due to wind or earthquake are important in sizing ... tructures for Seismic and Wind Load cases ..... First Street, Santa Ana, CA and was built using modern.
SEAOC 2012 CONVENTION PROCEEDINGS
Structural Optimization in Steel Moment and Braced Frame Structures for Seismic and Wind Load cases Rakesh Pathak, Ph.D., Software Research Engineer, Allen Adams, S.E., Senior Product Manager, Seth Guthrie, S.E., Technical Support Manager, Bulent Alemdar, Ph.D., P.E., Senior Software Research Engineer, Raoul Karp, S.E., Director of Product Management, Bentley Systems Inc., Carlsbad, CA Abstract This article presents a virtual work based optimization method for steel moment and braced frame structures subjected to wind and seismic loading. Virtual work based structural optimization concepts are generally used in limiting drift and building period by redistributing the steel in a given lateral force resisting system so that it resists the forces and displacements more efficiently. Limiting lateral drift and period is important in reducing damage occurring in structural and nonstructural elements, lateral accelerations and also in reducing second-order P-Delta Delta effects. If this method is used with complete understanding, rstanding, a cost reduction of structural steel framing in the building can be achieved in a couple of design and analysis iterations. This paper also presents a brief literature review on virtual work based optimization, the theoretical concepts behind tthis procedure and how it may be effectively used on design projects. The procedure has already been implemented in Bentley’s RAM Structural System Frame Drift software which is used for example studies.
Introduction The building codes mandate structural al designers to control the drift due to wind and seismic loading. This is done by providing sufficient amount of stiffness to the lateral force resisting system. So typically if drift serviceability controls in a moment frame or braced frame structural design then the engineer ends up modifying member sizes. This always happens in the design of tall buildings where lateral loading due to wind or earthquake are important in sizing the members. The sizing of members for optimum design is an iterative procedure. The iterative nature of the problem makes it challenging as it may not be obvious which
members need bigger section size (i.e. more steel) or vicevice versa (i.e. less steel). Also, just adding or removing the same amount of steel from two different members m in a structure may not have a similar impact to the drift or natural period. This is why it is critical to identify members where adding more steel has more impact on control parameters. A trial and error procedure may work but may not result in an optimal distribution of steel in the structure. In this scenario, virtual work based concepts and tools can be utilized relatively efficiently to achieve an optimal distribution of steel in a structure. The use of virtual work to optimize structural design ign is not new but its potential is much underutilized in engineering practice. The use of formal optimization techniques like topology optimization on a design project are computationally and theoretically challenging in themselves and may not always be cost effective. The virtual work concepts are familiar to engineers compared to more formal optimization techniques which so far have been more prevalent among mechanical or aerospace engineers or to engineers with advanced degrees. The simplicity of the virtual work method for optimizing steel weight in a given structure with respect to a given control parameter is, in the authors opinion, opinion unparalleled. This paper presents these concepts along with simplified example problems. These concepts are later applied a to a case study building which is designed, analyzed and optimized using RAM Structural System (RSS) software (2012). The paper also presents a brief history of application of strain energy and virtual work to structural engineering problems.
