Combined Stress. Axial Stress. Axial vs. Eccentric Load Combined Stress Interaction Formulas

Transcription

1 Architecture 324 Structures II Combined Stress Axial vs. Eccentric Load Combined Stress Interaction Formulas from Man und Frau den Mond betrachtend by Caspar David Friedrich Alte Nationalgalerie, Berlin University of Michigan, TCAUP Structures II Slide 1/29 Axial Stress Loads pass through the centroid of the section, i.e. axially loaded Member is straight Load is less than buckling load University of Michigan, TCAUP Structures II Slide 2/29

2 Eccentric Loads Load is offset from centroid Bending Moment = P e Total load = P + M Interaction formula University of Michigan, TCAUP Structures II Slide 3/29 Combined Stress Stresses combine by superposition Values add or subtract by sign University of Michigan, TCAUP Structures II Slide 4/29

3 Example 1. Determine external reactions University of Michigan, TCAUP Structures II Slide 5/29 Example 2. Determine internal member forces: Axial and Flexural 3. Determine axial and flexural stresses 4. Use interaction formula to determine combined stresses at key locations (e.g. extreme fibers) University of Michigan, TCAUP Structures II Slide 6/29

4 Rafters Flexure + Axial Francesco on 2x4 Find normal and axial components of the load Axial = 22.5 lbs Normal = 39 lbs (sin60) = 39 lbs A (cos60) = 22.5 lbs B b a 45 lbs C 60 University of Michigan, TCAUP Structures II Slide 7/29 Rafters Flexure + Axial 39 Calculate axial stress P = 22.5 lbs A = 5.25 in 2 f c = 22.5 / 5.25 = 4.28 psi Calculate flexural stress f b M = PL/4 = 39 lbs x 96 /4 = lbs c = 1.5 /2 = 0.75 I y = 3.5(1.5 3 )/12 = in 4 = 963 (0.75)/0.984 = psi Add the stresses for max. compression The flexure stress can also be found using the projected length. Or f = fc + fb = psi M = PL/4 = 45lbs x /4 = lbs f b = 963 (0.75)/0.984 = psi University of Michigan, TCAUP Structures II Slide 8/29

5 Second Order Stress P Delta Effect P With larger deflections this can become significant. 1. Eccentric load causes bending moment 2. Bending moment causes deflection, 3. P x causes additional moment Δ University of Michigan, TCAUP Structures II Slide 9/29 Other Examples Columns with side loading Trusses loaded on members Moment frames University of Michigan, TCAUP Structures II Slide 10/29

6 Other Examples University of Michigan, TCAUP Structures II Slide 11/29 Pop Quiz University of Michigan, TCAUP Structures II Slide 12/29

7 Example Problem Determine internal member forces, axial and flexural, on the bottom chord of the truss University of Michigan, TCAUP Structures II Slide 13/29 Example (cont.) 1. Determine external end reactions of the whole truss. 2. Use FBDs of each member to find load distribution to each joint. University of Michigan, TCAUP Structures II Slide 14/29

8 Example (cont.) 3. Use vector addition to solve for the axial force in members University of Michigan, TCAUP Structures II Slide 15/29 Example (cont.) 4. Draw a FBD to show both axial and flexural loading. 5. Calculate the axial and flexural stress. University of Michigan, TCAUP Structures II Slide 16/29

9 Example (cont.) 6.Use interaction equations to calculate combined stress. 7.Check stress against allowable limits. University of Michigan, TCAUP Structures II Slide 17/29 Combined Stress in NDS University of Michigan, TCAUP Structures II Slide 18/29

10 Tension + Flexure NDS Equations CASE 1. Tension is critical. Eq * no C L CASE 2. Flexure is critical. Eq ** no C V University of Michigan, TCAUP Structures II Slide 19/29 Tension + Flexure University of Michigan, TCAUP Structures II Slide 20/29

11 Compression + Flexure (Flatwise bending + compression) University of Michigan, TCAUP Structures II Slide 21/29 Combined Stress in NDS 1. Determine load per stud 2. Use axial load and moment to find actual stresses fc and fb 3. Determine load factors 4. Calculate factored stresses 5. Check NDS equations University of Michigan, TCAUP Structures II Slide 22/29

12 Combined Stress in NDS 1. Determine load per stud 2. Use axial load and moment to find actual stresses fc and fb University of Michigan, TCAUP Structures II Slide 23/29 Combined Stress in NDS 3. Determine load factors (bending) University of Michigan, TCAUP Structures II Slide 24/29

13 Combined Stress in NDS 4. Calculate factored stresses Bending Stress University of Michigan, TCAUP Structures II Slide 25/29 Combined Stress in NDS 4. Calculate factored stresses Bending Stress University of Michigan, TCAUP Structures II Slide 26/29

14 Combined Stress in NDS 3. Determine load factors (compression) University of Michigan, TCAUP Structures II Slide 27/29 Combined Stress in NDS 4. Calculate factored stresses Compression Stress University of Michigan, TCAUP Structures II Slide 28/29

15 Combined Stress in NDS Combined Stress Calculation University of Michigan, TCAUP Structures II Slide 29/29