The CHEVRON EFFECT Not an Isolated Problem Patrick J. Fortney 1 and William A. Thornton 2 ABSTRACT

Transcription

1 The CHEVRON EFFECT Not an Isolated Prolem Patrick J. Fortney 1 and William A. Thornton ABSTRACT When vertical races connect concentrically to frame eams away from the eam-column joint, these concentrically oriented races are referred to as V-type or inverted V-type raced frames. It is also common to refer to these types of raced frames as chevron raced frames, or mid-span races. The races are commonly connected to the frame eam using gusset plates. Typically, these gusseted connections are analyzed and designed considering only the effect of the race forces on the region of the eam within the connection region. This is a reasonale approach when the summation of the vertical components of the race forces is zero. However, when the vertical components result in a non-zero net vertical force (also referred to as an unalanced force), the connection should not e analyzed and designed as if it were isolated from the frame. The eam span and the location of the work point along the span of the eam must e considered in order to fully understand the impact of the race forces on the frame eam. If these forces are not considered properly during final eam size selection, the eam may e significantly undersized, requiring expensive eam we and flange reinforcement. In this paper, the effect of the race forces on the eam in this type of raced frame configuration is referred to as the chevron effect. This paper presents a method for determining the distriution of race forces within the connection and also the impact of the race force distriution on the frame eam. To illustrate the chevron effect, the mechanism analysis required y the AISC seismic provisions is presented. The discussion illustrates the importance of considering the entire frame when evaluating the impact of the race forces on the eam, and the potentially unconservative results when evaluating the connection as if it were isolated from the frame. 1 Patrick J. Fortney, Ph.D., P.E., S.E., P.Eng., Cives Engineering Corporation, pfortney@cives.com William A. Thornton, Ph.D., P.E., NAE, Cives Engineering Corporation, thornton@cives.com

2 INTRODUCTION When vertical races connect concentrically to frame eams away from the eam-column joint, these concentricallyconfigured races are referred to as V-type or inverted V-type raced frames. It is also common to refer to these types of raced frames as chevron raced frames, or mid-span races. The races are commonly connected to the frame eam using gusset plates. Typically, these gusseted connections are analyzed and designed considering only the effect of the race forces on the portion of the eam within the connection region. This is called designing the connection in isolation, and is a reasonale approach when the summation of the vertical components of the race forces is zero. However, when the vertical components result in a non-zero net force, the connection should not e analyzed and designed as if it were isolated from the frame. The eam span and the location of the work point along the span of the eam must e considered in order to fully understand the impact of the race forces on the frame eam. In this paper, the effect of the race forces on the eam in this type of raced frame consideration is referred to as the chevron effect. This paper presents a method for determining the distriution of race forces within the connection and also the impact of the race force distriution on the frame eam. To illustrate the chevron effect, the mechanism analysis required y the AISC seismic provisions is presented. The discussion illustrates the importance of considering the entire frame when evaluating the impact of the race forces on the eam, and the potentially unconservative results when evaluating the connection as if it were isolated from the frame. Inverted V-Type Configuration Frame Column, Typical Frame Beam, Typical V-Type Configuration Two-Story X-Brace Configuration Figure 1: V-type, inverted V-type and two-story X-raced frame configurations Concentric raced frame structures can e set up in various configurations. Braces can frame to eam-column joints, to various locations along the height of the frame column and various locations along the span of the frame eam. The discussion presented in this paper focuses on a concentric race configuration referred to in AISC (AISC 341- /5

3 1) as V-type or inverted V-type configurations, also known as chevron races or mid-span races. In the V- type configuration, two races connect to the top side of the frame eam somewhere along the clear span of the frame eam away from the eam-column joint. In the inverted V-type configuration, two races connect to the ottom side of the frame eam somewhere along the clear span of the frame eam away from the eam-column joint. In some cases, the configuration is such that the races form a two story X-race in a manner where oth V- type and inverted V-type races connect to the intermediate frame eam level. Figure 1 shows these three types of chevron configurations. There are two common types of gusseted connections used in the types of race configurations shown in Figure 1. A comined gusset which is one plate that is used to connect oth races to the eam, or when geometry permits, a single gusset can e used to connect each race to the eam individually. Figures a and show these two common types of gusset connections. The discussion presented in this paper focuses on comined gussets. However, it is important to recognize that the issues addressed in this paper apply equally to chevron races connected with single gussets. The only significant difference etween the two types of gussets is how the forces acting at the gusset-toeam interface are calculated. FRAE BEA FRAE BEA e BRACE 1 COBINED GUSSET SINGLE GUSSET SINGLE GUSSET BRACE (a) comined gusset BRACE 1 BRACE () single gusset Figure : Representative sketches of comined and single chevron gusset plates Unlike connection design for races that frame to a eam-column joint, where the Uniform Force ethod (UF) is typically used to distriute race forces through the connection, the force distriution in a chevron race connection can e determined using any type of distriution that satisfies static equilirium. Part 13 of the AISC anual (1) provides comprehensive guidance how to distriute forces in race connections for races that frame to a eamcolumn joint. However, there is very little pulished work on how to distriute race forces in other types of race configurations such as V-type and inverted V-type race configurations (see example prolem II.C-5 of the AISC Design Examples anual, v.14.1). A method for doing so is presented in this paper. The impact of the force distriution on the frame eam must also e considered. A thorough treatment on this topic is also presented. 3/5

