$-------------------------------------------------------------------------------
$                       RIGID FORMAT No. 1, Static Analysis
$                       Thermal Bending of a Beam (1-10-1)
$ 
$ A. Description
$ 
$ This problem demonstrates the solution of a beam subjected to a thermal
$ gradient over the cross-section. Two end conditions are solved, clamped-free
$ and clamped-pinned end conditions.
$ 
$ An equivalent linear gradient in the normal direction was used for the input
$ data. However, the actual temperatures at points on the cross-section were
$ input on the TEMPRB card in order to produce correct stresses. The beam was
$ subdivided into 14 variable lengths for maximum efficiency.
$ 
$ B. Input
$ 
$ [???Fig. refs.]
$ 
$ C. Theory
$ 
$ For subcase 1, the effective temperature gradient, T', (see NASTRAN
$ Theoretical Manual) is:
$ 
$                  1
$        T'(x)  =  - integral from z integral from y  T(x,y,z)y dy dz        (1)
$                  I
$ 
$ where
$ 
$        I =   integral from z integral from y  y  dy dz                     (2)
$ 
$ Using the given temperature distribution the effective gradient is:
$ 
$                  3
$        T' =  T  x                                                          (3)
$               c
$                                              4
$ where T  is calculated to be 0.170054 deg./in  by substituting the temperature
$        c
$ distribution into Equation 1 and evaluating the expression:
$ 
$               1
$        T   =  -  integral from z integral from y  Cy  dy dz                (4)
$         c     I
$ 
$ Since the bar is not redundantly constrained the curvature at the center line
$ is:
$ 
$       2
$      d v                            3
$      ---  =  -alphaT'  =  -alphaT  x                                       (5)
$        2                         c
$      dx
$ 
$ The slope is:
$ 
$                                   2
$      dv                          d v           alpha     4
$      --  =  integral from 0 to x --- dx  =  -  ----- T  x                  (6)
$      dx                            2             4    c
$                                  dx
$ 
$ The deflection is:
$ 
$                                    dv          alpha     5
$      v(x)  =  integral from 0 to x -- dx  =  - ----- T  x                  (7)
$                                    dx           20    c
$ 
$ The moment, M, shear, V, and axial stress, å , are:
$                                             x
$                                                                    +
$               ( 2         )                                        |
$               (d v        )                                        |
$      M  =  EI (---  +  àT')  =  0                                  |
$               (  2        )                                        |
$               (dx         )                                        |
$                                                                    |
$            dM                                                      |       (8)
$      V  =  --   =  0                                               |
$            dx                                                      |
$                                                                    |
$                                                                    |
$      sigma (x,y)  =  E(epsilon  - alphaT) =                        |
$           x                   x                                    |
$                                                                    |
$                                              3   3                 |
$      E(alphayT' - alphaT)  =  Ealpha(T y - Cy ) x                  |
$                                       c                            |
$                                                                    +
$                                             6
$ where C = 1 has dimensions of degrees/length .
$ 
$ For subcase 2, with a simple support at x = 10.0, we calculate the deflection
$ due to subcase 1 and apply a constraint load P  to remove the deflection at
$ the end.                                      L
$ 
$                                   alphaT
$                3EI                      c   2
$      P   =  -  ---  v(L)  =  3EI  -------  L                               (9)
$       L          3                 20
$                 L
$ 
$ Note: Transverse shear deflection is neglected.
$ 
$ The deflections and slopes are the sum of the results for the two independent
$ loads as follows.
$ 
$                       P                alphaT
$                        L     2    3          c  5
$ deflection: v(x)  =  --- (3Lx  - x ) - ------- x   =
$                      6EI                20
$                                                                           (10)
$                      alphaT
$                            c    3    2      3   2
$                      ------- (3L  - L x - 2x ) x
$                       40
$ 
$                                 alphaT
$                      alphav           c     3     2       3
$ slope: alpha (x)  =  ------  =  -------  (6L  - 3L x - 10x )x             (11)
$             z        alphax      40
$ 
$ The net stress is the sum of the stress due to each load:
$ 
$                                     M y
$                              3  3    L
$ sigma (x,y) = Ealpha(T y - Cy )x  - ---  =
$      x                c              I
$                                                                           (12)
$                           3  3   3     2
$           Ealpha[(T y - Cy )x  - -- T L (L - x)y]
$                    c             20  c
$ 
$ 
$ where M  is the moment due to the constraint load.
$        L
$ 
$ D. Results
$ 
$ Tables 1 and 2 compare the analytical maximum value of displacement,
$ constraint force, element force, and stress to the maximum deviation of
$ NASTRAN in each category. All results are within 2.66%.
$ 
$    Table 1. Comparison of NASTRAN and Analytical Results, Clamped-Free Ends
$                                   (Subcase 1)
$              -------------------------------------------------------
$               CATEGORY       MAXIMUM         MAXIMUM
$                              ANALYTICAL      NASTRAN        PERCENT
$                              VALUE           DIFFERENCE     ERROR
$              -------------------------------------------------------
$              Displacement              -2              -4
$                            -1.1054 x 10     2.9424 x 10     2.66
$              -------------------------------------------------------
$              Constraint            0              *           *
$              Force
$              -------------------------------------------------------
$              Element               0              *           *
$              Force
$              -------------------------------------------------------
$              Element                   +3
$              Stress         5.1965 x 10         0.671         0.01
$              -------------------------------------------------------
$ 
$             * These results vary with the computer. The very small numbers are
$               essentially zero when compared to subcase 2 results.
$ 
$ 
$   Table 2. Comparison of NASTRAN and Analytical Results, Clamped-Pinned Ends
$                                   (Subcase 2)
$              -------------------------------------------------------
$               CATEGORY       MAXIMUM         MAXIMUM
$                              ANALYTICAL      NASTRAN        PERCENT
$                              VALUE           DIFFERENCE     ERROR
$              -------------------------------------------------------
$              Displacement              -3             -4
$                             4.3936 x 10     8.024 x 10      0.18
$              -------------------------------------------------------
$              Constraint                +2
$              Force         -2.2859 x 10     6.0841          2.66
$              -------------------------------------------------------
$              Element                   +2
$              Force          2.2859 x 10     6.0846          2.66
$              -------------------------------------------------------
$              Element                   +3
$              Stress         5.1965 x 10     4.4136 x 10     0.85
$              -------------------------------------------------------
$-------------------------------------------------------------------------------
