. Heat of Hydration. Introduction --------------------------------------------------------------------------------------- 57. Heat ransfer Analysis ------------------------------------------------------------------------- 58.3 heral Stress Analysis ----------------------------------------------------------------------- 65.4 Hear of Hydration Analysis Considering Construction Stages ---------------------- 69.5 Results of Hear of Hydration Analysis ----------------------------------------------------- 70. Material Models. Introduction --------------------------------------------------------------------------------------- 7. Yield Criteria -------------------------------------------------------------------------------------- 8 3. otal Strain Crack 3. Introduction --------------------------------------------------------------------------------------- 30 3. Basic Properties --------------------------------------------------------------------------------- 305 3.3 Loading and Unloading ------------------------------------------------------------------------ 308 3.4 Crack Strain ransforation ----------------------------------------------------------------- 3 3.5 Stiffness Matrix ---------------------------------------------------------------------------------- 34 3.6 Copression Models -------------------------------------------------------------------------- 30 3.7 ension Models ---------------------------------------------------------------------------------- 36 3.8 Shear Models ------------------------------------------------------------------------------------ 336 3.9 Lateral Influence -------------------------------------------------------------------------------- 337 4. Interface Nonlinear Behaviors Analysis 4. Introduction --------------------------------------------------------------------------------------- 344 4. Discrete Cracking ------------------------------------------------------------------------------- 346 4.3 Crack Dilatancy ---------------------------------------------------------------------------------- 35 4.4 Bond Slip ------------------------------------------------------------------------------------------ 36 iv
CONENS 4.5 Coulob Friction -------------------------------------------------------------------------------- 365 4.6 Cobined Cracking-Shearing-Crushing -------------------------------------------------- 368 5. Geoetric Nonlinear 5. Introduction --------------------------------------------------------------------------------------- 380 5. russ Eleent------------------------------------------------------------------------------------ 386 5.3 Plane Stress Eleent -------------------------------------------------------------------------- 388 5.4 Plate Eleent ------------------------------------------------------------------------------------ 39 5.5 Plane Strain Eleent -------------------------------------------------------------------------- 396 5.6 Axisyetric Eleent ------------------------------------------------------------------------- 397 5.7 Solid Eleent ------------------------------------------------------------------------------------ 400 6. Iteration Method 6. Introduction --------------------------------------------------------------------------------------- 403 6. Initial Stiffness Method ------------------------------------------------------------------------ 407 6.3 Newton-Raphson Method--------------------------------------------------------------------- 408 6.4 Arc-Length Method ----------------------------------------------------------------------------- 4 6.5 Displaceent Control -------------------------------------------------------------------------- 45 6.6 Convergence Criteria -------------------------------------------------------------------------- 47 6.7 Auto-Switching ----------------------------------------------------------------------------------- 49 7. Equation Solver 7. Introduction --------------------------------------------------------------------------------------- 40 7. Direct Method ------------------------------------------------------------------------------------ 4 7.3 Iterative Method --------------------------------------------------------------------------------- 45 7.4 Solver Characteristic --------------------------------------------------------------------------- 47 v
CONENS 8. Contact 8. Introduction --------------------------------------------------------------------------------------- 48 8. Contact Search ---------------------------------------------------------------------------------- 430 8.3 Function ------------------------------------------------------------------------------------------- 435 9. Fatigue 9. Introduction --------------------------------------------------------------------------------------- 437 9. Load Cycles -------------------------------------------------------------------------------------- 438 9.3 Mean Stress Effects ---------------------------------------------------------------------------- 44 9.4 Modifying Factors ------------------------------------------------------------------------------- 443 9.5 Rainflow-Counting Algorith ----------------------------------------------------------------- 449 9.6 Fatigue Analysis Procedure ------------------------------------------------------------------ 45 0. CFD Analysis (Coputational Fluid Dynaic) 0. Introduction --------------------------------------------------------------------------------------- 453 0. RANS equation and turbulence odel --------------------------------------------------- 454 0.3 Spatial discretization -------------------------------------------------------------------------- 457 0.4 Steady flow -------------------------------------------------------------------------------------- 459 0.5 Unsteady flow ----------------------------------------------------------------------------------- 460 0.6 Nuerical stability ----------------------------------------------------------------------------- 46 0.7 Coputational fluid analysis results ------------------------------------------------------ 463 vi
