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김경종
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where $Y_{i}$ is based on current stresses, computed by use of elastic coefficients, and temporarily updated for this step only. The revised $[E_{ep}]_{i-1}$ is $m[E]$ plus $(1 - m)$ times the right-hand side of Eq. 17.4-9; this reduces to
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$$
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[ \mathrm{E} _ {\mathrm{ep}} ] _ {i - 1} = [ \mathrm{E} ] - (1 - m) [ \mathrm{E} ] \left\{\frac {\partial Q}{\partial \sigma} \right\} _ {i - 1} \left\{\mathrm{C} _ {\lambda} \right\} _ {i - 1} ^ {T} \tag {17.5-8}
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$$
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Repeat steps 2 and 3a until convergence but without making the updates final (step 4 below) until convergence. In applying Eq. 17.5-2, use $(1 - m)\{\Delta\epsilon\}_{i}$ rather than $\{\Delta\epsilon\}_{i}$ , as $\{C_{\lambda}\}_{i-1}$ is zero for the elastic portion of the increment.
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3b. Optional. Without changing the load or recalculating $\{\Delta D\}_{i}$ , one can evaluate Eqs. 17.5-2 through 17.5-6 more accurately by dividing the increment $\{\Delta \epsilon\}_{i}$ into subincrements. After each such subincremental cycle, one updates $\{\sigma\}_{i}$ , $\{\alpha\}_{i}$ , and so on (Eqs. 17.5-10). Note that $\{C_{\lambda}\}_{i-1}$ is zero in elastic subincrements if the sampling point makes the elastic-to-plastic transition within the current load step.
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3c. Optional (but recommended). “Corrective” loads are introduced to prevent progressive drift, as discussed in connection with Figs. 17.2-5 and 17.2-6. Thus the next load increment is not simply $\{\Delta R\}_{i+1}$ ; rather, it is
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$$
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\{\Delta \mathbf {R} \} _ {i + 1} + \{\Delta \mathbf {R} _ {c} \} _ {i} \quad \text { where } \quad \{\Delta \mathbf {R} _ {c} \} = \{\mathbf {R} \} _ {i} - \sum \int_ {V _ {e}} [ \mathbf {B} ] ^ {T} \{\boldsymbol {\sigma} \} _ {i} d V \tag {17.5-9}
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$$
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and $\{R\}_{i}$ is the total externally applied load in cycle i. The summation spans all elements of the structure and expresses the loads that elements apply to nodes because they have stresses $\{\sigma\}_{i}$ . Stresses $\{\sigma\}_{i}$ are updated values (Eqs. 17.5-10).
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4. Update the solution:
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$$
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\{\mathbf {D} \} _ {i} = \{\mathbf {D} \} _ {i - 1} + \{\Delta \mathbf {D} \} _ {i} \quad \{\boldsymbol {\sigma} \} _ {i} = \{\boldsymbol {\sigma} \} _ {i - 1} + \{\Delta \boldsymbol {\sigma} \} _ {i} \tag {17.5-10}
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$$
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$$
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\{\alpha \} _ {i} = \{\alpha \} _ {i - 1} + \{\Delta \alpha \} _ {i} \quad (W _ {p}) _ {i} = (W _ {p}) _ {i - 1} + (\Delta W _ {p}) _ {i}
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$$
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5. Apply the next load increment and return to step 2.
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6. Stop when $\Sigma \{\Delta \mathbf{R}\}_i$ reaches the total applied load.
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Exercising steps 3a, 3b, and/or 3c permits load increments to be larger without increasing the error. Step 3c can be exercised repeatedly within a given load increment, either with or without updates in the structure matrix $[K_{r}]$ ; the effect is depicted in Figs. 17.2-3 and 17.2-4. If step 3c is used in this way there should be no test for unloading $(dF < 0)$ until these cycles are complete because intermediate cycles can give false indications. Use of step 3b within this cycling gives a three-level process: subincrements within iterations within load increments.
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Flow rules often allow little or no volume change. Thus, if plastic strains become extremely large, the response becomes nearly incompressible, and fully integrated elements may encounter numerical difficulties associated with locking of the mesh. Use of selective reduced integration is recommended.
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<!-- source-page: 542 -->
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<details>
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<summary>text_image</summary>
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R
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1
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2
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3
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E
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E
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E
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D
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</details>
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Figure 17.5-1. Load versus displacement plot for a representative d.o.f. D in a multi-d.o.f. model, showing convergence of the initial stiffness method.
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Initial-Stiffness Method. The step-by-step procedure remains as outlined in Section 17.3. We need make only a few modifications in step 5 to allow for the multidimensionality of stress. Specifically, in place of Eq. 17.3-6, we use Eqs. 17.5-1 through 17.5-6 to obtain plastic strain increments and update the yield criterion and the flow rule. Fraction m, if used, is given by Eq. 17.5-7. These calculations and updates are performed at each sampling point that sustains plastic strains. Equation 17.3-7 is replaced by
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$$
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\{\Delta \mathbf {R} _ {s} \} = \sum \{\Delta \mathbf {r} _ {s} \} \quad \text { where } \quad \{\Delta \mathbf {r} _ {s} \} = \int_ {V _ {e}} [ \mathbf {B} ] ^ {T} [ \mathbf {E} ] \{\Delta \boldsymbol {e} ^ {p} \} d V \tag {17.5-11}
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$$
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Note that sampling points that unload in a computational cycle $(\Delta F < 0)$ make no contribution to $\{\Delta \mathbf{R}_s\}$ .
