""" - The source code is developed by Marin Grubišić https://github.com/mgrubisic at University of Osijek, Croatia. - The numerical model with the associated analysis was described in detail by Prof. Michael Scott in the Portwood Digital blog https://portwooddigital.com/2021/01/03/sensitivity-training/ - Run the source code in your favorite Python program and should see following plot. """ import time import sys import numpy as np import matplotlib.pyplot as plt import openseespy.opensees as ops # +===============================================================================+ # | OpenSees Header | # +===============================================================================+ nSpaces = 90 OpenSeesHeader = {"header_00": " ", "header_01": nSpaces * "=", "header_02": "OpenSees -- Open System For Earthquake Engineering Simulation", "header_03": "Pacific Earthquake Engineering Research Center (PEER)", "header_04": "OpenSees " + ops.version() + " 64-Bit", "header_05": "Python " + sys.version, "header_06": " ", "header_07": "(c) Copyright 1999-2021 The Regents of the University of California", "header_08": "All Rights Reserved", "header_09": "(Copyright and Disclaimer @ http://www.berkeley.edu/OpenSees/copyright.html)", "header_10": nSpaces * "=", "header_11": " ", } for i in OpenSeesHeader.keys(): print(OpenSeesHeader[i].center(nSpaces, " ")) def title(title="Title Example", nSpaces=nSpaces): header = (nSpaces-2) * "-" print("+" + header.center((nSpaces-2), " ") + "+") print("|" + title.center((nSpaces-2), " ") + "|") print("+" + header.center((nSpaces-2), " ") + "+") # +===============================================================================+ # | Units | # +===============================================================================+ inch = 1.0 # define basic units kip = 1.0 sec = 1.0 ft = 12*inch lb = kip/1000 ksi = kip/inch**2 psf = lb/ft**2 LunitTXT = "inches" # (Length) define basic-unit text for output FunitTXT = "kips" # (Force) define basic-unit text for output TunitTXT = "seconds" # (Time) define basic-unit text for output # +===============================================================================+ # | Define some functions | # +===============================================================================+ def run_gravity_analysis(steps=10): """ Run gravity analysis (in 10 steps) """ ops.wipeAnalysis() ops.system("BandGeneral") ops.numberer("RCM") ops.constraints("Transformation") ops.test("NormDispIncr", 1.0E-12, 10, 3) ops.algorithm("Newton") # KrylovNewton ops.integrator("LoadControl", 1/steps) ops.analysis("Static") ops.analyze(steps) title("Gravity Analysis Completed!") # Set the gravity loads to be constant & reset the time in the domain ops.loadConst("-time", 0.0) ops.wipeAnalysis() def run_sensitivity_pushover_analysis(ctrlNode, baseNodes, dof, Dincr, max_disp, SensParam, IOflag=False): """ Run pushover analysis with sensitivity """ ops.wipeAnalysis() start_time = time.time() ops.loadConst("-time", 0.0) title("Running Displacement-Control Pushover Sensitivity Analysis ...") testType = "NormDispIncr" # EnergyIncr tolInit = 1.0E-8 iterInit = 10 # Set the initial Max Number of Iterations algorithmType = "Newton" # Set the algorithm type ops.system("BandGeneral") ops.constraints("Transformation") ops.numberer("RCM") ops.test(testType, tolInit, iterInit) ops.algorithm(algorithmType) # Change the integration scheme to be displacement control # node dof init Jd min max ops.integrator("DisplacementControl", ctrlNode, dof, Dincr) ops.analysis("Static") ops.sensitivityAlgorithm("-computeAtEachStep") # automatically compute sensitivity at the end of each step if IOflag: print(f"Single Pushover: Push node {ctrlNode} to {max_disp} {LunitTXT}.