The plastic moment capacity of a composite cross-section is calculated in the program on the following basis (BS 4.4.2):

Transcription

1 COMPUTERS AND STRUCTURES, INC., BERKELEY, CALIFORNIA SEPTEMBER 2002 COMPOSITE BEAM DESIGN BS Technical Note Composite Plastic Moment Capacity for Positive Bending This Technical Note describes how the program calculates the positive bending moment capacity for a composite section assuming a plastic stress distribution. Overview The plastic moment capacity of a composite cross-section is calculated in the program on the following basis (BS 4.4.2): Concrete is assumed to be stressed to a uniform compression of 0.45 f cu over the full depth of concrete on the compression side of the plastic neutral axis (PNA) (BS a). Concrete is assumed to have no tensile strength. The structural steel member is assumed to be stressed to its design strength p y either in tension or in compression for Class 1 (Plastic), Class 2 (Compact) and Class 3 (Semi-Compact) sections (BS b). Class 4 (Slender) sections are not designed by the program. For sections under the influence of high shear, the web is ignored in calculating the plastic moment capacity (BS 5.3.4). The longitudinal reinforcement is ignored in the program for calculating plastic moment capacities for both positive and negative moment. This is conservative. The effect of partial composite connections is considered in computing the plastic moment capacity for positive moment. Figure 1 illustrates a generic plastic stress distribution for positive bending. Note that the concrete is stressed to 0.45 f cu and the steel is stressed to p y. The distances y p and y c are measured from the bottom of the beam bottom flange (not cover plate) to the plastic neutral axis (PNA) and the bottom of the concrete compression block, respectively. The illustrated plastic stress Technical Note E-CB-AISC-ASD Page 1 of 14

2 distribution is the basic distribution of stress used by the program when considering a plastic stress distribution for positive bending. Note that if the metal deck ribs are parallel to the beam, the concrete in the ribs is also considered. a 0.45f cu C Conc Plastic neutral axis (PNA) y p z p yc T Steel C Steel p y p y Beam Section Beam Elevation Plastic Stress Distribution Figure 1: Generic Plastic Stress Distribution for Positive Bending Figure 2 illustrates how the program idealizes a steel beam for calculating the plastic stress distribution. Two different cases are shown, one for a rolled section and the other for a user-defined section. The idealization for the rolled section considers the fillets whereas the idealization for the user-defined section assumes there are no fillets because none are specified in the section definition. Although not shown in Figures 1 and 2, the deck type and orientation may be different on the left and right sides of the beam as shown in Figure 2 of Technical Note Effective Width of the Concrete Slab Composite Beam Design. For a rolled steel section, the fillets are idealized as a rectangular block of steel. The depth and width of this rectangular block are given by: k depth = k - T (Rolled) The rectangular block, k width, is: k width = (A s - 2BT - td) / 2k depth (Rolled) Overview Page 2 of 14

3 B e k T top D p t c k width B top k depth t d D k width k depth k T bot T cp B bot Idealization for Rolled Section T top D p t c B top t T bot D s B cp B e d D s D B cp T cp B bot Idealization for User-Defined Section Figure 2: Idealization of a Rolled Section and a User-Defined Section used for Calculating the Plastic Stress Distribution Overview Page 3 of 14

