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Pr Concrete shrinks as it hardens. In addition. LL Pa 2 Dead load pressure. If necessary adjust this value depending on your judgment.. DL Pa 3 Ceiling load and other attachments below the slab Identify the uniform floor pressure Pa to be carried by the slab.. DL Pa Calculate the factored moment M Compute the weight of slab. This load may consist of: Compute the required main bar spacing: The slab is to carry a uniform live load of 7. Temperature bars: See Page 74 for the required steel ratio..
The slab has a length of 4 m with both ends continuous. The slab is reinforced with 12 mm tension bars 1! Steel covering is 20 mm. The beam is required to carry a factored moment of kN-m. The beam may also be L-shaped if it is located at the end of a slab. If however it covers the web as shown in Figure 3. If it falls within the flange as shown in Figure 3. Location of neutral axis bw b. From the strain diagram shown: Note that the strain in concrete is taken as 0.
The grade steel as shown in Figure 3. Location of neutral axis c As shown in Figure 3. In T-beam construction. Where flanges ofT-beam construction are in tension. It also states that in T-beams where the web is in tension. The intention of this section is to minimize the possibilities of flexural cracks that will occur at the top face of the flange due to negative moments. The following code requirements shall be applied for this case: In checking for maximum p Pmax.
For beams with slab on one side only. For other T-beams. For isolated beam Assume that the entire flange is in compression and solve for Mu1: Compressive force in concrete.
As1 fv Mul o. As max. Assume steel yields if. Area of compression flange.. Solve for z: The beams are 4. The slab thickness is mm. T-beam with the following properties: Flange area. SSOmm llOmm. Assume steel yields.
Continuous compression bars are also helpful for positioning stirrups and keeping them in place during concrete placement and vibration. M"1 is the couple due to compression concrete and the part of the tension steel As1.
The spacing of these ties shall not exceed 16Iongittidinal bar dimeters. According to Sectibn 5: Various tests show that compression reinforcement also prevents the. Deformed wire ot welded wire fabric ofequivalent area is allowed. Aside from these reasons. Compression reinforcement is pro': Solve for Pmax and M.
On the other. I According to Section 5. For this reason. This stress must always be checked. If the compression steeL yields. Solve for. Solve for the stres! Solve fQr a and c: Solvefor cby quadraticformula. Solve for M.. Solve for a. Ot yield. Beams with web depth that exceed The total area of longitudinal skin reinforcement in both faces need not exceed one-half of the required flexural tensile reinforcement. Solve for a and c: The size of the beam is limited to x riun. Steelccrirering from bar centroid to outermost fiber is mm for both tension and compression bars..
Mu ni. Solving for a and c: Figure 3. Compression steel area. Use fy. Stress of compressiOn steel.. Problem 3. The beam is reinforced with six mm bottom bars and two mm top bars located 65 mm from the top of the beam. Assume concrete over measured from bar centroid to be 70 mm in compression and 80 mm in tension.. Shear failures are very dangerous especially if it happens before flexure failure because they can occur without warning.
Shear failure in beams Without stirrup. To avoid shear failure. Stirrups prevent this occurrence especially if they are closely spaced as in Figure 4.
Shear span p. Shear failure may be a diagonal tension failure. Shear span p Splitting or True Shear Failure Splitting failure occurs when the shear span is less than the effective depth d.
Diagonal Tension Failure Diagonal tension failure usually occur when the shear span is greater than 3d or 4d. For a beam that does contain web reinforcement. Vc Eq. When Mm as computed by Eq. Vc shall not be taken greater than lf' Vc. Quantity V11 dj M11 shall not be taken greater than 1. Ag Eq. The shear strength provided by the stirrups is given by the following but shall not be taken greater than d.
In the foregoing. Quantity N. For nonprestressed members. Stirrups and other bars or wires used as shear reinforcement shall extend to a distance d from extreme compression fiber and shall be anchored at both ends to develop the design yield strength of reinforcement.
Where shear reinforcement is required. For m. Such tests shall simulate effects of different settlement. Spacing of stirrups: Calculate the factored shear force V. Calculate the shear strength provided by concrete. Vc If v. For prestressed member. See Figure 4. When V. Beam width. Concrete strength. Use q. The nominal shear capacity of the section is given by: I Note: We cannot check for maximum spacing because bw is not given.
