• Effective Bearing Length of Crane Mats


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    • Abstract: This crane mat design method is the most straightforward. Once the load from the crane has been calculated, whether an. outrigger ... foundation analysis are shown graphically in Fig. 3. We see. that the actual bearing pressure between the mat and the soil. is ...

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Effective Bearing Length of Crane Mats
David Duerr, P.E.
2DM Associates, Inc., Consulting Engineers
Houston, Texas
INTRODUCTION written in equation form as follows. The basic arrangement
is illustrated in Fig. 1. Note that Eqs. 7 and 8 are written for
Crane mats are used to distribute the high concentrated
the design of timber crane mats. The term d in Eq. 7 and the
loads from mobile cranes over a relatively large ground area
coefficient 1.5 in Eq. 8 are not used for the design of steel
so that the soil is loaded at tolerable bearing pressures. This
mats.
has been common construction industry practice for many
decades. Although crane mats are most commonly made of P +W
Areqd = (1)
heavy timbers, fabricated steel mats are occasionally used qa
under large cranes or when soil conditions are poor.
Areqd
The analysis of a crane mat requires a determination of the Lreqd = (2)
B
length of the mat that actually bears on the soil and contributes
to the support of the crane. At working loads, this is a relatively Lreqd - C
Lc = (3)
simple "beam on an elastic foundation" problem. However, 2
such a solution may not produce a realistic result due to the
nonlinearity of the soil as the ultimate bearing capacity is P
q= (4)
approached. Further, the elastic properties of the soil needed Lreqd B
to perform such an analysis are not often available.
M=
( qB ) L2
c (5)
The purpose of this paper is to develop a practical means 2
of calculating the effective bearing length of a crane mat that
is based on readily available values and that produces an M
fb = Fb (6)
acceptably safe and reliable result. Bd 2 6
V = ( qB ) ( Lc - d ) (7)
CURRENT PRACTICE
1.5V
Engineers in construction presently use a number of fv = Fv (8)
different approaches to design crane mats. The two most Bd
common of these methods are described here. where:
P = crane load applied to one mat;
Mat Length Based on Soil Bearing Capacity
W = self-weight of the mat;
This crane mat design method is the most straightforward.
Once the load from the crane has been calculated, whether an P
outrigger load or a crawler track pressure, the required crane
mat area is calculated by dividing the crane load plus the Lc Crane Mat
weight of the mat by the allowable ground bearing pressure.
Divide this area by the width of the mat and we have the
required effective bearing length. This mat length is then used
to calculate bending and shear stresses in the mat, based on
the assumption of a uniform pressure equal to the crane load
divided by the bearing area acting upward on the bottom of qt
the mat. If the actual stresses are equal to or less than the Lreqd or Leff
allowable stresses, the mat is acceptable. This method can be
Copyright 2010 by 2DM Associates, Inc. Fig. 1. Simple Crane Mat Arrangement
Presented at the Crane & Rigging Conference 1
Houston, Texas May 26 27, 2010
qa = allowable ground bearing pressure; be assumed, Eqs. 9 through 14 solved, and then Leff adjusted
Areqd = required mat bearing area; as necessary to satisfy the equalities of Eqs. 12 and 14. Note
B = mat width; that the design is complete when either the bending stress
Lreqd = required effective bearing length of the mat; or the shear stress reaches its allowable value. When the
C = bearing width of the track or outrigger pad; mat is made of the most common hardwood species and the
Lc = cantilevered length of the mat; allowable ground bearing pressure is not unusually high,
q = ground bearing pressure due to P; bending strength usually governs the mat design. However,
M = bending moment in the mat; both shear and bending must always be checked.
d = mat depth (or thickness);
fb = bending stress due to M;
Comments on These Design Methods
Fb = allowable bending stress;
V = shear in the mat; Both of these crane mat design methods are in popular
fv = shear stress due to V; and, use and give adequate results. There is one important short-
Fv = allowable shear stress. coming in the way these calculations are commonly applied
in practice. Neither shows how close a particular design is
to reaching its load carrying limit. The first method loads the
Mat Length Based on Mat Strength
soil to its allowable bearing capacity and then shows that the
This method is the reverse of the first method. Here, the stresses in the mats are something less than their allowable
effective bearing length Leff of the mat is assumed initially values. The second method loads the mats to the allowable
and is then adjusted until the resulting bending stress or bending or shear capacity and then shows that the ground
shear stress reaches the corresponding allowable stress. The bearing pressure is something less than the allowable pressure.
ground bearing pressure is then computed using this effective
bearing length. If the actual pressure is equal to or less than We can examine this problem by way of an example.
the allowable ground bearing pressure, the mat is acceptable. Consider a load of 100,000 pounds applied to a mat by an
Again, we can write the method in equation form. As above, outrigger pad that is 24 inches wide along the length of the
Eqs. 13 and 14 are written for the design of timber crane mats. mat. This pad is supported at the middle of a 12" x 4' x 20'
timber crane mat. The allowable ground bearing pressure for
Leff - C the site is 3,000 psf. Allowable stresses for the mat design are
Lc = (9) Fb = 1,400 psi and Fv = 200 psi. The mat is checked by both
2
methods in Table 1.