Eigenvalue analysis for analyzing the dynaic behavior of structures is also referred to as free vibration analysis. For a syste without daping and with no excitation, the otion equation is the nd order linear differential equation (5.). Mu&& ( t) + Ku( t) = 0 (5.) where, K : Stiffness atrix of structure M : Mass atrix of structure u(t) : Displaceent vector of structure ü(t) : Acceleration vector of structure If the displaceent vector u is assued to be a linear cobination of ode-shape-vectors, defined by the ode shape atrixφ, and the cobination factors for the selected odes are defined by a vector of tie-functions Y ( t), we can replace the displaceent vector u= ΦY ( t). By substituting the expression for u in equation (5.) the following equation is obtained: idasfea MΦY+KΦY && = 0 (5.) he tie function, Y( t) is defined as. { n } Y( t) = y ( t) L y ( t) L y ( t) (5.3) where n is the total nuber of degrees of freedo in the syste. When we assue that the cobination factors y (t) are haronic functions in tie y ( ) t : cos( wt+ b)
Analysis and Algorith Manual t he second derivative y (t) with tie of the haronic function can be written as inverse ultiplication with a constant factor λ = ω of the original function y (t) : t y ( t) =- y ( t) l (5.3.) Subsequently we can use this assuption to transfor the equation (5.3) into, (- MΦΛ+ KΦY ) = 0 (5.4) where the atrices Λ and Φ are forulated as below. idasfea Λ él ê ê ê ù ú ú ú, = l ê ú ê ê ë O O ú l ú nû [ f f f ] n l = w (5.5) Φ= L L (5.6) he equation (5.4) ust be valid for every haronic function, that eans for every y ( t ), it can be transfored into. Kf - l M f = 0 (5.7) Equation (5.7) is an eigenvalue proble, which ust satisfy the condition of equation (5.8) and fro this condition the free vibration odes can be calculated. l K- M = 0 (5.8) Equation (5.8) has n nuber of solutions (eigenvalues) l, l,, l that satisfy the condition n (5.8) and equal to the nuber of degrees of freedo of the finite eleent odel. Usually the eigenvalues are ordered such that λ is the sallest eigenvalue. For each eigenvalue, l a corresponding eigen ode, f exists. Since the ass atrix, M and the stiffness atrix, K in
equation (5.7) are syetrical, the eigenvalue, l and the eigen ode, f are real nubers. Because the ass atrix is positive definite, and the stiffness atrix is positive sei-definite l ³ 0. As such the circular frequency, w under the condition of undaped free vibration is a real nuber. A structure vibrates in the eigen ode shapes of f without external excitation, and the velocities are circular frequencies, w ( radian / tie ). he velocities of a structure are expressed in ters of natural frequencies, f ( cycle / tie ) or natural periods, ( tie / cycle ). he relationship betweenw, =, f f and is given below. w f = (5.9) p Generally the eigenvalue, l represents the ratio of the strain energy to the kinetic energy for the -th ode shape, and the odes are referred to as st ode, nd ode,, n-th ode fro the sallest ratio onward. Fig. 5-() shows the fr ee vibration odes of a cantilever fro the st to 3 rd ode. idasfea u u u t t t
Analysis and Algorith Manual MIDAS calculates indices of the dynaic properties of a structure such as odal participation factors, effective odal asses and odal direction factors. he values of the directional odal participation factors are calculation by equation (5.0), which are used for response spectru analysis or tie history analysis of a structure subjected to seisic loads. idasfea where, fmx fmy fmz G X =, G =, G = Y Z f Mf f Mf f Mf fmrx fmry fmrz G RX =, G =, G = RY RZ f Mf f Mf f Mf (5.0) : Modal participation factors in the GCS X, Y and Z GX, GY, GZ translations for the -th ode : Modal participation factors in the GCS X, Y and Z GRX, GRY, GRZ rotations for the -th ode : Directional vectors, which retain unit values only for the X, Y, Z translations in the X, Y and Z directions : Directional vectors, which retain unit values only for the RX, RY, RZ rotations in the respective X, Y and Z directions f : th ode shape he directional values of the odal participation asses for each ode are calculated according to equation (5.). Since the calculation includes the signs (positive or negative) of the odes, the values can be 0 depending on the ode shapes. he su of the directional participation asses for all the odes is equal to the total ass of the structure in each corresponding direction. General seisic design codes require that the su of the odal participation asses included in the analysis of a structure in each direction be at least 90% of the total ass. his is intended to include ost of the ajor odes, which influence the analysis results.
