# Terzaghi's Bearing Capacity Equations

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1 Terzaghi's Bearing Capacity Equations Bearing capacity equation Bearing capacity factors Bearing capacity Chart Example 1: trip footing on cohesionless soil Example 2: quare footing on clay soil Example 3: Circular footing on sandy clay Terzaghi s bearing capacity theory Figure 1.1: hear stresses based on Terzaghi s soil bearing capacity theory Based on Terzaghi s bearing capacity theory, column load P is resisted by shear stresses at edges of three zones under the footing and the overburden pressure, q (=γd) above the footing. The first term in the equation is related to cohesion of the soil. The second term is related to the depth of the footing and overburden pressure. The third term is related to the width of the footing and the length of shear stress area. The bearing capacity factors, Nc, Nq, Nγ, are function of internal friction angle, φ. Terzaghi's Bearing capacity equations: trip footings: Qu = c Nc + γ D Nq γ B Nγ [1.1] quare footings: Qu = 1.3 c Nc + γ D Nq γ B Nγ [1.2] Circular footings: Qu = 1.3 c Nc + γ D Nq γ B Nγ [1.3] 2/15/2009 Page 1 of 23 ce-ref.com

2 Where: C: Cohesion of soil, γ : unit weight of soil, D: depth of footing, B: width of footing Nc, Nq, Nr: Terzaghi s bearing capacity factors depend on soil friction angle, φ. Nc=cotφ(Nq 1) [1.4] Nq=e 2 (3π/4-φ/2)tanφ / [2 cos2(45+φ/2)] [1.5] Nγ=(1/2) tanφ( Kpr /cos 2 φ -1) [1.6] K pr =passive pressure coefficient. (Note: from Boweles, Foundation analysis and design, "Terzaghi never explained..how he obtained Kpr used to compute Nγ") Table 1: Terzaghi s Bearing Capacity Factors φ Nc Nq Nr /15/2009 Page 2 of 23 ce-ref.com

3 Figure 2 Terzaghi s bearing capacity factors Example 1: trip footing on cohesionless soil Given: oil properties: oil type: cohesionless soil. 2/15/2009 Page 3 of 23 ce-ref.com

4 Cohesion: 0 (neglectable) Friction Angle: 30 degree Unit weight of soil: 100 lbs/ft 3 Expected footing dimensions: 3 ft wide strip footing, bottom of footing at 2 ft below ground level Factor of safety: 3 Requirement: Determine allowable soil bearing capacity using Terzaghi s equation. olution: From Table 1 or Figure 1, Nc = 37.2, Nq = 22.5, Nr = 19.7 for φ = 30 degree Determine ultimate soil bearing capacity using Terzaghi s bearing capacity equation for strip footing Qu = c Nc + γ D Nq γ B Nγ = *2* *100*6*19.7 = lbs/ft 2 Allowable soil bearing capacity, Qa = Qu / F.. = / 3 = 3470 lbs/ft lbs/ft 2 Example 2: quare footing on clay soil Given: oil type: Clay oil properties: Cohesion:2000 lbs/ft 2 Friction Angle: 0 (neglectable) Unit weight of soil: 120 lbs/ft 3 Expected footing dimensions: 6 ft by 6 ft square footing, bottom of footing at 2 ft below ground level Factor of safety: 3 Requirement: Determine allowable soil bearing capacity using Terzaghi s equation. olution: From Table 1 or Figure 1, Nc = 5.7, Nq = 1.0, Nr = 0 for φ = 0 degree Determine ultimate soil bearing capacity using Terzaghi s bearing capacity equation for square footing Qu = 1.3 c Nc + γ D Nq γ B Nγ = 1.3*1000* *2*1+ 0 = 7650 lbs/ft 2 Allowable soil bearing capacity, Qa = Qu / F.. = 7650 / 3 = 2550 lbs/ft lbs/ft 2 2/15/2009 Page 4 of 23 ce-ref.com

