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lecture22

# lecture22 - Lecture 22 Biaxial Columns Design CVEN 444...

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Lecture 22 -  Biaxial Columns  Lecture 22 -  Biaxial Columns  Design Design July 30, 2003 CVEN 444

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Lecture Goals Lecture Goals Short Column Biaxial Design  Slender Column Design
Biaxial Bending and Axial  Biaxial Bending and Axial  Load Load Ref. PCA Notes on ACI 318-95 Unaxial bending about y-axis

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Biaxial Bending and Axial  Biaxial Bending and Axial  Load Load Ref. PCA Notes on ACI 318-95 The biaxial bending moments M x = P*e y M y = P*e x
Approximate Analysis  Approximate Analysis  Methods Methods Use Reciprocal Failure surface S 2 (1/P n ,e x ,e y ) The ordinate 1/P n on the surface S 2 is approximated by ordinate 1/P n on the plane S’ 2 (1/P n e x ,e y ) Plane S 2 is defined by points A,B, and C.

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Approximate Analysis  Approximate Analysis  Methods Methods P 0 = Axial Load Strength under pure axial compression (corresponds to point C ) M nx = M ny = 0 P 0x = Axial Load Strength under uniaxial eccentricity, e y (corresponds to point B ) M nx = P n e y P 0y = Axial Load Strength under uniaxial eccentricity, e x (corresponds to point A ) M ny = P n e x
Approximate Analysis  Approximate Analysis  Methods Methods Design: P u M uy , M ux P u , P u e x , P u e y

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Approximate Analysis  Approximate Analysis  Methods Methods P n = Nominal axial load strength at eccentricities, e x & e y Limited to cases when 0 0y 0x n 0 0y 0x n n 1 1 1 1 1 1 1 1 1 P P P P P P P P P - + - + = g c n 1 . 0 A f P
Biaxial Bending in Short  Biaxial Bending in Short  Columns Columns 1) Calculate P 0 2) Calculate P 0y ( P n for e = e x , e y = 0 ) 3)Calculate P 0x ( P n for e x = 0, e = e y ) 4) Calculate P n (from Bresler’s Formula ) Analysis Procedure: Reciprocal Load Method Bresler’s Formula: Steps: 0 0y 0x n 1 1 1 1 P P P P - + 2245

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Biaxial Bending in Short  Biaxial Bending in Short  Columns Columns where, φ = 0.65 n u P P φ
Biaxial Column Example Biaxial Column Example The section of a short tied column is 16 x 24 in. and is reinforced with 8 #10 bars as shown. Determine the allowable ultimate load on the section φ P n if its acts at e x = 8 in. and e y = 12 in. Use f c = 5 ksi and f y = 60 ksi.

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Biaxial Column Example Biaxial Column Example Compute the P 0 load, compression with no moments ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 2 2 st 0 c g st st y 2 2 n0 0 8 1.27 in 10.16 in 0.85 0.85 5 ksi 24.0 in. 24.0 in. 10.16 in 10.16 in 60 ksi 2198.4 k 0.8 2198.4 k 1758.7 k A P f A A A f P rP = = = - + = - + = = = =
Biaxial Column Example Biaxial Column Example Compute P nx , by starting with e y term and assume that compression controls. Check by Compute the nominal load, P nx and assume second compression steel does not contribute assume small ( 29 y 2 2 12 in. 21.5 in. 14.33 in. 3 3 e d = = = n c s1 s2 P C C C T = + + -

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Biaxial Column Example Biaxial Column Example The components of the equilibrium equation are: Use similar triangles to find the stress in the steel, f s ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 c 2 s1 2 s s 0.85 5 ksi 16 in. 0.8 54.4 3.81 in 60 ksi 0.85 5 ksi 212.4 kips 3.81 in 21.5 in.
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