The Interstorey Drift Sensitivity Coefficient in Calculating Seismic Building Frames

Transcription

1 Journal of Geological Resource and Engineering (6) - doi:.765/8-9/6.. D DAVID PBISHING The Interstorey Drift Sensitivity Coefficient in Calculating Seismic Building Frames J. M. Martínez Valle, J. M. Martínez Jiménez and P. Martínez Jiménez. Deartment of Mechanics, eonardo Da Vinci Building, niversity of Córdoba, Córdoba 47, Sain. Deartment of Alied Physics, Albert Einstein Building, niversity of Córdoba, Córdoba 47, Sain Abstract: In calculating the seismic resonse of a building, the Sanish Instructions NCSE- and CTE, aragrah.7.7 (also EROCODE 8 aragrah. art -), establish that if for all storeys the interstorey drift sensitivity coefficient, ξ, is less than or equal to., then it will not be necessary to consider the effects of the nd order ( P effects). In this aer the authors review this claim, because even for., increases of the bending moment at the ends of the columns due to the inclusion of second order effects can account for between 5% and 4% of its value for static service loads. This is significant since most adverse effects are shown in the lower height buildings (u to 5 floors) which it is recisely the range in which most of the housing stock of Sain is located. Finally, the authors delimit the coefficient for buildings of lesser height (u to 5 floors), roosing to lower it generally to.6. Key words: Seismic analysis, interstorey drift sensitivity coefficient.. Descrition of the Problem and the Building sed as an Examle The NCSE- [] and CTE [] Sanish Instructions are instructions that allow us to study dynamic effects in different structural elements. Sain is a country that has different rovinces in which dynamic studies are imortant and mandatory. In this article, it is analyzed the interstorey drift sensitivity coefficient, and consider whether or not the nd order effects are imortant in building structures to dynamic stresses. This coefficient is defined in Eurocode 8 [], but one of the conditions of whether or not to consider the effects of the nd order of coefficient, is formulated identically in Sanish Instructions NCSE and the CTE. The interstory drift sensitivity coefficient deends on the total gravity load at and above the story considered in the seismic design situation P, and the design interstory drift, evaluated as the difference of the average lateral dislacements d s at the to and bottom of the story under consideration d, which is Corresonding author: Martinez Valle J. M., Ph.D., assistant rofessor, research fields: structural dynamics, non linear finite elements for lates and shells. directly roortional, and the total seismic story shear F and interstory height h, which is inversely roortional. To analyze the roblem, we take a building tye, as shown in Fig., and study the influence of nd order effects using the corresonding theoretical model we develo in the body of the article. The examle in Fig. is, a regular rectangular concrete building of dimension 5 5 m with 5 columns arranged to give rise to 5 frames in each direction with sacing of 5.5 m. For the analysis, we adot the hyotheses that it might have, 4, 7, and floors, the loadings considered are for medium duty use, and that the location of the building is in an average Sanish earthquake zone. The section of the beams of the floor is..4 m (Fig. ). The dead load is the corresonding to a.8 m thick slab, including flooring and ceiling. Overloading is the aroriate use of ublic saces with chairs and tables. The basic seismic acceleration is. g and the coefficients of terrain features and contribution are C =.4 and K =. With these dimensions, loads and frequently used materials (concrete HA- or 5 and reinforcement steels B-5) the obtained mechanical loading

