PSI - Issue 78

Marco Bonopera et al. / Procedia Structural Integrity 78 (2026) 1143–1150

1145

2018]. v tot ( a ) can instead be expressed by Eqs. (3a) and (3b) in Bonopera et al. (2018). As n=NL 2 /EI approaches 0 (axially unloaded beam), the limit of such equations yields the first-order deflection v I ( a ) ( x ), according to the Euler – Bernoulli theory, which can be expressed by Eqs. (4a) and (4b) in Bonopera et al. (2018). The shear beam theory is often adopted to model the behavior of structures both for stability or dynamic simulations (Challamel 2006). If the transverse shear deformability is considered along the simply supported PC girder-bridge (Han et al. 1999; Bažant and Cedolin 2010), the first-order deflection v I,shear ( a ) ( x ), based on the Timoshenko theory, can be described by the following equation (Timoshenko 1946):

AG 

for 0  x  L .

(1)

( ) a v x v x = ( ) a I, shear I ( ) ( )

( ) a M x

( )

+

I

Here the numerical factor for a rectangular cross-section α =1.5, with which the average shearing stress must be multiplied in order to achieve the shearing stress at the centroid of the cross-sections. M I ( a ) ( x ) is the bending moment in correspondence of the cross-section in which the deflection v I,shear ( a ) ( x ) is assumed. Elastic shear modulus G = E / [2 (1+  )]. Vice versa, a compressed member of length L can allow taking the second-order shear effects into account in a simply supported PC girder-bridge prestressed by a straight tendon which is in contact with the surrounding cross section [Fig. 1(a) in Bonopera et al. 2018]. This configuration characterized the experiments illustrated in Section 3. Indeed, a simply supported PC girder-bridge specimen was subjected to different values of post – tensioning ( N ) exerted by an eccentric straight tendon. Notably, the cross-sectional area of such a tendon was assumed to be unchanged after deformations. Accordingly, the total deflection v tot ( a ) ( x ), expressed by Eqs. (3a) and (3b) in Bonopera et al. (2018), can properly be yielded by considering the shear deformation and, specifically, multiplying the first-order deflection v I,shear ( a ) ( x ) [Eq. (1)] by the factor of the second-order effects, k ( u ) (Timoshenko and Gere 1961) reported as follows: ( ) shear shear ( ) ( ) ( ) tot, shear I, shear I, shear 3 shear 3 tan ( ) ( ) ( ) ( ) a a a u u v x v x k u v x u − = = for 0  x  L . (2)

Here we have the coefficient u shear which, in turn, is furnished by the following formula:

π 2

N

,

(3)

u

=

shear

N

crE,shear,1

where the first-order critical buckling load, N crE,shear,1 , considers the shear deformation and takes the following expression for a simply supported PC girder-bridge of length L (Bažant and Cedolin 2010):

N

.

(4)

crE,1

N

=

crE,shear,1

1 ( +

)

crE,1 N GA

0

The Euler buckling load N crE,1 = π o = A / m . m is the correction coefficient which assumes the nonuniform distribution of the shear stresses throughout the cross-sectional area A . For a rectangular cross-section, m =1.2. When the PC girder-bridge in Fig. 1 in Bonopera et al. (2018) meets all the requirements of the Timoshenko theory, it is subjected to higher small-deflections and rotations compared to the Euler – Bernoulli beam model. The total deflection v tot,shear ( a ) ( x ) [Eq. (2)] can well be approximated by multiplying the first-order deflection v I,shear ( a ) ( x ) [Eq. (1)] by the magnification factor of the second-order shear effects, 1/(1 – N / N crE,shear,1 ) (Timoshenko and Gere 1961; Bažant and Cedolin 2010 ) as illustrated as follows: 2 EI / L 2 , whilst the parameter A