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Example of the 45˚ Cone Method

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Note: The following information is meant only as a quick reference. The reader should refer to ACI 355.1R-91 and ACI 349 Appendix B, 1985 for complete information.
Excerpts from “State of the Art Report on Anchoring to Concrete” ACI 355.1R-91
(3.2.3.2) When the embedment of an anchor or group of anchors is insufficient to develop the tensile strength of the anchor steel, a pullout cone failure of the concrete is the principal failure mode. When the spacing of anchors or location of an edge interferes with the development of the full cone strength of an anchor, its capacity will be reduced.

In ACI 349 Appendix B, (ACI committee 349, 1985) the angle of the failure cone of headed and expansion anchors is assumed as 45˚.

The following formulas have been developed to describe behavior of headed studs, expansion, and undercut anchors.

Anchor Tension Load
Concrete Cone Failure (3.2.3.2)
ACI 349 Appendix B, limits the tensile capacity of cone failure of an anchor or group of anchors, to a uniform stress on the stress cone surface of the anchors.

(3.2)

= Strength reduction factor
= .85 for uncracked concrete
= .65 in zone of potential cracking
A
= The summation of the projected areas of individual stress cones minus the areas of overlap and of any area, or areas, cut off by intersecting edges (in2)
Fu
= Tensile capacity of the concrete cone (lbs.)
f 'c
= Concrete Compressive strength (psi)
Minimum Edge Distance (3.2.3.2)
With respect to the minimum edge distance, tests reported a direct relationship between anchor load and side cone failure. The equation below is a more correct design boundry for the edge distance for headed anchors.

(3.6)

Fut
= ASTM - specified tensile strength of the anchor bolt (kips)
m
= Minimum edge distance (in.)
f 'c
= Concrete Compressive strength (psi)
Side Cone Bursting (3.2.3.2)
The average failure load for a side cone (bursting) failure is given as:

(3.7)

m
= Actual edge distance (in.)
Fu
= Failure load of side cone bursting (kips)
f 'c
= Concrete Compressive strength (psi)
Splitting Failure (3.2.3.2)
With respect to minimum edge distance, the following criteria is proposed to preclude splitting failure occurring at a load lower than the capacity of the concrete cone failure or pullout. This equation is valid for anchor spacing S ≥ 2 in.

(3.21)

m
= Minimum edge distance (in.)
D
= Anchor bolt diameter (in.)
ld
= Embedment depth to the bottom of the anchor (in.)
Anchor Shear Loading
Steel Failure (3.2.5.1)
Steel failure usually occurs after relatively large displacements and is most common for deep embedments, lower strength steels and large edge distances. The failure load depends on the steel area and the steel strength and given by:

(3.22)

N
= Where the factor N takes account of the steel “shear” strength and has the range .6 to .7
As
= Tensile stress area (in2)
fut
= Ultimate tensile strength (psi)
Fu
= Ultimate shear strength of steel (lbs.)
Edge Failure (3.2.5.2.1)
For all types of anchors loaded in shear toward an adjacent, free edge and exhibiting a concrete failure, the failure load is influenced by the concrete tensile strength, the edge distance m and the stiffness of the anchor. Another influencing factor is the embedment depth. The Failure surface has conical shape that may radiate from the embedded end of the anchor for shallow embedments or from the upper part of the anchorage for deep embedments.
ACI 349 appendix B, Commentary gave a design shear strength of:

(3.23)

= .85
f 'c
= Concrete compressive strength (psi)
m
= Distance from anchor to free edge (in)
= 3.14
Vu
= Concrete shear breakout capacity (lbs.)
Minimum Edge DIstance Shear (3.2.5.2.1)
ACI 349, Appendix B further recommends a minimum side cover or edge distance m required to preclude edge failures, be calculated by:

(3.24)

D
= Anchor diameter (in.)
Fut
= Anchor ultimate tensile load (lbs.)
f 'c
= Concrete Compressive strength (psi)
m
= Minimum edge distance for shear loading (in.)
Critical Spacing (3.2.5.2.1)
Calculating the failure load of single fastenings situated in a corner or in narrow members. Klinger, Mendonca, and Malik (1982) recommend a critical (minimum) edge spacing of:

(3.29)

D
= .9
m
= Critical minimum edge spacing (in)
D
= Anchor diameter (in)
Fut
= ASTM - specified tensile strength of the anchor bolt (kips)
f 'c
= Concrete Compressive strength (psi)
Combined Tension and Shear (3.2.5.2.2)
The behavior of anchors under combined tension and shear loading lies in between the behavior under tension or shear loading, and for a given depth of embedment, is dependent on the angle of the loading.
To calculate the failure load under combined tension one approach is the straight-line function.

(3.32)

Ta,Va
= Applied tensile and shear load, respectively
Tu,Vu
= Ultimate tensile and shear load, respectively
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