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In the one-dimensional problem of Fig. 17.3-2a, the applied load must be carried by every element. The initial-stiffness method then converges slowly (Fig. 17.3-3b). In conditions of multiaxial stress, the yielding of one element is accompanied by a transfer of load to other elements, and the initial-stiffness method converges in fewer iterations (Fig. 17.5-1).
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# 17.6 NONLINEAR DYNAMIC PROBLEMS
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When the frequency of excitation exceeds about one-third the structure's lowest natural frequency of vibration, inertia becomes important and the problem is dynamic rather than quasistatic. Of the methods of response history analysis discussed in Chapter 13, direct time integration methods are usually the most effective for nonlinear problems. In this section, explicit and implicit methods are briefly discussed with particular emphasis on nonlinearities due to plasticity. Most remarks, however, are true of material nonlinearities in general. Direct integration methods that also include geometric nonlinearities are discussed [17.24]. It is assumed that the reader is familiar with Sections 13.9 through 13.14.
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Explicit Methods. Treatment of nonlinearities by explicit methods is usually straightforward, accurate, and effective. All remarks of Section 13.10 are valid, but elaboration is necessary for nonlinear problems, as follows.
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Explicit methods require that the internal force of each element, $\{r^{int}\}_{n}$ , be calculated before the new displacement $\{D\}_{n+1}$ can be computed. Element-by-element calculation of $\{r^{int}\}_{n}$ using Eq. 13.2-7 requires that element stresses $\{\sigma\}_{n}$ be known. For linear problems, $\{\sigma\}_{n} = [E][B]\{D\}_{n}$ in which $\{D\}_{n}$ is known. For plasticity, stress increment $\{d\sigma\} \approx \{\Delta\sigma\}$ can be computed from the strain incre-
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<!-- source-page: 543 -->
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ment, $\{\Delta \epsilon\} = [\mathbf{B}](\{\mathbf{D}\}_n - \{\mathbf{D}\}_{n-1})$ , and the constitutive law, Eq. 17.4-8. Hence, the stress at time $n \Delta t$ is given by $\{\sigma\}_n = \{\sigma\}_{n-1} + \{\Delta \sigma\}$ and $\{\mathbf{r}^{\mathrm{int}}\}_n$ can be obtained from Eq. 13.2-7.
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As with linear problems, the accuracy of an explicit solution is usually assured when the time-step stability criterion is satisfied. In Chapter 13, stability criteria are cited for several explicit methods as applied to linear problems, for example, Eqs. 13.10-8, 13.10-14, and 13.12-6 with $\beta = 0$ . Extensive computational experience suggests that these criteria are also valid for nonlinear problems provided that one uses the instantaneous value of $\omega_{max}$ , which is a function of material properties, element geometry, and mesh geometry.
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Many materials display “softening” behavior in which the tangent modulus decreases with increasing stress or strain level. Examples of such materials are ductile and brittle solids. For these materials, it can often be shown that the instantaneous value of $\omega_{max}$ will not exceed the $\omega_{max}$ for linearly elastic response [17.25]. Thus, a time step that is stable for purely elastic response will also be stable for nonlinear response. Alternatively, it is possible to change $\Delta t$ during a problem solution by continuously monitoring $\omega_{max}$ through the use of an element frequency bound. However, for most problems the possible increase in $\Delta t$ is small, as $\omega_{max}$ often continues to be governed by elements of the mesh that experience little or no plastic deformation.
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A material that displays “stiffening” has a tangent modulus that exceeds its initial modulus. In such a material $\omega_{max}$ may increase, making the stability criterion more restrictive. In problems with geometric nonlinearity, $\omega_{max}$ may increase or decrease. In these situations it is usually necessary to monitor $\omega_{max}$ through the use of element bounds during the course of a computational solution [17.24].
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Numerical instability is usually easy to detect in linear problems because the solution grows without limit. In nonlinear problems, with elastic–plastic or other energy-dissipating materials, extra energy introduced into the system by the numerical instability may be dissipated by plastic work or some other irreversible mechanism so that it is possible for the instability to be arrested $[17.26]$ . An “arrested instability” is often difficult to detect because the solution, although in error by 10% to 100% or more, may appear to be reasonable.
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Energy Balance Check. In the analysis of nonlinear dynamic problems by explicit methods, it is usually advisable to perform an energy balance check to help assure stable and accurate computation. Ideally, the energy at time $(n + 1)\Delta t$ in a system should satisfy the equation
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$$
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W _ {n + 1} ^ {\text {int}} + T _ {n + 1} = W _ {n + 1} ^ {\text {ext}} \tag {17.6-1}
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$$
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where W represents work and T represents kinetic energy. Physically, Eq. 17.6-1 states that the work of external loads is converted to kinetic energy and to energy either stored elastically or dissipated by plastic deformations. The separate terms in Eq. 17.6-1 are explained as follows.