P
q= (10)
Leff B Table 1. Design Method Comparison - Example 1
Soil Bearing Mat Strength
M=
( qB ) L2
c (11) Capacity Method Method
2
Mat Weight, W 4,000 lbs 4,000 lbs
M
fb = = Fb (12) Areqd (Eq. 1) 34.67 ft2
Bd 2 6
Lreqd (Eq. 2) 8.67 feet
V = ( qB ) ( Lc - d ) (13)
Lc (Eq. 3) 3.33 feet
1.5V Assumed Leff 14.48 feet
fv = = Fv (14)
Bd
Lc (Eq. 9) 6.24 feet
P +W
qt = qa (15) q (Eq. 4; Eq. 10) 2,885 psf 1,727 psf
Leff B
M (Eq. 5; Eq. 11) 769,231 lb-in 1,612,800 lb-in
where:
fb (Eq. 6; Eq. 12) 668 psi 1,400 psi
Leff = effective mat bearing length;
V (Eq. 7; Eq. 13) 26,923 lbs 36,184 lbs
qt = actual ground bearing pressure; and,
all other terms are as previously defined. fv (Eq. 8; Eq. 14) 70 psi 94 psi
Note that this method is iterative. A value of Leff must qt (Eq. 15) 3,000 psf 1,796 psf
2
Both methods show that the mat design is acceptable, but indication of the utilization of the mat strength and the soil
the design margin is not obvious. The Soil Bearing Capacity bearing capacity. This method uses as input only values that
method shows that the applied ground bearing pressure are routinely available.
is equal to the allowable bearing pressure and that the mat
bending stress is 668 psi, or 48% of the allowable bending
Effective Length Calculation Method
stress. The Mat Strength method shows that the mat is loaded
to its allowable stress and that the ground bearing pressure is Consider first the bending strength of the mat. We wish
1,796 psf, or 60% of the allowable bearing capacity. to determine the effective bearing length Leff at which both
the allowable bending strength of the mat and the allowable
Now let's repeat this analysis using a load of 135,256 ground bearing pressure of the soil are reached. This can
pounds. All other values remain the same. The results of this be done by expressing q in terms of qa (Eq. 16) and then
exercise are shown in Table 2. writing Eq. 11 in terms of this expression for q, Eq. 9, and the
allowable moment of the mat Mn (Eq. 17). By rearranging
Table 2. Design Method Comparison - Example 2 terms, Eq. 17 can be written as Eq. 18. The last term of this
equation can be shown to be trivial, so Eq. 18 reduces to Eq.
Soil Bearing Mat Strength 19, which is a quadratic equation in which the quantity in the
Capacity Method Method first set of parentheses is a, the quantity in the second set is b,
Mat Weight, W 4,000 lbs 4,000 lbs and the quantity in the third set is c. The standard solution is
shown in Eq. 20.
Areqd (Eq. 1) 46.42 ft2
W
Lreqd (Eq. 2) 11.60 feet q = qa - (16)
Leff B
Lc (Eq. 3) 4.80 feet
2
" W % B " Leff ! C %
Assumed Leff 11.60 feet M n = $ qa ! (17)
# Leff B ' 2 $
& # 2 ' &
Lc (Eq. 9) 4.80 feet
q (Eq. 4; Eq. 10) 2,914 psf 2,914 psf ( qa B ) L2 + ( -2qa BC - W ) Leff
eff (
+ qa BC 2 + 2CW - 8 M n )
2
M (Eq. 5; Eq. 11) 1,612,800 lb-in 1,612,800 lb-in C W
- =0 (18)
fb (Eq. 6; Eq. 12) 1,400 psi 1,400 psi Leff
V (Eq. 7; Eq. 13) 44,317 lbs 44,317 lbs
( qa B ) L2 + ( -2qa BC - W ) Leff
eff (
+ qa BC 2 + 2CW - 8 M n )
fv (Eq. 8; Eq. 14) 115 psi 115 psi
=0 (19)
qt (Eq. 15) 3,000 psf 3,000 psf
-b b 2 - 4 ac (20)
Here we see that the two methods converge at the load Leff L
2a
where both the mat strength and the soil bearing capacity limits
are reached. This second example shows us that the capacity where L is the actual length of the mat.
limit of this mat on this soil is a crane load of 135,256 pounds.