4 In chevron race connections, the algeraic sum of the vertical components of the race forces can have a significant impact on the shear and moment distriution in the frame eam. When the sum of the vertical components of the race forces is non-zero, the eam shear and moment distriution along the span of the eam are highly dependent on the span of the eam as well as the location of the work point along the span of the eam. Furthermore, the maximum eam shear and moment can e potentially underestimated or overestimated if the impact of the race forces is evaluated as if the connection is isolated from the frame. aximum eam shear and moment may also e located outside of the connection region of the eam. This impact on the eam is referred to in this paper as the chevron effect, and will e discussed in detail. There are various reasons why the summation of the vertical components of the race forces is non-zero. The most common reason involves mechanism analysis as required in seismic raced frame analysis and design. It is also possile to have a non-zero vertical component summation when races are permitted to resist gravity loads simultaneous with a lateral load analysis. This paper presents the following: 1. A procedure for determining an admissile force distriution within the connection,. The current typical method for determining eam shear, 3. Distriution of forces acting on the eam-to-gusset interface, 4. Beam shear and moment distriution, a. The effect of the span of the eam,. The effect of the location of the work point along the span of the eam, 5. The chevron effect, 6. A rule of thum for estimating the moment acting at the gusset-to-eam interface, and 7. Actual Design Prolem Example prolems to support the discussion are provided throughout the paper. It s worth noting that in this type of work, calculated values part of the solutions are typically shown using three significant figures. However, in order to have eam shear and moment diagrams close nicely, the authors have chosen to present values with higher numer of significant figures than would typically e presented. 1. AN ADISSIBLE CHEVRON BRACE FORCE DISTRIBUTION When generating an admissile force distriution in the connection, a control section must e selected. The method presented in this paper assumes a horizontal control section which is taken at the edge of the gusset that interfaces with the eam. 4/5

5 Horizontal Control Section This method first evaluates the forces acting on the horizontal edge of the gusset adjacent to the frame eam. This section is referred to as section a-a. See Figure 3a for geometry and parameters used. Once the forces acting on section a-a are determined, a vertical section located at one-half of the gusset length, L g is cut. This section is referred to as section - (see Figures 3d and 3e). Each half of the gusset is evaluated. For each half-gusset ody, the forces acting on the horizontal edge are taken as one-half of the forces acting on section a-a. The moment acting on section a-a is applied to the horizontal edge of the ody as a couple and is taken as a-a / L g, as shown in Figures 3 and 3c. Figures 3d and 3e show the free ody diagrams of each the half-gusset odies. Note that the analysis considers race evels, race forces, and the effects of any eccentricities that may exist in oth the horizontal and vertical directions. The eccentricity,, accounts for variations etween race 1 and race evels and the vertical components of the race forces. The eccentricity resulting from the horizontal components delivered y the gusset to the eam flange is accounted for with the parameter e. The sign convention used assumes that a race force component is positive when acting to the right in the horizontal direction and when acting upward in the vertical direction. A clockwise moment is considered to e positive. It is important to recognize that this is not the only way this analysis can e approached. The equations derived from statics for the forces and moments acting on sections a-a and - using the approach shown in Figure 3 will e derived in their entirety. Referring to Figure 3a, the vertical eccentricity parameter, can e written as, 1 L L (1) Note that is positive when to the left of the work point. Forces Acting on Section a-a Referring to Figure 3, equations for the forces and moment acting on section a-a can e written using the three equations of equilirium. In these equations, the suscripts 1 and refer to the race forces from the left and right races, respectively. The suscripts H and V refer to forces acting in the horizontal and vertical directions, respectively. The suscript refers to the work point, and the suscript a-a refers to section a-a as shown in Figure 3. 5/5

6 L 1 L e a a h/ h H 1 H V 1 V L g/ L g/ Lg Sign Convention H (+) i V (+) i (+) i (a) geometry, parameters, and sign convention H 1 V 1 H V L 1 L a -(V +V ) -(H +H ) 1 a-a a e h/ h N = eq L g/4 L g/4 -(V 1+V ) N = a-a eq L g -(H +H ) a H 1 V 1 H V L 1 L a a-a L g 1 e h/ h L g/ L g/ L g/ L g/ L g L g H 1 V 1 () forces and moment on section a-a N = eq a-a -1/(V +V ) -1/(H +H ) (d) forces on left half of gusset 6/5 (c) equivalent forces on section a-a L g N = a-a eq L g H H 1-1/(V +V ) L / g L g/4 V 1 H 1 V 1 1 e h/ h (e) forces on right half of gusset Figure 3: Free ody forces on critical horizontal and vertical gusset sections V H V L /4 g H L / g -1/(H +H ) V 1 e h/ h

7 F H H H H H ( H H ) F V V V V V ( V V ) ( V V ) ( H H ) e w. p. ( V V ) ( H H ) e In summary, the forces and moments acting on section a-a are as given in Equations, 3, and 4. H H H () V V V (3) ( V V ) ( H H ) e (4) Force Acting on Section - (left half of the gusset) Referring to Figure 3d, equations for the forces and moment acting on section - at the left half of the gusset can e written using the three equations of equilirium. As discussed previously, note that the forces acting on the horizontal section of the left half gusset are taken as one-half of the total forces acting on section a-a. Also, the moment a-a is converted to a couple acting at L g /4 of the gusset on oth the left and right halves of the gusset. In the following derivations, the suscript 1 refers to forces and moments acting on section - due to the race force from race 1. Refer to Figure 3d. 1 F H H H H H 1 H H H H F V V V V V 1 1 L V V V V L 1 1 g g h L 1 L g g 1 h H e V ( ) ( ) V V H H L 4 4 Lg h h ( V V ) ( H H ) VH e The couple, N eq, of the moment, a-a, shown in Figures 3d and 3e is given in Equation in Equation 5. g 7/5