éf ù éf ù éf ù * ë MXû * *, ë MY = = û, = ë MZû X Y Z f Mf f Mf f Mf M M M éf ù éf ù éf ù * ë MRXû * *, ë MRY = = û, = ë MRZû RX RY RZ f Mf f Mf f Mf M M M (5.) where, M, M, M * * * X Y Z M, M, M * * * RX RY RZ : Modal participation asses in the GCS X, Y and Z translations for the -th ode : Modal participation asses in the GCS X, Y and Z rotations for the -th ode he directional coefficient for each ode represents the ratio of the directional participation ass to the total participation ass for the corresponding ode. MIDAS uses the Lanczos and subspace iteration ethods which are suitable for analyzing large systes of eigenvalue probles such as equation (5.7). idasfea
Analysis and Algorith Manual he Lanczos iteration ethod finds approxiate eigenvalues using a tri-diagonal atrix,, which can be found by defining the Krylov subspace span{ V, V,..., V k }. In order to effectively apply the Lanczos iteration ethod to the free vibration eigenvalue proble as expressed in equation (5.7), the eigenvalue, l ust be replaced by l = s+ / q. his is referred to as a shift-invert technique. s is the expected first eigenvalue. he Lanczos iteration calculation process applying the shift-invert technique is described in below. k idasfea - Assue an initial value V for the block vector in case of first iterative calculation. - Multiply the ass atrix Uk = MV k - Solve linear siultaneous equations ( K- sm) Wk = U k - Orthogonalize W k - Calculate atrix C k - Orthogonalize * W k - Noralize the block vector W = W - V B * k k k- k- C = V MW * k k k W = W - V C ** * k k k k W = V B ** k k+ k MIDAS uses the block vector, V for effectively calculating eigenvalues. his is referred to as k the Block Lanczos ethod. he block tri-diagonal atrix,, which takes place in the process of iterative calculations above, is as follows: k éc B ù ê ú êb C O ú k = ê O O O ú ê ú ê O Ck- Bk- ú ê ú ë Bk- Ck û (5.)
If k is used to solve the eigenvalue proble, l = s + / q. * * y = q y, * * * k * l can be obtained using * l is an approxiate value of the original eigenvalue proble of Eq. (5.7). If N b is defined to be the block size of V, the size of increases to as uch as k k Nb and converges to l, as the nuber of iterations increases in the process of iterative calculations. * he approxiate value, f of the eigenode, f can be obtained by the equation below, which * converges with l. * l [ ] f = V V V V (5.3) * * 3... k y he convergence criterion for eigenvalues and eigenodes calculated fro the Lanczos iteration ethod is noted below. Kf -l Mf * * * K e Where,. eans -nor. MIDAS uses e -6 =. 0. (5.4) idasfea
Analysis and Algorith Manual he Subspace ethod searches for a set of N s orthogonal vectors X k that expand over the subspace Ek that is defined by the eigen odes é ëf f.. f N ù sû For large systes the nuber of N s is uch saller than the nuber of degrees of freedo in the finite eleent odel. he iterative process of calculating the set of vectors X is explained below. k idasfea - Assue a first set of N s vectors X and call this X k - Calculate vector Y k with length N s by solving linear set of equations KYk = MX k - Calculate the projection of stiffness in E K k - Calculate the projection of ass E M k = Y K Y k+ k k = Y MY k+ k k - Solve projected eigenvalue proble of size N s Kk+ Qk+ = Mk+ Qk+ Λ k+ by using classical eigenvalue solution techniques * * * é k+ y y y ù N Q = ë... s û Λ k+ * él ù ê * ú l = ê ú ê O ú ê ú * êë ln ú sû Calculating X k+ by Xk+ = Yk Qk+ and repeating these steps until individually converge to N s eigenvalues and N s eigen odes of the syste * ln and X k, * l l, X k éf f... f ù ë Ns û (5.5) If Xk is coposed of N s vectors, the sizes of Kk and the nuber of iterations in the process of iterative calculations. M k are Ns Ns calculated by the subspace iteration ethod is evaluated by the change ofl., irrespective of he convergence of eigenvalues * l -l ( k+ ) ( k ) ( k+ ) l e (5.6)
wo types of ass atrices can be used in an eigenvalue analysis, which are the consistent ass atrix and the luped ass atrix. When the consistent ass atrix is applied, this generally results in higher eigenvalues copared to the theoretical values. Whereas using the luped ass atrix results in saller eigenvalues. It is known that using the consistent ass atrix results in better convergence than using the luped ass atrix depending on the nuber of eleents. he drawback however is that the consistent ass atrix requires ore calculations and eory (see figure 5-() ). idasfea When calculating eigen frequencies using the Lanczos iteration ethod in MIDAS two
