Problem Statement 

 

 

 

 

 Figure 1. Schematic of a simply supported beam under load.

 Coupling dP and kinematics generates the Euler-Bernoulli beam theory statement

            dP:                                                               (1)

where y is vertical displacement, E is Young’s modulus, I is cross-section area moment of inertia, p(x) is a distributed load and P is a point load.

 Admissible boundary conditions at x = 0 and x = L include

            deflection:                                          y(x)

            slope:                                                

            moment:                                                                                              (2)

            shear:                                    

      The FE implementation of GWSh for (1) produces

            GWSh =

                                                                           (3)

 The Euler-Bernoulli beam theory possession of a fourth derivative requires the FE basis in (3) to support two derivatives. Thus a minimum of a quadratic polynomial is required.

      One can modify this problem statement such that a linear basis can be used. Noting the definition of moment M can be written as a displacement differential equation, separating dP and kinematics leads to a pair of conservation statements

             dP:                                                                          (4)

            kinematics:  (5)

For two degrees of freedom per node, i.e, {Q}e = {M, Y}eT the associated GWSh now produces the familiar syntactical form

            {WS}e = {DIFF}e{Q}e - {b}e                                                                                  (6) 

      The eligible boundary conditions remain as stated in (2), and the linear FE basis is adequate for this formulation. Recalling the template syntax statement 

            {WS}e = (const)(avg.data)e{dist.data}e(det)[FE matrix]{Q or data}e                        (7)

 the template pseudo-code essence for (6) is 

            {WS(M)}e = ( ) ( ) { } (-1) [A211] {M} – ( ) ( ) { } (1) [A200] {P} -Pd+ + BC

            {WS(Y)}e = (EI) ( ) { } (-1) [A211] {Y} + ( ) ( ) { } (1) [A200] {M} + BC             (8)

       Recall now the Timoshenko beam theory statement

             DP:                                                        (9)

            kinematics:                                                    (10)

 where r is transverse plane rotation, w is vertical displacement, E is Young’s modulus, I(x) is area moment of inertia of the beam cross-section for A(x), p(x) is a distributed load, P is a point load, G is shear modulus and k is a theory constant.

       For two degrees of freedom per node, the GWSh produces the familiar expression

             {WS}e = {DIFF}e{Q}e - {b}e                                                                                (11)

 The eligible boundary conditions are constraints on r and w, or their normal derivatives and the linear FE basis is again adequate. The template pseudo-code for uniform I for (9)-(10) is

             {WS(R)}e = (EI) ( ) ( ) {-1} [A211] {R} - (kG) (A) { } (0) [A201] {W}

+ (kG) (A) { } (1) [A200] {R} + BC

            {WS(W)}e = (kG) (A) { } (-1) [A211] {W} + (kG) (A) { } ( ) [A201] {R} +        (12)

+ ( ) ( ) { } (1) [A200] {P} +P{d} + BC

      The GWSh for the Timoshenko beam theory produces an algorithm that is “too stiff.” The use of under integration introduces an artificial diffusion mechanism, which produces a more accurate solution on a coarse mesh. From the lecture, under integration of the offending term in (10) produces the additional diffusion term 

            {WS(R)}e = {WS(R)}e + (kG/12) (A) { } (1) [A211] {R}                                     (13)

 The following data specification completes the problem statement assuming AISI 1018 cold-rolled steel. 

b

6 in.

h

3 in.

l

48 in.

E

29.7E+06 psi

I

54 in4

G

11.6 E+06 psi

k

0.866