Literature Review The concept of strain energy and virtual work can be found in any structural mechanics text book. The virtual force method is generally presented in the context of deriving
1
displacements due to a set of forces on beams, columns or trusses. These displacements are presented as a sum of strain energy contributions from various sources of member flexibilities. The text books generally do not explicitly emphasize the importance of these sources of flexibility which play a critical role in structural optimization. Vilvasakis and DeScenza (1983) presented the virtual work method application on a 10 story framed system subjected to lateral loading. They used an iterative approach, where in each iteration they modified member properties based on the ratio of maximum displacement at control point to the target displacement at that point. Baker (1990) presented a sizing technique for multistory steel frame building using Lagrange multiplier constraint. In a second paper Baker (1991) presented the theoretical basis for stiffness optimization methods for frame systems. In both the papers presented by Baker the key idea applied in weight optimization is to obtain the same energy density for all the structural elements. This however is based on the assumption that resisting moment of inertia in the direction of loading can be defined as a linear function of the cross-sectional area of the member. Baker in his second paper also presented charts of moment of inertia vs. Area for various AISC W-sections used in moment frames. It is noted that for most of the W-sections moment of inertias can be written as a linear function of their crosssectional areas. The only exception to the case was W14 which was only linearized in certain regions and presented to be used in that form. Henige (1991) shows how virtual work and strain energy principles can be applied to limit natural periods and achieve optimum member sizes. The proposed approach targets high rise design where limiting accelerations become critical over structural drift. Charney (1990a and 1990b) developed postprocessor software named DISPAR which utilizes virtual work and strain energy concepts. The DISPAR stands for DISplacement PARticipation and was developed for practicing engineers to automatically resize and reanalyze a given structure. The reanalysis in DISPAR works on the assumption that member forces do not change significantly when members are resized. To include the exact forces resulting due to member resizing, the program requires results from updated structures analysis. The unique and powerful feature of this postprocessor is the capability to include true joint flexibility when optimizing for drift. The true joint deformation contribution is generally missing in the analytical models unless some kind of mechanical joint model is used. Charney (1993) addresses how an engineer may achieve economy of steel frame buildings by understanding member contributions. The author introduced the idea of sensitivity indices which is defined as member displacement contribution per unit volume. The sensitivity index is similar to energy density discussed in Baker (1990 and 1991) and is defined as change of displacement contribution per unit change in volume of the member. The paper discussed various methods on how one may identify
2
members which need resizing based on their sensitivity index. The paper presented practical example and techniques on how one may achieve a lighter weight truss and moment frame in a few design iterations. The virtual work methods discussed above consider only one constraint when members are optimized. However, a resizing done during optimization may disqualify a member against another design criterion which may be thought of as another constraint. To account for multiple constraints using above methods, the engineer has to update the member size, and then reanalyze and redesign the structure manually considering other constraints in the problem. Chan and Grierson (1993) considered this problem in their research and developed an automatic procedure which reanalyzes and redesigns repetitively considering multiple drift constraints for tall steel buildings. In his later research Chan et al. (1994, 1995 and 1996) addressed optimization conditions for tall steel building subjected to drift and strength constraints. Their work used extensive use of Lagrange multipliers and they noted that in order to optimize a structure for multiple drift constraints, each member must contribute the same weighted amount of virtual strain energy densities in resisting the drifts at control points. This observation is similar to observations of Charney and Baker which targets to achieve same energy density for all the elements in a structure in order to achieve an optimum structural weight. This optimum strain energy distribution however is only achievable in theory because in real practice there are many factors which restrict member sizes and design.
Virtual Work and Strain Energy Concepts This section presents theoretical concepts of virtual work which may be applied to structural optimization problems. The virtual work method is one of the ways to calculate displacements in a structure due to a certain loading also often referred to as real loading to differentiate from the virtual loading which actually does not exist. The virtual work method which is used in computing displacements is called principal of virtual forces and it requires the computation of internal and external work done in the presence of real loading and a virtual force at the location and direction of interest. The virtual force is assumed to be present when the real load is applied and often the magnitude of virtual force is set to unity. This is why sometimes principal of virtual forces is also referred to as unit force method. This paper presents virtual work optimization concepts using few examples below. In addition, one may also refer to Charney (1993) for some application examples to truss and frame problems.