Thus, the crane load of 100,000 pounds in the first example A solution of Leff must also be made based on the shear
loaded the mat/soil combination to 74% of its capacity. This strength of the crane mat. Again, this discussion and the
cannot be seen in the calculations summarized in Table 1. equations following are based on the methods for timber
design. Appropriate modifications must be made for the design
Although these commonly used crane mat design methods of steel crane mats. Here, Eq. 13 is written in terms of Eqs. 9
generally yield results that are acceptably safe, they do not and 16 and the allowable shear of the mat Vn (Eq. 21). This
provide an indication of the percent utilization (or demand/ equation is then rewritten in quadratic form (Eq. 22) where the
capacity ratio) of the mat/soil combination. quantity in the first set of parentheses is a, the quantity in the
second set is b, and the quantity in the third set is c. Eq. 20 is
used to solve for Leff.
MAT EFFECTIVE BEARING LENGTH CALCULATION
A practical method of crane mat design can be derived " W % " Leff ! C %
Vn = $ qa ! B ! d' (21)
that is based upon the current practice, but that gives an # Leff B ' $
& # 2 &
3
( qa B ) L2 + ( -2Vn - qa BC - 2qa Bd - W ) Leff
eff
is 10,000 psf and a factor of safety of 2.00 is to be applied,
giving us an allowable ground bearing pressure of 5,000 psf.
+ (WC + 2Wd ) = 0 (22) As before, the allowable stresses for the mat design are 1,400
psi in bending and 200 psi in shear. The results of the mat
Last, a limit of the effective bearing length based on design are shown in Table 3.
deflection is proposed. Examination of numerous design
examples using only the criteria of bending and shear strength Table 3. Example Mat Design Results
shows that some mats exhibit excessive deflections (greater
than one inch) on softer soils. Therefore, we should limit the Mat Weight, W 4,000 lbs
effective bearing length based on the stiffness of the mats. Leff (Eq. 19) 9.41 feet controlling value
This is a more difficult criterion to define, since there isn't
a well defined deflection limit state as exists for bending and Leff (Eq. 22) 11.81 feet
shear. A deflection limit of 0.75% of Lc is suggested, based on Leff (Eq. 25) 13.54 feet
an examination of numerous mat designs.
Lc (Eq. 9) 3.70 feet
The deflection of a crane mat is commonly calculated by
q (Eq. 10) 4,652 psf
treating the mat as a cantilever beam of length Lc and loaded
by an upward uniform pressure equal to q. We can express M (Eq. 5) 1,530,575 lb-in
this in equation form as Eq. 23.
fb (Eq. 6) 1,329 psi 95% of Fb
( qB ) L4 V (Eq. 7) 50,288 lbs
!= c (23)
8 EI
fv (Eq. 8) 131 psi 65% of Fv
where:
qt (Eq. 15) 4,758 psf 95% of qa
D = vertical deflection;
I = moment of inertia of the mat; and, The mat behavior may be treated as elastic for our
all other terms are as previously defined. purposes. Soil may be treated as elastic at allowable load
levels, but is very nonlinear as the ultimate bearing capacity is
This deflection criterion will only control the effective approached. Thus, we will use two different analysis methods
bearing length with softer soils. Examination of such designs to evaluate this design.
shows us that q 0.9 qa. If we let D = 0.0075 Lc and use
this approximation for q, we can easily solve Eq. 23 for Lc The mat behavior at the working load can be analyzed as
(Eq. 24) and Leff (Eq. 25). a beam on an elastic foundation (Young and Budynas, 2002).
In addition to the values already discussed, we must also know
0.06 EI the modulus of elasticity E of the timbers and the modulus of
Lc = 3 (24) subgrade reaction ks of the soil. E may be taken as 1,200,000
0.9 ( qa B )
psi for the hardwood species commonly used for crane mat
construction. Bowles (1996) suggests that a practical value
Leff = 2 Lc + C L (25) of ks in kips per cubic foot is 12 qult where qult is the ultimate
bearing capacity in kips per square foot. Alternately, ks in
The smallest value of Leff based on the moment and shear pounds per cubic inch is qult / 144 where qult is the ultimate
strength analyses and the deflection analysis is taken as the bearing capacity in pounds per square foot. The basis of this
effective bearing length of the crane mat. The mat and the soil approximation is illustrated in Fig. 2.
