SME Annual Meeting 

Feb. 25-Feb. 28, 2007, Denver, CO 

1 Copyright © 2007 by SME

APPLICATION OF PARAMETRIC COLUMN ANALYSIS TO EVALUATE ECCENTRIC LOADING CONDITIONS ON PROP 

SUPPORT PERFORMANCE 

D. Gearhart, NIOSH, Pittsburgh, PA 

Abstract 

Full-scale tests are conducted on mine roof supports using 

protocols developed by the National Institute for Occupational Safety 

and Health (NIOSH) using the unique capabilities of the Mine Roof 

Simulator (MRS). These protocols simulate the loading conditions that 

the support will be exposed to in an underground mine to determine 

the performance capabilities and limitations of the support. Some tests 

are conducted with rigid boundary conditions, creating uniform contact 

and near ideal axial loading conditions. These tests are used to 

determine the ultimate performance potential of a support, including 

the strength, stiffness, and stability. For prop supports where stability 

is a critical design issue, tests are implemented to induce buckling by a 

modified end condition, eccentric contact, or biaxial loading. It is 

important to determine the cpacity of the support design when the 

support is subjected to unpredictable loading conditions. This paper 

examines the effect of end conditions and load profiles on prop support 

performance and applies a new analysis methodology based on 

eccentric loading to determine a factor of safety for load capacity. 

Recommendations for installation practices that minimize eccentricity 

and therefore preserve capacity are also provided. 

Introduction 

The evaluation of standing supports at the National Institute for 

Occupational Safety and Health’s (NIOSH) Mine Roof Simulator (MRS) 

has been an ongoing research effort with a significant record of 

success. The objective of this research is to eliminate hazards and 

injuries that result from roof falls in underground mines, and to reduce 

or eliminate material handling injuries that occur during installation of 

these support systems. In order to continue the advance toward these 

objectives, a complete understanding of the interaction between the 

support and the strata is necessary. One component of that 

relationship is the behavior or response of the standing supports. The 

MRS is a unique biaxial press that can replicate the loads and 

displacements that standing supports are subjected to in the 

underground environment. 

The protocol used at the MRS for testing standing supports first 

establishes the baseline performance under controlled loading 

conditions. This initial phase of testing is extremely beneficial because 

design flaws or defects are exposed during the laboratory 

investigation, rather than during an extended and uncontrollable 

underground trial. This reduces the exposure of miners to unproven 

support technology and avoids product development delays for the 

manufacturers. 

Underground, variable boundary conditions can significantly 

decrease the capacity of prop supports from that measured under 

uniform loading conditions in the laboratory. These eccentric loading 

conditions are replicated during the second phase of the MRS testing 

protocol, which is intended to reveal design and performance 

limitations of the support product under adverse, non-axial loading. 

While the full-scale tests conducted in the MRS effectively reveal 

performance limitations for a support product, an analysis methodology 

must be developed to identify critical design parameters and quantify 

the limitations of the support within its full operational range. 

For structural columns, historical analysis methods classify their 

design by length and use different equations to predict the buckling 

stress. In this paper, an alternative analysis method is proposed that 

allows for determination of prop support capacity based on a stability 

parameter and an eccentricity factor. The stability parameter is 

dependent on the geometric and material properties of the structure. 

The eccentricity factor is dependent on the boundary conditions and 

load application. This analysis method can provide a direct measure of 

the strength performance of the supports once the stability and 

eccentricity are determined. Finally, prop support capacity can be 

optimized with installation practices that reduce the eccentricity of the 

loading. 

MRS Testing Protocol 

The testing protocol used to evaluate standing support in the 

MRS is designed to establish the baseline performance for a support, 

and then to induce realistic loading conditions that can degrade the 

capacity. Tests with modified boundary conditions and biaxial loading 

are used to evaluate the effect of eccentric loads on performance. All 

tests are conducted in the displacement-controlled mode of operation 

(Barczak 2000). 

 Loading Induced Through Vertical Displacement - This 

loading profile applies a uniaxial vertical displacement to the 

support and is used to measure the baseline performance. 

These test results determine the strength, stiffness, roof and 

floor bearing pressures, and general stability of the support. 

On new support concepts that are under development, 

design flaws are frequently discovered during these tests. 

The rate of load application depends on the stiffness or 

yielding characteristics of the specimen. 