Analysis and Algorith Manual frequencies, f and f can be specified by the user for defining the range of interest of eigen frequencies. he range of interest of frequencies is used in the calculation to define the expected eigenvalues to s = ( p f ) by the shift-invert technique. Fig. 5-(3) shows the range and sequence of calculating eigenvalues for cobinations of f and f. If N f is defined as the nuber of eigenvalues to be calculated, the nuber of each calculated eigenvalues is deterined based on f and f as, - If f= f : he N f frequencies closest to f are calculated. - If f< f : he N f frequencies closest to [ f, f] are calculated. - If f> f : he N f frequencies closest to f within the range, [ f, f] are calculated. idasfea 4 f = f 3 f f f 3 4 f f f 4 3 f he default values of f and f, in the range of interest are f= f= 0. When eigenvalues are calculated using an iteration ethod, there is a possibility that higher eigenvalues are found before the lower eigenvalues. MIDAS provides functionality, which
prevents that frequencies are skipped. When applying the Lanczos iteration ethod the vector W k is calculated by solving the following set of equations: ( K- sm) Wk = U k (5.7) In order to solve the equations, the atrix is decoposed by the atrices nuber of negative values in the diagonal ters in the atrix, D is calculated. n LDL and the he nuber represents the nuber of eigenvalues, l that are saller thans. For exaple, if s = l + d and the atrix is decoposed, we can find eigenvalues that are saller than nuber of eigenvalues saller than ln f ln f N f + d. If the + d is greater than N f, the process of the Lanczos iteration continues until all the oitted eigenvalues are converged. At this point, d should satisfy the relationship below. l < l + d < l + (5.8) N N N f f f his condition to δ is called the Stur-sequence check. When MIDAS perfors a Stur sequence check, the size of the Lanczos block vector, V is set to N = 7 to speed up the calculations. k For liiting the coputational effort MIDAS also provides an option to skip the Stur sequence check. b idasfea When a free-vibration analysis of a structure with insufficient boundary conditions is perfored, K will be a singular atrix and the calculations cannot be progressed withs = 0. In such a case, the shift-invert technique is applied with a negative value fors. In this case the calculated eigenvalues will include a 0 for every rigid ode. MIDAS predicts the expected first ode frequency, l using the diagonal ters of the atrices K and M and setss =- l.
Analysis and Algorith Manual he subspace iteration ethod in MIDAS allows the user to set the size, k N s of the subspace, X, the nuber of axiu iterations, N and the convergence criterion, e in Eq. (5.6). he actual size of subspace, s s _ 0 f f I N s used for the calculations is as follows: N = ax{ N,in( N, N + 8)} (5.9) Where, N s _ 0 is the user input N s value. he default value for the axiu nuber of iterations is N I = 30. he default value for the convergence criterion is e -6 =.0 0. idasfea
When a structure is subjected to a dynaic load, the behavior is described by the dynaic otion equation given by (6.). where, Mu&& ( t) + Cu& ( t) + Ku( t) = F( t) (6.) In case there is no external excitation F( t ) = 0 and the daping atrix C = 0, the equation (6.) will reduce to the free vibration equation (5.). However, if F( t) is an external force (or external displaceent, velocity, acceleration, etc.), which varies with tie, the equation (6.) becoes a forced vibration analysis proble. For this type of probles MIDAS provides a odal superposition ethod as well as a direct integration ethod.. M : Mass atrix C : Daping atrix K : Stiffness atrix F ( t) : Dynaic load vector u ( t), u& ( t) & u&& ( t) : Displaceent, Velocity & Acceleration vector idasfea
Analysis and Algorith Manual he odal superposition ethod akes use of the orthogonality of eigen odes and decoposes the equation (6.) into independent ode equations. his ethod assues that the daping atrix is coposed of a linear cobination of the ass atrix and the stiffness atrix. he displaceent vector is obtained by a linear cobination of a selected nuber n of orthogonal eigen odes as expressed in equation (6.). n u( t) FY( t) fy (6.) = =å i= j By substituting equation (6.) into equation (6.), we get i idasfea MΦY&& ( t) + CΦY& ( t) + KΦY( t) = F( t) (6.3) Multiplying equation (6.3) byf (-th ode shape), the equation (6.4) is obtained. f MΦY&& ( t) + f CΦY& ( t) + f KΦY( t) = f F( t) (6.4) he ass and stiffness atrices can be orthogonalized to eigen odes as, f Mf = d i j ij ij f Kf = k d i j ij ij (6.5) where, d : Krönecker Delta ( i= j ; d =, i¹ j ; d = 0 ) ij ij Using the above orthogonality, the dynaic otion equation (6.3) can be decoposed into n independent equations. f Mf Y&& ( t) + f Cf Y& ( t) + f Kf Y( t) = f F( t) ( =,,...,n) (6.6) ij he above n independent decoposed equations thus becoe dynaic otion equations for a single degree of freedo in a general coordinate syste.
