# Combined Stress - Open Michigan

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 2/9 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 3/9 Eccentric Loads Load offset from centroid

M=Pe Total load = P + M Interaction formula P Mc f A I fa fb 1.0 Fa Fb University of Michigan, TCAUP

Source: University of Michigan, Department of Architecture Structures II Slide 4/9 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 5/9 Example Source: University of Michigan, Department of Architecture Determine external reactions University of Michigan, TCAUP Structures II Slide 6/9 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 7/9 Rafters

Flexure + Axial P Mc f A I 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 30 f = fc + fb = 717.4 psi

45#(sin60) = 39# Or 90 M = PL/4 = 45# x 83.14 /4 = 936 - # fb = 963 (0.75)/0.984 = 713.14 45# 45#(cos60) = 22.5# 60 University of Michigan, TCAUP Structures II

Slide 8/9 Other Examples University of Michigan, TCAUP Structures II Slide 9/9 Second Order Stress P Delta Effect P 1. Eccentric load causes bending moment 2. Bending moment causes deflection,

3. P x causes additional moment 4. The additional P moment causes additional deflection, 1 University of Michigan, TCAUP Structures II Slide 10/9 Treehouse Form Exploration University of Michigan, TCAUP

Structures II Slide 11/19 Treehouse Details Pinned at ends Radius bent at joints Continuous sections Welded seams

University of Michigan, TCAUP 12/19 Structures II Slide Treehouse Details Pinned at ends

Radius bent at joints Continuous sections Welded seams University of Michigan, TCAUP /19 Structures II Slide 13 Treehouse Erection

3 sections Trucked to site Erected with mobile crane One day process University of Michigan, TCAUP 19 Structures II Slide 14/ Treehouse Erection

3 sections Trucked to site Erected with mobile crane One day process University of Michigan, TCAUP 19 Structures II Slide 15/ Treehouse

Finished University of Michigan, TCAUP 19 Structures II Slide 16/ Treehouse Finished University of Michigan, TCAUP 19

Structures II Slide 17/ Treehouse Finished University of Michigan, TCAUP 19 Structures II Slide 18/ Treehouse

Finished University of Michigan, TCAUP 19 Structures II Slide 19/

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