The loads of the building are transferred to the soil through foundations. The sizes type of the foundation should be chosen so as,

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2 The loads of the building are transferred to the soil through foundations. The sizes type of the foundation should be chosen so as, to ensure structural safety of the foundation (no damage at the foundation), to ensure that there is no settlement (no excess of the bearing capacity of soil), to provide economy. The first step for the design of foundation is correct determination of soil properties (i.e. by soil survey at the site).

3 The stress distribution under the foundation depends on, type of loading (loads transferred through the foundation), type of soil, rigidity of the foundation. It is important to ensure that the max. stress under the foundation is smaller than the bearing capacity (allowable stress) of the soil (σ z,max. σ z,allow. ).

4 The types of foundations, 1. Shallow foundations a. Wall Footings: a continuous strip along the length of wall having a width larger than the wall thickness.

5 The types of foundations, b. Independent Isolated Column (Single) Footings: isolated rectangular or square slabs under the column. The length of all sides from the column face to the edge of foundation may be different (due to presence of the bending moments). They are reinforced along both directions and economical solution for relatively small loads.

6 The types of foundations, c. Combined Column Footings: support two or more columns. They are used to provide a uniform stress distribution over the soil. If the geometric center of the footing slab and location of the resultant forces transferred by the columns coincide, this uniform stress distribution can be achieved. They may also be used if the columns are too close to each other.

7 The types of foundations, d. Strap Footings: transfers the loads of two or more columns/shear walls to the ground safely. The straps may be either one-way or two-way. They are much more effective compared to isolated columns, especially when subjected to earthquake loads (i.e. When considerable bending moments are transferred by the column). Besides, these foundations prevent relative settlement, which may be a problem in case of isolated footings. One-way strap footings Two-way strap footings

8 The types of foundations, e. Raft (Mat) Foundations: When the allowable bearing capacity of the soil is low and/or the loads that are transferred from the building are high, the use of raft foundations may be necessary. The loads from the columns/shear walls are distributed over a large area of soil and relative support settlements are prevented. Also, if the ground water level is high, then mat foundation with water proofing may be ideal. The mat foundation may be assumed as an inverted floor system. L (1/4~1/5)L

9 The analysis and design of mat foundations are performed by assuming these systems: a. Rigid b. Flexible Rigid Foundation Assumption: It is assumed that the rigidity of the foundation is very high compared to the rigidity of soil. The stress distribution under the foundation will then depend on the magnitude and type of loads transferred by the structure and weight of the foundation itself. (Case I) (Case II)

10 Rigid Foundation Assumption: Case I: If the resultant of vertical loads from the structure passes close to geometric center of the mat and if the mat plate is thick, then the effect of bending moments may be ignored and stress distribution may be assumed as uniform. σσ zz,mmmmmm = NN dd AA N d : Total vertical loads from the structure A: Area of the mat foundation M dx : Moment along x direction (N d x eccentricity along y-direction) M dy : Moment along y direction (N d x eccentricity along x-direction) I x : Moment of inertia of mat area about x-axis (Case I) I y : Moment of inertia of mat area about y-axis Case II: If the eccentricity (distance between the resultant of vertical loads and geometric center of the mat) may not be ignored, then the stress distribution changes linearly along both axes. Then the maximum stress may be calculated as: σσ zz,mmmmmm = NN dd AA + MM ddxx yy mmmmmm II xx + MM dddd yy mmmmmm II xx (Case II) y max : max. distance from the geometric center, along y-direction x max : max. distance from the geometric center, along x-direction

11 Flexible Foundation Assumption: If the the mat foundation is not rigid (low thickness and no beams), then uniform or linear stress distribution may not be valid. In this case, finite element approach should be used. In this approach, elastic springs are defined under the mat foundation, in order to represent the response of elastic soil (spring coefficient is defined for each different type of soil).

12 In finite element analysis, the mat foundation should be meshed into parts which have approximately 1 m 2 area. The multiplication of spring coefficient (k o ; unit: kn/m 3 ) and max. vertical displacement (δ z,max., found by analysis) will then give the max. stress under the foundation (σ z,max. = k o δ z,max. ). k o unit: kn/m 3 δ z,max. unit: m σ z,max. unit: kn/m 2 Check: σ z,max. σ z,allow.