Subsequently we rearrange equation (6.6) where, ( t) Y && (t ) + xw Y & f F (t ) + wy (t ) = (6.7) f Mf f Cf x w = f Mf w f Kf = f Mf x : th ode daping ratio w : th ode natural frequency q ( t), q& ( t) & q&& ( t) : th ode general displaceent, velocity and acceleration Displaceent response in the general coordinate syste is obtained by equation (6.8). -x (0) (0) wté xwq + q ù q( t) = e êq(0)coswdt+ sinwdtú ë wd û t -xw ( t-t ) + P ( t ) e sin wd( t t ) dt w ò - 0 D (6.8) idasfea where, w = w - x D he displaceent response of a structure is obtained by substituting the general displaceent of each ode, which is obtained by the single degree of freedo equation (6.8), into the syste equation (6.). he accuracy of the displaceent response in the odal superposition ethod depends on the selected odes used for analysis. he odal superposition ethod is ost widely used by structural analysis progras, and it is effective in linear dynaic analyses of large structures. However, this ethod cannot be used for nonlinear dynaic analysis; nor can it be used in case daping devices are included and their properties cannot be assued on the basis of a linear cobination of stiffness and ass.
Analysis and Algorith Manual Using the direct integration ethod is one in which the total analysis tie range is sub-divided into a nuber of finite tie steps, and nuerical integration of the dynaic equilibriu equation is perfored at each tie step. his ethod can be applied to systes reflecting nonlinearity of stiffness and/or daping. Since the direct integration ethod evaluates the dynaic otion equation at every tie step, the analysis tie increases with the nuber of tie steps. idasfea A variety of ethods can be applied for nuerical integration. MIDAS uses the average acceleration ethod of the Newark-β ethod in which the acceleration u&& ( t) in the tie range t t t + i < < i is assued to be constant at the average of i u&& and u&& i+ in Eq. (6.9). u&& i + u&& i+ u&& ( t) = = const. (6.9) Consequently, the velocity and displaceent at t = t i+ are expressed as, u&& + u&& u& i i+ i+ = u& i + Dt (6.0) u&& + u&& ui+ = ui + uid t+ Dt 4 & i i+ (6.) Expressing the equations (6.0) and (6.) with the integration variables of the Newark-β ethod, ( g) u& = u& + - D tu&& + gdtu&& (6.) i+ i i i+ i i t æ ö u + = u +D u& i + ç -b D t u&& i + bdt u&& i+ (6.3) è ø
where, b =0.5, g =0.5. Rearranging the equation (6.3) gives the following expression for the acceleration at the end of the tie-step, ì æ ö ü u&& i = i i t i b t + íu - u -D u& - - D i b t + ç u&& ý (6.4) D î è ø þ Substituting equation (6.4) into equation (6.) and rearranging it, gives the following expression for the velocity at the end of the tie-step, g g æ g ö æ g ö u& t i + = u b t i + - u + b t i ç - u& + - D b i ç u&& (6.5) D D è ø è bø i he equations (6.4) and (6.5) are substituted into the dynaic otion equation, which is rearranged for the displaceent response ui+ at the end of the increent as follows: idasfea æ g ö ç M+ C+K u èbdt bdt ø i+ ìï æ ö üï ìï g æg ö æ g ö üï = F+ Mí u i + u& i + ç - u&& iý+ Cí ui + ç - u& i + ç - Dtu&& iý ïî bdt bdt è b ø ïþ ïî bdt èb ø è b ø ïþ (6.6) Substituting the displaceent u i+ at the tie t i + as defined in equation (6.6) into equations (6.4) and (6.5), the velocity and acceleration at the end of the increent as function of status at the beginning of the increent only can be obtained. Rayleigh daping expressed in equation (6.7) is used for daping in the direct integration