8 N eq a a (5) L g In summary, the forces and moment acting on section - from the perspective of the left half of the gusset are as given in Equations 6, 7, and 8. 1 H H H H (6) 1 1 V V V V (7) 1 1 L g Lg h h ( V V ) ( H H ) VH e (8) Forces Acting on Section - (right half of the gusset) Referring to Figure 3e, equations for the forces and moment acting on section - at the right half of the gusset can e written using the three equations of equilirium. As discussed previously, note that the forces acting on the horizontal section of the right half gusset are taken as one-half of the total forces acting on section a-a. In the following derivations, the suscript refers to forces and moments acting on section - due to the race force from race. Refer to Figure 3e. 1 F H H H H H 1 H H H H 1 F V V V V V L V V V V L g g h L 1 Lg 1 h H e V ( V V ) ( H H ) L 4 4 Lg h h ( V V ) ( H H ) V H e 8 4 g In summary, the forces and moments acting on section - from the perspective of the right half of the gusset are as given in Equations 9, 1, and H H H H (9) 8/5

9 V V V V (1) L g Lg h h ( V V ) ( H H ) V H e 8 4 (11) EXAPLE PROBLE 1 - Brace Force Distriution Figure 4 shows the chevron connection geometry and dimensions. The force distriutions acting on sections a-a and - will e determined using Equations 1-4 and 6-8, respectively. Equation 5 will e used to calculate the couple of the moment, a-a. 5' " 3' " ' " " W18x143 1' " 93 4 " HSS8x8x5/8 P=47 kips (T) 8 11 HSS6x6x1/ P= kips (C) Figure 4: Example Prolem 1 geometry and dimensions The variales for Example Prolem 1 are shown elow. Note the signs of the component race forces, and the calculation of. Using the assumed sign convention, a component force acting to the right in the horizontal direction, or upward in the vertical direction, is positive. The vertical eccentricity parameter,, is calculated as shown in Equation 1. In this solution, the forces acting on section - are calculated using the left-half gusset ody (Equations 6, 7, and 8). Note that the right-half gusset ody (Equations 9, 1, and 11) can e used just as easily giving the same results. 9/5

10 L 7.63", L 44.56", L 6.6" g e 9.75", h.75" 1 (44.56" 6.6") 9.5" H kips; H kips V 5.8 kips; V 15. kips Forces Acting on Section a-a H ( H H ) ( ) H 5 kips V ( V V ) ( ) V 75.8 kips ( V V ) ( H H ) e ( )(9.5) ( )(9.75) 4,195 kip-in " a 5 4, " a 76 5' " Section a-a " " " 93 4 " ' " 1' " 51 1' " 93 4 " Section - (left) Section - (right) Figure 5: FBDs for Example Prolem 1 1/5

11 Forces Acting on Section - 1 H H 1 H1HH1 1 H 1 ( ) ( 338.6) 87.5 kips V V1 V1 V V1 L 1 ()(4,195) V 1 ( ) ( 5.8) 69.1 kips 7.63 L h h 1 ( V1V) ( H1 H) V1H1e ,195 1 ( ) ( ) ( 5.8)(9.5) 8 4 ( 338.6)(1.13) kip-in The free ody diagrams are shown in Figure 5.. CURRENT ETHOD USED FOR BEA SHEAR DETERINATION Typically, the shear imparted to the eam y the race force distriution is evaluated as if the connection is isolated from the frame. The eam span and the location of the work point along the span of the eam are not considered. Consider the joint shown in Figure 6, where the race evels are equal, the magnitude of the race forces are equal, and one race is in tension while the other is in compression. Using the procedure presented previously, the forces acting at the gusset-to-eam interface are given in Figure 7. Without considering the span of the eam or the location of the work point along the span of the eam, the shear in the eam, V eam, is constant etween the two points of applied load and would e taken as the a-a couple of 6.94 kips ( a-a is k-ft, L g =3 - ). The moment in the eam would e taken as one-half of the area under that shear gradient which would e that given in Equation. It s worth noting that evaluating the shear demand on the eam would typically e a consideration to determine if the eam required a we douler in the connection region. Beam moment in the connection region is not typically considered. The eam moment in the connection region is calculated as given in Equation. VeamLg eam () 4 It is no coincidence that Equation is equivalent to one-half of the summation of the horizontal components of the race forces times one-half the eam depth as given y Equation 13. Figure 8 shows the eam shear and moment distriution resulting from the forces acting on section a-a. 11/5

12 eam Hie (13) 3'-" 1'-7" 1'-7" W16x57 11" " 1 k 1 k Figure 6: Equal race evels and forces in a compression-tension race arrangement 1'-7" 9 " 9 " " 7.71 k 7.71 k 6.94 k 6.94 k W16x k 6.94 k 7.71 k 7.71 k 1 k 1 k Figure 7: Forces acting on section a-a for connection shown in Figure 6 With the type of configuration and forces shown in Figures 6-8, the impact of the race forces on the eam is determined to e V eam = 6.94 kips, and eam = 48.4 kip-ft, when the connection is evaluated as if it is isolated from the frame. Is this isolated evaluation adequate? Should the span of the eam, and the location of the work point, e considered? Before addressing these questions, we first have to consider how the forces acting on section a-a will e assumed to e distriuted for the evaluation of the load effects on the eam. /5

