Equilibrium (Plasticity) Truss Model - Unified Theory of Concrete Structures - page 35

Figure 2.6 Equilibrium in torsion

is located on the dotted loop shown in Figure 2.6(a). This dotted loop is defined as the center
line of the shear flow, which may or may not lie in the mid-depth of the wall thickness.
A 2-D wall element ABCD is isolated and shown in Figure 2.6(b). It is subjected to pure
shear on all four faces. Let us denote the shear stress on face AD as τ 1 and that on face BC

as τ 2 . The thicknesses at faces AD and BC are designated h 1 and h 2 , respectively. Taking

equilibrium of forces on the element in the longitudinal direction we have

τ 1 h 1 = τ 2 h 2 (2.43)

Since shear stresses on mutually perpendicular planes must be equal, the shear stresses on
face AB must be τ 1 at point A and τ 2 at point B. Equation (2.43), therefore, means that τ h on

face AB must be equal at points A and B. Since we define q h as the shear flow, q must be

equal at points A and B. Notice also that the two faces AD and BC of the element can be selected
at an arbitrary distance apart without violating the equilibrium condition in the longitudinal
direction. It follows that the shear flow q must be constant throughout the cross-section.
The relationship between T and q can be derived directly from the equilibrium of moments
about the

= τ

axis. As shown in Figure 2.6 (a), the shear force along a length of wall element
d t is q d t . The contribution of this element to the torsional resistance is q d t ( r ), where r is the
distance from the center of twist (

axis) to the shear force q d t . Since q is a constant, integration
along the whole loop of the center line of shear flow gives the total torsional resistance:

q

T = r d t (2.44)