Issue 75

P. Grubits et alii, Fracture and Structural Integrity, 75 (2026) 124-156; DOI: 10.3221/IGF-ESIS.75.10

  1 2 W C    p

R R

 

dV

(7)

ijkl ij

kl

To control the extent of plastic deformation, an upper limit 0 p W is prescribed on the allowable complementary plastic work p W . This constraint is enforced by relating the limit directly to the actual residual stress distribution. Specifically, it is assumed that:

R R ij ij   

(8)

This identification yields a conservative and practical estimate, enabling the formulation of a tractable constraint on allowable plastic work. The resulting condition that limits plastic dissipation takes the form:

1 2

R R dV W  

W C  

 

(9)

0

p

ijkl ij

kl

p

0

Elastic and elasto-plastic limit analysis based on residual forces To establish a load-dependent response, assume that the surface traction 

 , i t x q applied to the body occupying volume

V , is defined as:   ,

    0 i i q t x m t q x 

t  



, 0

(10)

where   m t is a time-dependent load multiplier that increases monotonically, and   0 i q x is a time-independent surface load distribution defined over the surface q S . In the case of truss structures with constant structural weight 1 n s i i i G Al     where  is the material density, i A is the cross-sectional area, and i l is the length of the i -th bar, the loading expression can be reformulated in terms of the externally applied nodal load vector 0 P and a prescribed reference load P , such that:

0 m  P P

(11)

Throughout the loading history, the maximum value of the load multiplier m for which all bar elements remain within the elastic range is denoted by el m . Consequently, the corresponding elastic limit load el P is given by:

el el m  P P

(12)

Up to this point, no plastic deformations occur in any of the individual members, and the structural response remains entirely elastic. Accordingly, the elastic internal force vector corresponding to the applied external load 0 P can be expressed as:

1   N F G K P 1 el T 

(13)

0

where F denotes the flexibility matrix, G is the geometric matrix, and K is the global stiffness matrix. However, once the load exceeds

el P , plastic deformations begin to develop in certain members and continue to evolve

pl P are reached:

pl m and its corresponding load level

until the plastic collapse multiplier

128