13 1'-7" 9 " 9 " " 7.71 k 7.71 k 6.94 k 6.94 k W16x k SHEAR 48.4 k-ft OENT 48.4 k-ft Figure 8: Beam shear and moment distriution for the isolated connection shown in Figure 6 3. GUSSET-TO-BEA INTERFACE FORCE DISTRIBUTION For the analysis and design of the gusset and the gusset-to-eam weld, the normal forces acting on the eam are assumed to e distriuted uniformly along the gusset-to-eam interface length. The horizontal forces, H a-a, are assumed to act on the interface eccentrically with a lever arm equal to e, as discussed previously. When a normal force, V a-a is present, the normal force is assumed to e distriuted uniformly along the gusset-to-eam interface length. The moment at the interface, a-a, is assumed to act as distriuted tension/compression normal forces equal to the couple of a-a (see Equation 5) divided y one-half the gusset length, L g / (see Equation 14). Figure 9 shows the force distriutions that are typically assumed for the gusset plate design and the design of the gusset-to-eam weld. Lg L 1 L Lg oment per Unit Length L 1 L Induced y Ha-a L g e e H a-a Beam e Beam L g V a-a L g V a-a L g Lg Lg 4 a-a L g Lg Lg +/- +/- 4 a-a L g Figure 9: Interface forces uniformly distriuted 13/5

14 n eq Lg 4 Lg Lg (14) The forces and moment acting on the gusset-to-eam interface are treated as externally applied loads to evaluate eam shear and moment. The distriutions of these interface forces can e uniformly distriuted as shown in Figure 9. However, using these distriutions can e tedious when evaluating eam shear and moment distriution. To simplify the eam analysis, it is recommended that the resultant force like those shown in Figure 5 e used to evaluate the eam. The following is an example to illustrate the differences etween the two methods of eam evaluation (i.e., distriuted method versus resultant method). 5' " 3' " ' " W18x143 NODE 1 NODE a a " 1' " HSS8x8x5/8 P=47 kips (T) 8 15'-6" 9'-6" 5' (a) Gusset connection shown in Figure 5 in context of a frame 11 HSS6x6x1/ P= kips (C) Ha-a e Ha-a e V a-a a-a Lg V a-a a-a Lg () Beam model: resultant force loading and oundary conditions Figure 1: Gusset connection shown in Figure 5 in context of a frame 14/5

15 5' " ' " ' " " W18x143 NODE 1 a NODE 85.3 k/ft.88 k/ft " Edge of Gusset 11' " ' " 5' +/-4.44 k/ft k/ft Edge of Gusset ' " 7' " 8.83 SHEAR DIAGRA (kips) '.41' OENT DIAGRA (kip-ft) Figure 11: Beam shear and moment with uniformly distriuted loads acting on interface Suppose the connection presented in Example Prolem 1 is part of a frame with a eam spanning 5, and the work point is located 15-6 from the left support, as shown in Figure 1a. The forces acting on the gusset-to-eam interface can e assumed to e distriuted uniformly along the length of the interface, as shown in the loading diagram in Figure 11, or as resultant forces acting at the centroids of the two half gusset odies, as shown in Figure. The eam shear and moment distriutions along the length of the eam are shown in Figures 11 and for each method, respectively. As can e seen in the two figures, the eam shear and moment gradients are a little different as a result of the types of loads. Both loading conditions produce the same maximum eam shear and the maximum eam shears are located at the same locations. The maximum moment, for this example, is aout 37% larger when the resultant loads are used, and occurs within the connection region. The conservativeness in the maximum moment calculation using the resultant method can e attriuted to two facts: 1. The eam end reactions are the same regardless of which method is used. However, the area under the shear gradient is larger using the resultant loads ecause the length of the gradient interval is longer. For example, the 15/5

16 left support reactions using oth methods is 8.83 kips, as can e seen in Figures 11 and. The distance from the left support to the change in loading in Figure 11 is the distance to the left edge of the gusset (11-9 7/16 ) assuming distriuted load; the distance from the left support to the change in loading in Figure is the distance to the centroid of the left half gusset ody (13-3 1/8 ). The difference etween the two distances is ¼ of the gusset length, L g. Thus, the longer the gusset, the more conservative the moment calculation will e when using resulting loads.. The concentrated moment using the resultant method is more conservative relative to distriuting the moment over the entire length of the gusset. ' " ' " " W18x143 NODE 1 a NODE " Location of Resultant Forces Acting on Left Half Gusset Body ' " ' " Location of Resultant Forces Acting on Right Half Gusset Body 8' " 5' 8.83 SHEAR DIAGRA (kips) OENT DIAGRA (kip-ft) Figure : Beam shear and moment with resultant forces acting at interface Given that using resultant loads gives the same eam end reactions and maximum eam shears (for most cases), that the calculated moment will always e conservative, and that the resultant load method is far simpler relative to assuming distriuted loads, resultant loads will e used throughout this discussion and in the example prolems presented, except for eam we douler plate detailing where the distriuted loads are used. Figure 1 shows the general eam model that will e used to determine eam shear and moment distriution for eam evaluations. 16/5

17 3'-" 1'-7" 1'-7" W16x57 NODE 1 NODE 8." 11" HSS6x6x 1 kips (C) HSS6x6x 1 kips (T) 14' 14' 8' (a) Connection located at mid-span 3'-" 1'-7" 1'-7" 8." W16x57 NODE 1 NODE 11" HSS6x6x 1 kips (C) HSS6x6x 1 kips (T) ' 6' 8' () Connection located 6 from right support Figure 13: V i = at different locations along eam span Note that oth methods give the same maximum eam shear in this example. However, it should e noted that when races frame to oth the top and ottom of the eam, and the parameter is non-zero, it is possile that the resultant load method will produce a slightly larger maximum eam shear. The resultant load method will e conservative when comparing required eam shear strength to availale eam shear strength. However, when the required eam shear strength exceeds the availale eam shear strength, the distriuted load method should e used to determine required we douler thickness as well as the extent of the required eam we douler plate. An example of how to detail a eam we douler plate is provided in Part 4 of Example Prolem 3 presented later in this paper. 4. BEA SHEAR AND OENT DISTRIBUTION As discussed previously, the shear and moment imparted to the frame eam y the race forces is typically evaluated as if the connection joint is isolated from the frame. Is this a valid approach, or does the span of the eam and the location of the work point along the span of the eam need to e considered when evaluating the race load 17/5