JiB t J7': OK Using vertical U stirrup: Vc] v. Stirrups are not needed For V. For V. The dead load shown includes beam weight. Do not consider movement of load. VII v. The depth of a bracket or corbel at its outer edge shall not be less than one-half of the required depth d at the support. The shear span a is the distance from the point of load to the face of support.
With this method. The area of the shear bars shall not be less than 0.
At front face of bracket or corbel. Due to the rough surface at a crack. Concrete placed against hardened concrete not intentionally roughened Concrete placed monolithically.
An equal compression load develops in the concrete at the confined crack The reinforcement rriust be properly developed to prevent pull-out.. Concrete anchored to as-rolled structural steel by headed studs or by reinforcing bars see Sec.. The area of. Concrete placed against hardened concrete with surface intentionally roughened as specified in Sec According to Section Linear interpolation is permitted when partial sand replacement is used..
For non-prestressed members: For prestressed members: If a concentrated torque occurs within this distance. For prestressed members. For non-prestressed members. For hallow sections: For solid sections: The total longitudinal reinforcement including tendons at each section shall resist the factored bending moment at that section plus an additional concentric longitudinal tensile force equal to A1 fyt.
In Prestressed Beams: It shall be permitted to take 8 equal to: The most restrictive requirements for reinforcement spacing and placement must be met.
The spacing of the longitudinal reinforcement including tendons shall satisfy the requirements in Section A closed cage of welded wire fabric with transverse wires perpendicular to the axis of the member..
Minimum Torsion Reinforcement A degree standard hook around a longitudinal bar. Acp A1. Details of Torsional Reinforcement In non-prestressed beams. Closed stirrups or closed ties. There shall be at least one longitudinal bar or tendon in each comer of the stirrups.
Fer of p. Shear failure of beam. The longitudinal bars or tendons shall be inside the stirrups. Use mm U stirrups and assumef.
If there is slipping of steel with respect to surrounding concrete. Figure 5. The steel. BOND In reinforced concrete. For this to happen there must be absolutely no slippage of the bars in relation to the surrounding concrete. It's like a man playing tug of war holding the tip of the rope. No matter how strong that man and that rope is. Under initial loading.
Development length is a function of bar diameter db. If the average bond stress at ultimate is u. Near failure. The Code provides the basic development length ldb for various situations.
The values provided by the code have to be modified for different condition. Hooks may be used in developing bars in tension only.
JTc I 1. Development length may be As required reduced where reinforcement in a flexural member As provided is more than that required by analysis by a factor: Condition Modification Factor. Ore than that required by analysis b Spirals and Ties.
For determining the appropriate modification factors. Condition a Excess Reinforcement. Reinforcement shall extend beyond the point at which it is no longer required to resist flexure for a distance equal to the. Reinforcement enclosed within spiral reinforcement not less than 6mm diameter and not more than mm pitch or within 10 mm ties and spaced at not more than mm on center Modification Factor. Reinforcement IIJ. At simple supports and at points of inflection.
Excess stirrup area Av shall be not less than Adequate anchorage shall be provided for tension reinforcement in flexural members where reinforcement stress is not directly proportional to moment. M" Ld Flexural reinforcement shall not be terminated in a tension zone unless one of the following conditions is satisfied: In beams.
Continuing reinforcement shall have an embedment length not less than the development length ld beyond the point where bent or terminated tension reinforcement is no longer required to resist flexure.
Value of MnfV. Mn is nominal moment strength assur: Development length for positive moment on simply supported beam. Development length for negative moment ln2 Figure 5. Recommended bar details for continuous beams. Hooked-bar detail for development of standard hook.
Standard hooks I I ' r Critical section 65 mm min.. The achlal developine. Splicing may be done by welding. OI' most frequently by lapping bars. For this case. If the reinforcing bar has anjy otl1er than MPa. J -1 Lap splices of deformed bars and deformed wir'e in tension shail be Lap splices shall riot be used for bars larger than 32 mm except as provided by: A full mechanical ' ': A or Bsplice See Section Should the concrete strength f c less than 20 MPa.