A stack consisting of two steel columns of square crosssection is subjected to lateral loads as shown in Fig. 1. The example considers three optimization problems for a fixed volume of material and shape of the cross-section: (1) minimize horizontal drift at the top, (2) minimize inter-story drift, and (3) minimize the fundamental period of vibration. The column material is considered mass-less and for vibration analysis lumped unit mass is considered at the column tops. We ignore shear deformations in the discussion here for the sake of simplicity. To minimize drift δ1 we will formulate the drift in terms of internal strain energy of the system. One may note here that drift may directly be computed by matrix analysis approach but by using virtual work and strain energy we get a more detailed picture of the structural behavior. We assume that a virtual load Ѵ in the direction and location of δ1 is present when the real loads P1 and P2 are applied and the structure is in equilibrium. The external work done by this set of forces is given in Eq. (1). The internal work can be derived as a sum of work done by virtual forces on real strains and work done by real forces on real strain as shown in Eq. (2). The subscripts ‘r’ and ‘v’ are used for denoting real and virtual quantities respectively. The stresses and strain in the equation are one dimensional as we are ignoring shear deformations and the loading generates only flexural deformations. A closer look at Eq. (1) and (2) reveals that there is an obvious one to one correspondence between realreal terms and virtual-virtual terms. We get Eq. (3) when we equate the virtual terms from (1) and (2). This equation plays a central role for drift minimization problem at top of column stack. Eq. (4) rewrites right hand side of Eq. (3) in-terms of internal forces which is another friendlier form of the equation for frame analysis. Two things to note in Eq. (4) are: (a) each integral is the displacement contribution of a column due to flexural deformations and (b) this problem is statically determinate so drift is only a function of moments of inertia. The moments of inertia of the two columns change when the material is removed uniformly from one column and added to another. This is a very simple example where one can easily see how drift varies when material is redistributed between two columns. Table 1 presents loading and member properties used in this example. Table 1: Member properties for column stack P1 (kips) P2 (kips) E (kips/in2) H (in) VT (in3) 24000 120 10 15 30000 Eq. (4) is solved in a loop using different volumes of material in top (TC) and bottom (BC) columns. We start with 4000 in3 and 20,000 in3 steel in top (V1) and bottom (V2) columns respectively and increase the ratio of volumes (V1/V2) from 0.20 to 4, hence keeping the total volume (VT) as constant. Fig. 2-a presents drift δ1 and contributions to this drift from
top (δ11), bottom (δ21) columns plotted against volume ratios (VR) of the two columns. Fig. 2-b presents plots of sensitivity indices of top (SI1) and bottom (SI2) columns plotted against VR of the two columns. The sensitivity index is defined as displacement contribution per unit volume of the material. The minimum drift of 0.249” occurs at the VR of 1.591. This VR is the same at which the sensitivity index of the two columns intersect each other i.e. both columns displacement contribution per unit volume are equal. It must be noted that minimum displacement does not happen when displacement contributions from the two members are equal but instead when sensitivity indices for two columns are equal. This is an important observation and a mathematical proof of this is given in Appendix A.
Figure 1: A stack of two columns
=
1 1 + + Ѵ 2 2
1
2
=
1
2
+ +
+
(1)
(2)
3
Ѵ =
+
(3)
1 % !" !" = ! Ѵ & #$ % !" !" + !' #$ &
(4)
δ1 δ11 δ21
δ (in) 0.5
0 0
1
2 V1/V2
3
4
Figure 2-a: Drift at top of column stack Vs VR 1.5 SI1 Sensitivity Index (SI) x 1000
=
1 1 + + Ѵ − Ѵ
2 2
1 % !" !" − = ! Ѵ & #$ % !" !" + !' #$ &
1.5
1
other steps for computing internal stresses or forces are the same except this time the virtual stresses include the effect of negative virtual loading. The final formula for inter-story drift in terms of individual member flexure contribution is shown in Eq. (6) which is exactly the same as Eq. (4) except the moments in two columns due to virtual loading are different.
(5)
(6)
Fig. 4-a present’s the plot of inter-story drift against the VR of top and bottom columns. The plot also includes the contribution of top (δ1(1-2)) and bottom (δ2(1-2)) columns to inter-story drift (δ1-δ2). It is noted from the results that a VR of 1.155 gives the minimum inter-story drift the value of which is zero. However, for inter-story drift this minimum happens when the displacement contributions from two columns are equal in magnitude and not when their sensitivity indexes are the same. This observation is contrary to the one we noted for drift minimization earlier. The plot of sensitivity index with respect to VR is given in Fig. 4-b. The two sensitivity indexes are found to be equal at the VR of 1.101 which is very close to the point at which inter-story drift is zero.
SI2 1
0.5
0 0
1
2 V1/V2
3
4
Figure 2-b: Sensitivity index Vs VR To compute member contributions to inter-story drift we again do a virtual work analysis but this time we apply virtual loads as a couple as shown in Fig. 3. We rewrite the equation (Eq. 1) for external work done in order to accommodate work done by additional virtual force and this is shown in Eq. (5). As may be seen, the last two terms of this equation represents the work done by virtual loads causing inter-story drift. The
4
Figure 3: Virtual load pattern for inter-story drift
* ) ⋮ *, * ) ⋮ *,
1.5 δ1-δ2 1
δ1(1-2) δ2(1-2)
δ (in)
0.5 0 0
1
2
3
4
-0.5 -1 -1.5
Figure 4-a: Inter-Story drift Vs VR
0.5 Sensitivity Index (SI) x 1000
SI1
0.3
0.2
0.1
0 1
2 V1/V2
3
4
Figure 4-b: Sensitivity index Vs VR
A similar virtual work approach may be taken for fundamental period optimization. We know for a lumped mass linear structural system the eigen vectors and eigen values are associated using the relationship shown in Eq. (7). The eigen values (ω21…. ω2N) correspond to structural frequencies and eigen vectors represent mode shapes in which the structure tends to vibrate if its eigen frequency is excited. * ) ⋮ *,
⋯ *, 0 ⋯ ⋱ ⋮ /) ⋮ ⋱ ⋯ *,, 12 ⋯ * ⋯ *, ⋱ ⋮ / = ) ⋮ *, ⋯ *,,
0, ⋮ / 0,,
A mode shape matrix may also be viewed as a set of deflections for a given structural system and a corresponding force matrix causing that deflected shape may be obtained by multiplying mode shape matrix by global stiffness matrix as shown in Eq. (8).