 Loading Induced Through Biaxial Displacement - The 

biaxial loading profile is used to further evaluate the stability 

of the support when subjected to eccentric loads. These 

tests are typically conducted with ratios of 3:1 or 2:1 vertical 

to horizontal displacement. The performance of the support 

during this test protocol can be acutely affected by the 

boundary conditions on the support. 

Boundary Conditions 

Baseline tests use full, uniform contact between the ends of the 

support and the upper and lower platens of the simulator. This 

minimizes eccentricity since it maintains the best alignment of the load 

vector with the axis of the support throughout the test. Any eccentricity 

that exists or develops in this test condition is the result of material 

anomalies, misalignment of prop elements, or ends that are not 

perpendicular to the axis of loading. The addition of headboards, 

footboards, and prestressing devices are included in the second phase 

of testing to introduce potential eccentricity that affects the strength, 

stiffness, and stability of the support system. The objective is to 

evaluate the support under controlled eccentric load conditions to 

determine the performance limitations. 

Performance Assessment Criteria 

Support designs are assessed relative to three primary 

performance factors: (1) strength, (2) stiffness, and (3) stability. These 

three parameters are evaluated for standing supports that are 

subjected to displacements in the laboratory that realistically simulate 

the underground conditions. 

 Strength - The strength of a roof support generally refers to 
SME Annual Meeting 

Feb. 25-Feb. 28, 2007, Denver, CO 

2 Copyright © 2007 by SME 

its ultimate capacity. The strength of prop supports is 

dependent on the stability of the structure and the

eccentricity of the loading. All supports are tested to failure

to determine the peak capacity of the support. 

 Stiffness - Stiffness is a measure of how quickly a passive

support develops its load carrying capacity. The stiffness is

determined by measuring the change in support load as a

function of the applied displacement. 

 Stability - Stability is a measure of the capacity of the

support to maintain equilibrium under changing load 

conditions. The stability of a support structure is affected by

several parameters. These include the following: (1) 

characteristics and material properties of the specimen, (2) 

aspect or slenderness ratio of the support, (3) boundary 

conditions established at the roof and floor contact, (4) 

orientation of the load application, and (5) design of the yield

mechanism. 

Crib versus Prop Performance Assessment 

The behavior differences between crib and prop supports can be

traced directly to the stability of the structures. If the structure is very

stable, or resistant to buckling like a crib, the material properties 

control the strength and stiffness. The performance of these structures 

is characterized by support capacity equal to the material strength

times the loading area. However, if the structure is susceptible to 

buckling, then stability controls the performance. Prop supports 

behave like columns that are susceptible to buckling, therefore the

failure stress is much less than the material yield strength. In general, 

most mining props have slenderness ratios of less than 60, placing

them in the range where buckling behavior is the result of a 

combination of stability and elastic material response. 

Prop Support Performance Analysis 

Historically, columns have been classified by length, and the

buckling strength of long, slender columns is estimated by the Euler 

critical stress, which is well below the material compressive yield 

strength. 

2

c r ρ

π σ E ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ = ×

(1)

Where: σcr = Euler’s critical stress, psi

E = modulus of elasticity, psi and 

ρ = slenderness ratio. 

It is interesting to note that the load bearing capacity of long,

slender columns is dependent on the elasticity of the material and the

area moment of inertia, not on the compressive yield strength

(Timoshenko 1949). Short and intermediate columns, which are

typical of mine props, are more stable and their capacity is limited by 

the material strength. The capacity of these columns can be estimated 

using empirical design equations like the JB Johnson formula. 

⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡ ⎟

⎠

⎞ ⎜

⎝

⎛ = −

2

c r y C

ρ

2

1 σ σ 1

(2)

Where: σcr = JB Johnson critical stress, psi

σy = material yield strength, psi 

ρ = slenderness ratio, psi and 

C = critical slenderness ratio. 

The relationship among the column buckling load, moment and

deflection can also be described by the Universal Column Formula 

(Dishongh, 2002) based on the concept of combined stress from axial 

loading and bending. The Euler and JB Johnson curves are shown in 

figure 1. 