4 For welded sections, the fillets are non-existent. However, for the purpose of plastic moment capacity calculation, the depth and width of the rectangular blocks of fillets are taken as the following. This definition of the fillets for welded, user-defined sections allows them to be treated under the same framework as the rolled sections. k depth = 0 (Welded) k width = t (Welded) The basic steps in computing the positive plastic moment capacity are as follows: Determine the maximum compressive force that can be generated in concrete and steel for full and partial composite connection. Determine the size of concrete stress block, a, and the location of the bottom of the stress block, y c. Determine the location of the PNA in steel, y p. Calculate the plastic moment capacity, M p. Maximum Compressive Force in Concrete The program determines the location of the PNA by comparing the maximum possible compressive force that can be developed in the concrete with the maximum possible tensile force that can be developed in the steel section (including the cover plate, if applicable). The depth of the stress block is determined from the concrete compressive force in plastic condition. The location of the PNA and the depth of the compression block are heavily influenced by the partial composite connection ratio PCC. The maximum concrete force, F conc,max, that can be generated in a composite deck is calculated differently depending on whether the deck ribs are parallel or perpendicular to the beam. If the deck ribs are perpendicular to the beam, F conc,max is calculated as follows (BS ). Note that the maximum concrete force has contribution from the left and right sides of the beam. Those contributions are treated separately because they may be different. F conc,max = [0.45 f cu B e (D s D p )] left + [0.45 f cu B e (D s D p )] right (BS ) Maximum Compressive Force in Concrete Page 4 of 14

5 If the deck ribs are parallel to the beam, the contributions of the ribs as well as the contributions from the slabs are considered. In such cases, F conc,max is calculated as follows (BS ): F conc,max = 0.45 f cu B e t c + b r s D r p left f cu B e t c + b D r s r p right The maximum steel force, F steel,max, that can be generated in a composite beam is calculated differently depending on whether there is cover plate or not. F steel,max = A s p y (with no cover plate) (BS ) F steel,max = A s p y + B cp T cp p ycp (with cover plate) (BS ) In the preceding expressions, A s is the total area of steel section alone. For welded sections, A s is computed from plate dimensions. For rolled sections, A s is given in the section definition. In practical cases, especially when the shear connection between the slab and the steel beam is partial, the force in the concrete will not attain F conc,max, and the force in the steel section will not attain F steel,max. Assuming that the partial composite connection ratio is PCC, the maximum concrete force and total steel tensile force will be equal to, which is given by the following equation: = PCC min{f conc,max, F steel,max } The value of PCC ranges between 0 and 1. For full composite connection, PCC is 1 and is the minimum of maximum concrete force and maximum steel tensile force. In such cases, if F conc,max is greater than F steel,max, y p will be equal to the full depth of the beam d and the depth of compression block will be smaller than D s. For full composite connection and if F steel,max is greater than F conc,max, y p will be less than h and the depth of the compression block will be equal to D s. For partial composite connection, y p is always less than D, and the depth of the compression block is always less than D s. In full or partial composite connection, both the concrete compression force and steel tensile force will always to be equal to. The location of the Maximum Compressive Force in Concrete Page 5 of 14

6 plastic neutral axis, y p, depth of the compression block, a, and plastic moment capacity, M p, are calculated from this condition. Depth of the Compression Block The required depth of the compression block, a, is the depth of the concrete that is required to develop the concrete compression force equal to. The definition of is given in the previous section of this Technical Note. For the calculation of the required depth of the compression block, it is assumed that the concrete is stressed to a uniform compression of 0.45 f cu over the full depth of concrete on the compression side (BS a) and concrete is assumed to have no tensile strength. The longitudinal reinforcing bars are ignored. Once the required depth of compression block is determined, the location of the bottom of the compression block, y c, is also determined. For simple cases when the deck on the left and right sides of the beam have the same dimensions, y c can be calculated as follows: y c = D + D s - a For simple cases when the deck on the left and right sides of the beam have the same thicknesses and the same rib depths, the calculation of a and y c is simple. This calculation is also simple when there is only one slab on either the left or right side of the beam. However, the program considers the general condition where the slabs on the left and right sides are different. In such cases, the compression block may include part of the slab on either side or both sides, full slab and part of the ribs on either side or both sides. Also note that if the deck ribs are perpendicular to the beam, the ribs do not contribute to the compression block. The deck ribs may orient differently, parallel or perpendicular to the beam, on the two sides of the beam. Those geometric variations make the calculation of a and y c. The program handles these generalities using an efficient iterative procedure. In the iterative procedure, the program starts with a small value of a and progressively increases its value until the compression in concrete based on the assumed compression block becomes equal to. If the concrete decks are the same on both sides, or if there is one concrete deck at either side, and if the block sizes are smaller than the slab thickness, the iterative procedure will converge in a single step. Depth of the Compression Block Page 6 of 14