⋯ ⋱ ⋯
*, ⋮ / *,,
⋯ ⋱ ⋯
*, ⋮ / *,,
(8)
Eq. (9) shows the external work done using the set of forces from Eq. (8) and as can be seen this is equal to the left hand side of Eq. (7) divided by 2. If the mode shapes are normalized we know from basic vibration theory that the generalized mass matrix (first three terms on the right hand side of Eq. (7)) is an identity matrix. Hence, it can also be said that frequency of a system in a particular mode is equal to square root of twice the external work done by the forces due to that mode shape as shown Eq. (10).
SI2
0
⋯ 0 ⋱ ⋮ / ⋯ , ⋯ 0 ⋱ ⋮ /
⋯ 4,
⋯ , ⋱ ⋮ / ) ⋮ , ⋯ ,, 0 ⋯ 0, * ⋱ ⋮ /) ⋮ =) ⋮ 12 ⋯ 0,, *,
V1/V2
0.4
⋯ *, ⋱ ⋮ /) ⋮ ⋯ *,, 0 ⋯ *, 4
⋱ ⋮ /) ⋮ ⋯ *,, 0
(7)
) ⋮ 0
⋯ ⋱ ⋯
1 = ) ⋮ 2 ,
4 ) ⋮ 0
⋯ ⋱ ⋯
0 ⋮ / , ⋯ , * ⋱ ⋮ / ) ⋮ ⋯ ,, *,
0 62 ⋮ / = 5 ⋮ 4, 0
(9)
⋯ ⋱ ⋯
0 ⋮ 7 62,
(10)
The derivation so far has included all the structural frequencies but for sake of simplicity we now only concentrate on the lowest frequency (ω1). We do this with respect to our column stack example to demonstrate how frequency is associated with internal strain energy. However, the discussion here on is equally applicable to other modes, as well as other structures. We derive internal work equation using the member forces resulting due to external load vector generated for the first mode (column 1 from matrix in Eq. (8)). Eq. (11) presents the internal work equation in terms of member moments. The subscript ‘m’ represents member
5
;
% 8 !" : : > ! A : #$ : = & @ : 4 = :=
% 8 !" @ : : =+ ! @ #$ ? 9< & B =
2C
% 8
!" ! #$ DE G
% 8 !" + F& ! #$ F&
(11)
15 10 5 0 0
(12)
1
2
3
4
V1/V2
Figure 5-a: Fundamental frequency square Vs VR 2 SI1
(13)
We use Eq. (12) to compute square of frequency (ω2) at different VR of materials in column stack and these are plotted in Fig. 5-a along with top (ω21) and bottom (ω22) column contribution. It is noted that the maximum fundamental frequency occurs at the VR of 0.881. A look at the sensitivity indices plot (Fig. 5-b) shows that sensitivity indexes are equal at the VR of 0.881 i.e. at the location of maximum fundamental frequency. The frequency contributions however are equal at a different VR of 0.838. In the three examples we discussed so far we observed that the optimum responses we were interested in happened at a different distribution of material in the column stack i.e. the minimum drift at VR of 1.591, the minimum inter-story drift at VR of 1.155 and minimum fundamental period at VR of 0.881. There are obvious reasons for these differences and the challenge for the engineers comes in when they have to account for multiple control parameters like these in the structural analysis and design of a real problem. The above three control parameters may be common in design of tall buildings which are susceptible to larger drift, inter-story drift and accelerations due to high wind.
6
ω2 ω21 ω22
20
Sensitivity Index (SI) x 1000
% 1 % 8 !" 8 !" = ! + !' 2 & #$ #$ &
25
ω2 (radian2/sec2)
force due to calculated modal loads. Since internal work done must be equal to external work done, we use this to formulate the fundamental frequency and time period in terms of individual member contributions as shown in Eqns. (12) and (13) respectively.