2 The Interstorey Drift Sensitivity Coefficient in Calculating Seismic Building Frames Fig. Rectangular lant of building of size 5 m 5 m with the situation of the columns. lanar structure of rigid nodes, connected rigidly at both ends. A very aroriate methodology for this deduction is based on the consideration of the equilibrium of the slice in the deformed geometry. Alternatively, we can consider nonlinear exressions of deformations of a beam in bending (valid for small deformations and moderate or large dislacements) [4]. In this aer, we choose the former for its better descrition of the roblem. In the following aragrahs we exose the non-linear theory of beams including shear deformation (first order shear deformation theory) which is the basis for the descrition of the roblem. et us consider a beam whose initial neutral axis and deformed neutral axis initial imerfections are reresented as in Fig. a. As a consequence of geometric initial imerfections, the centerline of the beam is deflected by the amount. The transverse sections are rotated at an angle. Consider a slice of this beam of length dx, unstressed axial, bending and shear, and its geometry deformed as in Fig. b, where the sloe of the centerline is given by the shift v along the axis relative to one side, and on the other is given by () We can exress the sum as a fraction loading of the dislacement v and consequently, we obtain () Fig. Simle shear model for the frame A-B under the assumtions of, 4, 7, and storeys. amounts of a ermanent reinforcement are quite normal (e.g., , diameter of the concrete reinforcing bar, for columns of 5 5 cm on two floors, or - for columns, cm in seven storeys). If we use the bending theory of thin beams, where M = E I v and establish the equilibrium of the slice into the deformed geometry, we obtain (). Geometric Stiffness Matrix of First Order of Beams of d Frames In order to consider the nd order effects (effects), we must know the stiffness matrix, including the axial force in local coordinates of a beam belonging to a Fig. Deformed configuration of a beam (left), equilibrium of a slice of the beam in its deformed configuration (right).

3 The Interstorey Drift Sensitivity Coefficient in Calculating Seismic Building Frames (4) The coefficient gives us an idea, as a ercentage, of the level of imerfections on the de-formed configuration. Solving for Q in Eq. (), and deriving and substituting in Eq. (), we reach the following differential equation. (5) In studying free transverse oscillations P =, such that the equation reduces to This is usually written as = (6) N/(E I), as 4 d v d v, whose general solution is given by the 4 dx dx following exression: A B v= cos( x) sen( x)+cx+d (7) Integration constants A, B, C, and D are rovided by our boundary conditions: vx v v, x v v, x v x (8) where v i and i, are resectively the deflection and rotation of the end i. If we articularize Eq. (7) on the values x = and x =, we obtain: A v +D A B v cos( ) sen( )+C +D (9) = B +C A = sen( ) B cos( )+C which we can write in matrix form, as follows: the form () then, if we incororate the horizontal dislacements u and u from the beam ends, and we use the usual notation in the theory of structures in which dislacements and rotations of the ends of the beam are exressed by ex d u ey d v e = () ex u d ey d v e the equation of the deformed beam is: A B v= cos( x) sen( x)+c x+d= A cos( x) sen( x) = x B C D ex d ey d cos( x) sen( x) e () x ex d ey d e According to Eq. (), the matrix written as: =, can be -ω (-cos(ω )) ω sin(ω ) ω (-sin(ω )+ω cos(ω )) ω (-+cos(ω )+ω sin(ω )) ω sin(ω ) -cos(ω ) -cos(ω )-ω sin(ω ) -sin(ω )+ω cos(ω ) ω (-cos(ω ) -ω sin(ω ) ω (sin(ω )-ω ) ω (-cos(ω ) -ω sin(ω ) (-cos(ω ) -cos(ω ) sin(ω )-ω (4) () If we suose that the inverse matrix is artitioned in If the loads at the ends of the beam along the x-axis x are designated by, the matrix k viga can be deduced from the bending moment exressions ( M ), shear force (Q) and axial force (N) by

4 4 The Interstorey Drift Sensitivity Coefficient in Calculating Seismic Building Frames ex ex E A ex ex? d d ) ey dv d v (Q) x= =- = [ N E I ] x = dx dx + (5) ( M ) x= =-M =EI( v ) x= ey dv d v (Q) x= = = [ N E I ] x = dx dx (6) ( M ) x= =M =E I( v ) x= Substituting in the above equations the derivatives of the deflection v, calculated from Eq. (), and articularizing at x equals zero or, as aroriate, we obtain: Relacing the four submatrices of order, in the above Eq. () and oerating, we obtain the equation of beam loads movements, viga = k vigad viga in local coordinates: m m ex ey e ex ey e ij dex dey = kviga e (7) dex ey d e with k viga equal to (8) or in a more comact form Where k viga k viga viga viga (9) and is the beam stiffness matrix in local coordinates, including the influence of the axial force N. et us note that we have identified, for the differentiation of linear stiffness matrix k viga, by the stroke located on k, and d which deends, Via, on the value of the axial force on the beams (N). Aarently, the exression of the stiffness matrix of the beam exressed in Eq. (8) k viga, is quite different from the known k viga, without the inclusion of the axial force. However, we can find a similar exression when we consider that ω -cos(ω ) tg = () sin(ω )