Hooks shall not be used to develop bars in compression. It shall be permitted to use K. Basic development lerigth lab shall be -permitted to be multiplied by Reinforcement in excess of that required by analysis Jf the concrete strength [c is 21 MPa. A reinforced. Use Ld. Ld11 The beam is made. If this condition had not been satisfied.
L Assume that the beam carries a uniformly distributed load throughout. Basic development length for 25 mm bars in tension: The following table shows how the moment is affected as tl,le. The code Section 5. The theoretical length of each layer on each side "from the centerline is. For barb: Total A. Extension, the larger value of: For bar c: Calculate the basic development length fo;: A rectangular beam is reinforced with mm top bars to resist tensile forces. If the height of the column is less than three times its least lateral dimension, it may be considered as short compression blocks or pedestal.
Pedestals may be designed without reinforcement with a maximum permissible compressive strength of 0. If the compressive strength is greater than this value, the pedestal will have to be designed as a reinforced concrete short column. The load of the short columns depends on the dimension and the strength of the material of which it is made.
If the length of the column is increased, the chances that it will fail by lateral buckling will be increased. Columns that fail by buckling are called long. P-delta Moment When a column is subjected to priman; moments M, such as those caused by applied loads or joint rotation, the axis of the member deflects laterally. This deflection causes additional moment applied. If the secondary moment becomes too large, the column is said to be long column and it is necessary to design its section for the sum of both primary and secondary moments.
Tied and spiral columns are the most common forms. Either type may be circular, octagonal, square, or rectangular section.
Tied columns may also be L, T or other irregular shape. To counter the effect of possible eccentricities, the nominal strength P, is multiplied by 0. Finally, the ultimate axial load capacity of the column P" is 0Pu, where 0 is 0. Clear distance between longitudinal bars shall be not less than 1. Vertical spacing of ties shall be the smallest of the following: Section 5.
Typical tie arrangement. Where longitudinal bars are located around the perimeter of a circle. Ast shall not be less than 0. UseJO-mm diameter ties for mm bars or smaller and at least 12 mm in size for 36 mm and bundled longitudinal bars. The minimum number of longitudinal bars is 6. Limits of reinforcement for spiral columns Section 5. For cast-in-place construction. Clear spacing between spirals shall not exceed 75 rnrn. Splices of spiral reinforcement shall be lap splices of 48db but not less than mrn or welded.
Spiral column The axial load capacity of a spiral column is given by '. The minimum spiral percentage is given by: In columns where all the concrete is under compression. If floor space is not a problem. For multi-story buildings. All axial load strength not assigned to concrete of a composite member should be developed by direct connection to the structural steel shape. Strength of a composite member is computed for the same limiting j. Onditions applicable to ordinary reinforced concrete members.
Any axial load strength assigned to concrete of a composite member should be transferred to the concrete by members or brackets in direct bearing on the composite member concrete. Composite columns. According to Sec. Design yield strength of structural steel core should be the specified minimum yield strength for grade of structural steel used but not to exceed MPa. Specified compressive strength of concrete f c should be not less than 17 MPa. Lateral ties should extend completely around the structural steel core.
Longitudinal bars located within the spiral should be not less 0. Specified compressive strength of concrete [c should be not less than 17 MPa. Welded wire fabric of equivalent area is permitted. Longitudinal bars located within the spiral may be considered in computing A.
Spiral reinforcement should conform to Sec. A longitudinal bar should be located at every corner of a rectangular cross section. Vertical spacing of lateral ties should not exceed 16 longitudinal bar diameters. Longitudinal bars located within the ties should be not less than 0.
Composite columns with spiral and tie reinforcement. Longitudinal bars located within the ties may be considered in computing Ast for strength but not in computing It for evaluation of slenderness effects.
Use 28 mm main bars and 10 mm ties. The ideal value is from 0. Use 22 nun main bars and 10 m. Use mm main reinforcement and mm spiral with 30 mm steel covering. Properties of Wl4 x Verify also if this section complies with the requirement of the Code. Use f.. II oad. Properties of W14 x Repeat Problem 6. Use 30 mm steel cover. The following failures are possible under the combined axial and bending loads.
The load capacity given by Eq. Bars on tensile side yield at the same time concrete on compression side crushes at 0. Balanced loading condition.