Analysis and Algorith Manual ethod. Rayleigh Daping: C = a0m+ ak (6.7) where, a 0 & a : Proportional constants for ass and stiffness for daping calculation By substituting the equation (6.7) into equation (6.6), the dynaic otion equation is expressed as, idasfea ìï æ ag ö æ ag ö üï íç + M+ ç + Kýu ïî èbdt bdtø èbdt ø ïþ ìï æ ö üï =F+ Mí u i + u& i + ç - u&& i + adý+ akd ïî bdt bdt è b ø ïþ g æg ö æ g ö where, D = ui + ç - u& i + ç - Dtu&& i bdt èb ø è b ø i+ (6.8) he solutions for the dynaic otion equation that are found through tie history analysis are the relative displaceent u ( t), the relative velocity u& ( t) and the relative acceleration u&& ( t). When a structure is subjected to dynaic loads such as ground acceleration, the absolute response of the structure is obtained by adding the relative response and the ground response as expressed in equation (6.9). u&& + u&& : Absolute acceleration g, i+ i+ u& + u& : Absolute velocity (6.9) g, i+ i+ u + u : Absolute displaceent g, i+ i+
where, u&& g, i+, u& g, i+ & u g, i+ are ground acceleration, velocity & displaceent respectively. In MIDAS, ground velocity and displaceent due to ground acceleration are calculated by using the linear acceleration ethod (equation 6.0), which are applied to calculating the absolute response in the odal superposition and direct integration ethods. u&& - u&& u& g, i+ = u& g, i +D tu&& g, i + Dt Dt u&& u u u u 6 g, i+ g, i - u&& g, i+ g, i 3 g, i+ = g, i +D t& g, i + D t && g, i + Dt Dt (6.0) idasfea
Analysis and Algorith Manual In MIDAS, the following daping ethods are used depending on the ethod of dynaic analysis. Selection of daping for tie history analysis by response spectru and odal superposition: - Modal - Mass & Stiffness Proportional (Rayleigh daping) idasfea Selection of daping for tie history analysis by direct integration: - Mass & Stiffness Proportional (Rayleigh daping) he daping atrix in Rayleigh Daping is coposed of the linear cobination of the ass atrix and the stiffness atrix of a structure as shown in Fig. (6-b). If the daping integer and the natural frequency w r for the r th ode and the daping integerxs and the natural frequency w s for the s th ode are given, the Rayleigh daping atrix is expressed below. Notice that the r th and s th odes represent the two ain odes of the structure. x r C= a M+ a K (6.) 0 æ a ö 0 xi = ç + a wi èwi ø (6.) where, ( ) w w x w - x w a =, r s r s s r 0 ( ws -wr ) a = ( x w - x w ) s s r r ( ws -wr )
x i Mass Proportional C= a 0 M a0 xi = w i C= a K a wi x i = Stiffness Proportional w w w3 w4 Natural frequencies w i idasfea For odal daping, the user directly defines the daping ratio for each ode. Modal daping can be used for tie history analysis, by response spectru analysis and the odal superposition ethod. When the response spectru analysis and odal superposition ethod are used, the kineatic equation of the structure is decoposed by odes to which the odal daping ratios defined by the user are applied.