are then evaluated based on the usual assumption of a uniform
bearing pressure q (Eq. 10) between the mat and the soil over Using these values, the results of the beam on an elastic
the effective bearing area. foundation analysis are shown graphically in Fig. 3. We see
that the actual bearing pressure between the mat and the soil
The performance of this design method can be examined is greatly variable, not uniform as assumed in the standard
by sizing a crane mat using the method and then performing design methods. However, the peak bearing pressure due to P
a failure analysis to determine the actual capacity provided. is only about 5% greater than that given by the design method,
Consider a standard 12" x 4' x 20' hardwood timber crane which is not significant. The actual bearing length is shown as
mat centrally loaded by a 24" wide pad. The applied load 15'-1", markedly greater than the effective bearing length of
is 175,000 pounds. The ultimate bearing capacity of the soil 9'-5" calculated using the design method.
4
qult P = 460,000 lbs.
Crane Mat
Simplified approximation of
load-displacement behavior
q Actual load-displacement curve
Linear Nonlinear
behavior behavior Peak GBP =
qult 10,000 psf
Xmax ks =
Xmax
Fig. 4. Bearing Pressure Curve at 460,000 pounds
Displacement
and the moment in the mat is 3,660,397 pound-inches, which
Fig. 2. Modulus of Subgrade Reaction gives a bending stress of 3,177 psi, or 227% of the allowable
(based on Bowles 1996, Fig. 9-9) bending stress. The allowable bending and shear stresses for
hardwood timbers are based on a nominal design factor of
For the failure analysis, we can use the program 2.1 or greater (ASTM 2000, ASTM 2006), so we can see that
FADBEMLP, a nonlinear analysis program that is packaged the proposed method provides a strength design factor on the
with the Bowles (1996) text. This program treats the beam as order of 1.75 for this example.
elastic and the soil as elastic/perfectly plastic (Fig. 2). That is,
the soil behavior is linearly elastic up to the ultimate bearing Given the soil conditions on many construction sites, the
capacity and then perfectly plastic thereafter. The value of use of the approximation for the modulus of subgrade reaction
ks = 12 qult discussed above is based on the assumption that discussed here is questionable. Excavated and backfilled
the ultimate bearing capacity is reached at a deformation of areas, compacted surface layers, and other deviations from a
about one inch. As the bearing pressure in the center of the homogeneous soil mass all serve to increase the uncertainty
mat reaches qult, the bearing pressure curve takes on the shape with which soil elastic properties can be determined. Analyses
shown in Fig. 4, for which the applied load is 460,000 pounds, such as those done here for reference are usually not practical
about 2.6 times the design load of 175,000 pounds. since the needed soil elastic property, the modulus of subgrade
reaction, is generally not known reliably. Thus, although
We can see from the shape of the pressure curve that the the tools are available to perform such an analysis, the
soil still has additional support capability at this load. The questionable input renders the results similarly questionable.
moment in the mat at this load is 5,685,360 pound-inches,
which gives a bending stress of 4,935 psi. This indicates that
Comments on Support Deflection
the mat will likely fail before the ultimate bearing capacity of
the soil is reached. The deflection calculation discussed in the preceding
section provides some insight into the behavior of the mat,
If we look at the mat/soil behavior at 350,000 pounds, but is not an accurate calculation of the mat/soil deformation
twice the design load, we find that the soil is still in the elastic under load. This is due to the use of the idealized values of q
range. At this load, the peak soil bearing pressure is 9,783 psf and Lc in Eq. 23. More likely deflections can be investigated
using the beam on an elastic foundation approach with the
P = 175,000 lbs. approximate value of ks previously discussed.
Crane Mat Analyses of dozens of mat/soil combinations with crane
loads set equal to the maximum value permitted by the design
method proposed here show that the actual displacement
of the mat will differ, sometimes significantly, from that
computed using Eq. 23. However, these displacements do not
vary much as a group.
