Equivalent Nodal Load for Nonprismatic Beam Element (Slab with Drop Panel)

Adisorn Owatsirwong, D.Eng.

This equivalent nodal force is opposite to Fixed-end reactions and written in the element force calculation as follows.

{Fe} = [ke]{u} - {q}

{q} = Equivalent nodal force

When assembling the element stiffness, those {q} are summed into the global load vector {F}, i.e.

{F} + {Q}= [K]{u} [F] = Nodal applied load

{Q} = Sum of equivalent nodal load vector

[K] = The global stiffness of the structure

MATLAB Code (case qe = q)

function feq = eqv_load_drop(L, bw, EI, EI_drop, q)
% Equivalent nodal forces for haunch/drop-panel beam
% q is downward positive


    a = (L*q)/2;

    % Numerator of moment term

    num = q * ( EI_drop*L^3 ...

              - 4*EI*bw^3 + 4*EI_drop*bw^3 ...

              + 6*EI*L*bw^2 - 6*EI_drop*L*bw^2 );



    % Denominator

    den = 12 * ( 2*EI*bw - 2*EI_drop*bw + EI_drop*L );
   
    b =  num / den;
    % Assemble equivalent load vector (DOFs: w1, th1, w2, th2)

    feq = [ -a;

            -b;

            -a;

             b ];

end