5 Example 3: Circular footing on sandy clay Given: oil properties: oil type: sandy clay Cohesion: 500 lbs/ft2 Friction Angle: 25 degree Unit weight of soil: 100 lbs/ft3 Expected footing dimensions: 10 ft diameter circular footing for a circular tank, bottom of footing at 2 ft below ground level Factor of safety: 3 Requirement: Determine allowable soil bearing capacity using Terzaghi s equation. olution: From Table 1 or Figure 1, Nc = 17.7, Nq = 7.4, Nr = 5.0 for φ = 20 degree Determine ultimate soil bearing capacity using Terzaghi s bearing capacity equation for circular footing Qu = 1.3 c Nc + γ D Nq γ B Nγ = 1.3*500* *2* *100*10*5.0 = lbs/ft2 Allowable soil bearing capacity, Qa = Qu / F.. = 17985/ 3 = 5995 lbs/ft lbs/ft 2 Where we live. E 2/15/2009 Page 5 of 23 ce-ref.com

6 Meyerhof's general bearing capacity equations Bearing capacity equation for vertical load, inclined load Meyerhof's bearing capacity factors Chart for Bearing capacity factor Example 4: trip footing on clayey sand Example 5: Rectangular footing on sandy clay Example 6: quare footing with incline loads Meyerhof s general bearing capacity equations Vertical load: Qu = c Nc c Dc + γ D Nq q Dq γ B Nγ γ Dγ [1.7] Inclined load: Qu = c Nc c Dc Ic + γ D Nq q Dq Iq γ B Nγ γ Dγ Iγ [1.8] Where: Nc, Nq, Nr: Meyerhof s bearing capacity factors depend on soil friction angle, φ. Nc = cot φ ( Nq 1) [1.9] Nq = e πtanφ tan 2 (45+φ/2)] [1.10] Nγ = (Nq-1) tan (1.4φ) [1.11] c, q, γ: shape factors Dc, Dq, Dγ: depth factors Ic, Iq, Iγ: incline load factors Friction angle hape factor Depth factor Incline load factors Any φ c=1+0.2kp(b/l) Dc=1+0.2 Kp (B/L) Ic=Iq=(1-θ/90 ) 2 φ = 0 q=γ=1 Dq=Dγ=1 Iγ=1 φ10 q=γ=1+0.1kp(b/l) Dq=Dr=1+0.1 Kp (D/B) Iγ=(1-θ/φ) 2 C: Cohesion of soil γ : unit weight of soil D: depth of footing B, L: width and length of footing Kpr = tan 2 (45+φ/2), passive pressure coefficient. θ = angle of axial load to vertical axis Table 2: Meyerhof s bearing capacity factors φ Nc Nq Nr /15/2009 Page 6 of 23 ce-ref.com

7 Figure 2: Meyerhof s bearing capacity factors Example 4: trip footing on clayey sand Given: 2/15/2009 Page 7 of 23 ce-ref.com

8 oil properties: oil type: clayey sand. Cohesion: 500 lbs/ft 2 Cohesion: 25 degree Friction Angle: 30 degree Unit weight of soil: 100 lbs/ft 3 Expected footing dimensions: 3 ft wide strip footing, bottom of footing at 2 ft below ground level Factor of safety: 3 Requirement: Determine allowable soil bearing capacity using Meyerhof s equation. olution: Determine ultimate soil bearing capacity using Meyerhof s bearing capacity equation for vertical load. Passive pressure coefficient Kpr = tan 2 (45+φ/2) = tan 2 (45+25/2) = 2.5 hape factors: c=1+0.2kp(b/l) = 1+0.2*2.5*(0)=1 q=γ=1+0.1kp(b/l) = 1+0.1*2.5*(0) = 1 Depth factors: Dc=1+0.2 Kp (B/L) = 1+0.2* 2 (0) = 1 Dq=Dγ=1+0.1 Kp (D/B) = 1+0.1* 2.5 (3/3) = 1.16 From Table 2 or Figure 2, Nc = 20.7, Nq = 10.7, Nr = 6.8 for φ = 25 degree Qu = c Nc c Dc + γ D Nq q Dq γ B Nγ γ Dγ = 500*20.7*1*1 +100*3*10.7*1* *100*3*6.8*1*1.16 = lbs/ft 2 Allowable soil bearing capacity, Qa = Qu / F.. = / 3 = 5085 lbs/ft lbs/ft 2 Example 5: Rectangular footing on sandy clay Given: oil properties: oil type: sandy clay Cohesion: 500 lbs/ft 2 Friction Angle: 20 degree Unit weight of soil: 100 lbs/ft 3 Expected footing dimensions: 8 ft by 4 ft rectangular footing, bottom of footing at 3 ft below ground level. Factor of safety: 3 2/15/2009 Page 8 of 23 ce-ref.com