5 The Interstorey Drift Sensitivity Coefficient in Calculating Seismic Building Frames 5 and if we call to () where c and s are adimensional functions, called stability functions ω -sin(ω ) c= ; sin(ω )-ω cos(ω ) () which were originally derived by undquist and Kroll [5], and later develoed by Merchant [6] under different methodology and in another context, Fig. 4. We refer the new original aroach resented here [7], because we want to show the identity of rocedures and goals achieved over time, even though at times the results aear to be different and have been obtained by using very different methodologies. Oerating conveniently we found that the matrix Eq. (6), for a beam rigidly connected at both ends and belonging to a D frame, is written as e = k d e + k d e e = k d e + k d e () or in matrix form being k = ; k =( E A k ) T = E A e e E I 6 E I E A = k k e d (4) k k e d 6 E I 4 E I E I 6 E I 6 E I E I 4 k = (5) E I 6 E I 6 E I 4 E I Comaring these exressions with the known k ij, it is clear that the functions are simly multilier factors of the coefficients of the stiffness matrix of a beam without axial forces and can conveniently be exressed as functions of the relationshi between axial force N and the Euler critical load Ncr =, as shown in Fig. 5. Note that all are for N =, so that = for N =. k viga. Geometric Stiffness of a Column Belonging to Simle Shear Model If the beam is one of the grous of carriers of a frame modeled as a simle shear model, Fig. 6, and we take the global axes of the figure. Fig. 4 Stability Functions. Fig. 5 Grahical reresentation of the functions. k viga

6 6 The Interstorey Drift Sensitivity Coefficient in Calculating Seismic Building Frames Fig. 6 Simle shear model for a frame of storeys. then Eq. (8) can be written as (6) Therefore, we obtain (7) As for N =, the functions are referred indeterminate, and we refer to develo them using Taylor series and limiting the develoments to the first two terms, we obtain (8) If we roceed similarly for all matrix elements, the second addends constitute the geometric stiffness matrix of the first order. If we increase the terms of the develoment, the third summands constitute the

7 The Interstorey Drift Sensitivity Coefficient in Calculating Seismic Building Frames 7 geometric stiffness matrix of second order and so on. For the shear efforts at beam ends we have (9) Thus, the stiffness of the column of the media set of a frame modeled as a simle shear model is () where is the stiffness of each of the columns forming the grou and N is the area weight condition of all floors above the considered column. As already discussed, marks the level of initial imerfections, in ercentage terms, relative to the deformed column. To circumvent the aroach that involves limiting the Taylor series exansion of to the first two terms, and also based on the reresentation of Fig. 5, we can choose to use a very aroximate analytical exression for. If we call the function is given by, in which case the stiffness of the media set of columns of a frame modeled as a simle shear model is () Although the differences obtained by taking Eq. () or () are not significant, we have oted for the latter as being more accurate, even slightly, and is imlemented in the rogram for use with equal ease. 4. Results of Calculating the Building Taken as an Examle The following cases have been solved with the hel of a rogram written in Matlab (following the usual methodology for dynamic calculation of structures) whose key stes are: Calculation of the stiffnesses of the columns of the media set of a frame by Eq. (). Calculation of the interstorey drift sensitivity coefficients for each floor according to the equation: () where all the variables have been defined reviously; the subscrit refers to the stiffness and length of the column or grou of columns resectively. Calculation of the natural frequencies of the structural model. Calculation of the maximum values of the variables of the equations resulting from de-couling the initial system of differential equations, using the resonse sectrum roosed by the Sanish instruction NCSE. Calculation of the maximum values of the dislacements by the method of least squares. These maximum values of the dislacements will allow us to calculate the shear stresses and the resulting bending moments. Case A Two Storey Tye Building For the ratio of stiffness suosed in aragrah, the bending moments at the to of the column in the ground floor and at the end of the lintel for static loads are () The Interstorey Drift Sensitivity Coefficients,, and maximum deflections are in Table. The stiffness k of the ground floor columns grou is k = 7.8 KN/m and the increases of shear force and bending moment, as a result of considering the couling of axial bending, are (4) (5) The latter increase in the bending moment on the column reresents an increase of 6.% from the value of the moment due to gravity loads, and an increase of