7 Figures 3 and 4 show the internal forces for the conditions where the compression stress block lies in the slab and the deck rib, respectively, for a simple case where decks at the left and right sides are the same. Bottom of the compression block y c a C C 1 D Figure 3: Rolled Steel Section with PNA in Concrete Slab Above Metal Deck, Positive Bending (For User-Defined Welded Sections, Ignore the Fillets) Bottom of the compression block y c a C C 1 C C 2 Figure 4: Rolled Steel Section with PNA within Height of Metal Deck, Positive Bending (For User-Defined Welded Sections, Ignore the Fillets) Depth of the Compression Block Page 7 of 14

8 Location of the Plastic Neutral Axis in Steel The location of the PNA is located by the distance y p that is measured from the bottom of the beam bottom flange (not cover plate) to the PNA. For steel sections without cover plates, y p represents the depth of the tension zone of the steel section under plastic condition. The calculation of y p involves finding its value so that the total steel tension force becomes equal to, which is also equal to the compression force in concrete. The definition of is given previously in this Technical Note. In determining the value of y p, it is assumed that the structural steel is stressed to its design strength, p y, either in tension or in compression for all classes of sections, including Class 1 (Plastic), Class 2 (Compact), and Class 3 (Semi-Compact) (BS b). Class 4 (Slender) sections are not designed for composite beams. For sections under the influence of high shear, the web is ignored in calculating the plastic moment capacity (BS 5.3.4). The location of the PNA is heavily influenced by the partial composite connection ratio, PCC. If the PCC is 1 and F conc,max is greater than F steel,max, y p will be equal to the full depth of the beam D. If PCC is less than 1, or if PCC is 1 but F steel,max is greater than F conc,max, will be less than F steel,max, and the PNA will be below the top of the top flange. The location of the PNA can lie in any of the six following general locations depending on the relative value of and F steel,max. See Figures 5 to 10 for more details. Within the beam top flange. Within the beam top fillet (applies to rolled shapes from the program's section database only). Within the beam web. Within the beam bottom fillet (applies to rolled shapes from the program's section database only). Within the beam bottom flange. Within the cover plate (if one is specified). Note it is very unlikely that the PNA would be below the beam web but there is nothing in the program to prevent it. This condition would require a very large beam bottom flange and/or cover plate and a small PCC. Location of the Plastic Neutral Axis in Steel Page 8 of 14

9 For typical composite beams with equal flange and moderate PCC, the PNA would lie in the upper side of the web, in the top fillet, or in the top flange. Adding a cover plate would drag the PNA down. The program calculates the value of y p using an efficient procedure. The program starts with a value of y p equal to D and progressively decreases its value until the total tensile force in the steel section (including the cover plate if present) based on the assumed location of the PNA becomes equal to. In that procedure, if the location of the PNA is known to lie in any one of the six general locations described previously, the value of y p is determined directly. That means the value of y p can be obtained by at best six trials. The details of the expressions for different cases are given as follows: If = F steel,max, then, y p = D, else if (F steel,max 2 T top B top p y ) then, y p = D (F steel,max ) / (2 T top B top p y ), else if F steel,max 2 (T top B top + k depth k width ) p y then, y p = D T top (F steel,max 2 T top B top p y )/ (2 k width p y ), else if F steel,max 2 (T top B top + k depth k width + 2 t d) p y then, ( F ) steel, max 2TtopBtop py 2kdepthkwidth py Fstud y p = D T top k depth 2tp y, else if F steel,max 2 (T top B top + k depth k width + t d + k depth b width ) p y then, y p = D T top k depth d [F steel,max 2(T top B top + k epth k width + t d) ], [2 k width p y ] Location of the Plastic Neutral Axis in Steel Page 9 of 14