18 effects on the frame eam? If the algeraic sum of the vertical components of the race forces is zero, shear and moment imparted to the eam is independent of the span of the eam and the location of the work point. If the algeraic sum of the vertical components of the race forces is non-zero, the shear and moment imparted to the eam is dependent on the span of the eam and the location of the work point. Figures 13a and 13 show a W16x57 spanning 8. The race geometry and forces are such that the algeraic sum of the vertical components of the race forces is zero. Figure 13a has the work point located at mid-span of the eam while Figure 13 has the work point located 6 (for simplicity, the race evels are assumed unchanged to illustrate a point) from the right support. Figures 14 and 15 show the eam shear and moment diagrams for the two work point locations, respectively. Referring to Figures 14 and 15, it can e oserved that the eam shear and moment imparted to the eam is contained within the connection region. Beam shear and moment outside of the connection region is zero. If the connection was evaluated as if the joint is isolated from the frame, the eam shear and moment would e the same as that shown in Figures 14 and 15. Thus, the shear and moment imparted to the eam y the race forces is independent of the eam span and the location of the work point along the eam span. Therefore, when the algeraic sum of the vertical components of the race forces is zero, it is sufficient to evaluate the eam as if the connection is isolated from the frame. Historically, this is proaly the reason that this type of connection has een designed in isolation. 1'-7" 9 " 9 " 8." W16x57 NODE 1 a NODE '- " 1'-7" 13'- " 8' SHEAR DIAGRA (kips) 6.94 OENT DIAGRA (kip-ft) Figure 14: V i = ; eam shear and moment with work point at mid-span Now consider that the tension race force shown in Figure 13 is increased from 1 kips to 3 kips while the compression race force remains at 1 kips, as shown in Figures 16a and 16. The algeraic sum of the vertical components of the race forces is now non-zero. The summation of the vertical components of the race forces is 18/5

19 141. kips. Figures 16a and 16 show the geometry and forces for these cases. Figures 17 and 18 show the eam shear and moment diagrams for the case where the work point is at mid-span and when the work point is 6 from the right support. As can e seen in Figures 17 and 18, the location of the work point has an effect on how the eam shear and moment are distriuted along the length of the eam, and has an effect on the maximum eam shears and moments. Referring to the eam shear diagram in Figure 17, it can e seen that the eam shear within the connection region is equal to the couple of the moment, a-a (in this case,.1 kips), and the shear outside of the connection region is equal to one-half of the algeraic sum of the vertical components of the race forces (in this case, 7.6 kips). This may lead one to conclude that this is always the case when there is an unalanced vertical force and the work point is at mid-span of the eam. However, this is not always the case; there is one other parameter that must e satisfied. The centroid of the gusset interface must also e vertically aligned with the work point. In other words, the parameter must e zero. For the eam shear within the connection region to e equal to the couple, and the shear outside of the connection region to e equal to one-half of the unalanced force, the work point must e located at mid-span of the eam and the parameter must e equal to zero. This will also e true if is non-zero ut the centroid of the gusset happens to e vertically aligned with the mid-span work point. Simply put, this is only true when the resultant normal forces acting on the interface are symmetric aout the mid-span work point. Comparing the eam shear distriutions shown in Figures 17 and 18, it can e concluded that the eam shear and moment distriution, as well as the maximum shear and moment, are dependent on the location of work point along the span of the eam. If the eam shear and moment are evaluated as if the connection is isolated from the frame, the eam shear and moment would e determined to e.1 kips and 96.7 kip-ft, respectively, regardless of the eam span and the location of the work point along the span of the eam (refer to previous discussion). If the eam shear and moment are evaluated considering the frame, with the work point at mid-span, the maximum eam shear and moment is.1 kips and 1,3 k-ft, respectively. The eam shears are the same ecause the resultant normal forces are symmetric aout the work point. However, the eam moment is 1,3/96.7=1.6 times larger when the frame is considered. When the frame is considered and the work point is not at mid-span (6 from the right support in this case), the maximum eam shear and moment is 111. kips and k-ft (see Figure 18). The maximum eam shear is overestimated y.1/111.=1.1 times and the moment is underestimated y 674.4/96.7=7.97 times when the frame is considered. Furthermore, as can e seen in Figure 18, the maximum eam shear occurs outside of the connection region. This would not e noticed if the eam shear is evaluated as if the connection is isolated from the frame; a prolematic issue if one was to evaluate the need for we douler plates and the location where such reinforcement would e required. 19/5

20 1'-7" 9 " 9 " 8." W16x57 NODE 1 a NODE '- " 1'-7" 5'- " 8' SHEAR DIAGRA 6.94 OENT DIAGRA Figure 15: V i = ; eam shear and moment with work point off of mid-span 3'-" 1'-7" 1'-7" W16x57 NODE 1 NODE 8." 11" HSS6x6x 1 kips (C) HSS6x6x 3 kips (T) 14' 14' 8' (a) Connection located at mid-span 3'-" 1'-7" 1'-7" 8." W16x57 NODE 1 NODE 11" HSS6x6x 1 kips (C) HSS6x6x 3 kips (T) ' 6' 8' () Connection located 6 from right support Figure 16: V i (141. kips) at different locations along eam span /5