SI2 1.5
1
0.5
0 0
1
2
3
4
V1/V2
Figure 5-b: Sensitivity index Vs VR
Analysis Program and Case Study This study uses RAM Frame and RAM Frame Drift modules [13, 12] for the analysis of a case study structure. These programs use virtual work and strain energy based approaches and are ideal for human loop optimization process. The phrase “human loop optimization” is defined by Charney (1993) as a process in which design engineer is interactively involved in making decisions based on experience and engineering judgment. The drift module also has graphical tools and reanalysis capabilities which are helpful in understanding member behavior and how member resizing impacts the control displacement. The module however assumes that member resizing does not impact forces in them. One may always reanalyze the structure with the new member size by switching to RAM Frame analysis module. The RAM Frame program is very powerful and allows automatic creation of multiple seismic and wind load cases for analysis and design.
It also allows generation of multiple virtual load cases which can be associated with any real load cases later in the drift module. The virtual loads can be applied in different ways in order to understand different modes of behavior. The virtual loads may be applied to study (1) drift at a certain location in the structure, (2) inter-story drift, and (3) torsional rigidity of the structure for asymmetric buildings etc. The program presents detailed reports where member contributions from structural elements are always split into their six flexibility components i.e. axial, shear major, shear minor, flexure major, flexure minor and torsion. This break up of member contribution into six different components helps identify which flexibility component needs to be scaled up or down in order to achieve the control drift. For example, if flexural component of a member is a higher contributor to displacement at control point relative to other components then one may select section size with higher bending inertia to reduce the control displacement. One can see that resizing using this approach is very efficient as it conveys which component needs to be modified. The component contribution and total member contribution can also be viewed graphically. In the graphics view, the program colors the members based on their relative contributions. This visual representation helps the user to understand how the structure behaves overall and which members need resizing. The structural model of Xerox Center building is selected for the demonstration of practical usage of virtual work in structural optimization. The building is located at 1851 East First Street, Santa Ana, CA and was built using modern architectural style in 1988. The building has 16 floors and is used for office purposes. The building is defined as high rise structural type and lateral load resisting system consists of four special moment frames, two in each orthogonal direction. Fig. 6 depicts an image of the Xerox Center building and corresponding structural model as viewed in RSS. We demonstrate the use of virtual work on the framing running in the east-west direction. This is the framing which has five bays and is shown on the right most side in the RSS model shown in figure. The drift analysis of the structure is performed using equivalent lateral force procedure for seismic loads as prescribed in ASCE 7-10. The frame member sizes are used from the actual drawings. The building plan and elevation are regular except for one moment frame in the E-W direction which has a few frame beams not included at 1st and 2nd levels as may be seen in the elevation view of Fig. 10. The story heights for 1st and 2nd levels are 12.5’ and 17’ respectively. The next thirteen typical levels are 13.5’ each and roof is 14’ in height. The gravity and live loads are resisted by beams and girders which support a composite
floor deck. A surface dead load of 40 psf and reducible live load of 50 psf is assumed on typical levels.
Figure 6: Xerox Center, Santa Ana, California
According to the hazard maps from USGS the buildings short period and 1-second mapped spectral acceleration parameters are Ss = 1.390 g and S1 = 0.496 g. Due to lack of information on the soil characteristics at the location we assume a site class D. The seismic design category for the building is II and importance factor is 1. As the lateral load resisting system of this building is special steel moment resisting frame, its response modification factor (R) is 8 and deflection amplification factor (Cd) is 5.5. These details are provided as an input for this model and RAM Frame program automatically generates the loading pattern in the E-W direction as shown in Fig. 7. The diaphragms at all the levels are modeled as rigid except for the bottom most level which is considered flexible in the analysis. The joint flexibilities are included via centerline modeling. The P-Delta option using gravity loads is also turned on in the program during the analysis. We are only interested in the drift results for this study so we note these at control points at various levels. These control points are the same at all the levels and are defined at the center of mass co-ordinates of the top most level. It is noted that center of mass co-ordinate do not change much over the height of the building at all the rigid diaphragm levels. A horizontal drift plot over the height of the building is shown in Fig. 8. The bottom story is considered flexible so its drift is interpolated from 2nd and ground level drifts. An important point to note here is that the gravity loading and some of the other conditions are assumed for the analysis due to lack of availability of actual data used on the building. The results hence obtained do not reflect any correlation with the true buildings analysis or design. The main intent of using this building for example is to demonstrate the concepts on a
7
problem which depicts real world dimensions and member sizes.