9 Requirement: Determine allowable soil bearing capacity using Meyerhof s equation. olution: Determine ultimate soil bearing capacity using Meyerhof s bearing capacity equation for vertical load. Passive pressure coefficient Kpr = tan 2 (45+φ/2) = tan 2 (45+20/2) = 2. hape factors: c=1+0.2kp(b/l) = 1+0.2*2*(4/8)=1.2 q=γ=1+0.1kp(b/l) = 1+0.1*2*(4/8) = 1.1 Depth factors: Dc=1+0.2 Kp (B/L) = 1+0.2* 2 (4/8) = 1.14 Dq=Dγ=1+0.1 Kp (D/B) = 1+0.1* 2 (3/4) = 1.1 From Table 2 or Figure 2, Nc = 14.9, Nq = 6.4, Nr = 2.9 for φ = 20 degree Qu = c Nc c Dc + γ D Nq q Dq γ B Nγ γ Dγ = 500*14.9*1.2* *3*6.4*1.1* *100*4*2.9*1.1*1.1 = lbs/ft 2 Allowable soil bearing capacity, Qa = Qu / F.. = / 3 = 4406 lbs/ft lbs/ft 2 Example 6: quare footing with incline loads Given: oil properties: oil type: sandy clay Cohesion: 1000 lbs/ft 2 Friction Angle: 15 degree Unit weight of soil: 100 lbs/ft 3 Expected footing dimensions: 8 ft by 8 ft square footing, bottom of footing at 3 ft below ground level. Expected column vertical load = 100 kips Expected column horizontal load = 20 kips Factor of safety: 3 Requirement: Determine allowable soil bearing capacity using Meyerhof s equation. olution: Determine ultimate soil bearing capacity using Meyerhof s bearing capacity equation for vertical load. Passive pressure coefficient 2/15/2009 Page 9 of 23 ce-ref.com

10 Kpr = tan 2 (45+φ/2) = tan 2 (45+15/2) = 1.7 hape factors: c=1+0.2kp(b/l) = 1+0.2*1.7*(8/8)=1.34 q=γ=1+0.1kp(b/l) = 1+0.1*1.7*(8/8) =1.17 Depth factors: Dc=1+0.2 Kp (B/L) = 1+0.2* 1.7 (8/8) = 1.26 Dq=Dγ=1+0.1 Kp (D/B) = 1+0.1* 1.7 (3/8) = 1.05 Incline load factors: θ = tan -1 (20/100) = 11.3 Ic=Iq=(1-θ/90 ) 2 =(1-11.3/90) 2 = 0.76 Iγ=(1-θ/φ) 2 =(1-11.3/15) 2 =0.06 From Table 2 or Figure 2, Nc = 11, Nq = 3.9, Nr = 1.2 for φ = 15 degree Qu = c Nc c Dc Ic + γ D Nq q Dq Iq γ B Nγ γ Dγ Iγ = 500*11*1.34*1.26* *3*3.9*1.17*1.05* *100*8*1.17*1.05*0.06 = 8179 lbs/ft 2 Allowable soil bearing capacity, Qa = Qu / F.. = 8179 / 3 = 2726 lbs/ft lbs/ft 2 arth Bearing capacity from PT numbers One of most commonly method for determining allowable soil bearing capacity is from standard penetration test (PT) numbers. It is simply because PT numbers are readily available from soil boring. The equations that are commonly used were proposed by Meryerhof based on one inches of foundation settlement. Bowles revised Meyerhof s equations because he believed that Meryerhof s equation might be conservative. Meryerhof s equations: For footing width, 4 feet or less: Qa = (N/4) / K [1.12] For footing width, greater than 4 ft: Qa = (N/6)[(B+1)/B] 2 / K [1.13] Bowles equations: For footing width, 4 feet or less: Qa = (N/2.5) / K [1.14] For footing width, greater than 4 ft: Qa = (N/4)[(B+1)/B] 2 / K [1.15] Qa: Allowable soil bearing capacity, in kips/ft2. N: PT numbers below the footing. 2/15/2009 Page 10 of 23 ce-ref.com