10 else if F steel,max 2 (T top B top +2 k depth k width + t d + T bot B bot ) p y then, y p = D T top k depth d k depth [F steel,max 2(T top B top + 2k epth k width + t d)p y ], [2 B bot p y ] else, [F steel,max ] 2(T top B top + 2k epth k width + t d + T bot B bot )p y y p = [2 B cp p ycp ] [2 B cp p ycp ] Figures 5 through 10 show the internal forces for the conditions where the PNA lies in the six general locations of the steel sections. Those locations were described previously in this section of this Technical Note. In the figures, the rolled sections and welded sections are treated under uniform framework, even though there is no fillet in the welded section. For welded sections, the depth of the fillets should be considered as zero in all expressions. Also, Figures 6 and 8 should be ignored for welded sections. Location of the Plastic Neutral Axis in Steel Page 10 of 14

11 y p z p Plastic neutral axis (PNA) C F T T F T T K T T Web T K B T F B T C P Figure 5: Rolled Steel Section with PNA within Beam Top Flange, Positive Bending (For User-Defined Welded Sections, Ignore the Fillets) y p z p Plastic neutral axis (PNA) C F T C K T T K T T Web T K B T F B T C P Figure 6: Rolled Steel Section with PNA within Beam Top Fillet, Positive Bending (This Case Does Not Apply for Welded Sections) Location of the Plastic Neutral Axis in Steel Page 11 of 14

12 y p z p Plastic neutral axis (PNA) C F T C K T C Web T Web T K B T F B T C P Figure 7: Rolled Steel Section with PNA within Beam Web, Positive Bending (For User-Defined Welded Sections, Ignore the Fillets) C F T C K T y p z p Plastic neutral axis (PNA) C Web C K B T K B T F B T C P Figure 8: Rolled Steel Section with PNA within Beam Bottom Fillet, Positive Bending (This Case Does Not Apply for Welded Sections) Location of the Plastic Neutral Axis in Steel Page 12 of 14

13 C F T C K T y p z p Plastic neutral axis (PNA) C Web C K B C F B T F B T C P Figure 9: Rolled Steel Section with PNA within Beam Bottom Flange, Positive Bending (For User-Defined Welded Sections, Ignore the Fillets) C F T C K T y p z p Plastic neutral axis (PNA) C Web C K B C F B C CP T C P Figure 10: Rolled Steel Section with PNA within Cover Plate, Positive Bending (For User-Defined Welded Sections, Ignore the Fillets) Location of the Plastic Neutral Axis in Steel Page 13 of 14

14 Plastic Moment Capacity for Positive Bending After the depth of the compression block and the location of the PNA are known, the forces in all individual elements are computed using the design basis described in the Overview section of this Technical Note. In addition, the centroid of tension and compression forces can be determined. The plastic moment capacity is determined using statics. If the shear is high, the web of the steel section is ignored in computing the plastic moment capacity. In general, the forces in the following individual elements are considered. Concrete slab above the metal deck (left) Concrete slab above the metal deck (right) Concrete ribs in the metal deck (left) Concrete ribs in the metal deck (right) Steel in the beam top flange Steel in the beam top fillet Steel in the beam web Steel in the bottom fillet Steel in the bottom flange Steel in the cover plate Depending on the size of the concrete compression block, some of the forces in concrete can be zero, because concrete tensile strength is assumed to be zero. Also, depending on the location of the PNA, some of the forces in any of the six elements can be compressive and some can be tensile. However, the element in which the PNA will lie has been split into two parts: one involving tension and the other part involving compression. Because the total axial force over the whole composite section is zero, the moment can be computed using any axis. The program uses the bottom of the bottom flange as the reference axis for calculating the plastic moment capacity. Plastic Moment Capacity for Positive Bending Page 14 of 14