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The internal work, $W_{n+1}^{int}$ , represents the work done by nodal loads that are developed from straining of material and is given by
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$$
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W _ {n + 1} ^ {\text {int}} = W _ {n} ^ {\text {int}} + \int_ {n \Delta t} ^ {(n + 1) \Delta t} \dot {W} ^ {\text {int}} d t \tag {17.6-2}
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$$
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<!-- source-page: 544 -->
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Noting that $\dot{W}_n^{\mathrm{int}} = \{\dot{\mathbf{D}}\}_{n}^{T|}\mathbf{R}^{\mathrm{int}}\}_n$ and approximating the integral in Eq. 17.6-2 by the trapezoidal rule, we obtain
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$$
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W _ {n + 1} ^ {\text {int}} = W _ {n} ^ {\text {int}} + \frac {\Delta t}{2} \left(\left\{\dot {\mathbf {D}} \right\} _ {n} ^ {T} \left\{\mathbf {R} ^ {\text {int}} \right\} _ {n} + \left\{\dot {\mathbf {D}} \right\} _ {n + 1} ^ {T} \left\{\mathbf {R} ^ {\text {int}} \right\} _ {n + 1}\right) \tag {17.6-3}
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$$
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Equation 17.6-3 is appropriate for use with explicit methods that compute velocities at whole time steps (e.g., the Newmark method with $\beta = 0$ ). When velocities are known at half time steps, as in the central-difference method, then Eq. 17.6-3 can be written as
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$$
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W _ {n + 1} ^ {\text {int}} = W _ {n} ^ {\text {int}} + \frac {\Delta t}{2} \left\{\dot {\mathbf {D}} \right\} _ {n + 1 / 2} ^ {T} \left(\left\{\mathbf {R} ^ {\text {int}} \right\} _ {n} + \left\{\mathbf {R} ^ {\text {int}} \right\} _ {n + 1}\right) \tag {17.6-4}
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$$
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The external work, $W_{n+1}^{\mathrm{ext}}$ , represents the work of the externally applied loads and is given by
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$$
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W _ {n + 1} ^ {\mathrm{ext}} = W _ {n} ^ {\mathrm{ext}} + \int_ {n \Delta t} ^ {(n + 1) \Delta t} \{\dot {\mathbf {D}} \} ^ {T} \{\mathbf {R} ^ {\mathrm{ext}} \} d t \tag {17.6-5}
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$$
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Difference expressions for $W_{n+1}^{ext}$ can be obtained from Eqs. 17.6-3 and 17.6-4 by replacing superscript “int” by “ext.” The kinetic energy, $T_{n}$ , is given by
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$$
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T _ {n} = \frac {1}{2} \{\dot {\mathbf {D}} \} _ {n} ^ {T} [ \mathbf {M} ] \{\dot {\mathbf {D}} \} _ {n} \tag {17.6-6}
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$$
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or, if half-time-step velocities are known, by
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$$
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T _ {n} = \frac {1}{2} \left(T _ {n - 1 / 2} + T _ {n + 1 / 2}\right) \tag {17.6-7}
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$$
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To construct an energy balance, we note that, in general, Eq. 17.6-1 is not satisfied exactly. To measure the quality of a solution, we can use-
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$$
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W _ {n} ^ {\text {int}} + T _ {n} - \left| W _ {n} ^ {\text {ext}} \right| \leq e (W _ {n} ^ {\text {int}} + T _ {n} + \left| W _ {n} ^ {\text {ext}} \right|) \tag {17.6-8}
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$$
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where e is a tolerance and absolute magnitude bars are a precaution to avoid small negative values of $W^{ext}$ due to numerical errors. Terms within parentheses on the right-hand side of Eq. 17.6-8 represent the total energy in the system. The left-hand side is the energy error. A stable explicit computation should satisfy Eq. 17.6-8 with $e \leq 0.02$ [17.24]. If satisfaction of Eq. 17.6-8 requires $e \geq 0.05$ , even for models with hundreds of elements and using thousands of time steps, then instability should be suspected.
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<!-- source-page: 545 -->
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response will also be stable for nonlinear response. Hence, with $\rho = 7.4(10^{-4})$ lb-sec $^2$ /in. $^4$ , the highest element frequency is $(\omega_{\max})_e = 2\sqrt{E/\rho}/L = 8.054(10^5)$ rad/sec. Thus stable integration by the central-difference method requires $\Delta t \leq 2/(\omega_{\max})_e = 2.483(10^{-6})$ sec according to Eq. 13.10-14 with $\xi = 0$ .
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The computational procedure in Table 13.10-1 and the Fortran program in Fig. 13.10-3 were used except that Subroutine INTFOR in Fig. 13.10-3 was replaced by the code in Fig. 17.6-1 (which does not include the energy balance check of Eq. 17.6-8). Additional alterations to the main program in Fig. 13.10-3 include adding SIGMA(101) and SIGYLD(101) to the DIMENSION statement, adding the tangent modulus (ET) and initial yield strength ( $\sigma_{Y}$ in Fig. 17.3-1) to the READ statement, initializing the stress in each element to zero and the yield strength in each element to $\sigma_{Y}$ , modifying the tip load, the CALL INTFOR statement, and the WRITE statement.
