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The design load combinations are various combinations of the load cases for which the structure is required to be designed. The load combination factors are applied to the forces and moments obtained from the analyses cases to form the factored design forces and moments for the load combination.
The design load combinations are the various combinations of the analysis cases for which the structure is required to be checked. In this study, it is considered that the structure is only subjected to dead load (D), live load (L), and earthquake load(E), and earthquake forces are reversible. Consequently, the following load combinations are needed to be considered for the design of reinforced concrete frame;
1.4 D (ACI 9-1)
1.2 D + 1.6 L (ACI 9-2)
0.9 D + 1.0 L 1.0 E (ACI 9-5)
0.9 D 1.0 E (ACI 9-7)
B.2 Column Design
For the design of columns, the geometry of the reinforcing bar configuration adopted for each concrete column section has been defined to determine the column capacity. The design procedure for the reinforced concrete columns of the structure involves in calculating the uniaxial axial force-moment capacity and shear capacity. To determine uniaxial moment capacity of a concrete section, the uniaxial interacting diagram is generated for all of the different concrete section types of the model. The capacity ratio is calculated using the obtained interaction diagrams for the factored axial force and uniaxial bending moments obtained from the design load combinations at each station of the column. The column axial force-bending moment interaction diagram of concrete sections is numerically defined by a series of discrete points that are generated on the uniaxial interaction failure surface. The interaction surfaces are generated for the range of allowable reinforcement; 1 to 8 percent for Ordinary and Intermediate Moment Resisting Frames (ACI 10.9.1) and 1 to 6 percent for Special Moment Resisting Frames (ACI 18.104.22.168). A typical uniaxial interaction surface is shown in Figure A.1.
Figure B.1 Typical Axial Load-Moment interaction diagram.
The coordinates of the points on the uniaxial interacting diagram are determined by rotating a plane of linear strain on the concrete column section as shown in Figure B.2. The formulation of column capacity is based on the general principles of ultimate strength design (ACI 10.3). The maximum concrete strain (¥c) at the extremity of the section is limited to 0.003 (ACI 10.2.3) on the linear strain diagram. And the stress in the steel (given by the product of the steel strain (¥s) and the steel modulus of elasticity (Es)) is limited to the yield stress of the steel (fy) (ACI 10.2.4).
Varying linear strain plane
neutral axis direction
Figure B.2 Idealized strain distribution for generation of interaction surface.
The concrete compression stress block is assumed to be rectangular as shown in Figure A.3. The stress value of compression stress block is taken 0.85fc (ACI 10.2.7.1). The depth of the equivalent rectangular stress block, a, is computed with:
where, c is the depth of section in compression strain and,
Figure B.3 Idealization of stress and strain distribution in a column section.
The strength reduction factor ¦, is used for the generation of the interaction surface. The value of ¦ used in the interaction diagram varies from compression controlled section to tension controlled section according to the maximum tensile strain in the reinforcing bars at the extreme edge (¥t) (ACI 22.214.171.124).
The definition of compression controlled section is used for the sections when the tensile strain in the extreme tension steel in the section is equal to or less than the compression controlled strain limit at the time the extreme concrete strain in compression reaches its assumed strain limit of (¥c,max), which is 0.003. Sections are tension controlled when the extreme tensile strain in the steel bars is equal to or greater than 0.005, just as the extreme concrete strain in compression reaches its assumed strain limit of 0.003 (ACI 10.3.4). The sections are considered to be in a transition region between compression controlled and tension controlled sections according to the value of the maximum tensile strain in the reinforcing bars at the extreme edge (¥t) (ACI 10.3.4). The strength reduction factor ¦, is linearly interpolated between two values for the section within the transition region as shown in the following (ACI 9.3.2);
where, ¦ t = 0.90 for tension controlled sections, ¦ c = 0.65 for compression controlled column sections with tied reinforcement.
The maximum compressive axial load for tied column sections, ¦Pn, max, is calculated as;
¦ Pn(max) = 0.80 ª [0.85 f 'c (Ag - Ast ) + fy Ast ] (ACI 10.3.6.2)
The column bending moment capacity for each design load combination is found by using the obtained uniaxial interacting diagram at both end of the column member. By this way, the design point defined by the resulting axial load and bending moment set (obtained from load combinations) is determined whether it lies within the interaction surface or not. If the design point lies within the interaction surface, the column capacity is adequate, or else, the design point lies outside the interaction surface, the column is overstressed.
The shear design of columns involves the shear reinforcing for a particular column for a particular design load combination resulting from shear forces. Firstly, the factored forces, Pu and Vu, acting on the column section are determined to calculate the reinforcement steel required to carry the acting loads. In addition to the factored shear forces, the shear design of the columns is also based on the maximum probable moment strengths and the nominal moment strengths of the members or special and Intermediate moment resisting frames (Ductile frames) (IBC 2006, ACI 126.96.36.199, 21.12.3).
In the shear design of Special moment resisting frames (i.e., seismic design), in addition to the factored design loads, the shear capacity of the column is also checked for the capacity shear design force based on the maximum probable moment strengths of the column and beams framed to the column.