21 The effect of the location of the work point has een illustrated in Figures 17 and 18. The span of the eam also has an effect on the eam shear and moment distriution. Although not illustrated in this discussion, one can deduce that a change in eam span has an effect on the eam end reactions. Given that the eam shear and moment distriutions are a function of the eam end reactions, the span of the eam also affects the eam shear and moment distriution as well as the maximum eam shears and moments. 5. THE CHEVRON EFFECT When evaluating the race forces in a chevron raced frame sujected to lateral loads, the analysis will reveal that one race is in tension while the other is in compression. For static equilirium, the vertical components of the race forces will sum algeraically to zero. However, it is sometimes necessary, or required, to perform some type of mechanism analysis where in such a case, the algeraic sum of the vertical components of the race forces will e non-zero. One example of a mechanism analysis is that required in a seismic raced frame where the race in tension is assumed to reach the expected tensile strength of the race, while the race in compression is assumed to reach its uckling strength, or even a post-uckling strength. The impact of the race forces on the frame eam needs to e evaluated in either case. 1'-7" 9 " 9 " 8." W16x57 NODE 1 a NODE '- " 7.6 1'-7" '- " 8' SHEAR DIAGRA (kips) , OENT DIAGRA (kip-ft) Figure 17: V i (141. kips) eam shear and moment with mid-span work point 7.6 The following example prolem illustrates the chevron effect, and emphasizes the importance of accounting for the span of the eam as well as the location of the work point along the span of the eam. 1/5

22 1'-7" 9 " 9 " 8." W16x57 NODE 1 a NODE '- " 1'-7" 5'- " 8' 3.5 SHEAR DIAGRA (kips) OENT DIAGRA (kip-ft) Figure 18: V i (141. kips) eam shear and moment with work point off mid-span EXAPLE PROBLE The Chevron Effect with echanism Analysis For the chevron racing configuration shown in Figure 19, 1. Determine the force distriution in the connection for the race forces given in Tale 1. For this analysis only the forces acting on section a-a need to e determined using Equations 1-4. For an actual gusset design, the forces acting on section - are also required, ut not necessary for this example prolem.. Determine the eam shear and moment distriution along the span of the eam for each load case ased on the forces and moments acting on Section a-a determined in Part Compare the maximum eam shears and moments determined as if the connection was isolated from the frame to those values otained from the eam shear and moment diagrams. Assume that (KL) x and (KL) y for oth races is. Note that this length accounts for the pull-off dimensions at oth ends of each race. Note that typically, oth directions of lateral load would e considered. For this example, only the three load cases shown in Tale 1 will e considered. It is worth noting here that Load Cases and 3 are representative of the mechanistic analysis required y the AISC Seismic Provisions. /5

23 3' " 1'-8 " '-31 4 " 1.7" W1x83 NODE 1 NODE a a HSS9x9x 8 5 P ' 18' 3' HSS9x9x 8 5 P Figure 19: Connection geometry and dimension for Example Prolem " Tale 1: Load cases for Example Prolem Load Case *P 1 *P (kips) (kips) R y F y A g =+1,5 P =-1.14F cre A g = R y F y A g =+1,5 (.3)P =-33 *(+) = tension; (-) = compression Solution The variales for Example Prolem are shown elow. L 47.75", L.5", L 7.5" g e 1.7", h9.5" 1 (.5" 7.5") 3.375" Load Case 1 H 69.4 kips; H 43.5 kips V kips; V kips Figure shows the geometry and race forces for Load Case 1. From the data given in Figure, the forces acting on section a-a can e determined. Forces Acting on Section a-a H ( H H ) ( ) H kips 3/5

24 V ( V V ) ( ) V kips ( V V ) ( H H ) e ( )( 3.375) ( )(1.7) 7, kip-in The couple of a-a is: ()(7,195.57) Neq 31.4 kips Figure 1 shows the resulting interface forces and eam shear and moment distriutions. As can e seen in Figure 1, the maximum eam shear and moment are: V u,max u,max 31.4 kips 99.8 kip-ft Load Case Figure shows the geometry and race forces for Load Case. From the data given in Figure, the forces acting on section a-a can e determined. H 73. kips; H kips V 964. kips; V kips Forces Acting on Section a-a H ( H H ) ( ) H 1,33.98 kips V ( V V ) ( ) V kips ( V V ) ( H H ) e ( )( 3.375) ( )(1.7) 15, kip-in The couple of a-a is: ()(15,459.77) Neq kips Figure 3 shows the resulting interface forces and eam shear and moment distriutions. As can e seen in Figure 3, the maximum eam shear and moment are: V u,max u,max 6.9 kips 3,65 kip-ft 4/5

25 3' " 1'-8 " '-31 4 " 1.7" W1x83 NODE 1 NODE a a " HSS9x9x kips (T) ' 18' 3' HSS9x9x kips (C) Figure : Geometry and race forces for Example Prolem, LC1 1.7" W1x83 NODE 1 a NODE ' " 1' " 3' 16' " SHEAR DIAGRA (kips) V =331 kips n OENT DIAGRA (kip-ft) 99.8 Figure 1: Beam shear and moment distriution for LC1 =735 k-ft n =735 k-ft n 5/5

26 3' " 1'-8 " '-31 4 " 1.7" W1x83 NODE 1 NODE a a " HSS9x9x 5 8 1,5 kips (T) ' 18' 3' HSS9x9x kips (C) Figure : Geometry and race forces for Example Prolem, LC Load Case 3 Figure 4 shows the geometry and race forces for Load Case 3. From the data given in Figure 4, the forces acting on section a-a can e determined. H 73. kips; H 174. kips V 964. kips; V kips Forces Acting on Section a-a H ( H H ) ( ) H 897. kips V ( V V ) ( ) V 89.3 kips ( V V ) ( H H ) e ( )( 3.375) ( )(1.7), kip-in 6/5

