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Chapter 6 Special footings and beams on the elastic foundations

Chapter 6 Special footings and beams on the elastic foundations

6.1 Introduction This chapter will take up the design of several of the more complicated foundation members such as those required to support several columns in a line or from industrial loadings. When a footing supports a line of two or more columns, it is called a combined footing. A combined footing may have either rectangular or trapezoidal shape or be a series of pads connected by narrow rigid beams called a strap footing. The types of combined footing-Fig.6.1 Both the conventional "rigid"and the beam-on-the- foundation method of combined footing analysis will be presented

6.1 Introduction ▪ This chapter will take up the design of several of the more complicated foundation members such as those required to support several columns in a line or from industrial loadings. ▪ When a footing supports a line of two or more columns, it is called a combined footing. ▪ A combined footing may have either rectangular or trapezoidal shape or be a series of pads connected by narrow rigid beams called a strap footing. ▪ The types of combined footing-Fig. 6.1 ▪ Both the conventional “rigid” and the beam-on-thefoundation method of combined footing analysis will be presented

6.2 Rectangular combined footings Columns located off-center will usually result in a non-uniform soil pressure. To avoid the non-uniform soil pressure,an alternative is to enlarge the footing and place one or more of the adjacent columns in the same line on it. The footing geometry is made such that the resultant of the several columns is in the center of the footing area. This footing and load geometry allows the designer to assume a uniform soil pressure distribution

6.2 Rectangular combined footings ▪ Columns located off-center will usually result in a non-uniform soil pressure. ▪ To avoid the non-uniform soil pressure, an alternative is to enlarge the footing and place one or more of the adjacent columns in the same line on it. ▪ The footing geometry is made such that the resultant of the several columns is in the center of the footing area. ▪ This footing and load geometry allows the designer to assume a uniform soil pressure distribution

6.2 Rectangular combined footings The footing can be rectangular if the column that is eccentric with respect to a spread footing carries a smaller load than the interior columns. The basic assumption for the design of a rectangular combined footing is that it is a rigid member,so that the soil pressure is linear. The pressure will be uniform if the location of the load resultant coincides with the center of area. ◆ This assumption is approximately true if the soil is homogeneous and the footing is rigid

6.2 Rectangular combined footings ▪ The footing can be rectangular if the column that is eccentric with respect to a spread footing carries a smaller load than the interior columns. ▪ The basic assumption for the design of a rectangular combined footing is that it is a rigid member, so that the soil pressure is linear. ▪ The pressure will be uniform if the location of the load resultant coincides with the center of area. ▪ This assumption is approximately true if the soil is homogeneous and the footing is rigid

6.2 Rectangular combined footings In actual practice it is very difficult to make a rigid footing,for the thickness would have to be great. In recognition of the over-design using the conventional (or "rigid")method,current practice tends to modify the design by a beam-on- elastic-foundation analysis. ■ The conventional (or rigid)design of a rectangular combined footing consists in determining the location of the center of footing area.Next the length and width can be found

6.2 Rectangular combined footings ▪ In actual practice it is very difficult to make a rigid footing, for the thickness would have to be great. ▪ In recognition of the over-design using the conventional (or “rigid”) method, current practice tends to modify the design by a beam-onelastic-foundation analysis. ▪ The conventional (or rigid) design of a rectangular combined footing consists in determining the location of the center of footing area. Next the length and width can be found

6.2 Rectangular combined footings With these dimensions the footing is treated as a beam supported by the two or more columns,and the shear and moment diagrams are drawn. The depth,based on the more critical of the two-way action or wide-beam shear,is computed. Critical sections for two-way action and wide-beam shear are the same as for spread footings,i.e.,at d/2 and d,respectively,from the column face. With the depth selected,the flexural steel can be designed using the critical moments from the moment diagram. 喝

6.2 Rectangular combined footings ▪ With these dimensions the footing is treated as a beam supported by the two or more columns, and the shear and moment diagrams are drawn. ▪ The depth, based on the more critical of the two-way action or wide-beam shear, is computed. ▪ Critical sections for two-way action and wide-beam shear are the same as for spread footings, i.e.,at d/2 and d, respectively, from the column face. ▪ With the depth selected, the flexural steel can be designed using the critical moments from the moment diagram

6.2 Rectangular combined footings These beam-type members usually have both positive and negative moments,resulting in reinforcing steel in both the top and bottom of the footing. If we compute the short,or transverse,direction bending moments as for a rectangular spread footing,they will be in substantial error. ■ The reason is the soil pressure is larger near the columns,from their stiffening effect on the footing,and lesser in the zone between columns. That zone closest to and approximately centered on, the column is most effective. The author suggests that the effective zone should be about as shown in Fig.6.3

6.2 Rectangular combined footings ▪ These beam-type members usually have both positive and negative moments, resulting in reinforcing steel in both the top and bottom of the footing. ▪ If we compute the short, or transverse, direction bending moments as for a rectangular spread footing, they will be in substantial error. ▪ The reason is the soil pressure is larger near the columns, from their stiffening effect on the footing, and lesser in the zone between columns. ▪ That zone closest to , and approximately centered on, the column is most effective. ▪ The author suggests that the effective zone should be about as shown in Fig. 6.3

6.2 Rectangular combined footings The column loads are actually distributed over the column width but should always be taken as point loads. This assumption greatly simplifies the shear and moment computations,and the values at the critical locations are the same by either method. It should be self-evident that combined footings are statically determinate for any number of columns. With the column loads known and assuming a rigid footing,the resulting soil pressure q=>P/A.The problem then becomes that of a uniformly loaded continuous 周 beam with all the reactions (the columns)known 调

6.2 Rectangular combined footings ▪ The column loads are actually distributed over the column width but should always be taken as point loads. ▪ This assumption greatly simplifies the shear and moment computations, and the values at the critical locations are the same by either method. ▪ It should be self-evident that combined footings are statically determinate for any number of columns. ▪ With the column loads known and assuming a rigid footing, the resulting soil pressure q=∑P/A. The problem then becomes that of a uniformly loaded continuous beam with all the reactions (the columns) known

6.3 Design of trapezoid-shaped footings A combined footing will be trapezoid-shaped if the column that has too limited a space for a spread footing carries the larger load. ■ In this case the resultant of the column loads (including moments)will be closer to the larger column load,and doubling the centroid distance as done for the rectangular footing will not provide sufficient length to reach the interior column. The footing geometry necessary for a two- column trapezoid-shaped footing:

6.3 Design of trapezoid-shaped footings ▪ A combined footing will be trapezoid-shaped if the column that has too limited a space for a spread footing carries the larger load. ▪ In this case the resultant of the column loads (including moments) will be closer to the larger column load, and doubling the centroid distance as done for the rectangular footing will not provide sufficient length to reach the interior column. ▪ The footing geometry necessary for a twocolumn trapezoid-shaped footing:

6.3 Design of trapezoid-shaped footings The forming and reinforcing steel for a trapezoid footing is somewhat awkward to A= a+b place. 2 ■ With x'falling at a L 2a+b particular location and x'- 3 a+b defining the center of area,the dimensions a L and b have unique 3 2 values that require a simultaneous solution of Eqs.(6.1)and(6.2)

6.3 Design of trapezoid-shaped footings ▪ The forming and reinforcing steel for a trapezoid footing is somewhat awkward to place. ▪ With x’ falling at a particular location and defining the center of area, the dimensions a and b have unique values that require a simultaneous solution of Eqs.(6.1) and (6.2). 3 2 2 3 2 L x L a b L a b x L a b A + + = + =

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