Software Verification

Transcription

1 EXAMPLE FRAME MOMENT AND SHEAR HINGES EXAMPLE DESCRIPTION This example uses a horizontal cantilever beam to test the moment and shear hinges in a static nonlinear analysis. The cantilever beam has a moment (M y ) hinge and a shear (V z ) hinge at its fixed end. A vertical load, P, is applied to the cantilever and increased until the vertical tip deflection, U z, equals 2". Two models are used in the example. Model B applies a default auto subdivide hinge overwrite to the line object containing the hinge whereas Model A does not. Multiple states are saved for the analysis with the minimum number of saved states set to 6 and the maximum number of saved states set to 10. The tip deflection, U z, and tip rotation, R y, for several of the saved states (identified on the next page) are compared with independent hand calculated results. Important Notes: Bending and shear deformations are included in this example. Also, in frame hinges are only active in nonlinear static and nonlinear direct time history load cases. The hinges are ignored in all other types of load cases GEOMETRY, PROPERTIES AND LOADING Moment (kip-in) Z Y Moment hinge (M y ) and shear hinge (V X z ) 24" 12" Material Properties C E = 3,600 k/in 2 ν = 0.2 G= 1,500 k/in 2 C Moment Hinge Moment-Rotation P Shear (kips) " Section C-C Shear Hinge Force-Deformation Section Properties b = 12 in d = 18 in A = 216 in 2 I = 5,832 in 4 A v = 180 in 2 (shear area) Loading Increase P until the free end tip deflection in the Z direction is 2 inches 0.04 Plastic Rotation (radians) 0.4 Plastic Deformation (inches) EXAMPLE

2 TECHNICAL FEATURES OF TESTED Static nonlinear analysis of a frame structure with moment and shear hinges RESULTS COMPARISON Force P and free end vertical displacement and rotation are reported for the saved states corresponding to the points labeled 1, 2 and 3 on the cantilever beam force-tip deflection (P-U z ) diagram shown to the right. Independent results are hand calculated using the unit load method described on page 244 in Cook and Young 1985 together with basic deflection formulas and superposition. Force P (kips) Tip Deflection, U z (inches) Results Without Hinge Overwrite (Model B) Point Output Parameter Independent Percent Difference Force P (free end) kips % 1 U z (free end) in % R y (free end) rad % Force P (free end) kips % 2 U z (free end) in % R y (free end) rad % Force P (free end) kips % 3 U z (free end) in % R y (free end) rad % EXAMPLE

3 Results With Hinge Overwrite (Model B) Point Output Parameter Independent Percent Difference Force P (free end) kips % 1 U z (free end) in % R y (free end) rad % Force P (free end) kips % 2 U z (free end) in % R y (free end) rad % Force P (free end) kips % 3 U z (free end) in % R y (free end) rad % COMPUTER FILES: Example 1-026a, Example 1-026b CONCLUSION The results show an acceptable comparison with the independent results when the auto subdivide overwrite is used. Sharp vertical drops in hinge forces or moments may not always be realistic and can cause analysis convergence difficulties in more complicated models. For this reason Sap2000 automatically limits the steepness of vertical drops to one-tenth of the elastic stiffness of the element containing the hinge. When a frame object is subdivided the shorter elements have larger elastic stiffnesses which permit drops in hinge forces or moments that are closer to vertical. It is for this reason that model B, which has an automatically meshed element that is 1% of the total length, permits a steeper drop than model A. The drop is so steep that snap back (negative displacement increment) of the overall cantilever can be observed. Steep (nearly vertical) hinge drops should be avoided whenever possible. EXAMPLE

4 HAND CALCULATION EXAMPLE

5 EXAMPLE

6 EXAMPLE