Failure occurs by crusping of concrete. Columns will tend to bend under the action of moment. Bars in far side in tension but have not yielded. Large axial load with. Large axial load and small moment with the entire cross-section in compression Failure occurs by crushing of concrete with all bars in compression. Large moment. Failure initiated by yielding of tensile bars.
Failure occurs like a beam. LEM 7. It represents the location of the resultant force produced by the steel and concrete. The eccentricity of a column load is the distance from the load to the plastic centroid of the column. In locating the plastic centroid. For symmetrical sections. Large bending moment. X2 X If all steel yields. Another approach to solve for Pn is to have series of assumption for the values of c until equilibrium conditions are satisfied.
FiJr column with large eccentricity and when all steel has yielded: The following formulas can be applied for columns with two rows of reinforcements. Any combination of loading that falls inside the curve is satisfactory The diagram is made by plotting the axial load capacity of the column at A. Balanced loading occurs when the tension steel just reached its yield strain hiEs and the concrete is strained to 0.
See Problem 7. As a result. Interaction diagram. For every column there is always a balanced loading situation where an ultimate load Pb. In between the two lies the so-called balanced load condition where failure may be of either type. In computing the. In between the points A and C. Point B is called the balanced point. In reference to point D. Dividing the section into two rectangles. For concrete: Determine the plastic centroid of the column measured from the mm side. The center of the bars is located 65 mm from the column edge.
Calculate the balanced loading Pbn.. A rectanghlar column mm by mm is reinforced with mm diameter bars with three bars along each mm side. After successive trials. Interaction diagram From the diagram shown: JAg elh. MPa Figure J Agh. Pn Design a square tied column to carry a factored axial load of 2, kN and a factored Il10ment of kN-m. Use 25 mm bars to be placed uniformly around the faces of the column. With the column dimension known, the steel ratio can be computed.
Assume average compressive stress as 0. Since the bars must be placed uniformly around the faces, use mm bars with 3 bars in each face. This is t: Jle usual case of corner columns in buildings where beams or girders frame into the column from both directions only. Biaxial bending on circular columns would not be a problem due to polar symmetry of the column. If there is bending moment about both x and y axes, the biaxial moment and eccentricity can be computed from the following formulas: For column shapes other than circular ones, the analysis would be as shown in the figure below.
One could think how difficult to solve for Pu using the statics equation as presented in Problem 7. Such solution would lead to correct answer, but the mathematics involved is so complicated that the method is not a practical one, but with the aid of computer such is not a problem. The equation is: The Bresler equation works well as long as P" is at least as large as 0.
Should P, be less than 0. Using theinteraction diagram: For eccentricity ex: Calculate the nominal axial load capacity of the. The column is to carry a factored load of Such analysis should take into account influence of axial loads and variable moment of inertia on member stiffness and fixed-end moments. This is the situation of a short column. Figure 8. Stick of the same material and cross-sectional area subjected to different compressiv! To visualize the effect of slenderness.
As the slenderness increases. In lieu of this procedure. For rectangular compression members: Where column capitals or haunches are present. For compression members not braced against sidesway. For other shapes. Radius of Gyration Radius of gyration r may be taken equal to 0. EIILofbeams Eq. EI I L of the column meeting at that joint including the column in question. To use this chart. Alignment chart For columns for which the slenderness ratio lies between 22 and M2s Eq.
According to Sect. For frames not braced against sidesway. EI in Eq. Pr are the summations for all columns in a story. For braced. In calculation of P. For frames braced against sidesway. Cu shall be taken as 1. M2s in Eq. M2 b Eq. Find its minimum dimension such that slenderness may not be considered in the analysis.
The column length is 4. Calculate the magnified factored moment Me. The column is loaded in single curvature by the ultimate factored moments of kN-m at its top end and kN-m at its bottom end. The column carries factored axial load due to dead load of kN and kN due to live load. This problem did not mention on which side the moments were applied. M2b slenderness must be considered. See Page 21 Section See Page 21 Where 1.
Investigate the adequacy of the column for an unsupported length of a 3 meters. M2b Slenderness must be considered. Mlb 0. Computer calculation shows that the column will be adequate with mm bars on each side. The column is used to carry the following loads: EI may be computed conservatively as: MPa From the interaction diagram.