The shear capacity force in the column, Vu, is determined from consideration of the acting shear forces on the column section. Two different capacity shears are calculated for the column section. The first design force is based on the probable moment strength of the column, while the second is computed from the probable moment strengths of the beams framing into the column. The design strength is taken as the minimum of these two values, but never less than the factored shear forces obtained from the design load combinations.
(ACI 188.8.131.52, IBC 2006)
Where, Vec = the shear capacity force of the column based on the probable maximum flexural strengths of the two ends of the column, Veb = the shear capacity force of the column based on the probable moment strengths of the beams framing into the column. The shear capacity of the column, Vec, is calculated using the maximum probable flexural strength at the two ends of the column for existing factored axial loads. The shear capacity force is computed for clockwise rotation of the joint at one end of column and the associated counter-clockwise rotation of the other joint of column. These situations produce two shear capacity force, and the maximum of these two values is taken as the Vec. The maximum probable positive and negative moment capacities, Mpr+ and Mpr-, of the column under the influence of the axial force, Pu, are computed using the uniaxial interaction diagram in the corresponding direction. Therefore, Vec is the maximum of Ve1c and Ve2c.
MI+ and MI - = positive and negative probable maximum moment capacities at end I of the column using a steel yield stress value of α fy (α =1.25) and no reduction factor (Ø =1.0), MJ+ and MJ - = positive and negative probable maximum moment capacities at end j of the column using a steel yield stress value of α fy (α =1.0) and no reduction factor (Ø =1.0), and L= the clear span of the column.
The capacity shear of the column based on the flexural strength of the beams framing into the column, Veb, is computed using the maximum probable positive and negative moment capacities of each beam framing into the top joint of the column. Both clockwise and counter-clockwise rotations are considered separately, then the sum of the beam moments is calculated as a resistance to joint rotation. The capacity shear of the column due to the flexural strength of the beams in the columns is determined assuming that the point of inflection occurs at mid-span of the columns above and below the joints. The effects of load reversals are investigated and the maximum of the joint shears obtained from the two cases is selected as the capacity shear.
where, Ve1b = the column capacity shear for clockwise joint rotation, Ve2b = the column capacity shear for counter-clockwise joint rotation,
Mr1 = sum of beam moment resistances with clockwise joint rotations, Mr2 = sum of beam moment resistances with counter-clockwise joint rotations, and H = distance between the inflection points (this distance is equal to the mean height of the columns above and below the joint, the distance is taken as one-half of the height of the column at the bottom of the joint for the columns at top story).
In the shear design of columns, the required shear reinforcement in the form of ties is determined according to the principles of ultimate strength design. After the concrete shear capacity and required shear design capacity are determined, the area of shear reinforcement, Av and the spacing between ties s, are calculated to carry the factored shear forces obtained from the design load combinations safely.
The concrete shear capacity is calculated separately, for axial compression and axial tension cases. For axial compression loadings, the nominal shear carried by the concrete is given by,
and the nominal concrete shear capacity for axial tension loadings is given by,
where, Pu = the factored axial load at a section, Acv = the area of concrete used to determine shear stress (shown in Figure B.4), Ag = the gross area of concrete, bw = the width of the column section, d = the distance from compression face to tension reinforcement.
Direction of Shear Force
Figure B.4 Shear stress area Acv for rectangular column section.
The required shear reinforcement in the form of ties within a spacing s obtained from factored loads and probable member moment capacities is calculated by the following design procedure for rectangular columns.
For special moment frames, column ends require adequate confinement and adequate shear reinforcement in order to ensure column ductility and to prevent shear failure prior to the development of flexural yielding at the column ends. The rectangular hoop reinforcement is used as the transverse reinforcement in this design study and regulations given in ACI design code for special seismic moment frames are shown in Figure B.5.
larger of b or d
1/6 of clear span
Figure A.B5 Confinement requirements at column ends.
B.3 Beam Design
In the design of concrete beams, the beams are designed for flexure and shear to calculate the required areas of steel for flexure and shear based on the beam moments which are computed under the factored loads,. The beams are divided three equal parts along their span and the design flexural moments and the design shear forces are calculated for each part. The section of beams is considered as singly reinforced rectangular section.
The factored moments for each design load combination at a particular beam section (for three part of beam) are obtained by factoring the corresponding moments for different analysis cases. The moment capacities of beam sections (parts at both ends and the middle) are checked for the factored design moments obtained from all of the design load combinations. Positive moment capacity is produced by bottom steel and the negative moment capacity is produced by top steel. In such cases, the beam is always designed as a rectangular section.
The design procedure of beam sections is based on the simplified rectangular stress block (ACI 10.2) in Figure B.6. Furthermore, it is assumed that the net tensile strain of the reinforcing steel shall not be less than 0.005 (tension controlled) (ACI 10.3.4). All of the beams are designed ignoring axial force, because it is assumed that the design ultimate axial force does not exceed Ø(0.1fcAg ) (ACI 10.3.5).
The moment capacity of beams, Mu (for positive and negative moments), are calculated by following procedure.