The second important factor affecting the ultimate capacity of

columns is the end conditions. A column with pinned ends and a

slenderness ratio of 100 will theoretically fail at 30 percent of the 

material yield strength, while the same column with fixed ends provides 

a slenderness ratio of 50 and will fail at nearly 80 percent of the

material strength (see figure 1). This difference is because pinned end

columns cannot resist the rotational moments that occur when the

column begins to buckle, whereas the fixed end columns can resist

these moments. 

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

0 20 40 60 80 100 120 140 160 180 200

SLENDERNESS RATIO

STRENGTH RATIO (CRITICAL / MATERIAL) .

Euler critical stress

JB Johnson formula

Material strength

Pinned ends

Fixed ends

Strength Ratio at

Critical Slenderness Ratio

Figure 1. Empirical design equations for column stress. 

Parametric Column Analysis

Conventional analysis of buckling is based on an idealized 

“perfect” column where pure elastic buckling can be defined. An 

alternative parametric analysis method that simplifies the interaction

between strength and stability has been proposed (Wright 1999).

Parametric Column Analysis complements the evaluation of

underground support systems because: (1) it provides a quantifiable 

measure of the stability of the support and (2) the effect of 

unpredictable end conditions and eccentric loading that directly affect

the strength and stiffness of props is measured in this analysis. This 

approach assumes that "imperfections" in material properties or 

fabrication and installation practices cause eccentricity that invariably 

reduces the load carrying capacity predicted by the idealized elastic 

buckling theory. For example, prop supports are typically constructed

from multiple elements that work together to form a “composite” 

structure. The supports also have a variety of hardware or other 

materials used on the ends of the prop to establish contact and to

distribute the support loads to the mine roof and floor. These features

alter the ideal uniform loading conditions assumed in the conventional 

theory and create eccentric loading conditions that decrease the 

capacity of the support. 

Column engineering practices typically use a design load and a

factor of safety to develop a solution to ensure that the column will not 

fail. The values for the factor of safety and the assumed eccentricity

contribute to robust designs that meet these requirements. However,

the underground mining environment presents a more difficult problem

for the support system designer. The loads or deflections that will be 

applied to the supports underground are for the most part unknown, so

the requirements for the design are based on experience of what

works and what does not. For prop supports, eccentric loads and 

indefinite boundary conditions will reduce the capacity of the supports 

and these conditions must be properly considered in the final design

analysis. The proposed analysis method will assist in the development

of useful values for eccentricity that can be used in design to improve

the performance and reliability of prop support systems. 

The input values needed for this analysis of MRS test results fall 

into three categories: (1) material properties, (2) geometric parameters,

and (3) external loading conditions. The material properties are the 

modulus of elasticity and the compressive strength. The geometric 

parameters are the length, cross-sectional area, minimum area 

moment of inertia, and the maximum fiber distance. The properties of

the external load are the magnitude, boundary conditions and initial 

eccentricity. Using these values, the radius of gyration, slenderness 

ratio, Euler critical buckling stress and the direct compressive stress 
SME Annual Meeting 

Feb. 25-Feb. 28, 2007, Denver, CO

3 Copyright © 2007 by SME

are calculated. Finally, the parameters required for the analysis 

reduce to three dimensionless values: 

 Stability Parameter - The stability parameter, •, is the ratio 

of the direct stress caused by axial loading to the critical 

buckling (Euler) stress. The stability parameter is a measure

of the tendency towards stability failure. This parameter

cannot exceed unity. It can be calculated for any support

structure as shown in equation 3, where the maximum direct

stress value can be defined as the material compressive

strength. The advantage of the stability parameter for 

evaluation is that it is completely dependent on the material

properties and geometric dimensions of the structure and

does not require testing to be determine. 

critical

direct

σ

σ

ψ =

 (3) 

Where: Ψ = stability parameter, 

σdirect

 = sum of the axial and bending stress, psi and

σcritical

 = Euler critical buckling stress, psi. 

 Strength Parameter - The strength parameter, θ, is the ratio

of the direct compressive stress to the material compressive

stress. The strength parameter measures the ultimate load 

capacity of the prop without loss of stability. Where actual 

test results are applied, the strength parameter is calculated

as the ratio of the applied direct stress at failure to the

material strength. As such, this parameter cannot exceed 

unity. 

material

direct

σ

σ θ =

(4)

Where: θ = strength parameter, 

σdirect = sum of the axial stress and bending stress, psi

and 

σmaterial

 = material compressive strength, psi. 