250
Height (ft)
200
Building Height (ft)
250 200
100
150
50
100
0 0
1
2
50
3
4
5
6
Drift (in)
Figure 8: Building Drift in the E-W direction
0 20
40
60
80
Force (kips)
Figure 7: Lateral loading in the E-W direction
We use this building for two example studies using RAM Frame Drift module. (1) Fig. 9 plots the inter-story drift ratio over the height of the building. As may be observed that there is an increase in inter-story drift ratio between 2nd and 3rd levels using original sizes. We investigate the frame elements contributing to the inter-story drift ratio at this level and how it may be reduced, and (2) Also, it may be observed from Fig. 9 that the example building satisfies the drift limitation in the E-W direction by a huge margin so in the second problem we study the options to take out the material to reduce the structural weight and still satisfy the drift criteria. It is realized that some member’s code check for steel standard and seismic provisions may not work out after we change the sizes but to keep things simple we limit ourselves to drift and analysis modules only. The RAM Frame program however provides easy toggling between the analysis, code check and drift modules. The program provides features which let user easily change the size of a member and check them against the various code provisions. One should refer to Bentley’s RAM Frame Steel Post Processor manuals [14] for more detail on this.
To study the inter-story drift at 2nd and 3rd levels we create a virtual load case by applying a +100 kips at the center of mass location of 3rd level and -100 kips at the center of mass location of 2nd level in the E-W direction. We analyze the lateral load and virtual load case together in RAM Frame and then switch to the drift module. The drift module is a postprocessor to member force results obtained in the analysis module. It require user to create a real and virtual load pair for the assessment of displacement contribution of various members to the control parameter. In this particular case, the control parameter is inter-story drift between 2nd and 3rd levels. DRIFT LIMITATION 250
ORIGINAL
ITERATION 1
200 Height (ft)
0
8
150
150 100 50 0 0.000
0.001
0.002 0.003 Inter-Story drift ratio
0.004
Figure 9: Inter-Story drift ratio using original and modified member sizes
Figs. 10(a) shows the original member sizes on the moment frame in the E-W direction and 10(b) shows normalized displacement contribution per unit volume of members towards inter-story drift. As may be seen from the color scaling the beams and columns at level 2nd and 3rd are contributing the most to inter-story drift. This contribution tends to fade out as we go further up the levels; overall this is
an expected response. The Fig. 10(b) is not just depicting the normalized member contribution per unit volume but also tells the order in which members are working towards resisting that inter-story drift. It is observed that 3rd level column in the 4th column stack from the left is doing the most work followed by the outermost beam at the 2nd level. Also, we note that there is an uneven distribution of work done by these members and in order to get an optimum solution we must have a uniform distribution of work load. To achieve this we follow the rule of penalizing the members which do less work and rewarding the members which do more work. By penalizing a member we refer to material being taken from it and then giving it to the one doing more work i.e. rewarding that member. This is done to create a balance of work but there may be constraints involved. For example, for the column doing the most work in this example case we increase its size from W14x283 (A = 83.3 in2, I = 3840 in4) to W14x311 (A = 91.4 in2, I = 4330 in4) but one may also want to change the size of the column above it to be W14x311 in order to maintain the uniformity in sizes. As one may observe that the original design follows a pattern where interior columns at consecutive two levels are the same size. We follow the same pattern when we change the size here and modify interior columns at 3rd and 4th levels to be W14x311. We do a similar exercise for exterior columns at level 3rd and 4th and change their size from W14x233 (A = 68.5 in2, I = 3010 in4) to W14x257 (A = 75.6 in2, I = 3400 in4). To balance the change we did so far, now we look for the members which need to get penalized. We change the exterior column sizes at levels 5th and 6th from W14x211 (A = 62.0 in2, I = 2660 in4) to W14x193 (A = 56.8 in2, I = 2400 in4). We do a similar reduction for interior columns at these levels as well. Some of the beams are found to do relatively less work so we penalize these beams in the influence zone as well. All the beams between level 3rd to 8th are changed from W33x118 (A = 34.7 in2, I = 5900 in4) to W30x108 (A = 31.7 in2, I = 4470 in4). The beams at level 1st and 2nd are all assigned W33x141 (A = 41.6 in2, I = 7450 in4) which is a change from W36x231 (A = 68.1 in2, I = 15600 in4) and W33x130 (A = 38.3 in2, I = 6710 in4). One may note here that the right beam at the second level which was doing the second largest work is assigned a bigger size and inertia. These changes may seem overwhelming but the drift module provides an easy interface to change and update sizes on members while also keeping an eye on the displacement contributions due to the updated size. The only drawback is that it uses the old member forces to compute the member contributions. In general, small change in sizes are not anticipated to change the member forces significantly but if needed one can easily toggle to the analysis mode, run the analysis, and come back to the drift to see the updated contributions.