11 B: Footing width, in feet K = (D/B) 1.33 D: Depth from ground level to the bottom of footing, in feet. Example 7: Determine soil bearing capacity by PT numbers Given oil PT number: 10 Footing type: 3 feet wide strip footing, bottom of footing at 2 ft below ground surface. Requirement: Estimate allowable soil bearing capacity based on. olution: Meryerhof's equation K = (D/B) = *(2/3) = 1.22 Qa = (N/4) / K = (10 /4) /1.22 = 2 kips/ft2 Bowles equation: Qa = (N/2.5) / K = (10 /2.5) /1.22 = 3.3 kips/ft2 Example 8: Determine soil bearing capacity by PT numbers Given: oil PT number: 20 Footing type: 8 feet wide square footing, bottom of footing at 4 ft below ground surface. Requirement: Estimate allowable soil bearing capacity based on Meryerhof s equation. olution: Meryerhof s equation K = (D/B) = *(4/8) = 1.17 Qa = (N/6)[(B+1)/B]2 / K = (20/6)[(8+1)/8]2 /1.17 = 3.6 kips/ft2 Bowles equation: Qa = (N/4)[(B+1)/B]2 / K = (20/4)[(8+1)/8]2 /1.17 = 5.4 kips/ft2. Effect of water table on soil bearing capacity 2/15/2009 Page 11 of 23 ce-ref.com

12 When the water table is above the wedge zone, the soil parameters used in the bearing capacity equation should be adjusted. Bowles proposed an equation to adjust unit weight of soil as follows: γ e =(2H-D w )(D w /H 2 )γ m +(γ /H 2 )(H-D w ) 2 [1.16] Where γ e = Equivalent unit weight to be used in bearing capacity equation, H = 0.5Btan(45+φ/2), is the depth of influence zone, D w = Depth from bottom of footing to ground water table, γ m = Moist unit weight of soil above ground water table, γ = Effective unit weight of soil below ground water table. Conservatively, one may use the effective unit water under ground water table for calculation. Equation 1.16 can also used to adjust cohesion and friction angle if they are substantially differences. Example 9: Determine equivalent unit weight of soil to calculate soil bearing capacity with the effect of ground water table Given: Moist unit weight of soil above ground water table: 120 lb/ft 3. Moist content = 20% Friction angle, φ = 25 degree Cohesion of soil above ground water table: 1000 lb/ft 2. Cohesion of soil below ground water table: 500 lb/ft 2. Footing: 8 feet wide square footing, bottom of footing at 2 ft below ground surface. Location of ground water table: 6 ft below ground water surface. Requirement: Determine equivalent unit weight of soil to be used for calculating soil bearing capacity. olution: Determine equivalent unit weight: Dry unit weight of soil, γ dry = γ m /(1+ ω) = 120/(1+0.2) = 100 lb/ft 3. Volume of solid for 1 ft 3 of soil, V s = γ dry / (G s γ w ) = 100 / (2.65*62.4) = 0.6 ft 3. Volume of void for 1 ft 3 of soil, V v = 1-V s =1-0.6=0.4 ft 3. aturate unit weight of soil, γ sat = γ dry + γ w V v = *0.4=125 ft 3. Effective unit weight of soil = γ sat - γ w = =62.6 ft 3. Effective depth, H = 0.5B tan(45+φ/2) = 0.5*8*tan (45+30/2) = 6.9 ft Depth of ground water below bottom of footing, D w = 6-2 = 4 ft Equivalent unit weight of soil, γ e = (2H-D w )(D w /H 2 )γ m +(γ /H 2 )(H-D w ) 2 =(2*6.9-4)(4/6.9 2 )*100+(62.6/6.9 2 )(6.9-4) 2 2/15/2009 Page 12 of 23 ce-ref.com