13 The following mat foundation (for a 4-story building) will be modeled and analyzed by using SAP2000. The thickness of the mat is selected as 0.80 m. The local site class is defined as Z3. The soil type is hard clay and corresponding modulus of subgrade reaction (k o ) is assumed as kn/m 3. The effective depth may be taken as 0.75 m. Materials: C25/S420. Note that cantilever portion of the mat is assumed to have a length of 0.60 m. (away from the column face at all sides). The allowable stress of the soil will be taken as 200 kn/m 2. The analysis was performed considering column loads corresponding to «1.4G+1.6Q» combination for the analysis of building, which yields highest axial column loads. The axial column loads was taken from the analysis of 4-story building (SAP2000 modeling was demonstrated previously). Examples utilized: Examples on Page 143 and Page 642; Betonarme Yapıların Hesap ve Tasarımı; Doğangün, A.

14 The column axial loads at the base which were taken from «1.4G+1.6Q» load combination of the analysis for 4-story building are shown below. The mat foundation will be analyzed for these point loads at the column joints (Load Case: Dead). Note that each portion between the columns will be divided into 4 4 parts (meshing). This will provide approximately 1 m. dimension for each meshed part.

15 Change units to «kn, m, C» File > New Model > Quick Grid Lines

16 Right Click > Edit Grid Data > Modify/Show System Change distances between grid lines according to your plan dimensions! After clicking OK two times, you will see the modified dimensions (as shown below).

17 Define > Materials > Add New Material > (Region: User; Material Type: Other) > OK > «Material Property Data» will be changed as below: For C25 concrete, modulus of elasticity is kn/m 2. The poisson s ratio and coefficient of thermal expansion may be taken as 0.2 and 0, respectively. The weight per unit volume is taken as 25 kn/m 2 in order to include the self weight of the mat foundation in the analysis.

18 Define > Section Properties > Area Sections > Select «Shell» > Add New Section > (Type: Plate Thick; Material Name: C25; Thickness: 0.8) > OK (two times) Thickness for both membrane and bending is taken identical.

19 Click on xy view once again (although you see X-Y Plane view on the screen). Then Draw > Quick Draw Area > «draw the area members starting from the left bottom corner moving to the right each time» (note that «Section» should be the one that you have defined for the mat on the «Properties of Object» window). Set Display Options > Select «Labels» under «Areas» > you will see labels of each area. Areas 2, 3, 9, 10, 11, 12, 13, 14, 15, 16, 18, 19 will be meshed. The number of meshing along the x and y directions are shown below for each area label. Area Label Along x-dir. Along y-dir

20 For each area that will be meshed, select corresponding area first (you may select the areas that will be meshed identically at the same time). For example, the areas 2, 3, 18 and 19 will all be divided into 4 parts along x-direction and 1 part along y-direction. Select these areas at the same time. Edit > Edit Areas > Divide Areas > on the «Divide Selected Areas» select «Divide Area Into This Number of Objects» Along Edge from Point 1 to 2 = Along x-direction Along Edge from Point 1 to 3 = Along y-direction Do the same meshing (divide areas) operation for the other areas and you will see the following meshed area. But the labeling order of the areas is not proper as you see here. Therefore, this will be re-ordered in the next step.

21 Select all model Edit > Change Labels > in «Interactive Name Change» menu > Item Type: Element Labels Area; First Relabel Order: Y, Second Relabel Order: X > on the same menu > Edit > Auto Relabel > All in List > Click OK (twice).

22 Set Display Options > Un-select «Labels» under «Areas» > Select «Labels» under «Joints» > you will see labels of each joint. Note that the labels of joints are also not in order. «Select all» so that the same ordering will be done. Edit > Change Labels > in «Interactive Name Change» menu > Item Type: Element Labels Joint; First Relabel Order: Y, Second Relabel Order: X > on the same menu > Edit > Auto Relabel > All in List > Click OK (twice). At the end, the labeling of each joint will be changed as you see in the next figure.