The above discussion only refers to one of the moment frames in the E-W direction. A similar change is made to the other frame as well as it had symmetric sizing. Since, the sizes were changed at various levels we redo the analysis with real and virtual load cases. The new inter-story drift ratio obtained is compared with the previous one and a slight decrease is achieved from 0.00267 to 0.00260 as shown in Fig. 9. This is a 2.6% drop which is not significant but further similar iterations may be performed to achieve more reduction. The structural moment frame weight drop from 405 tons to 399 tons. This is 1.5% drop and further iterations may be used to get more reduction on the weight as well. In this particular example, the structure is way below the drift limit so it is not worth reducing the drift but the concept remains the same for cases where drift ratio is beyond the code-specified limitation. For the second study, we create a virtual load case by applying 100 kips at the center of mass location of roof level in the E-W direction. A plot of normalized contribution of various components to the drift at the location of virtual load is shown in Fig. 12. The plot show that major contribution to the roof drift is coming from member and joints, flexure and shear. In this example, we study the impact of following three different options (1) downsize all the interior columns, (2) downsize the interior beams, and (3) downsize both interior columns and beams. The interior columns are uniformly downsized by one size as shown in AISC steel tables. The interior beams at level 14th and 13th are changed from W27x94 to W27x84. The interior beams between levels 5th to 12th are changed from W30x99, W30x108 to W30x90. The interior beams at level 1st to 4th are changed from W33x130 to W33x118. Fig. 13 shows the plot of E-W drift ratios over the height of the building for the above three options. It is observed that option 1 and 2 reduce the structural steel weight by 16 and 12 tons respectively. However, option 1 has a very small impact on drift ratios compared to option 2 which is way higher at intermediate levels. This is primarily due to the fact that interior columns have less displacement participation per unit volume relative to interior beams. So taking even less material out of beams will have more impact on drift ratios compared to columns. For option 3, a total of 28 tons of steel saving is noted when both interior beams and columns are downsized which is a 7% reduction in steel weight.
9
10
(a) SMF E-W direction (original sizes)
(a) SMF E-W direction (changed sizes)
(b) Displacement Participation per unit Volume - Normalized Figure 10: SMF E-W direction (405 ton steel)
(b) Displacement Participation per unit Volume - Normalized Figure 11: SMF E-W direction (399 ton steel)
Flexure
Shear
Joint
Axial
Total Figure 12: Normalized Component Contributions using original sizes
Total per Unit Volume of Material Figure 12: continued
11
References
DRIFT LIMITATION ORIGINAL DOWNSIZING COLUMNS DOWNSIZING BEAMS DOWNSIZING BEAMS AND COLUMNS
250
Height (ft)
200 150 100 50 0 0.0000
0.0010
0.0020
0.0030
0.0040
Inter-Story drift ratio
Figure 13: Normalized Component
Summary The ideas presented in this article are not original and similar work has been done in the past. This is also evident from the literature review presented in this article. The virtual work based displacement participation concept provides an efficient approach for resizing and optimizing members. This article is an effort to revisit the virtual work based optimization concepts and demonstrates them on a practical building using Bentley’s RAM Frame drift module. The RAM Frame is an advanced lateral force analysis program and includes drift, steel standard and seismic code check modules. The program automatically generates the analytical model. The load cases and load combos are also trivially generated based on the code selection and other input parameters. The architecture of the program is ideal for “human loop optimization” as toggling between the analysis, code check and drift modules can be easily done.
Acknowledgements The authors would like to acknowledge Mr. Tom Culp of Culp & Tanner Structural Engineers for providing structural drawings of the Xerox-Center building used in the example study.