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The Fortran code in Fig. 17.6-1 generates the internal force vector for a model with elastic–plastic behavior. The yield function, called YLDFUN in Fig. 17.6-1, is obtained from the discussion following Eq. 17.4-13 as $F = |\sigma_{x}| - \sigma_{Y}$ , where $\sigma_{x}$ in element I is called SIGMA(I) in Fig. 17.6-1. Because of isotropic hardening, the stress required for renewed or continued yielding exceeds $\sigma_{Y}$ ; this stress is called $\sigma_{B}$ in Fig. 17.3-1b and SIGYLD(I) in Fig. 17.6-1. The code uses modulus E when operating on the linearly elastic part of the stress–strain diagram shown in Fig. 17.3-1 and modulus ET when operating on the plastic part of the diagram. When states of stress make the transition from elastic
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```prolog
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SUBROUTINE INTFOR(D,DOLD,FINT,SIGMA,X,SIGYLD,E,ET,CSA,NELE)
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IMPLICIT DOUBLE PRECISION (A-H,O-Z)
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DIMENSION D(1),DOLD(1),FINT(1),SIGMA(1),X(1),SIGYLD(1)
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NUMNOD=NELE+1
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C---- Zero internal force vector.
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DO 10 I=1,NUMNOD
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10 FINT(I)=0.
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C---- Loop over elements.
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DO 30 K=1,NELE
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RL=X(K+1)-X(K)
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C---- Compute strain increment.
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EPSOLD=(DOLD(K+1)-DOLD(K))/RL
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EPSNEW=( D(K+1)- D(K))/RL
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EPSINC=EPSNEW-EPSOLD
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C---- Compute stress increment (DSIGMA) assuming elastic response.
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DSIGMA=E*EPSINC
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C---- Check if DSIGMA satisfies yield function .LT. 0.
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YLDFUN=DABS(SIGMA(K)+DSIGMA)-SIGYLD(K)
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IF (YLDFUN.LT.0.) GO TO 20
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C---- If material was plastic before strain increment, and
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C---- YLDFUN .GE. 0., then entire strain increment is plastic.
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IF (DABS(SIGMA(K)).EQ.SIGYLD(K)) THEN
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DSIGMA=ET*EPSINC
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GO TO 20
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ELSE
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C---- Compute the portion of the strain increment that is elastic.
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RATIO=(SIGYLD(K)-DABS(SIGMA(K))/DABS(DSIGMA)
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DSIGMA=RATIO*EPSINC*E + (1.-RATIO)*EPSINC*ET
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ENDIF
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C---- Update stress and account for isotropic hardening.
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20 CONTINUE
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SIGMA(K)=SIGMA(K)+DSIGMA
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IF (DABS(SIGMA(K)).GT.SIGYLD(K)) SIGYLD(K)=DABS(SIGMA(K))
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C---- Assemble contribution into internal force vector.
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F1=-SIGMA(K)*CSA
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F2=+SIGMA(K)*CSA
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FINT(K)=FINT(K)+F1
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FINT(K+1)=FINT(K+1)+F2
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30 CONTINUE
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RETURN
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END
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```
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Figure 17.6-1. Fortran program to compute the internal force vector for a mesh of linear displacement bar elements with elastic-plastic, isotropic-hardening material.
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<!-- source-page: 546 -->
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to plastic, the fraction of the strain increment that is elastic is computed according to the procedure described in step 3a of Section 17.5. In the Fortran code, the elastic fraction of the strain increment, $m$ in Eq. 17.5-7, is called RATIO.
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The stress time-history results for the midpoint of element 20 (at $x = 9.75$ in.) are shown in Fig. 17.6-2 for $\Delta t = 2.4(10^{-6}) \sec (C_n = 0.966)$ and 83 time steps. Two separate stress waves arrive at different times. These correspond to an elastic wave traveling at speed $c_e = \sqrt{E / \rho}$ , which arrives first, followed by a plastic wave traveling at speed $c_p = \sqrt{E_t / \rho} = \frac{1}{2} c_e$ . Thus, at $x = 9.75$ in., an elastic wave arrives at $t = 0.0484$ msec and a plastic wave arrives at $t = 0.0968$ msec.
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The solution is devoid of noise until after the passage of the plastic wave. The reason is that during the initial passage of the elastic stress wave, numerical noise causes stress excursions above the initial yield stress. These excursions are transmitted at the plastic wave speed, and hence do not arrive at element 20 until approximately 0.11 msec has elapsed.
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Implicit Methods. Compared with explicit methods, implicit methods for nonlinear problems are less attractive in every respect save one—that is, the possibility of using large time steps permitted by the excellent stability properties of popular implicit methods. Nonlinearities present the same difficulty to both static and implicit dynamic solution algorithms: stiffness is a function of displacements, which are not known in advance. For example, in Eq. 13.11-5, $[K^{eff}]$ is a function of $\{D\}_{n+1}$ , which is unknown. Methods for addressing this difficulty are analogous to the tangent-stiffness and initial-stiffness methods described in Section 17.3 for nonlinear quasistatic problems.
