5.11 Shear forces

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Shear forces are the least reliable quantities in FE analysis. They easily oscillate

and tend to exhibit erratic behavior; see Fig. 5.33.

In a Kirchhoff plate model the shear forces are the third-order derivatives

of the unit deflections,

qx = −K(w,xxx +w,yyx ), qy = −K(w,xxy +w,yyy ) (5.78)

while in a mixed model they are the first-order derivatives of the bending

moments,

qx = mxx,x +mxy,y qy = myy,y +myx,x , (5.79)

and they are therefore often constant because in mixed methods mainly linear

functions are used to approximate the bending momente mxx,mxy,myy.

452 5 Slabs

Fig. 5.33. Distribution of shear forces in a slab near the supports

In a Reissner–Mindlin model the shear forces are proportional to the shearing

strains γx and γy, and thus proportional to the rotations θx, θy, and the

derivatives of w:

qx = K

1 − ν

2

￣λ

2 (θx + w,x ) qy = K

1 − ν

2

￣λ

2 (θy + w,y ) . (5.80)

In slabs, no shear reinforcement is necessary if the shear stresses remain

below some threshold limits like τ ≤ 0.5 MN/m2; see Fig. 5.34. Only at

certain critical points the shear stresses exceed these limits. But even then it

is questionable whether it is really necessary to provide shear reinforcement,

because while the numbers indicate a trend, the magnitude of the numbers

itself is dubious.

In Fig. 5.35 the distribution of the shear forces in a horizontal (qx) and a

vertical (qy) cross section in front of a wall is plotted. While the shear force qx

exhibits normal variability the shear force qy grows exponentially to a peak

value of 104 kN/m. At such points it is more appropriate to calculate an

equivalent punching strain, as in the case of the slab in Fig. 5.36. Nowadays

this is done routinely by most FE programs; see Fig. 5.37.

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