von buelow, peter - deep blue...second order stress “p delta effect” 1. eccentric load causes...
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University of Michigan
2009-01
ARCH 324 - Structures 2, Winter 2009
von Buelow, Peter Buelow, P. v. (2008, December 19). Structures 2. Retrieved from Open.Michigan - Educational
Resources Web site: https://open.umich.edu/education/tcaup/arch324-winter2009.
<http://hdl.handle.net/2027.42/64938>http://hdl.handle.net/2027.42/64938
Deep Blue deepblue.lib.umich.edu
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University of Michigan, TCAUP Structures II Slide 2/9
Architecture 324 Structures II
Combined Stress
• Axial vs. Eccentric Load • Combined Stress • Interaction Formulas
Source: Caspar David Friedrich from “Man und Frau den Mond betrachtend” 1830-35 Alte Nationalgalerie, Berlin
University of Michigan, TCAUP Structures II Slide 3/9
Axial Stress
• Loads pass through the centroid of the section
• Member is straight
• Load less than buckling load
• f = P/A
Source: University of Michigan, Department of Architecture
University of Michigan, TCAUP Structures II Slide 4/9
Eccentric Loads
• Load offset from centroid
• M = P e
• Total load = P + M
Interaction formula
Source: University of Michigan, Department of Architecture
University of Michigan, TCAUP Structures II Slide 5/9
Combined Stress • Stresses combine by superposition • Values add or subtract by sign
Source: University of Michigan, Department of Architecture
University of Michigan, TCAUP Structures II Slide 6/9
Example
Determine external reactions
Source: University of Michigan, Department of Architecture
University of Michigan, TCAUP Structures II Slide 7/9
Example
Determine internal member forces: Axial and Flexural
Determine axial and flexural stresses
Use interaction formula to determine combined stresses at key locations (e.g. extreme fibers)
University of Michigan, TCAUP Structures II Slide 8/9
Rafters Flexure + Axial
P = 22.5 # A = 5.25 in2
fc = 22.5 / 5.25 = 4.28 psi
M = PL/4 = 39# x 96″ /4 = 936 ″- # c = 1.5″ /2 = 0.75″ Iy = 3.5(1.53)/12 = 0.984 in4 fb = 963 (0.75)/0.984 = 713.14
f = fc + fb = 717.4 psi
Or
M = PL/4 = 45# x 83.14″ /4 = 936 ″- # fb = 963 (0.75)/0.984 = 713.14
90°
60°
30°
45#
45#(sin60) = 39#
45#(cos60) = 22.5#
University of Michigan, TCAUP Structures II Slide 9/9
Other Examples
University of Michigan, TCAUP Structures II Slide 10/9
Second Order Stress “P Delta Effect”
1. Eccentric load causes bending moment
2. Bending moment causes deflection, Δ
3. P x Δ causes additional moment
4. The additional PΔ moment causes additional deflection, Δ1 Δ
P
University of Michigan, TCAUP Structures II Slide 11/19
Treehouse
• Form Exploration
University of Michigan, TCAUP Structures II Slide 12/19
Treehouse
• Details – Pinned at ends – Radius bent at joints – ‘Continuous’ sections – Welded seams
University of Michigan, TCAUP Structures II Slide 13/19
Treehouse
• Details – Pinned at ends – Radius bent at joints – ‘Continuous’ sections – Welded seams
University of Michigan, TCAUP Structures II Slide 14/19
Treehouse
• Erection – 3 sections – Trucked to site – Erected with mobile crane – One day process
University of Michigan, TCAUP Structures II Slide 15/19
Treehouse
• Erection – 3 sections – Trucked to site – Erected with mobile crane – One day process
University of Michigan, TCAUP Structures II Slide 16/19
Treehouse
• Finished
University of Michigan, TCAUP Structures II Slide 17/19
Treehouse
• Finished
University of Michigan, TCAUP Structures II Slide 18/19
Treehouse
• Finished
University of Michigan, TCAUP Structures II Slide 19/19
Treehouse
• Finished