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Tangent-Stiffness (Implicit) Method. In the tangent-stiffness method, the internal force in the equations of motion, Eq. 13.9-2, is written as
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$$
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\{\mathbf {R} ^ {\mathrm{int}} \} _ {n + 1} = \{\mathbf {R} ^ {\mathrm{int}} \} _ {n} + [ \mathbf {K} _ {t} ] \{\Delta \mathbf {D} \} \tag {17.6-9}
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$$
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where
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$$
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\{\Delta \mathbf {D} \} = \{\mathbf {D} \} _ {n + 1} - \{\mathbf {D} \} _ {n} \tag {17.6-10}
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$$
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Combining Eqs. 17.6-9 and 17.6-10 with the equations of motion, Eq. 13.9-2, and the trapezoidal rule equations, Eqs. 13.11-3 and 13.11-4, we obtain
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$$
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[ \mathbf {K} ^ {\mathrm{eff}} ] \{\Delta \mathbf {D} \} = \{\mathbf {R} ^ {\mathrm{eff}} \} _ {n + 1} \tag {17.6-11}
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$$
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where
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$$
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[ \mathbf {K} ^ {\mathrm{eff}} ] = \frac {4}{\Delta t ^ {2}} [ \mathbf {M} ] + \frac {2}{\Delta t} [ \mathbf {C} ] + [ \mathbf {K} _ {t} ] \tag {17.6-12}
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$$
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and
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$$
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\{\mathbf {R} ^ {\text { eff }} \} _ {n + 1} = \{\mathbf {R} ^ {\text { ext }} \} _ {n + 1} - \{\mathbf {R} ^ {\text { int }} \} _ {n} + [ \mathbf {M} ] \left(\frac {4}{\Delta t} \{\dot {\mathbf {D}} \} _ {n} + \{\ddot {\mathbf {D}} \} _ {n}\right) + [ \mathbf {C} ] \{\dot {\mathbf {D}} \} _ {n} \tag {17.6-13}
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$$
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Note that $[K_{t}]$ must be predicted using $\{D\}_{n}$ (and possibly $\{\dot{D}\}_{n}$ if strain rate effects are important) and must be factored at least once each time step during nonlinear
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<!-- source-page: 547 -->
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<details>
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<summary>line</summary>
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| Time (milliseconds) | Exact | Central difference |
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| ------------------- | ----- | ------------------ |
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| 0.00 | 0 | 0 |
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| 0.05 | -40 | -40 |
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| 0.10 | -80 | -80 |
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| 0.15 | -80 | -80 |
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| 0.20 | -80 | -80 |
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</details>
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Figure 17.6-2. Stress versus time at x = 9.75 in. for a 40-element model of a 20-in. bar with elastic-plastic material using $\Delta t = 2.4(10^{-6})$ sec ( $C_{n} = 0.966$ ). Figure 13.10-2 applies, except that $P_{0} = 80{,}000$ lb.
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response. If $[K_{t}]$ is not an accurate prediction of the true tangent-stiffness matrix from time $n \Delta t$ to time $(n + 1) \Delta t$ ; then the solution of Eq. 17.6-11 for $\{\Delta D\}$ will be in error. The error in nodal forces—that is, the residual—is given by the imbalance in the equation of motion (Eq. 13.9-2) as
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$$
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\{\mathbf {R} ^ {\text { err }} \} = \{\mathbf {R} ^ {\text { ext }} \} _ {n + 1} - [ \mathbf {M} ] \{\ddot {\mathbf {D}} \} _ {n + 1} - [ \mathbf {C} ] \{\dot {\mathbf {D}} \} _ {n + 1} - \{\mathbf {R} ^ {\text { int }} \} _ {n + 1} \tag {17.6-14}
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$$
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where $\{R^{int}\}_{n+1}$ is computed using Eq. 17.6-9 and an improved tangent-stiffness matrix (i.e., $[K_{t}]$ obtained using $\{D\}_{n+1} = \{D\}_{n} + \{\Delta D\}$ , which is obtained by solving Eq. 17.6-11). Alternatively, $\{R^{int}\}_{n+1}$ can be computed element-by-element in the same way as for explicit direct integration. If measures are not taken to control the growth of $\{R^{err}\}$ , the solution will diverge in a manner similar to the instability displayed by explicit methods when the stability criterion is violated [17.26]. A computational procedure for the trapezoidal rule with error control is given in Table 17.6-1. Note that the procedure for error control is essentially the pseudo-load approach, described later in this section, applied within a time step. If enough iterations are performed within each time step to guarantee that $W^{err}$ of Table 17.6-1 is bounded for the entire solution, then the trapezoidal rule algorithm is unconditionally stable, although not necessarily accurate [17.27]. To assure accuracy, $W^{err}$ should be small.