12
[1] Baker, W.F., (1990). “Sizing Techniques for Lateral Systems in Multi-Story Steel Buildings”, Proceedings of the Fourth World Congress on Tall Buildings, Hong Kong, pp 857-868. [2] Baker, W. F., (1991). “Stiffness Optimization Methods for Lateral Systems of Buildings: A Theoretical Basis”, Electronic Computation drift constraints', [3] Chan, C. M. and Grierson, D. E. (1993). “An Efficient Resizing Technique for the Design of Tall Steel Building Subject to Multiple Drift Constraints”, International Journal of the Structural Design of Tall Buildings, 1993, Vol 2, 17-32.sign Tall Buildings, 1993, 2, 17-32 [4] Chan, C.M., Grierson D. E., and Sherbourne A. N. (1995). “Automatic Optimal Design of Tall Steel Building Frameworks”, Journal of Structural Engineering, Vol. 121, No. 5. [5] Chan, C. M. and Park, J. (1996). “Application of Structural Optimization to Practical Tall Steel Building Design”, Proceedings of the twelfth Analysis and Computation Conference, Chicago, Illinois. [6] Chan, C. M., Sherbourne, A. N., and Grierson, D. E. (1994). “Stiffness Optimization technique for 3D tall Steel Building Frameworks under multiple Lateral Loadings”, Engineering Structures, Vol 16, No. 8. [7] Charney, F. A. (1990a). “DISPAR for ETABS, User’s Manual”, Advanced Structural Concepts, Denver, Colorado. [8] Charney, F. A. (1990b). “DISPAR for SAP90, User’s Manual”, Advanced Structural Concepts, Denver, Colorado. [9] Charney, F. A. (1993). "Economy of Steel Frames through Identification of Structural Behavior" Proceedings of the National Steel Construction Conference, Orlando, Florida, 12-1 to 13-33. [10] Henige R. A. (1991). “Structural Optimization to Limit Natural Periods”, Electronic Computation [11] The RAM Structural System V8i. (2012). “RAM Manager Manual" Bentley Systems, Inc., Carlsbad, CA [12] The RAM Structural System V8i. (2012). “RAM Frame Drift Control Manual" Bentley Systems, Inc., Carlsbad, CA [13] The RAM Structural System V8i. (2012). “RAM Frame Manual" Bentley Systems, Inc., Carlsbad, CA [14] The RAM Structural System V8i. (2012). “RAM Frame Steel Post-Processors Manual" Bentley Systems, Inc., Carlsbad, CA [15] Velivasakis, E. E., and DeScenza, R. (1983). "Design Optimization of Lateral Load Resisting Frameworks" Proceedings of the Eight Conference on Electronic Computation, Houston, Texas.
Appendix A We use the concept of Lagrange multiplier to prove that the minimum drift at top story occurs when DISPAR per unit volume of the members are equal. The equality constraint used in Lagrange function is shown in Eq. (A-1) below. H , H " = H J + H J
(A-1)
The first drift function to be minimized is shown in Eq. (A2). The drift is represented as a sum of internal strain energy and is the same as Eq. (4) with virtual load set to unity. H , H " =
%
&
!" !" ! #$ % !" !" + ! #$ &
We see from Eqns. (A-7) and (A-8) that for local maxima or minima to exist for this problem, the Lagrange multiplier must be equal to DISPAR per unit volume of a column. Also, the DISPAR per unit volume of both the columns must be the same at local extreme. The sign on the second derivative of the Lagrange function may be used to determine if the local extreme condition will result in maximum or minimum. K=
12L 12L
= JHN JHN
(A-9)
(A-2)
The Lagrange function using Lagrange multiplier µ, Eqns. (A-1) and (A-2) is defined in Eq. (A-3). ʆ H , H , K" = H , H " + K H , H "
(A-3)
Before we delve into Eq. (A-3) any further let’s just assume that integral of product of moments can be represented as a constant. So we re-write Eq. (A-2) using the constants as shown in Eq. (A-4). This is true for this system because it is statically determinate. H , H " =
L L
+ #$ #$
(A-4)
Eqns. (A-5) and (A-6) writes moment of inertia of columns as a function of their cross-sectional areas. Please note that since the cross-section shape is assumed to be a square always during redistribution, the inertia functions turned out to parabolic. 1 $ = H
(A-5) 12 1
$ = H
(A-6) 12
Now, we differentiate Eq. (A-3) with respect to our variables A1 and A2 and set them to zero to find the condition for local maxima our minima of Eq. (A-2). M ʆ H , H , K" −12L = + KJ = 0 MH HN
M ʆ H , H , K" −12L
= + KJ = 0 MH
HN
(A-7)
(A-8)
13