Is the column satisfactory to support a P11 of kN and an Mnx of kN-m? The column is braced against sidesway with k factor of 1. Calculate the value of M.. The column is use in a frame braced against sidesway and is bent in single curvature. Reinforced concrete is the most suited material for footing for reinforced concrete and structural steel buildings. This type might be economical where two heavily loaded columns are so spaced that when designed for isolated footing would run into each other.
A wall footing. A combined footing is a longer rectangular slab strip that supports two or more individual columns. The footing for such a column can be combined with an interior column to fit within the property line.
For these types of footings. A floating. This kind of foundation is used where soil strength is low or where column loads are large but where piles or caissons are not used. These are widely used for columns with light load and are not closely spaced. The goal is to remove an amount of earth approximately equal to the building weight. An isolated or single-column footing. Isolated footings. Pile caps are slabs of reinforced concrete used to distribute column loads to group of piles.
Types of footing. Table 9. In the absence of soil investigation. Allowable Soil Pressure. This can be derived on the basis of test borings. Isolated poles for uses such as flagpoles or signs and poles used to support buildings which are not adversely affected by a mm motion at ground surface due to short term lateral loads may be designed using lateral bearing values equal to two times the tabulated values.
Sandy Clay. Massive Crystalline Bedrock 2. In no case shall the lateral sliding resistance exceed one half the dead load. OH and PT i. GM and GC. Sedimentary and Foliated Rock 3. Except as in Footnote 7 below. Silty Sand. For footings supporting a circular or regular polygon shaped columns orpedestal. Ys is the unit weight of soil above the footing. Yc he.
For isolated footings, the critical sections fot moment are located as follows: One-way footings are those, which are reinforced' in one ditection only, while two-way footings are reinforced in two directions. A 5 in center band width TotalA 5 inshortdirection where. The shear strength of slabs and footings in the vicinity of the columns, concentrated load, or reactions. Beam action one-way , where each critical' section.
For this case, the slab. One"way shear will very often control the depths for rectangvlar footings, whereas twocway shear normally controls the-depth of square footings. The critical sections for development of reinforcement may be assumed at the same locationas those of critical moment. At the base of the column, the permissible bearing strength of for either surfaces is 0 0.
The soil above ;j the footing has a weight of Use 25 l min main bars. This requires several cycles of trial and error procedure because its value affects the. Vc V, ;,. Number of mm bars: Minimum area of dowel or extension bar required by the Code: The column is to be supported by a rectangular.
Unfactored Load g qe. Allowable soil bearing capacity at the l: Assume the piles are adequate to resist the loads. Design the footing.. J apart on a 3 x 3 formation.
Soil bearing capacity. Since soil pressure cannot be in tension. OK Note: This type of footing seldom fail by two-way or punching shear. Use 20 mm main bars. Design the footingsuch that the soil pressure at the base is uniform. The base of footing is 1. Based on wide-beam shear: Effective soil pressure. Yconc hconc. Ysoil hsoil q. Considering 1 m strip: Steel Requirements i.
Pmin Pmax 0. Combined footings. Another situation is when the column is very near the property line. Combined footings support more than one column. In this manner. One situation where these footings may be used is when the columns are so close together so that isolated or individual footing would run into each other.
For column spacing inore than 4. As the distance between -such columns increases.. This is attained wh. The strap should be designe. The footing under the exterior column may be designed as a wall footing. Its depth is determined by the maximufl' bending moment. The interior footing should be designed as a single-column footing.
Allowable soil bearing pressure. Use the following qata: Use 25 nun. Due to complexi. Yc he c Ys-hs q. Usually two or more trials are made before getting its most reasonable. X Ashad'ed V.. Column 1: E Note: The Code did not specify the width to be considered in the design of transverse bars. It is usually assumed equal to column width plus 0.
Using 16 mm bars withf: Allowable soil bearing pressure.? Use the following data: Notations and Symbols OVerreinforced Design Chapter 4 - Shear and Diagonal Tension. Limits, of Stirrups Plastic Centroid Design of columns with. Axial Load Plus bending in Both Axes Types of Footing Critical Sections in Footings Load transfer from Columns to Footings Pesign of Square Footing View Full Document.
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