|
||||
|
||||
Initial-Stiffness (Implicit) Method. In the initial-stiffness method, the initial tangent-stiffness matrix of the structure is used throughout the analysis and corrective loads due to nonlinearities are transferred to the right-hand side of the equations solved at each iteration. With this method, internal forces are given by
|
||||
|
||||
$$
|
||||
\{\mathbf {R} ^ {\mathrm{int}} \} _ {n + 1} = [ \mathbf {K} ] \{\mathbf {D} \} _ {n + 1} - \{\Delta \mathbf {R} _ {s} \} _ {n + 1} \tag {17.6-15}
|
||||
$$
|
||||
|
||||
<!-- source-page: 548 -->
|
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|
||||
TABLE 17.6-1. COMPUTATIONAL PROCEDURE FOR DIRECT INTEGRATION OF MATERIAL NONLINEAR PROBLEMS BY THE TRAPEZOIDAL RULE METHOD WITH TANGENTIAL STIFFNESS. SUPERPOSED TILDE ( $^{-}$ ) DENOTES QUANTITIES OBTAINED FROM AN ESTIMATED $[K_{L}]$ .
|
||||
|
||||
1. Form [C] and [M] (also form $[\mathbf{K}_t]$ if $\{\mathbf{D}(t = 0)\} \neq \{0\}$ ).
|
||||
2. Set initial conditions $\{\mathbf{D}\}_{0} = \{\mathbf{D}(t = 0)\}$ and $\{\dot{\mathbf{D}}\}_{0} = \{\dot{\mathbf{D}}(t = 0)\}$ ; use Eq. 13.9-1 to compute $\{\ddot{\mathbf{D}}\}_{0} = [\mathbf{M}]^{-1}(\{\mathbf{R}^{\mathrm{ext}}\}_{0} - [\mathbf{C}]\{\dot{\mathbf{D}}\}_{0} - [\mathbf{K}]\{\mathbf{D}\}_{0})$ ; use Eq. 13.2-7 to compute $\{\mathbf{R}^{\mathrm{int}}\}_{0}$ ; $n = 0$ .
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||||
3. Form $[\mathbf{K}_t]$ and compute $[\mathbf{K}^{\mathrm{eff}}] = \frac{4}{\Delta t^2} [\mathbf{M}] + \frac{2}{\Delta t} [\mathbf{C}] + [\mathbf{K}_t].$
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4. Form $\{\mathbf{R}^{\mathrm{eff}}\}_{n + 1} = \{\mathbf{R}^{\mathrm{ext}}\}_{n + 1} - \{\mathbf{R}^{\mathrm{int}}\}_{n} + [\mathbf{M}] \left( \frac{4}{\Delta t} \{\dot{\mathbf{D}}\}_{n} + \{\ddot{\mathbf{D}}\}_{n} \right) + [\mathbf{C}]\{\dot{\mathbf{D}}\}_{n}$ .
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5. Solve $[\mathbf{K}^{\mathrm{eff}}]\{\Delta \bar{\mathbf{D}}\} = \{\mathbf{R}^{\mathrm{eff}}\}_{n + 1}$ for $\{\Delta \bar{\mathbf{D}}\}$ .
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6. Compute $\{\tilde{\mathbf{D}}\}_{n + 1} = \frac{2}{\Delta t}\{\Delta \tilde{\mathbf{D}}\} -\{\dot{\mathbf{D}}\}_{n}$ and $\{\tilde{\mathbf{D}}\}_{n + 1} = \frac{4}{\Delta t^2}\{\Delta \tilde{\mathbf{D}}\} -\frac{4}{\Delta t}\{\dot{\mathbf{D}}\}_{n} - \{\ddot{\mathbf{D}}\}_{n}$ .
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||||
7. Compute $\{\hat{\mathbf{R}}^{\mathrm{int}}\}_{n + 1} = \sum \{\hat{\mathbf{r}}^{\mathrm{int}}\}_{n + 1} = \sum \int [\mathbf{B}]^T \{\hat{\sigma}\}_{n + 1} dV$ using $\{\hat{\mathbf{D}}\}_{n + 1} = \{\mathbf{D}\}_n + \{\Delta \hat{\mathbf{D}}\}$ .
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||||
8. Compute $\{\mathbf{R}^{\mathrm{ext}}\} = \{\mathbf{R}^{\mathrm{ext}}\}_{n+1} - [\mathbf{M}]\{\dot{\bar{\mathbf{D}}}\}_{n+1} - [\mathbf{C}]\{\dot{\bar{\mathbf{D}}}\}_{n+1} - \{\bar{\mathbf{R}}^{\mathrm{int}}\}_{n+1}$ .
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||||
9. If $W^{\mathrm{err}} = \Delta t\{\dot{\mathbf{D}}_{[n + 1]}^{T}\{\mathbf{R}^{\mathrm{err}}\} >$ tolerance, add $\{\mathbf{R}^{\mathrm{err}}\}$ to $\{\mathbf{R}^{\mathrm{ext}}\}_{n + 1}$ and go to Step 4.