Analysis and Algorith Manual he accuracy of analysis depends greatly on the tie interval used in analysis. Iproper tie interval ay result in inaccurate solutions. In particular, the size of the tie interval is closely related to the axiu frequency of oveent of the structure. In general, one-tenth of the highest odal period under consideration is a reasonable value for the tie interval. In addition, the tie interval should be saller than that of the applied load. A dynaic load needs to sufficiently depict the change in the total loads. MIDAS linearly interpolates the excitation loads. idasfea
Response spectru analyses are generally carried out for seisic designs using the design spectra defined in design standards. Response spectru analysis assues the response of a ulti-degreeof-freedo (MDOF) syste as a cobination of ultiple single-degree-of-freedo (SDOF) systes. A response spectru defines the peak values of responses corresponding to and varying with natural periods (or frequencies) of vibration that have been prepared through a nuerical integration process. Displaceents, velocities and accelerations for the basis of a spectru. o predict the peak design response values, the axiu response for each ode is obtained first and then cobined by an appropriate ethod. Equation (7.) shows the dynaic otion equation for a structure subjected to a ground otion used in a response spectru analysis. M[ u&& ( t) + ru&& ( t)] + Cu& ( t) + Ku( t) = 0 g Mu&& ( t) + Cu& ( t) + Ku( t) =-Mru&& ( t) g (7.) idasfea where, M C K : Mass atrix : Daping atrix : Stiffness atrix r : Directional vector of ground acceleration u&& g ( t ) : ie history of ground acceleration u( t ), u& ( t ) & u&& ( t ) : Relative displaceent, velocity and acceleration If the displaceent, u( t) is expressed in ters of a cobination of odal displaceents using the eigen ode shapesφ, obtained fro undaped, free vibration analysis, equation (7.) is
Analysis and Algorith Manual applicable. u( t) = Φy( t ) (7.) Now we substitute equation. (7.) into equation (7.) and ultiply both sides byφ, resulting in Φ MΦy &&( t) + Φ CΦy& ( t) + Φ KΦy( t) =-Φ Mr u&& ( t ) (7.3) g he relationship below is found due to orthogonality of the eigen odes. i ( or or ) 0 ( i j) f M C K f = ¹ (7.4) j Accordingly, if the diensionless eigen ode shape ( Φ MΦ = ) for ass is applied to equation (7.3), we obtain the independent siultaneous differential equations for each ode. idasfea é ù éxw ù éw ù ê ú ê ú ê ú ê O ú ê O ú ê O ú ê ú && y t + ê ú y& t + ê ú y t =-Φ Mru&& t ê ú ê ú ê ú ê O ú ê O ú ê O ú ê ú ê x w ú ê w ú ë û ë n nû ë nû ( ) x w ( ) w ( ) g ( ) (7.5) And rearranging the expression for the -th ode in equation (7.5) results in. && y t x w y& t w y t u&& t ( ) + ( ) + ( ) =-G g ( ) G = f Mr (7.6) he odal participation factor, G in equation (7.6) is defined by the ultiplication of the diensionless ode shape for ass φ, the ass atrix M and the directional vector of ground acceleration r. he directional vector of ground acceleration contains unit value only for the degree of freedo in the direction of ground acceleration and all other coponents are equal to zero. he solution to the dynaic equilibriu equation of a structure under ground acceleration
action is obtained by solving the n equations of (7.6) and then cobining the in the sae way as equation (7.7). u( t) = Φy( t), u& ( t) = Φy& ( t), u&& ( t) = Φy && ( t ) (7.7) he concept of the dynaic equilibriu equation subjected to a ground acceleration action in response spectru analysis follows the equations (7.) to (7.7). In response spectru analysis, spectru functions are used to obtain the results for each ode rather than defining the seisic acceleration as a specific function. Generally, the spectru function eans the axiu value for each period in equation (7.5). he spectru function is defined in various design codes and specifications, which consider the probabilities of seisic accelerations, characteristics of regions and iportance of structures. he solutions to a single d.o.f syste corresponding to the -th ode like equation (7.6) are found by ultiplying the displaceent, velocity and acceleration obtained fro the spectru function by the corresponding odal participation factors as equation (7.8). Sa Sa Sd =, S = v w w y =G S, y& =G S, && y =G S d v a (7.8) idasfea he results for each ode are calculated by ultiplying the results of equation (7.7) by the ode shapes as in equation (7.9). u = f G S, u& = f G S, u&& = f G S (7.9) d v a where, S d : Spectral displaceent of th ode S v : Spectral velocity for of th ode S a : Spectral acceleration of th ode Since the analysis results for each ode pertain to only the axiu values, it is not possible to perfor linear cobinations as done in tie history analysis. herefore, the final results of a