\n") # Set some parameters tCurrent = ops.getTime() currentStep = 1 outputs = {"time": np.array([]), "disp": np.array([]), "force": np.array([]), } for sens in SensParam: outputs[f"sensLambda_{sens}"] = np.array([]), nodeList = [] for node in baseNodes: nodeList.append(f"- ops.nodeReaction({node}, dof) ") nodeList = "".join(nodeList) currentDisp = ops.nodeDisp(ctrlNode, dof) ok = 0 while ok == 0 and currentDisp < max_disp: ops.reactions() ok = ops.analyze(1) tCurrent = ops.getTime() currentDisp = ops.nodeDisp(ctrlNode, dof) if IOflag: print(f"Current displacement ==> {ops.nodeDisp(ctrlNode, dof):.3f} {LunitTXT}") # if the analysis fails try initial tangent iteration if ok != 0: print("\n==> Trying relaxed convergence...") ops.test(testType, tolInit/0.01, iterInit*50) ok = ops.analyze(1) if ok == 0: print("==> that worked ... back to default analysis...\n") ops.test(testType, tolInit, iterInit) if ok != 0: print("\n==> Trying Newton with initial then current...") ops.test(testType, tolInit/0.01, iterInit*50) ops.algorithm("Newton", "-initialThenCurrent") ok = ops.analyze(1) if ok == 0: print("==> that worked ... back to default analysis...\n") ops.algorithm(algorithmType) ops.test(testType, tolInit, iterInit) if ok != 0: print("\n==> Trying ModifiedNewton with initial...") ops.test(testType, tolInit/0.01, iterInit*50) ops.algorithm("ModifiedNewton", "-initial") ok = ops.analyze(1) if ok == 0: print("==> that worked ... back to default analysis...\n") ops.algorithm(algorithmType) ops.test(testType, tolInit, iterInit) currentStep += 1 tCurrent = ops.getTime() outputs["time"] = np.append(outputs["time"], tCurrent) outputs["disp"] = np.append(outputs["disp"], ops.nodeDisp(ctrlNode, dof)) outputs["force"] = np.append(outputs["force"], eval(nodeList)) for sens in SensParam: # sensLambda(patternTag, paramTag) outputs[f"sensLambda_{sens}"] = np.append(outputs[f"sensLambda_{sens}"], ops.sensLambda(1, sens)) # Print a message to indicate if analysis completed succesfully or not if ok == 0: title("Pushover Analysis Completed Successfully!") else: title("Pushover Analysis Completed Failed!") print(f"Analysis elapsed time is {(time.time() - start_time):.3f} seconds.\n") return outputs # +===============================================================================+ # | Define model | # +===============================================================================+ # Create ModelBuilder # ------------------- ops.wipe() ops.model("basic", "-ndm", 2, "-ndf", 3) # Create nodes # ------------ ops.node(1, 0.0, 0.0) # Ground Level ops.node(2, 30*ft, 0.0) ops.node(3, 0.0, 15*ft) # 1st Floor Level ops.node(4, 30*ft, 15*ft) ops.node(5, 0.0, 2*15*ft) # 2nd Floor Level ops.node(6, 30*ft, 2*15*ft) # Fix supports at base of columns # ------------------------------- ops.fix(1, 1, 1, 1) ops.fix(2, 1, 1, 1) # Define material # --------------- matTag = 1 Fy = 50.0*ksi # Yield stress Es = 29000.0*ksi # Modulus of Elasticity of Steel b = 1/100 # 1% Strain hardening ratio # Sensitivity-ready steel materials: Hardening, Steel01, SteelMP, BoucWen, SteelBRB, StainlessECThermal, SteelECThermal, ... ops.uniaxialMaterial("Steel01", matTag, Fy, Es, b) # ops.uniaxialMaterial("SteelMP", matTag, Fy, Es, b) # Define sections # --------------- # Sections defined with "canned" section ("WFSection2d"), otherwise use a FiberSection object (ops.section("Fiber",...)) colSecTag, beamSecTag = 1, 2 WSection = {"W18x76": [18.2*inch, 0.425*inch, 11.04*inch, 0.68*inch], # [d, tw, bf, tf] "W14X90": [14.02*inch, 0.44*inch, 14.52*inch, 0.71*inch], # [d, tw, bf, tf] } # secTag, matTag, [d, tw, bf, tf], Nfw, Nff ops.section("WFSection2d", colSecTag, matTag, *WSection["W14X90"], 20, 4) # Column section ops.section("WFSection2d", beamSecTag, matTag, *WSection["W18x76"], 20, 4) # Beam section # Define elements # --------------- colTransTag, beamTransTag = 1, 2 # Linear, PDelta, Corotational ops.geomTransf("Corotational", colTransTag) ops.geomTransf("Linear", beamTransTag) colIntTag, beamIntTag = 1, 2 nip = 5 # Lobatto, Legendre, NewtonCotes, Radau, Trapezoidal, CompositeSimpson ops.beamIntegration("Lobatto", colIntTag, colSecTag, nip), ops.beamIntegration("Lobatto", beamIntTag, beamSecTag, nip) # Column elements ops.element("forceBeamColumn", 10, 1, 3, colTransTag, colIntTag, "-mass", 0.0) ops.element("forceBeamColumn", 11, 3, 5, colTransTag, colIntTag, "-mass", 0.0) ops.element("forceBeamColumn", 12, 2, 