27 W1x83 NODE 1 a NODE " ' " 1' " 16' " 3' 67.9 V =331 kips n SHEAR DIAGRA (kips) V =331 kips n OENT DIAGRA (kip-ft) 3,65 3,4,987,45 =735 k-ft n Figure 3: Beam shear and moment distriution for LC The couple of a-a is: ()(,38.35) Neq kips Figure 5 shows the resulting interface forces and eam shear and moment distriutions. As can e seen in Figure 5, the maximum eam shear and moment are: V kips u u 5,881 kip-ft 7/5

28 3' " 1'-8 " '-31 4 " 1.7" W1x83 NODE 1 NODE a a " HSS9x9x 5 8 1,5 kips (T) ' 18' 3' HSS9x9x kips (C) Figure 4: Geometry and race forces for Example Prolem, LC3 Load V n n Case (kips) (k-ft) Tale : Beam shears and moments; summary of Example Prolem results Considering Frame Beam Span Connection Isolated From Frame and Work Point Location V u,max u,max (kips) (k-ft) V u,max V n u,max n V u,max u,max (kips) (k-ft) V u,max V n u,max n V u,max Within Connection Region or Outside of Connection Region W , W , O A summary of Example Prolem results are shown in Tale. Upon reviewing these results, three primary oservations can e made. 1. The maximum eam shear occurs within the connection region in LC1 and LC and outside of the connection region in LC3. Thus, if the span of the eam and the location of the work point are not considered when evaluating the eam shear, the maximum eam shear would not e captured for LC3.. The algeraic sum of the vertical components of the race forces is zero for LC1. For LC and LC3, the algeraic sums of the vertical components of the race forces are kips and -89. kips, respectively. Thus, the vertical components of the race forces for LC and LC 3 are unalanced. For LC, if the maximum eam shear is determined assuming the connection is isolated from the frame, the maximum eam shear is 8/5

29 determined to e kips. When the span of the eam and the location of the work point is considered for LC, the maximum eam shear is determined to e 6.9 kips; an overestimation of approximately 7.3%. For LC3, if the maximum eam shear is determined assuming the connection is isolated from the frame, the maximum eam shear is determined to e kips. When the span of the eam and the location of the work point is considered for LC3, the maximum eam shear is determined to e kips (and is located outside of the connection region); an overestimation of approximately 6.3%. For this example, the maximum eam shear is overestimated y 6.3% to 7.3%. Under different geometry and loading, it s quite possile to significantly underestimate or overestimate the maximum eam shear when the connection is evaluated as if it is isolated from the frame. 3. The eam moment is significantly underestimated when the connection is evaluated as if it is isolated from the eam when the vertical components of the race forces are unalanced. Referring to Tale, the ratios of availale flexural strength to required flexural strength for LC and LC 3 are. and., respectively. Thus, the actual eam moment demands for LC and LC3 are 4.9 and 8. times larger, respectively, than what would e determined if the span of the eam and location of the work point is not considered. 1.7" W1x83 NODE 1 a NODE ' " 1' " 16' " 3' V =331 kips n SHEAR DIAGRA (kips) V =331 kips n ,881 5,481 5,4 5,14 OENT DIAGRA (kip-ft) Figure 5: Beam shear and moment distriution for LC3 =735 k-ft n 9/5

30 It s important to reiterate that the eam shears and moments calculated for this example are ased on the race forces only. Load effects from other types of loads (e.g., dead, live, etc.) must e superimposed to get the total shear and moment demands on the eam. In almost all cases, the additional loads will increase the maximum eam shear and moments eyond those imparted to the eam y the race forces alone. 6. FINAL BEA SIZE SELECTION A Rule of Thum As demonstrated in the previous discussions, it is important to include race force effects when making final eam size selections. To account for the race force effects, the geometry of the connection must e known in order to calculate the gusset-to-eam interface forces. Typically, the connection geometry is not known at the time final eam size selection is made, and therefore can e prolematic. This is especially prolematic when connection design is delegated to a Contractor that is not the Engineer-of-Record for the design of the structure. To address this issue, the authors recommend a rule of thum for accounting for the race force effects. To approximate the race force effects on the eam, assume that the length of the gusset is approximately one-sixth of the eam span, and assume that the depth of the eam, d, in inches, is 75% of the span of the eam in feet. Thus, the approximate value for e can e taken as one-half of the approximated eam depth. These approximations for a trial eam size are given in Equations 15 and 16. These approximations are ratios averaged from twenty different chevron race connections taken from real connections designed y the authors over several years. The twenty different chevron connections were taken from a mix of different types of projects with varying types of races, evels, and race forces. L Lg, app (15) 6 e, app (in inches).375(span of the eam in feet) (16) With the length of the gusset and e approximated, the moment acting at the gusset-to-eam interface can e conservatively estimated using Equation 17. Equation 17 contains the term with the horizontal components of the race forces given in Equation 4. The couple of the moment acting on the gusset-to-eam interface can e estimated y dividing Equation 17 y the approximated gusset length. Equation 18 is the simplified expression for the approximated couple. The couples are placed at the centroids of the two half gusset odies (i.e, at L g, app / 4 in from each gusset edge; L g / apart). The direction of the couple should e considered to act in each direction to capture the worst case effect when comining the race force effects with other types of loads (e.g., dead, live, wind load, etc.)., app H1 H e, app EQ. 17 3/5