 Eccentricity Ratio - The eccentricity ratio, •, reflects the

severity of the imperfections or offset load. The value, as 

shown in equation 5, is the ratio of the actual eccentricity of

the applied load to the maximum eccentricity that would

cause no tensile stress by bending of the column. One

challenge in design is selecting a reliable value for the initial 

eccentricity as shown in figure 2. 

2 r

e y η × =

 (5) 

Where: η = eccentricity ratio, 

e = initial maximum eccentricity, in

y = extreme fiber distance, in 

r = radius of gyration, in. 

For the analysis presented in this paper, η is computed from Ψ

and θ to solve for the eccentricity, e. Beam deflection theory is used

to derive an equation that describes the relationship among these 

three parameters. For a straight column with a constant initial offset,

as shown in figure 2, the equation is:

⎟

⎠

⎞ ⎜

⎝

⎛ = + × × ψ 2

π 1 η sec θ

1

(6)

For analysis, this equation is rearranged to calculate η, thus: 

⎟

⎠

⎞ ⎜

⎝

⎛ ⎟ × × ⎠⎞ ⎜

⎝

⎛ = − ψ 2

π 1 cos θ

1

η

(7)

The strength-stability loci for different values of the eccentricity 

ratio are plotted on the chart shown in figure 3. The black diamond on 

the chart shows that for a support with a stability parameter of 0.75 and 

an eccentricity ratio of 0.2, the strength parameter is 0.5. This means 

that this prop will achieve only 50 percent of its ideal full load capacity.

The chart demonstrates the importance of minimizing the stability 

parameter and the eccentricity ratio to maximize the strength 

performance for prop design and installation. Therefore, it is

recommended that prop supports be designed with a stability factor •, 

less than 0.75 for their entire usable height range, and eccentric 

loading should be minimized. The following example will demonstrate

the Parametric Column Analysis method using MRS test data to

calculate the initial eccentricity. 

e

Figure 2. Initial eccentricity is the distance from the load axis to the

unloaded column axis.

 h = 0.01

h = 0.05

 h = 0.1

 h = 0.2

 h = 0.5

 h = 1

 h = 2

 h = 5

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.2 0.4 0.6 0.8 1.0

STABILITY PARAMETER

STRENGTH PARAMETER

?

?

?

?

?

?

?

?

Figure 3. Loci of strength parameter versus stability parameter by

eccentricity ratios. 

Parametric Column Analysis Computations 

This example of Parametric Column Analysis is presented to

demonstrate the evaluation of the performance of an 11-foot tall, 

adjustable-height, non-yielding prop of the design shown in figure 4.

The height-adjusting spindle is at its full extension of one foot. This 

specimen was tested in the MRS and the peak support capacity was

measured as 111.8 kips (55.9 tons). The prop failed by buckling in the

pipe section. The objective of this analysis is to demonstrate the

method for calculation of the initial eccentricity for this test condition,

which can then be used in subsequent evaluations of this prop design. 

The first step of the analysis is to calculate the stability parameter,

Ψ, using equation 3 for the prop at its ultimate capacity. The next step

is to calculate the strength parameter, θ, using equation 4 with the

direct applied stress (force/area) measured during testing, divided by 

the material strength. The values for the stability and strength

parameters are then used to calculate the eccentricity ratio, η, using 
SME Annual Meeting 

Feb. 25-Feb. 28, 2007, Denver, CO 

4 Copyright © 2007 by SME

equation 7, and finally the initial eccentricity, e, using equation 8. 

y

η r e

2 × =

(8)

Where: e = initial eccentricity, in, 

η = eccentricity ratio, 

r = radius of gyration, in, and 

y = extreme fiber distance, in. 

12"

132"

Figure 4. An adjustable height prop with detail of the spindle at the top

end of the prop. 

Full Prop Analysis: First assume that the spindle section of the

prop is stronger than the pipe section, and therefore will behave as an

extension of the pipe. With this assumption, the pipe will buckle at a 

critical load that is lower than the critical load of the spindle. 

Given: Total height (pipe and spindle): 132 in 

Pipe outer diameter: 3.563 in 

Pipe wall thickness: 0.254 in 

Material yield strength: 73 ksi 

Modulus: 29 E 6 psi 

The area moment of inertia, radius of gyration and slenderness 

ratio are needed to compute the Euler buckling strength for the

specimen.