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||||
10. If tolerance is satisfied, update histories $\{\mathbf{D}\}_{n+1} = \{\mathbf{D}\}_n + \{\Delta \mathbf{D}\}$ , $\{\mathbf{D}\}_{n+1} = \{\mathbf{D}\}_{n+1}$ , $\{\ddot{\mathbf{D}}\}_{n+1} = \{\ddot{\mathbf{D}}\}_{n+1}$ , $\{\mathbf{R}^{\mathrm{int}}\}_{n+1} = \{\hat{\mathbf{R}}^{\mathrm{int}}\}_{n+1}$ ; $n \leftarrow n + 1$ , go to Step 3.
|
||||
|
||||
where [K] is the initial tangent-stiffness matrix and $\{\Delta R_{s}\}$ is a vector of nodal loads computed so that the material constitutive law is satisfied. Combining Eq. 17.6-15 with the equations of motion, Eq. 13.9-2, and the trapezoidal rule equations, Eqs. 13.11-3 and 13.11-4, we obtain
|
||||
|
||||
$$
|
||||
[ \mathbf {K} ^ {\mathrm{eff}} ] \{\mathbf {D} \} _ {n + 1} = \{\mathbf {R} ^ {\mathrm{eff}} \} _ {n + 1} \tag {17.6-16}
|
||||
$$
|
||||
|
||||
where
|
||||
|
||||
$$
|
||||
[ \mathbf {K} ^ {\mathrm{eff}} ] = \frac {4}{\Delta t ^ {2}} [ \mathbf {M} ] + \frac {2}{\Delta t} [ \mathbf {C} ] + [ \mathbf {K} ] \tag {17.6-17}
|
||||
$$
|
||||
|
||||
and
|
||||
|
||||
$$
|
||||
\begin{array}{l} \left\{\mathbf {R} _ {n + 1} ^ {\text {eff}} = \left\{\mathbf {R} _ {n + 1} ^ {\text {ext}} + [ \mathbf {M} ] \left(\frac {4}{\Delta t ^ {2}} \left\{\mathbf {D} \right\} _ {n} + \frac {4}{\Delta t} \left\{\dot {\mathbf {D}} \right\} _ {n} + \left\{\ddot {\mathbf {D}} \right\} _ {n}\right) \right. \right. \\ + [ \mathbf {C} ] \left(\frac {2}{\Delta t} \{\mathbf {D} \} _ {n} + \{\dot {\mathbf {D}} \} _ {n}\right) + \{\Delta \mathbf {R} _ {s} \} _ {n + 1} \tag {17.6-18} \\ \end{array}
|
||||
$$
|
||||
|
||||
From Eq. 17.5-11, the corrective loads can be written as
|
||||
|
||||
$$
|
||||
\{\Delta \mathbf {R} _ {s} \} _ {n + 1} = \sum \int_ {V _ {e}} [ \mathbf {B} ] ^ {T} \{\sigma^ {\text {err}} \} d V \tag {17.6-19}
|
||||
$$
|
||||
|
||||
where $\{\sigma^{\mathrm{err}}\}$ is the linear-stress prediction less the actual stresses.
|
||||
|
||||
<!-- source-page: 549 -->
|
||||
|
||||
Before Eq. 17.6-16 can be solved for $\{\mathbf{D}\}_{n+1}$ , it is necessary to use information at time $n\Delta t$ , for example, from an estimate of $\{\mathbf{D}\}_{n+1}$ , so that $\{\Delta\mathbf{R}_s\}_{n+1}$ can be computed. If solution vector $\{\mathbf{D}\}_{n+1}$ is not in close agreement with the prior estimate (and usually it is not), then $\{\Delta\mathbf{R}_s\}_{n+1}$ must be recomputed using the improved $\{\mathbf{D}\}_{n+1}$ and the solution of Eq. 17.6-16 repeated. This process continues until convergence of $\{\mathbf{D}\}_{n+1}$ and $\{\Delta\mathbf{R}_s\}_{n+1}$ . As an example, consider a uniform linear-displacement bar element with the material shown in Fig. 17.3-3a. If $\epsilon_B$ is the strain obtained from the predicted $\{\mathbf{D}\}_{n+1}$ , then $\sigma_C = E\epsilon_B$ is the stress that would result if material behavior were linearly elastic. Stress $\sigma_C$ , which violates the material's constitutive law, is implied by the term $[\mathbf{K}]\{\mathbf{D}\}_{n+1}$ in Eq. 17.6-15. To satisfy the constitutive law, we compute a “corrective” stress $\sigma_C - \sigma_B$ . The corresponding corrective load in the element is then $A(\sigma_C - \sigma_B)$ and the corrective internal load vector for the element is $\{\Delta\mathbf{r}_s\}_{n+1} = A(\sigma_C - \sigma_B) | -1 - 1|^T$ .
|
||||
|
||||
The principal advantage of this method is that $[K^{eff}]$ need be formed and factored only once. Corrective loads $\{\Delta R_{s}\}_{n+1}$ are estimated using only information available at time $n \Delta t$ . Since this estimate is rarely satisfactory, several iterations must usually be performed within a time step to improve $\{\Delta R_{s}\}_{n+1}$ so that the material constitutive law is satisfied. For problems with mild nonlinearities, this method can be more economical than the tangent-stiffness method. When nonlinearities are severe, widespread, or both, this method often demonstrates very poor convergence or divergence and may require a time step that scarcely exceeds a Courant number of unity.
|
||||
|
||||
Remarks on Implicit Methods. Solution of nonlinear problems by implicit methods is not easy and there are many pitfalls. Convergence is usually the major difficulty. It is usually good practice to repeat a solution using a smaller time step. Obviously the results of analyses using a different time step should show good agreement if one is to have faith in them.
