Additional Internal Load Contributions

The non-linear strain measures for the most general comprehensible case, involving finite displacements but small rotations of the beam, are derived in Section 7.1 and given by Eq. (7.15). In accordance with the calculus of variations, see Funk [77], the virtual variant of these strain measures reads 

As shown here, the strains may be split into the linear part Se(x) and the non-linear part S (x). The prior corresponds to the variation of the linear strain measures as they are obtained for the thin-walled beam in Section 7.2 and given by Eq. (7.31). Further on, the internal loads vector N x,t) of the beam can be subdivided into the portion N x,t), associated with the initial configuration, and the portion N x,t) related to the superposed deformation. Then the virtual strain energy based on the general formulation of 

In the last line of Eq. (8.40), Eq. (8.39) is introduced and expressions are multiplied out. The first term represents the linearized virtual strain energy (t). The initial internal loads vector N x,t) can be determined in advance, while the vector N x,t) of the other internal loads needs to be substituted with the aid of a constitutive relation. Therefore, the second term is free of non-linear products, while the third term contains such products and, consequently, will be neglected. Such a second-order theory corresponds to the equilibrium of the slightly deformed system and contributes the virtual work of initial stresses Thus, the virtual work of internal mechanical 

The internal loads vector N(x, t) constitutes the mechanical part of the combined internal loads L x, t) of the beam as given by Eq. (8.3). Correspondingly, the initial internal loads vector N x,t) and the associated initial external loads vector n x, t) have the following components  

As outlined above, the initial internal loads need to be known and, for this purpose, may be determined with the aid of the first-order theory developed so far. Due to the absence of non-linear strains related to both rotation and warping, only normal and transverse forces, as well as bending moments, have to be obtained. In accordance with the equilibrium equations and natural boundary conditions of Eq. (8.36), these can be expressed as 

---

The criteria of admissibility for the virtual displacements have been discussed in Section 3.4.2. As rigidity has been assumed in the case at hand, the occurring displacements do not cause strains. Therefore, virtual strains do not exist and, consequently, there are no contributions of internal loads to the virtual work. As expected, the virtual work of external impressed loads is identical to the term in the static principle of virtual displacements. The accelerated motion results in the additional term representing the virtual work of the loads of inertia. In general, the principle may be formulated as follows ... 

Sulfur vulcanization leads to a variety of cross-link structures as shown in Figure 1. All the sulfur does not result in cross-links some of it remains as pendent accelerator polysulfide groups and internal cyclic polysulfides. These alternative structures do not contribute to load bearing or strength properties and are more prevalent in unaccelerated or weakly accelerated vulcanization systems. Additional heating can also reduce the polysulfide rank of the cross-links. In some elastomers, this leads to a larger number of cross-links. However, in natural mbber or its synthetic polyisoprene equivalent, the overall result is a loss of cross-links, especially at temperatures over 160°C. 

Several equations for the concrete contribution can be found in the relevant literature. However, this equation describes the design shear resistance of a RC member without shear reinforcement. It is important to note that this definition for the concrete contribution is different from the definition of State r given above in this section. To determine Erc, the load at the first shear crack should be calculated. Additionally, all the usual design verifications for RC (failure of the concrete struts, shift of moment line, etc.) have to be considered. For ductility reasons, the member should have a minimum internal shear reinforcement ratio, otherwise strengthening is not recommended. 

---