Shear Strength

Concrete Shear Strength

The shear stress (f_v) is computed as follows:

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At the left end or right end of a member where sections are located less than a distance d from the face of the support, they will be designed for the same shear, V, as that computed at a distance d.  For LRFD, substitute d_v for d.  Note that in LRFD C5.13.3.6.1/C5.12.8.6 (2^nd Paragraph), the design section may be taken at a distance dv past the end of the haunch.

In the case of Vee-Bottom culverts, where the bottom slab is thickened at the walls to provide a “fish bottom”, the appropriate additional thickness (at a distance d from the wall) is used to calculate the shear capacity of the bottom slab.  The bottom slab member is still assumed to be prismatic with a thickness equal to that at mid-span for flexural design at all locations.

For the Standard Specifications, the allowable concrete shear stress is computed as follows:

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Per LRFD 5.14.5.3/5.12.7.3, the provisions of 5.8/5.7 apply to fill depths less than 2 feet for both the sidewalls and the slabs.  Eriksson Culvert calculates the shear resistance of the concrete using the simplified approach as allowed in LRFD 5.8.3.3/5.7.3.3, and assumes a beta value of 2.0 (except when the member thickness is greater than or equal to 16 inches, see below).  This leads to the calculation of V_c and V_c,max as follows (V_p is equal to zero as this is a non-prestressed member):

Where the member thickness is equal to or greater than 16 inches or the member is in tension, the program uses the iterative approach in Appendix B5 for the calculation of beta used in Eq. 5.8.3.3-3/5.7.3.3-3, due to the lack of transverse reinforcing assumed in the design algorithm.  A value is assumed for theta, which is used in Eq. B5.2-1.  From this we can use Table B5.2-1 to look up a new value for theta.  This process is repeated until successive values for theta are approximately the same.  Once the calculations for theta have converged, theta is used in Table B5.2-1 to obtain a value for beta.

The proper selection of the governing provision in LRFD can summarized by the following flow chart:

4-sided structure

3-sided structure

Note that the iterative beta method (beta calculated using Appendix B) is actually usable in all situations, regardless of the member thickness or location.

For most culverts, moment and axial force demands are calculated on a preliminary estimate of each member thickness.  The area of reinforcing steel is determined based on moment demand and the corresponding section depth is checked for shear capacity.  If the depth is insufficient the loop re-iterates and this process is performed until all requirements are met. 

However, when the section depth is greater than 16 inches the flexure reinforcing steel affects the calculation of the shear strength and causes the strain to vary.  If the usual design approach (as detailed in the preceding paragraph) is followed then every iteration increases the member depth, reduces the flexural steel, and increases the strain.  This increase in strain reduces the provided shear stress.  The increase in strength in shear from every depth increment is overcome by the decrease in Beta which causes the loop to not converge and continue until the maximum shear strain is met or the maximum number of iterations is performed.  This process produces inefficient design results in which large member thicknesses are produced.     

Therefore the design of flexural steel reinforcing for members greater than 16 inches are controlled by the demands of shear and not moment.  The new process used by Eriksson Culvert for such member proceeds as follows:

1. Once the member thickness reaches 16 inches, a preliminary estimate of the reinforcing steel is based on moment demand.  Then the area of steel is used to calculate the shear capacity and checked against the shear demand.  If the preliminary estimate of the area of reinforcing steel is unsatisfactory the program enters a loop inside the shear calculations to increase the area of steel until this requirement is met.  This area of steel should be limited to around 2.0 in2/foot for practical purposes (note that this upper limit can be adjusted by the user).   If this limit of the reinforcing steel cannot be satisfied then the program increases the member thickness, recalculating the structural steel demand, and re-starts the shear demand loop.

2. When the member thickness is fixed, the program then checks the steel reinforcing and member thickness for moment demand requirements.

3. Proceed to check all other requirements such as service stress limits.

For fill depths greater than 2 feet, the horizontal slabs the provisions of Article 5.14.5.3/5.12.7.3 shall apply.  Note that this section uses d_e instead of d_v.

In all specifications, Mu can either be taken as that which occurs at the critical section for flexure design, or as that moment that is present at the critical section for shear (‘corresponding moment’).

For CHBDC, V_c can be found from the following:

The value of β can be determined using one of the methods in Section 8.  Eriksson Culvert uses the general method.

Shear Stirrups

Shear stirrups are only taken into account during Analysis mode.  In Design mode, the program assumes that the shear is resisted by the concrete only.

LRFD

The capacity of the defined shear steel area can be determined by:

where theta is assumed to be 45 degrees

The program then checks for the maximum capacity of the shear steel using:

Finally, the program finds the maximum spacing of the shear stirrups, which depends on the shear stress in the concrete, which can be calculated by:

Standard Specification

The capacity of the defined shear steel area can be determined by:

The program then checks for the maximum capacity of the shear steel using:

Finally, the program finds the maximum spacing of the shear stirrups.

CHBDC

Per Section 7.8.8.2.3, if V_c is less than V_u at any section, shear stirrups shall be designed to satisfy: