Sensitivity and Reliability Analysis of Nonlinear Frame Structures Michael H. Scott Associate Professor School of Civil and Construction Engineering Applied Mathematics and Computation Seminar April 8, 2011 M. H. Scott (OSU) Sensitivity and Reliability AMC Seminar, 2011 1 / 34
Outline 1 Structural Response Sensitivity Analysis Governing Equilibrium Equations Computing Response Sensitivity Sensitivity Verification 2 Structural Reliability Analysis Problem Statement Computing Reliability Sensitivity at Design Point Cantilever Example Reinforced Concrete Frame Example M. H. Scott (OSU) Sensitivity and Reliability AMC Seminar, 2011 2 / 34
Sensitivity Overview Governing Equilibrium Equations Structural Equilibrium Balance of external loads and internal resisting forces P f (t) = P r (U f (t)) (1) P f (t) time-varying external loads (given) U f (t) time-varying nodal displacements (unknown) P r resisting forces (by selection and assembly) M. H. Scott (OSU) Sensitivity and Reliability AMC Seminar, 2011 3 / 34
Sensitivity Overview Governing Equilibrium Equations Residual Equilibrium Equations Solve residual equilibrium using Newton-Raphson algorithm where R f (U f (t)) = P f (t) P r (U f (t)) = 0 (2) U f = ( ) 1 K j T Rf (U j f ) (3) K j T = P r U f (4) or any other suitable stiffness matrix according to other root finding algorithms (modified, quasi, accelerated Newton). Regardless, use increment to update displacement U j+1 f = U j f + U f (5) M. H. Scott (OSU) Sensitivity and Reliability AMC Seminar, 2011 4 / 34
Sensitivity Overview Governing Equilibrium Equations Parameterized Equilibrium Equations Let θ be an uncertain parameter of the external load or the finite element model P f (t, θ) = P r (U f (t, θ), θ) (6) Nodal displacements, U f, depend on all θ, whether of the load or FE model Resisting forces depend explicitly on a θ of the FE model and implicitly on all θ via U f Load vector, P f, depends explicitly on a θ of the external load Typical properties θ represents: steel yield stress, concrete strength, member cross-section dimensions, nodal coordinates, etc. M. H. Scott (OSU) Sensitivity and Reliability AMC Seminar, 2011 5 / 34
Sensitivity Overview Computing Sensitivity Sensitivity to Uncertain Parameters Two approaches to determine U f / θ, the sensitivity of structural response to changes in θ 1 Finite Differences re-run analysis with perturbed parameter value, e.g., forward finite difference U f θ U f (θ + θ) U f (θ) θ 2 Direct Differentiation develop analytic expressions for sensitivity by differentiating governing equations of the finite element analysis (7) M. H. Scott (OSU) Sensitivity and Reliability AMC Seminar, 2011 6 / 34
Sensitivity Overview Computing Sensitivity Direct Differentiation Method (DDM) Differentiate equilibrium equations wrt parameter, θ P f θ = P r U f U f θ + P r θ (8) Uf Solve for nodal displacement sensitivity, U f / θ ( U f θ = P f K 1 T θ P ) r θ Uf (9) At end of a simulation time step: Conditional derivative, P r / θ Uf, assembled from element contributions in same manner as P r Factorized K T from last Newton-Raphson iteration. Or reform and factorize K T if using Newton-like iteration. M. H. Scott (OSU) Sensitivity and Reliability AMC Seminar, 2011 7 / 34
Sensitivity Overview Computing Sensitivity Verification Compare with forward finite difference U f (θ + θ) U f (θ) lim = U f θ 0 θ θ (10) Repeat analysis with perturbed parameter If finite difference approximation converges to analytic sensitivity as θ decreases, then DDM is correct M. H. Scott (OSU) Sensitivity and Reliability AMC Seminar, 2011 8 / 34
Sensitivity Overview Sensitivity Verification Verification Example Slender cantilever with material and geometric nonlinearity Cross-Section 0.5 m Stress-Strain Bilinear 0.1 m 000000000000000000000 111111111111111111111 000000000000000000000 111111111111111111111 000000000000000000000 111111111111111111111 U(t) 5.0 m P(t) = P max sin(t) Discretize with five frame finite elements (linear curvature, constant axial deformation approximation) Corotational formulation for large displacements Numerically integrate stress-strain response over cross-section E = 2.0e8 kpa, σ y = 4.1e5 kpa, 2% kinematic strain-hardening One load cycle with P max = 1710 kn (5 times yield load) M. H. Scott (OSU) Sensitivity and Reliability AMC Seminar, 2011 9 / 34
Sensitivity Overview Sensitivity Verification Load-Displacement Response 2000 Load, P (kn) 1000 0 1000 2000 4 2 0 2 4 Displacement, U (m) Material yield at about 400 kn load Tension stiffening at about 1 m displacement Elastic unloading and reverse cyclic yielding M. H. Scott (OSU) Sensitivity and Reliability AMC Seminar, 2011 10 / 34
Sensitivity Overview Sensitivity Verification Sensitivity to Yield Stress (θ σ y ) DDM verification and response envelope for strength parameter ( U/ θ)θ (m) 6 4 2 0 2 FFD, θ = 0.001θ DDM FFD Load, P (kn) 2000 1000 0 1000 U ± ( U/ θ)(0.1θ) 4 0 0.2 0.4 0.6 0.8 1 Time, t 2000 4 2 0 2 4 Displacement, U (m) M. H. Scott (OSU) Sensitivity and Reliability AMC Seminar, 2011 11 / 34
Sensitivity Overview Sensitivity Verification Sensitivity to Elastic Modulus (θ E) DDM verification and response envelope for stiffness parameter ( U/ θ)θ (m) 0.3 0.2 0.1 0 0.1 FFD, θ = 0.001θ DDM FFD Load, P (kn) 2000 1000 0 1000 U ± ( U/ θ)(0.1θ) 0.2 0 0.2 0.4 0.6 0.8 1 Time, t 2000 4 2 0 2 4 Displacement, U (m) M. H. Scott (OSU) Sensitivity and Reliability AMC Seminar, 2011 12 / 34
Sensitivity Overview Sensitivity Verification Sensitivity to Cross-Section Depth (θ d) DDM verification and response envelope for local geometric parameter ( U/ θ)θ (m) 10 5 0 5 FFD, θ = 0.001θ DDM FFD Load, P (kn) 2000 1000 0 1000 U ± ( U/ θ)(0.1θ) 10 0 0.2 0.4 0.6 0.8 1 Time, t 2000 4 2 0 2 4 Displacement, U (m) M. H. Scott (OSU) Sensitivity and Reliability AMC Seminar, 2011 13 / 34
Sensitivity Overview Sensitivity Verification Sensitivity to Cantilever Length (θ L) DDM verification and response envelope for global geometric parameter ( U/ θ)θ (m) 10 5 0 5 FFD, θ = 0.001θ DDM FFD Load, P (kn) 2000 1000 0 1000 U ± ( U/ θ)(0.1θ) 10 0 0.2 0.4 0.6 0.8 1 Time, t 2000 5 0 5 Displacement, U (m) M. H. Scott (OSU) Sensitivity and Reliability AMC Seminar, 2011 14 / 34
Problem Statement Structural Reliability Analysis Assess uncertain structural response in terms of probability of failure under prescribed events, or limit states Define scalar valued limit state function in terms of a vector of random variables, x g(x) = { 0 fail > 0 safe (11) Each random variable in x Has a prescribed PDF, CDF, mean, st.dev, etc.; Generally, correlated and non-normal; and Maps to one or more parameters, θ, of the load or FE model M. H. Scott (OSU) Sensitivity and Reliability AMC Seminar, 2011 15 / 34
Problem Statement Probability of Failure The probability that the limit state function evaluates to fail over all possible realizations of the random variables in x p f = f X (x)dx (12) g(x) 0 f X (x) is the joint PDF of all random variables Integrate over failure domain g(x) = 0 x n Contours of f X (x) SAFE FAIL x 1 g(x) = 0 M. H. Scott (OSU) Sensitivity and Reliability AMC Seminar, 2011 16 / 34
Problem Statement Monte Carlo Simulation Generate N random realizations of x using joint inverse CDF N i=1 p f (g(x(i) ) 0) N Captures nonlinearities of limit state function Computationally intense: N >= 1e5 for accurate p f Difficult to obtain sensitivity of failure to random variables (13) x n g(x) = 0 SAFE FAIL x 1 g(x) = 0 M. H. Scott (OSU) Sensitivity and Reliability AMC Seminar, 2011 17 / 34
Problem Statement First Order Approximation Linearize at design point, x, on failure surface closest to M x, mean value of x g(x) = 0 M x x x n SAFE FAIL x 1 g(x) = 0 x is solution to constrained optimization problem min x M x 2 (14) g(x)=0 Distance from mean to failure surface is measure of reliability Higher order approximations of failure surface also possible M. H. Scott (OSU) Sensitivity and Reliability AMC Seminar, 2011 18 / 34
Problem Statement Standard Normal Space To find design point, transform all random variables to space of uncorrelated, standard normals, u, with joint PDF ( 1 φ(u) = exp 1 ) (2π) n/2 2 ut u Find design point, u, in u-space (15) min u 2 (16) g(u)=0 Rotational symmetry of u-space: p f from inverse normal CDF 3 0.2 2 φ(u) 0.15 0.1 0.05 0 un 2 2 0 2 0 2 2 3 u n u 1 3 2 1 0 1 2 3 u 1 M. H. Scott (OSU) Sensitivity and Reliability AMC Seminar, 2011 19 / 34 1 0 1
Computing Reliability HLRF Iteration Find design point, u, by Hasofer-Lind-Rackwitz-Fiessler (HLRF) iteration Start at u 0 (usually at mean point 0), then u m+1 = ( α T u m + g(x) u g where α = ug u g Converges to a point where with the reliability index Probability of failure from normal CDF ) α (17) (18) u = βα (19) β = α T u (20) p f = Φ( β) (21) M. H. Scott (OSU) Sensitivity and Reliability AMC Seminar, 2011 20 / 34
Convergence of HLRF Iteration Design point, u, where Computing Reliability or magnitude is less than a specified tolerance u α T u α = 0 (22) u α T u α 2 tol (23) u n α α T u m α u m α T u m α u m u 1 g = 0 M. H. Scott (OSU) Sensitivity and Reliability AMC Seminar, 2011 21 / 34
Failure Sensitivity Sensitivity Vector The vector α is unit normal to failure surface α = ug u g In addition to providing search directions for HLRF iteration, component α i indicates The nature of random variable i in u-space (α i > 0 load, α i < 0 resistance) The importance (via magnitude) of random variable i relative to other random variables in u-space Transform to original x-space in order to determine nature and importance of random variables in problem domain Depends on α and Jacobian of x-u transformation M. H. Scott (OSU) Sensitivity and Reliability AMC Seminar, 2011 22 / 34
Failure Sensitivity Computation of Sensitivity Vector All measures of sensitivity and importance depend on the gradient of the limit state function u g = x g x u x/ u Jacobian of x-u transformation (24) For a limit state function defined in x-space in terms of finite element response, U f, which depends on an uncertain parameter θ, the chain rule gives x g = g U f U f θ θ x g/ U f analytic derivative of limit state function U f / θ nodal response sensitivity of FE model θ/ x mapping of random variable to FE model parameter (25) M. H. Scott (OSU) Sensitivity and Reliability AMC Seminar, 2011 23 / 34
Failure Sensitivity Direct Differentiation Analytical U f / θ in gradient of limit state function Pros: Cons: Requires as many limit state function evaluations as HLRF iterations Compute sensitivity as analysis proceeds Use factorized stiffness matrix, K T, for each parameter of FE model Not subject to round off error (beyond that of FE analysis) Difficult to implement Must be implemented for all objects present in FE analysis when path-dependency is present M. H. Scott (OSU) Sensitivity and Reliability AMC Seminar, 2011 24 / 34
Failure Sensitivity Finite Differences Approximate U f / θ in gradient of limit state function Pros: Cons: Easy to implement Applicable for free to all objects in FE analysis Requires more limit state function evaulations than HLRF iterations Re-run analysis for each parameter perturbation Reform and factorize tangent stiffness matrix, K T, for each parameter perturbation at each iteration Subject to round off error based on parameter perturbation M. H. Scott (OSU) Sensitivity and Reliability AMC Seminar, 2011 25 / 34
Cantilever Example Cantilever Reliability Determine probability that peak displacement exceeds 4.0 m g(x) = 4.0 U x 1 LN(4.1e5, 0.05) MPa maps to yield stress, σ y x 2 LN(2.0e8, 0.1) MPa maps to elastic modulus, E x 3 N(1710, 0.15) kn maps to applied load, P max x 4 N(5.0, 0.02) m maps to cantilever length, L x 5 N(0.5, 0.04) m maps to cross-section depth, d 2000 Load, P (kn) 1000 0 1000 Limit State 2000 5 0 5 Displacement, U (m) M. H. Scott (OSU) Sensitivity and Reliability AMC Seminar, 2011 26 / 34
Cantilever Example Reliability Results HLRF converges in 7 iterations with both approaches to U f / θ: DDM 7 evaluations of limit state function Finite Difference 42 evaluations of limit state function Reliability index, β = 1.983 Probability of failure, p f = Φ( 1.983) = 2.368% Random Variable, i µ i xi α i 1 σ y 4.1e5 kpa 3.897e5 kpa -0.2317 2 E 2.0e8 kpa 1.996e8 kpa -0.005413 3 P max 1710 kn 2032 kn 0.6393 4 L 5.0 m 5.100 m 0.5008 5 d 0.5 m 0.4787 m -0.5356 M. H. Scott (OSU) Sensitivity and Reliability AMC Seminar, 2011 27 / 34
Cantilever Example Mean and Design Point Response Cantilever load-displacement response at realization of random variables in x Load, P (kn) 3000 2000 1000 0 1000 2000 Mean Design Pt Limit State 3000 5 0 5 Displacement, U (m) M. H. Scott (OSU) Sensitivity and Reliability AMC Seminar, 2011 28 / 34
Reinforced Concrete Frame Limit state 2% roof drift RC Frame Example g(x) = 0.02(8.3) U 7 (26) 7 (9) 8 (10) 9 (2) (4) (6) 3.7 m 4 (7) 5 (8) 6 (1) (3) (5) 4.6 m 1 2 3 7.3 m 7.3 m Outer column sections Inner column sections Girder sections b=610 mm b=610 mm h=610 mm h=610 mm Bar area = 700 mm 2 h=690 mm b=460 mm Bar area = 1000 mm 2 M. H. Scott (OSU) Sensitivity and Reliability AMC Seminar, 2011 29 / 34
RC Frame Example Material Properties Capture axial-moment interaction of reinforced concrete using numerical integration of stress-strain response over cross-sections E=200,000 MPa f y =420 MPa α=0.05 σ f y E 1 1 αe ǫ f c =28 MPa f c =36 MPa f cu =33 MPa ǫ c =0.002 ǫ c =0.005 ǫ cu =0.06 ǫ cu =0.02 σ ǫ cu ǫ c ǫ ǫ cu ǫ c σ ǫ -f y f c f cu f c (a) (b) (c) M. H. Scott (OSU) Sensitivity and Reliability AMC Seminar, 2011 30 / 34
RC Frame Example Uncertainty Modeling Total of 142 random variables For each member: Reinforcing steel material properties (LN and 0.6 group correlation) Concrete material properties (LN and 0.6 group correlation) Cross-section width and height (N and no group correlation) Reinforcing bar area (N and no group correlation) For column members only: Cover concrete thickness (N and no group correlation) For all joints: X and Y coordinates (N and no group correlation) Deterministic gravity and lateral loads M. H. Scott (OSU) Sensitivity and Reliability AMC Seminar, 2011 31 / 34
RC Frame Example Pushover Load-Displacement At mean and design point realizations HLRF converges in 9 iterations: β = 2.76, p f = 0.29% DDM 9 limit state function evaluations Finite difference 1200+ limit state function evaluations M. H. Scott (OSU) Sensitivity and Reliability AMC Seminar, 2011 32 / 34
RC Frame Example Importance Ranking Highest importance on depth of members framing in to interior joint Steel yield stress and cover thickness also rank high Using γ importance vector instead of α as it accounts for correlation among random variables M. H. Scott (OSU) Sensitivity and Reliability AMC Seminar, 2011 33 / 34
RC Frame Example Conclusions DDM is accurate and efficient alternative to finite differences in assessing structural response sensitivity Fewer function evaluations in reliability analysis Other applications in optimization and system identification Ongoing work: Live load reliability analysis of bridge girders Particle finite element reliability analysis of fluid-structure interaction M. H. Scott (OSU) Sensitivity and Reliability AMC Seminar, 2011 34 / 34
References T. Haukaas and M. H. Scott. Shape sensitivities in the reliability analysis of nonlinear frame structures. Computers and Structures, 84(15 16):964 977, 2006. M. H. Scott and F. C. Filippou. Exact response gradients for large displacement nonlinear beam-column elements. Journal of Structural Engineering, 133(2):155 165, February 2007. M. H. Scott and T. Haukaas. Software framework for parameter updating and finite-element response sensitivity analysis. Journal of Computing in Civil Engineering, 22(5):281 291, 2008. M. H. Scott (OSU) Sensitivity and Reliability AMC Seminar, 2011 34 / 34