|
||||
|
||||
Of the tangent-stiffness and initial-stiffness methods, the tangent-stiffness method has better convergence properties, but is often much more expensive. However, the initial-stiffness approach sometimes does not converge. A hybrid solution strategy consisting of the initial-stiffness approach with occasional stiffness matrix reformation can be effective and is often used in practice. This strategy can be particularly effective when combined with the inverse-Broyden method of Fig. 17.2-8 to update the initial and revised stiffness matrices.
|
||||
|
||||
# 17.7 A PROBLEM HAVING GEOMETRIC NONLINEARITY
|
||||
|
||||
Consider the plane cantilever beam shown in Fig. 17.7-1a. We seek the quasistatic deflections produced by loads P and $M_{L}$ . We assume that the beam is slender and that its material is linearly elastic at all times. For small deflections, linear theory is adequate; for example, the root moment is $M_{0} = PL_{T} + M_{L}$ because moment arm $L_{T}$ is almost independent of load. For larger deflections, the moment arm H of force P is less than $L_{T}$ , and $M_{0} = PH + M_{L}$ , where H depends on P and $M_{L}$ . In such a problem, the nonlinearity is called geometric nonlinearity. The name implies that deformations significantly alter the location or distribution of loads, so that equilibrium equations must be written with respect to the deformed geometry, which is not known in advance.
|
||||
|
||||
<!-- source-page: 550 -->
|
||||
|
||||
|
||||
|
||||
<details>
|
||||
<summary>text_image</summary>
|
||||
|
||||
H
|
||||
y, v
|
||||
M_L
|
||||
P
|
||||
L_T
|
||||
x, u
|
||||
</details>
|
||||
|
||||
(a)
|
||||
|
||||
|
||||
|
||||
<details>
|
||||
<summary>text_image</summary>
|
||||
|
||||
P > P_cr
|
||||
P < P_cr
|
||||
L_T
|
||||
</details>
|
||||
|
||||
(b)
|
||||
Figure 17.7-1. (a) Cantilever beam under tip loading. (b) Column under axial load.
|
||||
|
||||
To solve this problem we can use plane frame elements—that is, straight elements that have two nodes, six d.o.f., axial stiffness AE, and bending stiffness EI. The equilibrium configuration under a given loading will be obtained by the Newton–Raphson method of Fig. 17.2-3. The particulars given in what follows allow us to solve geometrically nonlinear problems of various plane structures that can be modeled by plane frame elements. Other elements [17.31] and other solution algorithms can also be used, but the same physical concept applies to all: we seek a displacement state in which the deformed structure is in equilibrium with loads applied to it [17.28, 17.29].
|
||||
|
||||
A Computational Algorithm. A structure—for example, one of those in Fig. 17.7-1—is divided into finite elements in the usual way. We can begin with zero initial displacements, $\{D\}_{0} = \{0\}$ , or with a better estimate for $\{D\}_{0}$ if one is available. A load $\{R\}$ is applied and the corresponding $\{D\}$ is sought by iteration as follows. Computational details are explained subsequently.
|
||||
|
||||
1. Form the tangent-stiffness matrix $[K_{i}]_{i}$ of the structure in the current configuration $\{D\}_{i}$ (which is $\{D\}_{0}$ initially, $\{D\}_{1}$ after the first computational cycle, etc.).
|
||||
2. Use displacements $\{D\}_{i}$ to form resisting loads $\{R_{R}\}_{i}$ , which are loads applied to structure nodes by the deformed elements. (In the first computational cycle, these loads are zero if $\{D\}_{0} = \{0\}$ .)
|
||||
3. Solve for displacement increments $\{\Delta \mathbf{D}\}_{i+1}$ and update the configuration to $\{\mathbf{D}\}_{i+1}$ :
|
||||
|
||||
$$
|
||||
\{\mathbf {D} \} _ {i + 1} = \{\mathbf {D} \} _ {i + 1} + \{\Delta \mathbf {D} \} _ {i + 1} \quad \text { where } \quad \{\Delta \mathbf {D} \} _ {i + 1} = [ \mathbf {K} _ {t} ] _ {i} ^ {- 1} (\{\mathbf {R} \} + \{\mathbf {R} _ {R} \} _ {i}) \tag {17.7-1}
|
||||
$$
|
||||
|
||||
Net loads $\{R\} + \{R_{R}\}_{i}$ are an imbalance between externally applied and internally generated nodal loads. The load imbalance drives the structure toward a configuration that reduces the imbalance. In an equilibrium configuration, the imbalance is zero.
|
||||
|
||||
4. Check for convergence; for example, see if $\|\Delta D_{i+1}\| < e\|D_{i+1}\|$ , where e is a small number chosen by the analyst. If not converged, return to step 1.
|
||||
|
||||
Clearly, many alternatives are possible within this overall strategy. Modified Newton–Raphson cycles can be invoked by updating $[K_{t}]$ only occasionally. Or, the inverse-Broyden method can be used. A final load level $\{R\}$ can be approached
|
||||
Reference in New Issue
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