13 = 93.4 lb/ft 3. Where we live. Earth ACE 7-98 EIMIC LOAD CALCULATION Contents: ACE 7-98 Equivalent lateral force procedure Example 1: Bearing wall systems with ordinary reinforced masonry shear wall Example 2: Building frame systems with ordinary steel concentric braced frame Example 3: Building frame systems with ordinary reinforced concrete shear wall ACE 7-98 Equivalent lateral force procedure 1. Determine weight of building, W. 2. Determine 0.2 second response spectral acceleration, s from Figure (a) or (c), (e), (f), (g-1), (h-1), (i), and (j) 3. Determine 1 second response spectral acceleration, 1 from Figure (b), or (d), (f), (g-2), (h-2), (i) and (j) 4. Determine ite class from Table Determine site coefficient, F a, from Table a. 6. Determine site coefficient, F v, from Table b 7. Determine adjusted maximum considered earthquake spectral response acceleration parameters for short period, M and at 1 second period, M1. M = F a s (Eq ) M1 = F v 1 (Eq ) 8. Determine deign spectral response acceleration parameters for short period, D and at 1 second period, D1. D =(2/3) M (Eq ) D1 =(2/3) M1 (Eq ) 9. Determine Important factor, I, from Table 9.1.4, 10. Determine eismic design category from Table Determine Response modification factor, R. from Table and check building height limitation 12. Determine seismic response coefficient from Eq C s = D / (R/I) 13. Determine approximate fundamental period from Eq T = C T h n 0.75 where h n is the height of building above base. 2/15/2009 Page 13 of 23 ce-ref.com

14 C T is building period coefficient, for moment resisting frame of steel, 0.03 for moment resisting frame of concrete and eccentrically braced steel frame 0.02 for all other building. 14. Determine Maximum seismic response coefficient Eq C smax = D1 / [(R/I) T] 15. Determine minimum seismic response coefficient from Eq C smin = D I 16. If it is design category E or F, or 1 is equal or grater than 0.6g, calculate minimum seismic response coefficient from Eq C smin = / (R/I) 17. Determine eismic response coefficient based on result of steps 12 to 16 and calculate seismic base shear from Eq for strength design or load and resistance factor design. V = C s W 18. For service load design, multiply the seismic base shear by 0.7 V s = 0.7 V Example 1: Bearing wall systems with ordinary reinforced masonry shear wall Given: Code: ACE 7-98 Equivalent lateral force procedure Design information: Weight of building, W=500 kips 0.2 second response spectral acceleration, s = second response spectral acceleration, 1 = 0.1 oil profile class: E Bearing wall systems with ordinary reinforced masonry shear wall Building category I Requirement: Determine seismic base shear olution; ite coefficient, F a = 2.5 ite coefficient, F v = 3.5 Design spectral response acceleration parameters M = F a s = (Eq ) M1 = F v 1 = 0.35 (Eq ) D =(2/3) M =0.42 (Eq ) D1 =(2/3) M1 =0.233 (Eq ) eismic design category C from Table a, category D based on Table b. Use category D. From Table , Bearing wall system with ordinary reinforced masonry shear wall is NP (not permitted). Need to change structural system. 2/15/2009 Page 14 of 23 ce-ref.com

15 Example 2: Building frame systems with ordinary steel concentric braced frame Given: Code: ACE 7-98 Equivalent lateral force procedure Design information: Weight of building, W = 500 kips 0.2 second response spectral acceleration, s = second response spectral acceleration, 1 = 0.1 Building frame systems with ordinary steel concentric braced frame Building category I Building height: 30 ft Requirement: Determine seismic base shear olution: ite coefficient, F a = 2.5 ite coefficient, F v = 3.5 Design spectral response acceleration parameters M = F a s = (Eq ) M1 = F v 1 = 0.35 (Eq ) D =(2/3) M =0.42 (Eq ) D1 =(2/3) M1 =0.233 (Eq ) eismic design category C from Table a, category D based on Table b. Use category D. From Table , Building frame system with ordinary steel concentric braced frame is 160 ft Response modification factor, R = 5 Important factor, I = 1 (Table 9.1.4) eismic response coefficient (Eq ) C s = D / (R/I) = Fundamental period (Eq ) T = C T h n 0.75= Maximum seismic response coefficient (Eq ) C smax = D1 / [(R/I) T] = Minimum seismic response coefficient (Eq ) C smin = D I = eismic base shear V = C s W = 42 kips eismic base shear in service load, V s = 0.7 V = 29 kips Example 3: Building frame systems with ordinary reinforced concrete shear wall Given: 2/15/2009 Page 15 of 23 ce-ref.com