23 Select all. Assign > Joint > Restrains > on «Joints Restrains» window select only Translation 1, Translation 2 and Rotation about 3 > OK (two times) This will restrain movement along x and y direction, and bending about z axis (torsional movement). Actually rigid supports are placed to restrain these movements.

24 Select all. Assign > Area > Area Springs > on «Assign Spring To Area Object Face» window, Spring Type: Simple, Spring Stiffness per Unit Area: (kn/m 2 ), Simple Spring Resists: Tension and Compr., Area Object Face: Bottom, Spring Tension Direction: Parallel to Area Object Local Axes (3, meaning along positive z-direction), Options: Replace Existing Springs (may be selected). This will add a spring at the bottom face of each area (directed towards positive z-direction). These springs represent resistance of soil which was defined as a spring constant equal to k o =20000 kn/m 2.

25 The following point loads are the axial loads of columns for «1.4G+1.6Q» load combination at the base level. These axial loads are taken from the analysis of the building which was demonstrated previously). Joint Vertical Point Load (kn) After selecting the related joint, Assign > Joint Loads > Forces > Load Pattern Name: DEAD, Foce Global Z: (in the negative z-direction, along gravitational direction) > OK. Repeat this operation to define all loads defined in the next table.

26 Save Project and Run Analysis (you may not run for Modal Load Case; just Dead Load Case is enough). In order to see the vertical displacements (settlement), choose «Show Deformed Shape». On «Deformed Shape» window, choose «Draw displacement contours on area objects». «Area contour component» will be Uz (vertical displacement along z-axis). You will see the change of settlement on the mat foundation area in x-y view (following figure). The maximum settlement is at the bottom left corner, which is (δ z,max. ) m. (negative, in the downward direction). The maximum stress: σσ zz,mmmmmm. = kk oo δδ zz,mmmmmm. = kkkk mm 3 = kkkk mm 2 < kkkk 200 mm 2 SAFE!

27 In order to see the distribution of bending moment about x-axis, choose «Display», «Show Forces/Stresses», «Shells». On «Member Force Diagram» window, choose «M11» as «Component». The reinforcement along y-direction will be determined based on this bending moment. Due to positive moment, there will be tension at the lower side of the foundation. Due to negative bending moment, there will be tension at the upper side of the foundation. Therefore you may determine the bottom and top reinforcement (along the y-direction) by considering highest negative and positive bending moments (about the x-axis), respectively. AA ss,bbbbbbbbbbbb = MM dd + AA ss,tttttt = ff yyyy jj ll dd MM dd ff yyyy jj ll dd Positive sign convention according to SAP2000 (for shell members) As an example, the demonstrated analysis has the highest negative bending moment about x-axis with a value of kn.m (as shown in the next figure). Bending moment about x-axis: M 11 =M xx

28 In order to see the distribution of bending moment about y-axis, choose «Display», «Show Forces/Stresses», «Shells». On «Member Force Diagram» window, choose «M22» as «Component». The reinforcement along x-direction will be determined based on this bending moment. Bending moment about y-axis: M 22 =M yy You may determine the bottom and top reinforcement (along the x-direction) by considering highest negative and positive bending moments (about the y-axis), respectively. AA ss,bbbbbbbbbbbb = MM dd + AA ss,tttttt = ff yyyy jj ll dd MM dd ff yyyy jj ll dd Positive sign convention according to SAP2000 (for shell members) As an example, the demonstrated analysis has the highest negative bending moment about y-axis with a value of kn.m (as shown in the next figure).

29 The mat foundation is reinforced by longitudinal bars at the upper and lower portions of the foundation, along both x- and y-axis (mesh type reinforcement). These longitudinal reinforcements are calculated regarding the bending moments, as explained in the previous slides. The free edges of the foundation is reinforced by hairpin bars (no calculation). The upper reinforcements are held in place by means of rebar chairs (no calculation). The additional punching reinforcement against punching (if required) is as follows: Reference: Website «