Analysis and Algorith Manual response spectru analysis are obtained by a odal cobination of the analysis results for each ode in equation (7.9). MIDAS enables us to analyze response spectru in any direction on the X-Y plane and in the Z- direction in the global coordinate syste. he user can choose one of the 3 ethods for odal cobination, which are ABS (Absolute Su), SRSS (Square Root of the Su of the Squares) and CQC (Coplete Quadratic Cobination) ethods. ABS (ABsolute Su) Rax = R + R + + Rn (7.0) idasfea SRSS (Square Root of the Su of the Squares) Rax = é ër + R + + R ù n û (7.) CQC (Coplete Quadratic Cobination) é ù ax =êåå N N R Ririj R jú (7.) ë i= j= û 3 i j i + rijx rij 8 xx ( x ) rij = (- r ) + 4 r ( + r ) + 4( + ) r ij xx i j ij ij xi x j ij 8 x ( + r ) r r = ( x = x = x ) 3 ij ij ij i j (- rij ) + 4 x rij ( + rij ) wi r = w > w w ij j i j 0 rij rij = ( i= j )
where, R ax : Peak response R i r ij w i,w j : Peak response (spectru value) of i th ode : Eigenvalue ratio of j th ode to i th ode : Eigenvalues of i th and j th odes, x x i j : Daping ratios of i th and j th odes In equation (7.), when i = j, then ρ ij = regardless of the daping ratio ( x i, x j ). If the daping ratio becoes zero (0), both CQC and SRSS ethods produce identical results. he ABS ethod produces the largest cobination values aong the three ethods. he SRSS ethod has been widely used in the past, but it tends to overestiate or underestiate the cobination results in the cases where the values of natural frequencies are close to one another. As a result, the use of the CQC ethod is increasing recently as it accounts for probabilistic inter-relations between the odes. Exaple idasfea If we now copare the natural frequencies and displaceents for each ode for a structure having 3 DOF s with a daping ratio of 0.05, the results fro the applications of SRSS and CQC are as follows: Natural frequencies w = 0.46, w = 0.5, w =.4 3 Response spectru value for each ode: for j-th ode) D ij (displaceent coponent of i-th degree of freedo é 0.036 0.0 0.09ù é ê 0.0 0.044 0.005 ú ëd ijù= û ê - - ú êë 0.049 0.00-0.07úû
Analysis and Algorith Manual SRSS ethod results é0.04ù Rax = ér R R ù ê 3 0.046 ú ë + + û = ê ú êë 0.05úû CQC ethod results r = r= 0.3985 r3 = r3= 0.006 r3= r3 = 0.0080 idasfea é0.046ù Rax = ér R R3 rr R r3r R3 r3rr ù ê 3 0.04 ú ë + + + + + û = ê ú êë 0.053úû Coparing the two sets of displaceents for each degree of freedo, we note that the SRSS ethod underestiates the agnitude for the first degree of freedo but overestiates the value for the second degree of freedo relative to those obtained by CQC. hus, the SRSS ethod should be used with care when natural frequencies are close to one another.
idasfea S7- S6 Sx = ( x - 6) + S 6 7-6 In response spectru analysis, results for each ode are calculated by using spectru functions. Generally, the spectru functions are coposed of the axiu values of tie history analysis results obtained fro equation (7.6). Once the daping ratiox and the seisic acceleration varying with tie u&& g ( t) are defined, solutions to equation (7.6) can be found according to the natural periods of the structure. Fig. 7-() shows how the spectru functions are deterined using a plot of the results of displaceent, velocity and acceleration vertically against the periods of the
Analysis and Algorith Manual structure horizontally. Spectru functions used in response spectru analysis are generally provided by various design codes. MIDAS generates spectru functions used in seisic analysis according to a selected design codes by siply entering the dynaic factor, foundation factor, zoning factor, iportance factor, response odification factor, etc. Since linear or log scale interpolation is used to deterine spectru values corresponding to the natural periods of the structure, it is recoended that the data in the region of rapid changes be closely defined. And the range of the spectru function ust cover the range of the axiu and iniu periods calculated fro the eigenvalue analysis. MIDAS uses the axiu or iniu value of the spectru function if the periods of eigenvalue analysis exceed the range of the inputted range of the spectru function. idasfea