16 Code: ACE 7-98 Equivalent lateral force procedure Design information: Weight of building, W = 1000 kips 0.2 second response spectral acceleration, s = second response spectral acceleration, 1 = 0.15 oil profile class: C Building frame systems with ordinary reinforced concrete shear wall Building category II Building height: 40 ft Requirement: Determine seismic base shear olution: ite coefficient, F a = 1.2 ite coefficient, F v = 1.65 Design spectral response acceleration parameters M = F a s = 0.6 (Eq ) M1 = F v 1 = 0.25 (Eq ) D =(2/3) M =0.4 (Eq ) D1 =(2/3) M1 =0.17 (Eq ) eismic design category C from Table a, category C based on Table b. Use category C. From Table , Building frame system with ordinary steel concentric braced frame is 160 ft Response modification factor, R = 5 Important factor, I = 1 (Table 9.1.4) eismic response coefficient (Eq ) C s = D / (R/I) = 0.1 Fundamental period (Eq ) T = C T h n 0.75 = Maximum seismic response coefficient (Eq ) C smax = D1 / [(R/I) T] = Minimum seismic response coefficient (Eq ) C smin = D I = eismic base shear V = C s W = 100 kips eismic base shear in service load, V s = 0.7 V = 70 kips General: Design of Unreinforced Masonry Wall Unreinforced masonry walls are often used as load bearing or non-loading interior wall in one story building. Although it is called unreinforced, the masonry wall still needs to 2/15/2009 Page 16 of 23 ce-ref.com

17 be reinforced with joint reinforcements. In addition, ordinary and detailed plan masonry walls are allowed as shear walls in seismic design category A & B. But it still needs to meet code required minimum reinforcement requirements. By definition of ACI 530 ection 1.6, the tensile strength of masonry is considered, but the strength of reinforcing steel is neglected. Load on masonry walls: 1. Vertical dead load, live load, snow load, etc. 2. Lateral load from wind, seismic, earth pressure etc. tresses in concrete masonry wall: 1. Compressive stress from vertical load 2. Compressive stress from flexural moment due to lateral load. 3. Tensile stress from flexural moment due to lateral load, eccentric moment, etc. Design requirements: When the wall, pilaster, and column is subjected to axial compression and flexure. 1. The maximum compression stress shall satisfy the following equation f a /F a + f b /F b 1 (ACI 530 Eq. 2-10) 2/15/2009 Page 17 of 23 ce-ref.com

18 Where, f a is compressive stress from axial load, f b is compressive stress from flexure; F a and F b, F v are allowable compressive stress and tensile stress calculation from equation below: For member with h/r 99: F a = (1/4) f m {1-[h/(140r)] 2 } (ACI ) For member with h/r > 99: F a = (1/4) f m (70r/h) 2 (ACI ) Where h is effective height of wall, column or pilaster, r is radius of gyration, fm is compressive strength of masonry Allowable compressive stress from flexure: Fb = (1/3) f m (ACI ) 2. The maximum axial force P shall satisfy the following equation P (1/4) P e (ACI 530 Eq. 2-11) Where Pe is calculated as P e = [π 2 E m I/h 2 ][ *e/r) 2 (ACI ) Where E m is elastic modulus of masonry, I is moment of inertia, h is the height of wall, column or plaster, e is eccentricity of axial load, r is radius of gyration. 3. The tensile stress due to flexure shall not exceed the value listed in ACI 530 Table as shown below: Allowable flexural tension stress for hollow core concrete masonry unit, psi. Masonry Mortar Portland cement/lime or mortar cement Masonry cement or air entrained Portland cement/lime Normal to bed joints Ungrouted hollow units Grout hollow units /15/2009 Page 18 of 23 ce-ref.com

19 Parallel to bed joints in running bond Ungrouted or partially grout hollow units Fully grouted hollow units Example 1: Design of an interior unreinforced load bearing masonry wall Design data: Roof dead load: 20 psf Roof live load: 20 psf Tributary width: 30 ft Height of wall: 12 ft Normal width of wall: 8 in Assume minimum eccentricity: 0.8 in eismic load from ACE 7: 4 psf Requirement: Check if an 8 in unreinforced masonry wall is adequate olution: Axial load per foot width of wall from roof, P = (20 psf+20 psf)*30 psf = 1200 lb/ft Eccentricity: e = 0.8 in Eccentric moment: M c = 1200*0.8/12 = 80 lb/ft eismic moment: M s = 5 psf * (12ft) 2 /8 = 90 lb-ft/ft Moment per foot width of wall, M = ( ) lb-ft/ft = 170 lb-ft/ft Width of wall: in Cross section area: A = 42.8 in 2. Moment of inertia: I = in 4. ection modulus: = 86.8 in 3. Radius of gyration: r = 2.78 in Check flexural tensile stress: fb = M/ = 170*12/86.8 = 23.5 psf Less than allowable tensile stress 25 psi for ungrouted hollow unit O.K. Check compressive stress: Weight of wall at mid-height: W = 55 psf * 6 ft = 330 lb/ft Axial compressive stress: f a = (P+W)/A = 35.8 psi/ft lenderness ratio: h/r = 12*12/2.78 = 51.7 < 99 Use 1900 psi concrete masonry units with type mortar, Compressive strength of concrete masonry, f m = 1500 psi F a = (1/4) f m {1-[h/(140r)] 2 } = psi 2/15/2009 Page 19 of 23 ce-ref.com