4, colTransTag, colIntTag, "-mass", 0.0) ops.element("forceBeamColumn", 13, 4, 6, colTransTag, colIntTag, "-mass", 0.0) # Beam elements ops.element("forceBeamColumn", 14, 3, 4, beamTransTag, beamIntTag, "-mass", 0.0) ops.element("forceBeamColumn", 15, 5, 6, beamTransTag, beamIntTag, "-mass", 0.0) # Create a Plain load pattern with a Linear TimeSeries # ---------------------------------------------------- ops.timeSeries("Linear", 1) ops.pattern("Plain", 1, 1) # +===============================================================================+ # | Define Sensitivity Parameters | # +===============================================================================+ # /// Each parameter must be unique in the FE domain, and all parameter tags MUST be numbered sequentially starting from 1! /// ops.parameter(1) # Blank parameters ops.parameter(2) ops.parameter(3) for ele in [10, 11, 12, 13]: # Only column elements ops.addToParameter(1, "element", ele, "E") # "E" # Check the sensitivity parameter name in *.cpp files ("sigmaY" or "fy", somewhere also "Fy") # https://github.com/OpenSees/OpenSees/blob/master/SRC/material/uniaxial/Steel02.cpp ops.addToParameter(2, "element", ele, "fy") # "sigmaY" or "fy" or "Fy" ops.addToParameter(3, "element", ele, "b") # "b" title("Model Built") # Run analysis with 10 steps # ------------------------- run_gravity_analysis(steps=10) # +===============================================================================+ # | Define nodal loads & Run the analysis | # +===============================================================================+ # Create nodal loads at nodes 3 & 5 # nd FX FY MZ ops.load(3, 1/3, 0.0, 0.0) ops.load(5, 2/3, 0.0, 0.0) max_disp = 20*inch pushover_output = run_sensitivity_pushover_analysis( ctrlNode=5, baseNodes=[1, 2], dof=1, Dincr=(1/25)*inch, max_disp=max_disp, SensParam=ops.getParamTags(), IOflag=False) # +===============================================================================+ # | Plot results | # +===============================================================================+ rows, columns = len(ops.getParamTags())*2, 1 grid = plt.GridSpec(rows, columns, wspace=0.25, hspace=0.25) fig = plt.figure(figsize=(10, 10)) ParamSym = ["E", "F_y", "b"] ParamVars = [Es, Fy, b] def plot_params(): plt.rcParams['axes.axisbelow'] = True plt.xlim(xmin=0.0, xmax=max_disp) plt.grid(True, color="silver", linestyle="solid", linewidth=0.75, alpha=0.75) plt.tick_params(direction="in", length=5, colors="k", width=0.75) for s in range(0, rows): # Create subplots # --------------- if s < 3: plt.subplot(grid[s]), plot_params() plt.plot(pushover_output["disp"], pushover_output["force"], "-k", linewidth=2.0, label="$U$") plt.plot(pushover_output["disp"], pushover_output["force"] + pushover_output[f"sensLambda_{s+1}"] * ParamVars[s], "--k", linewidth=1.5, label=f"$U + (\partial V_b/\partial {ParamSym[s]}){ParamSym[s]}$") plt.plot(pushover_output["disp"], pushover_output["force"] - pushover_output[f"sensLambda_{s+1}"] * ParamVars[s], "-.k", linewidth=1.5, label=f"$U - (\partial V_b/\partial {ParamSym[s]}){ParamSym[s]}$") plt.fill_between(pushover_output["disp"], pushover_output["force"] + pushover_output[f"sensLambda_{s+1}"] * ParamVars[s], pushover_output["force"] - pushover_output[f"sensLambda_{s+1}"] * ParamVars[s], color='grey', alpha=0.15) plt.ylabel(r"$V_b$ [kip]") if s == 0: plt.title(r"Displacement Control", fontsize=14) if s == 1: plt.ylabel(r"Base Shear, $V_b$ [kip]") plt.legend(fontsize=9) if s >= 3: plt.subplot(grid[s]), plot_params() plt.plot(pushover_output["disp"], pushover_output[f"sensLambda_{s-2}"] * ParamVars[s-3], "-.k", linewidth=2.0, label="DDM") plt.fill_between(pushover_output["disp"], pushover_output[f"sensLambda_{s-2}"] * ParamVars[s-3], color='grey', alpha=0.15) plt.ylabel(f"$(\partial V_b/\partial {ParamSym[s-3]}){ParamSym[s-3]}$ [kip]") if s == 5: plt.xlabel(r"Roof Displacement, $U$ [in]") plt.legend() s += 1 # Save figure # ----------- plt.savefig("SteelFrameSensitivity2D_v1.png", bbox_inches="tight", pad_inches=0.05, dpi=300, format="png") plt.show()