31 N eq, app g, app app, app ()(6) H1 H e, H1 H (.375 L) L L ( in. / ft.) L eq, app N.375 H H (18) Referring to Equation 4, the moment at the gusset-to-eam interface, a-a, has two terms; the first term is a function of and the second term is a function of e. The proposed method presented here for approximating the moment at the gusset-to-eam interface does not consider any potential vertical misalignment of the work point with the centroid of the gusset interface. That is, the first term of Equation 4, (V 1 +V ), is not accounted for in the approximation. Upon close examination of Equation 4, it can e seen that the two terms may e the sum of the two terms or the difference of the two terms. Each term has the possiility of eing positive or negative. When the signs of each parameter are such that the moment is the difference etween the two terms, the approximated moment will e overestimated. When the signs of each parameter are such that the two terms are additive, the approximated moment will e underestimated. This is not a significant concern. Generally, the term is a relatively small percentage of the total moment acting at the gusset interface. Additionally, the approximated gusset length given in Equation 15 will generally underestimate the actual gusset length resulting in a relatively larger couple. Thus, the rule of thum presented here will provide a reasonaly conservative estimate to e used for eam size selection. 7. ACTUAL DESIGN PROBLE EXAPLE PROBLE 3 Accounting for race forces when sizing eam Gravity Loads D = 118 psf (includes all self-weight and all other superimposed dead loads) L = 5 psf (non-reducile) The triutary width of the frame eam is 8. Lateral Loads The load effects on the races from a wind load analysis are given in Figure 6. The race forces given are LRFD loads and are used with LC6 shown elow. Load Cominations Evaluate only Load Cominations and 4 from ASCE 7-1, as given elow. LC : 1.D+1.6L LC 4: 1.D+.5L+1.W (note that L is less than 1 psf) 31/5

32 Deflection Limits for Frame Beam (gravity) D+L: L/4 L: L/36 When checking deflection, assume that the clear span of the eam is from column centerline to column centerline. Prolem Statement A partial elevation of the raced frame is shown in Figure 6. As can e seen in the figure, the geometry and race forces are given. The race forces shown are load effects from a wind analysis. 1. Calculate the design gravity load on the eam for LC and LC 4 given aove.. ake a eam selection neglecting the load effects of the race forces acting at the gusset-to-eam interfaces a. Provide a eam size that satisfies the strength, deflection, and drift requirements given aove. Do not include the effect of the race force distriutions acting at the gusset interfaces, and assume the eam spans from column to column (i.e., the races are not present to carry gravity load effects). Calculate the race force distriutions at section a-a for the races aove and elow the frame eam. c. Draw the eam shear and moment diagrams that include oth the LRFD gravity loads and forces acting on the eam imparted y the race force distriutions. d. Compare the required eam shears and moments to the availale eam shears and moments otained in Parts and c. 3. ake a eam selection including the load effects of the race forces acting at the gusset-to-eam interfaces a. Determine a trial eam size that satisfies the strength, deflection, and drift requirements given aove. Include the effect of the race force distriutions using the rule of thum recommended previously.. Using the trial eam size selected in Part 3a, calculate the race force distriutions at section a-a for the races aove and elow the frame eam. c. Draw the eam shear and moment diagrams that include oth the LRFD gravity loads and forces acting on the eam imparted y the race force distriutions. d. Compare the required eam shear and moment to the availale eam shear and moment 4. Assume that a connection designer is faced with a design scenario such as that shown in Figure 3 where the required eam shear and moment exceed the availale eam shear and moment. Calculate the required we douler thickness and length of the we douler for the eam and loading shown in Figure 3 (i.e., the eam used in Part of this prolem). Evaluate the we douler for each of the following two load distriution conditions: a. using the resultant forces method, and. using the distriuted forces method. 3/5

33 Figure 6: Partial frame elevation for Example Prolem 3 Triutary width of frame eam is 8 Note that the availale eam moment is also exceeded in this scenario which should e addressed in some manner. However, this issue is not covered here. Use the race force load case shown in Figure 7. In the following solution, the authors have estalished gusset plate geometry ased on the trial e values estalished for each part of the prolem. SOLUTION Part 1 Calculate design eam gravity load It is given in the prolem statement that the triutary width of the frame is 8. The design gravity loads for LC and LC 4 are: 33/5

34 LC w w D L u, u, (118 psf )(8 ft) 3.3 k/ft 1, ls/kip (5 psf )(8 ft) 1.4 k/ft 1, ls/kip w 1.D1.6 L (1.)(3.3) (1.6)(1.4) w 6. k/ft LC 4 w w D L u,4 u,4 (118 psf )(8 ft) 3.3 k/ft 1, ls/kip (5 psf )(8 ft) 1.4 k/ft 1, ls/kip w 1.D.5 L (1.)(3.3) (.5)(1.4) w 4.66 k/ft Figure 7: Connection geometry and race forces for Example Prolem 3 Part a Size eam for load determined in Part 1; include deflection check. Do not include race forces 34/5

35 The required eam shears and moments are: V V u, u, V V u, u, u,4 u,4 u,4 u,4 wu,l (6. k/ft)(6 ft) 8.6 kips wu,l (6. k/ft)(6 ft) k-ft wu,l (4.66 k/ft)(6 ft) 6.6 kips wu,l (4.66 k/ft)(6 ft) k-ft LC governs the design for strength. The plastic section modulus, Z, required to resist u, is: (54 k-ft)( in/ft) Zreq 14 in (.9)(5 ksi) The moment of inertia required for the deflection limits is: 5wL i 5wL i 384EI 384 I D L I L 4 4 i Ii i Ei (5) (6)() 1, 8 in (6)() (384)(9, ) (5) (6)() 573 in (6)() (384)(9, ) Therefore, I>1,8 in 4. Thus, for a trial eam size, select a eam that satisfies the following requirements. V 8.6 kips u Z u req 54 k-ft 14 in I 1,8 in Try a W1x83 eam n n k-ft 54 k-ft OK V 331 k V 8.6 k OK I u u 4 4 1,83 in I 1, 8 in OK 3 35/5