8 8 The Interstorey Drift Sensitivity Coefficient in Calculating Seismic Building Frames Table Interstory drift sensitivity coefficient for story frame. Interstory Drift Sensitivity Coefficient ξ =.75 ξ =.9 Without consideration to axial force Couling of bending moment and axial forces % comared with the sum of the value of the moments due to static loads more dynamic loading. However, increases incurred as a result of considering a tyical level of initial imerfections are not significant. Case B Four Storey Tye Building For the ratio of stiffness suosed in aragrah, the bending moments at the to of the column in the ground floor and at the end of the lintel for gravity loads are Table Interstory drift sensitivity coefficient for 4 story frame. Interstory Drift Sensitivity Coefficient ξ =.79 ξ =.79 ξ =.86 ξ4 =.9 Without consideration to axial force Couling of bending moment and axial forces Case C Seven Story Tye Building For the ratio of stiffness suosed in aragrah, the bending moments at the to of the column in the ground floor and at the end of the lintel for gravity loads are (9) (6) The Interstory Drift Sensitivity Coefficients,, and maximum deflections are in Table. The stiffness k of the ground floor columns grou is k = 4,555 kn/m and the increases of shear force and bending moment, as a result of considering the couling of axial bending, are The Interstory Drift Sensitivity Coefficients,, and maximum deflections are in Table. The stiffness k of the ground floor columns grou is k = 45,999 kn/m and the increases in bending moment and shear forces, as a result of considering the couling of axial deflection, are (4) (7) (8) This latest increase of the bending moment on the column reresents an increase of.59% from the value of the moment due to gravity loads, and an increase of 5.7% from the sum of the value of the moments due to gravity loads more dynamic loading. However, increases incurred as a result of considering a tyical level of initial imerfections are not significant. (4) This latest increase of the bending moment on the column reresents an increase of 8.8% over the value of the moment due to gravity loads, and an increase of.7% from the sum of the value of the moments due to static loads more dynamic loading. However, increases incurred as a result of considering a tyical level of initial imerfections are not significant. Case D Ten Story Tye building. For the ratio of stiffness suosed in aragrah, the

9 The Interstorey Drift Sensitivity Coefficient in Calculating Seismic Building Frames 9 Table Interstory drift sensitivity coefficient for 7 story frame. Interstory Drift Sensitivity Coefficient ξ =.9 ξ =. ξ5 =.79 Without consideration to axial force ξ =.5 ξ4 =.9 ξ6 =.86 ξ7 =.9 Couling of bending moment and axial forces bending moments at the to of the column in ground floor and at the end of the lintel for gravity loads are (4) The Interstory Drift Sensitivity Coefficients,, and maximum deflections are in Table 4. The stiffness k of the ground floor columns grou is k =,5 KN/m and the increases in bending moment and shear forces, as a result of considering the couling of axial deflection, are (4) (44) This latest increase of the bending moment on the column reresents an increase of.87% from the value of the time due to gravity loads, and an increase in.5% from the sum of the value of the moments due to gravity loads more dynamic loading. However, increases incurred as a result of considering a tyical level of initial imerfections are not significant. 5. Dimensioning Interstory Drift Sensitivity Coefficient In the seismic calculation of a building, Sanish Instructions NCSE- and therefore, the CTE, Table 4 Interstory Drift Sensitivity Coefficient for Story Frame. Interstory Drift Sensitivity Coefficient ξ =. ξ =.9 ξ =.6 ξ4 =.6 ξ5 =. Without consideration to axial force ξ6 =.5 ξ7 =.7 ξ8 =.79 ξ9 =.86 ξ =.9 Couling of bending moment and axial forces establish in aragrah.7.7 that: As long as the collase of the head of the building does not exceed two er thousand of the height, it is not necessary to consider the nd order effects. On the other hand, it will not be necessary to consider the nd order effects, in line with that set by Eurocode Nº 8 (Paragrah- Part -), if for all storeys the Interstory Drift Sensitivity Coefficient is less than.. As we can see, what is actually limiting the destabilizing moment is that the column is less than % of the stabilizing moment. Because the criterion of the % limit seems to be arbitrary, we roose to relace it with another limit that is more in line with the structural reality. If we assume a square section column of side b, for frequently used materials and for the dimensions set out in aragrah, the relationshi between the critical load and the axial calculation is, so that we obtain N calc =.75 N crit for b =. m, and for b =.5 m (corresonding to buildings of lesser heights), we have N calc =.55 N crit. Accordingly, we adot