¥€ = 0.003
Figure A.B6 Idealization of stress and strain distribution in a beam section.
where, the depth of the compression block, a,
and the depth of the compression block, a,should satisfy the following conditions,
where, As = area of tension reinforcement, a = the depth of compression block, amax = maximum allowed depth of compression block, bw = the width of the beam section, c = the depth to neutral axis, d = the distance from compression face to tension reinforcement, fy = specified yield strength of flexural reinforcement, fc′ = specified compressive strength of concrete, β1 = factor for obtaining depth of compression block in concrete, ¥c,max = maximum usable compression strain allowed in extreme concrete fiber (0.003), ¥s,min = minimum tensile strain allowed in steel rebar at nominal strength for tension controlled behavior (0.005).
The minimum flexural tensile steel required in a beam section shall not be less than the following two limits;
(ACI 184.108.40.206, ACI 10.5.1)
The flexural steel in a beam section is limited to the maximum given by
For Special moment resisting concrete frames (seismic design), the beam design satisfies the following additional conditions. At any end (support) of the beam, the beam positive moment capacity (i.e., associated with the bottom steel) would not be less that 1/2 of the beam negative moment capacity (i.e., associated with the top steel) at that end (ACI 220.127.116.11). Neither the negative moment capacity nor the positive moment capacity at any of the sections within the beam would be less than 1/4 of the maximum positive or negative moment capacities of any of the beam end (support) stations (ACI 18.104.22.168).
Figure B.7 Reinforcement requirements for flexural members of Special Seismic Moment Frames.
The shear reinforcement is designed for each design load combination at three part along the beam span. For shear design, the effects of axial forces on the beam section are neglected. For Special and Intermediate moment frames (ductile frames), the shear design of the beams is also based on the maximum probable moment strengths and the nominal moment strengths of the members, respectively, in addition to the factored design. In the design of Special moment resisting concrete frames (i.e., seismic design), the capacity shear force, Vp , is calculated from the maximum probable moment capacities of each end of the beam and the gravity shear forces. The procedure for calculating the design shear force in a beam from the maximum probable moment capacity is the same as that described for a column earlier in this section.
The design shear force is then given by (ACI 22.214.171.124, IBC 2006);
where, Vp, is the capacity shear force obtained by applying the calculated maximum probable ultimate moment capacities at the two ends of the beams acting in two opposite directions. VD+L is the contribution of shear force from the in-span distribution of dead and live loads with the assumption that the ends are simply supported.
where, MI - = the moment capacity at end I, with top steel in tension, using a steel yield stress value of αfy and no reduction factors (Ø = 1.0), MJ+ = the moment capacity at end J, with bottom steel in tension, using a steel yield stress value of αfy and no reduction factors (Ø =1.0), MI + = the moment capacity at end I, with bottom steel in tension, using a steel yield stress value of αfy and no reduction factors (Ø = 1.0), MJ - = the moment capacity at end J, with top steel in tension, using a steel yield stress value of αfy and no reduction factors (Ø = 1.0), L= the clear span of beam.
The moment strengths are determined using a strength reduction factor of 1.0 and α is equal to 1.25 (ACI 2.1, R126.96.36.199).
After the design shear forces are calculated, the required shear reinforcement is calculated by the following design procedure for three parts of the beam.
The allowable concrete shear capacity is given by;
At the ends of flexural members, where plastic hinges are likely to form, adequate confinement is required in order to ensure sufficient ductility of the members under reversible loads. Furthermore, the transverse reinforcement is required at the end of beam to assist the concrete in resisting shear and to maintain lateral support for the reinforcing bars. For flexural members of special moment frames, the transverse reinforcement requirements are given in Figure B.8.
Figure B.8 Transverse reinforcement requirements for flexural members of Special Seismic Moment Frames.
In the shear design of the beam and column members, the diameter of shear reinforcements (ties) is taken as 8 mm, if the shear capacity of member or the shear constraints doesn't satisfy the requirements, the the diameter of shear reinforcements (ties) is taken as 10 mm.
Additionaly, for Special moment resisting frames, at a particular joint for a particular column direction the ratio of the sum of the beam moment capacities to the sum of the column moment capacities should satisfy following condition (ACI 188.8.131.52);
ΣMnc = sum of the nominal flexural strengths of columns framing into the joint, evaluated at the faces of the joint; individual column flexural strength is calculated for the associated factored axial force, ΣMnb = sum of nominal flexural strengths of the beams framing into the joint. The capacities are calculated with no reinforcing overstrength factor α, α = 1, and with no Ø factors (Ø = 1.0).
Spacing of lateral supports for a beam shall not exceed 50 times the width of beam,. Minimum thickness of beam, h should satisfy following conditions (; the length of beam).
for one end continuous
for both ends continuous
Detailing of beams is arranged in respect of requirements on ACI 315-99. For non-perimeter beams, typical details in code, the development lengths of bars used at spans 0.3 times of adjacent spans. The length of reinforcement bars are calculated according to these specifications, and hook lengths at the end of some reinforcement bars is taken as 12db (db; the diameter of bar).
The development lengths,, for reinforcement bars at column sections are computed with following formula.
where, the material properties coefficients, ™t ,™e , ¬ are taken as 1.0.