20 Allowable flexural strength: Fb = (1/3) f m = 500 psi Combined stress equation: f a /F a + f b /F b = < 1 O.K. Check axial force: Elastic modulus, Em = 900 fm = 1.35x10 6 psi P e = [π 2 E m I/h 2 ][ *e/r) 2 = 1.23x10 5 lb (1/4) P e = 3.09 x 10 4 lb > = 1530 lb O.K. World tallest building Concrete Masonry Design Mechanical properties of concrete masonry: Topics: Properties of concrete masonry units Mortar Grout Reinforcement Bond types Compressive strength of concrete masonry, f m Modulus of elasticity of concrete masonry Masonry control joints General: The mechanical properties of concrete masonry wall, pilaster, column, lintel etc. depends on the properties of concrete masonry units, mortar, grout, reinforcement and how the units were arranged. Properties of concrete masonry units: Material: Portland cement, Hydrated lime, Pozzolans, Normal weight or light weight aggregates. Dimension of commonly used Concrete masonry units: Nominal size 8x4x16 8x8x16 8x12x16 Actual size 7-5/8 x3-5/8 x15-5/8 7-5/8 x7-5/8 x15-5/8 7-5/8 x12-5/8 x15-5/8 Face Wall thickness Cross section (in 2 /ft) ection modulus (in 4 /ft) Moment of inertia (in 3 /ft) 3/ / Radius of gyration (in) 1-1/ /15/2009 Page 20 of 23 ce-ref.com

21 Type of commonly used hollow core concrete blocks : Mortar: Material: Portland cement/masonry cement water, lime, sand, and admixtures Mixes: Type M,, N and O. depend on proportion of mixes. Compressive strength of cues for mortar types are as follows: Mortar Types Average compressive strength at 28 days (psi) M N 750 O 350 Grout: Ingredients: Portland cement, fine aggregates, coarse aggregate, Lime. Mix proportions: depending on its strength, a commonly use mix is one part of cement with 1/10 of lime, three parts of fine aggregates, and 2 parts of coarse aggregate with maximum aggregate size limits by the grout space. Reinforcement: Joint reinforcement: Ladder type or truss type, usually, 9 or 10-gage wires. Cell reinforcement: rebars same as concrete reinforcement. 2/15/2009 Page 21 of 23 ce-ref.com

22 Bond types: Running bond and stack bond. Compressive strength of concrete masonry, f m The compressive strength of masonry varies with the type of mortar and the strength of units. There are two methods to determine the strength of masonry during construction. One is based on prism test; the other is based on values specified in the codes. The values list in International Building Code, 2003, Table are as follows: Net area compressive strength of concrete masonry units (psi) Net area compressive strength of masonry f m (psi) Type M or mortar Type N mortar /15/2009 Page 22 of 23 ce-ref.com

23 Modulus of elasticity of concrete masonry: E m = 900 f m (ACI 530 ection ) Thermal expanson coefficients: k t = 4.5 x 10-6 in/in/ o F hrinkage coefficient: Masonry made of non-moisture controlled concrete masonry units: k m = 0.5s 1 Masonry made of moisture controlled concrete masonry units: k m = 0.15s 1 Where s 1 = 6.5x10-6 in/in. Masonry control joints: Masonry control joints are used to allow expansion and shrinkage of masonry wall, minimize random cracks, and distress. pacing of masonry control joints recommended in the commentary of ACI is 25 ft or 3 times of wall height. It also recommends placing control joints at returns and jambs of openings. hear key may be used to control wall movement in the out-of-plan direction. World tallest building. Tai 2/15/2009 Page 23 of 23 ce-ref.com

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