10 The Interstorey Drift Sensitivity Coefficient in Calculating Seismic Building Frames N calc <.75 N crit (45) nder this assumtion, coefficient, defined in section,, gives (46) Keeing the definition for the interstory drift sensitivity coefficient and by taking into account Eq. (7), we can write From which we deduce, we obtain. So, (47) and as (48) By virtue of Eq. (46), it must be satisfied that.75, i.e.,.75. From which.6? we deduce.667 (49) which is consistent with the results obtained in the ractical cases A and B of section Conclusions In the seismic calculation of a building, Sanish Instructions NCSE- and thus, the CTE, establish in aragrah.7.7 that: As long as the collase of the head of the building does not exceed two er thousand of the height, it is not necessary to consider the effects of nd order. On the other hand, it will not be necessary to consider the nd order effects (s effects), in line with that set by Eurocode 8 (Paragrah- Part -), if for all storeys the interstory drift sensitivity coefficient fulfills: (5) We show in this discussion, based on the current state of knowledge, it is not justified to ignore the nd order effects ( P effects): () Because it is not justified that the sectral modal analysis be simlified, because a high-level rogram such as MATAB can very easily address a multimodal sectral analysis study for frames, the simle model of shear or associating the nodes of the structure of the inertial roerties of the frame beams and erforming a calculation of stiffness. () Because the inclusion of nd order effects (effects), does not add any concetual effort nor comlicate the calculation. Additionally, if we follow the line of thought of Bazant [8] and truly think that the equilibrium, static or dynamic loading, occurs in the deformed geometry, it is logical that the calculation includes this. () Because we show that although the Interstory Drift Sensitivity Coefficient does not reach the value of., increases of the bending moment at the ends of the columns resulting from the refinement of the calculation including the second order effects, can account for between 5% and 4% of its value for static service loads. (4) Because most adverse effects are shown in lower height buildings (u to 5 floors), which is recisely the range in which most of the housing stock of Sain is located, to be in the highest limit allowed by the various lanning regulations (PGO). (5) As in section 5, we limited the Interstory Drift Sensitivity Coefficient to.667 for buildings of lower height (u to 5 floors), so we roose that it is generally lowered to.6 instead of. as roosed in Eurocode 8 and the Sanish Instruction NCSE-. References [] NCSE-: Norma de Construcción Sismorresistente, Parte General y Edificación. Es-aña, Ministerio de Fomento, 4. [] Código Técnico de la Edificación. Esaña, Ministerio de Fomento, 4. [] NE ENV 998-: Euroean Committee for Standardization, Eurocode 8. Design of Structures for Earthquake Resistance. Part : General Rules, Seismic Actions and Rules for Buildings, Brussels, 5. [4] ivesley, R. K Matrix Methods of Structural Analysis. Oxford: Pergamon Press.

11 The Interstorey Drift Sensitivity Coefficient in Calculating Seismic Building Frames [5] undquist, E. E., and Kroll, W. D Extended Tables of Stiffness and Carry-Over Factor for Structural Members under Axial oad. angley Field, VA.: angley Memorial Aeronauti-cal aboratory. [6] Merchant, W The Failure oad of Rigidly Jointed Frameworks as Influenced by Stability. The Structural Engineer : [7] Martínez-Jiménez, J. M., and Martínez-Valle, J. M.. Diseño y Cálculo Elástico de los Sis-temas Estructurales, Tomo II: Inestabilidad y andeo de estructuras, líneas de influencia y cálculo dinámico, [8] Bazant, Z. P., and Cedolin,. 99. Stability of Structures (Elastic, Inelastic, Fracture, and Damage Theories), Oxford: Oxford niversity Press.