Peak GBP =
4,892 psf
This study examined 12" thick by 4' wide timber mats of
Bearing Length = 15'-1" oak, Mora and EmtekTM, one or two layers thick, supported on
soil with qult varying from 2,000 psf to 16,000 psf. The elastic
Fig. 3. Bearing Pressure Curve at the Design Load analysis deformations differed from the values calculated
5
using Eq. 23. The greatest differences occurred at the lowest now provide computer programs that compute the support
values from Eq. 23. However, all of the elastic analysis loads for their products for a given lift configuration. These
deformations for this group of mat/soil combinations were in tools should be used wherever possible.
the range of about 1/2" to just over 3/4". This indicates that the
mat deflections calculated using Eq. 23 are not a true indicator When sizing mats for an outrigger-supported crane,
of the mat behavior, but that the method developed here will consideration must be given to the size of the outrigger pad.
provide reasonable and consistent support to the crane. For example, an 18" diameter pad will bear on only two
timbers of a mat made up of 12" x 12" timbers and the tie
This study also examined the structural design factor rods that hold the mat together are not necessarily capable of
provided by the proposed method. The actual design factor distributing this concentrated load to the other timbers. In
was found to vary significantly, with a greater bending strength such a case, the mat should be checked considering only the
design factor in the mats occurring in conjunction with the two timbers on which the pad bears. Alternately, a steel plate
lower allowable ground bearing capacities. The example on can be used to distribute the outrigger load to all four timbers.
the previous page noted a design factor of about 1.75. This
value was found to be at the low end of the design factor The allowable stresses for timber design are taken as
range observed in the study. However, given this reasonable 1,400 psi for bending and 200 psi for shear in the example
lower bound strength design factor and the consistently small problems. These are practical values for mats made of the
vertical deflections of the mats under operating loads, the types of hardwoods popularly used in the U.S. for crane mat
calculation method defined here is shown to be practical for construction. However, the actual allowable stresses used
most crane mat design applications. must be appropriate for the species and condition of the
timbers under consideration.
Foundation Stiffness
Last, the allowable ground bearing pressure qa for the
A guide related to relative stiffness of the mat to the soil is site must be determined by a qualified engineer. Values of qa
the value of lLeff as defined in Eq. 26 (Bowles 1996), used to used for the design of foundations for permanent structures
distinguish between a rigid and a flexible foundation. are often based on a factor of safety of at least 3.00. Support
of a mobile crane does not require consideration of long-
ks BL4
eff
term settlements and many of the uncertainties associated
Leff = 4 (26) with the design of permanent structures do not exist for a
4 EI
crane installation. Thus, a lower factor of safety for qa may
The mat is considered to be a rigid foundation for values be appropriate. A value of 2.00 was used in the example
of lLeff less than p/4 = 0.79 and a flexible foundation for problems.
values of lLeff greater than p = 3.14. The mat example of
Table 3 gives us a value of lLeff equal to 2.01. Examination
CONCLUSION
of a range of mat design problems using qult from 2,000 psf to
16,000 psf generally shows values of lLeff in the range of 1.9 This paper presents a practical method for calculating
to 2.7 for a single layer of standard hardwood mats or 2.7 to the effective bearing length of a crane mat that is loaded by a
3.2 for a single layer of high strength (EmtekTM) timber mats. single outrigger or crawler track. The principles upon which
The lower values of lLeff occur at the higher values of qult. this derivation is based can be used to expand this approach to
a crane mat with two loads.
This calculation of lLeff should be considered as a guide
only when exercising engineering judgment in the solution of The true behavior of the mat/soil combination is more
a design problem. The value of lLeff is not to be used as a complex than is implied by the standard calculation approach.
design criterion due to the uncertainty with which ks is known. A more theoretically "exact" approach is usually not practical
due to the difficulty in determining the elastic properties
of the soil. As a result, it is sometimes necessary to apply
Notes on Practical Application
engineering judgment in the solution of a crane support design
The purpose of this paper is the derivation of a practical problem. Because of this potential need, it is necessary that
means of calculating the effective bearing length of a crane users of this material possess the engineering background and
mat. A few comments are offered here with respect to the practical experience required to exercise this judgment.
application of this material to the design of crane installations.
REFERENCES
Calculation of the crane loads to be supported must be
done with reasonable accuracy. Many crane manufacturers ASTM International (2006), D 245-06 Standard Practice
6
for Establishing Structural Grades and Related Allowable SI CONVERSION FACTORS
Properties for Visually Graded Lumber, West Conshohocken,
Following are conversion relationships between USCU
PA.
and SI for the quantities used in this paper. The standard
abbreviations for the SI units are shown in parentheses.
ASTM International (2000), D 1990-00 Standard
Practice for Establishing Allowable Properties for Visually-
1 inch = 25.4 millimeters (mm)
Graded Dimension Lumber from In-Grade Tests of Full-Size
Specimens, West Conshohocken, PA.
1 foot = 0.304 800 meter (m)
Bowles, J.E. (1996). Foundation Analysis and Design,
1 pound = 0.453 592 kilogram (kg)
5th ed., The McGraw-Hill Companies, Inc., New York, NY.


Use: 0.0479