Area:

( ) 2 2 2 3.563 3.055 2.640 in

4

π A = × − =

Area moment of inertia: 

4

4 4

3.635 in

2

3.055

2

3.563

4

π I =

⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡ ⎟

⎠

⎞ ⎜

⎝

⎛ ⎟ − ⎠⎞ ⎜

⎝

⎛ =

Radius of gyration: r (3.635 in/2.64 in 1.173 in 2 = =

End condition factor: K = 0.5 (Column ends are fixed)

Effective length: Leff

 = 132 in x 0.5 = 66 in 

Slenderness ratio: 

56.258

1.173in

66 in

ρ = =

Critical stress: 

90.4 ksi

56.258

π σ 29 E 6

2

c r ⎟ = ⎠

⎞ ⎜

⎝

⎛ = ×

Stability parameter: 

0.807

90.4ksi

73ksi

ψ = =

Spindle Analysis: Next, assume that the spindle is weaker than 

the pipe section and that the spindle will fail first. 

Given: Spindle height: 12 in 

Outer diameter: 2.000 in 

Wall thickness: 1.000 in (Solid section) 

Material yield strength: 73 ksi 

Modulus: 29 E 6 psi 

Using the same calculations as before: 

Area:

( ) 2 2 2 2.000 0.000 3.142 in

4

π A = × − =

Area moment of inertia: 

4

4 4

0.785 in

2

0.000

2

2.000

4

π I =

⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡ ⎟

⎠

⎞ ⎜

⎝

⎛ ⎟ − ⎠⎞ ⎜

⎝

⎛ = ×

Radius of gyration:r ( ) 0.785 in / 3.142 in 0.500 in 4 2 = =

End condition factor: K = 2.0 (One end free) 

Effective length: Leff = 12 in x 2.0 = 24 in 

Slenderness ratio: 

48

0.500in

24 in

ρ = =

Critical stress: 

124.2 ksi

48

π σ 29 E 6

2

c r ⎟ = ⎠

⎞ ⎜

⎝

⎛ = ×

Stability parameter: 

0.588

124.2 ksi

73ksi

ψ = =

Component Failure Assessment: The values calculated for the

stability parameter, Ψ, indicate that the whole prop is less stable than

the spindle section (Ψprop = 0.807 > Ψspindle = 0.588 ), meaning that the

pipe will fail at a lower load. This was confirmed by the laboratory test

where the buckling occurred in the pipe section. 

Determine Load Eccentricity: Using the laboratory load

measurement, calculate the value of the strength parameter, θpipe, for 

the pipe: 

( ) 0.580

73,000psi

111,800lbs 2.640in

σ

σ θ

2

material

direct

pipe = = ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ =

Using equation (7), η calculated as 0.115. 

⎟

⎠

⎞ ⎜

⎝

⎛ × × ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ ⎟ = − ⎠

⎞ ⎜

⎝

⎛ × × ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

= − 0.807

2

π 1 cos 0.580

1

ψ 2

π 1 cos θ

1

η

Using equation (8), solve for the eccentricity, e: 

( ) 0.089in

3.563 2

0.115 1.173

y

η r e

2 2

= × = × =

Compute Spindle Capacity: Using this same initial eccentricity

for the spindle; compute the strength factor for the spindle. 

θ 0.502

0.588

2

π ψ 1 0.356 sec 2

π 1 η sec θ

1

0.356

0.500

0.089 1

r

e y η

spindle

2 2

=

⎟

⎠

⎞ ⎜

⎝

⎛ ⎟ = + × × ⎠

⎞ ⎜

⎝

⎛ = + × ×

= × = × =

Analysis Summary: The results of these calculations indicate

that under laboratory conditions, the combined misalignment of prop
SME Annual Meeting 

Feb. 25-Feb. 28, 2007, Denver, CO 

5 Copyright © 2007 by SME 

components, material flaws and imperfect boundary conditions 

resulted in an initial eccentricity of 0.089 inches from the central axis of

the prop. The spindle buckling load under these conditions is

calculated to be 115.1 kips, which is slightly higher than the buckling 

load for the pipe, 111.8 kips, at this eccentricity. This shows that the

value for the initial eccentricity can be determined from analysis of

MRS test results and that this parametric method can be used to

assess prop support designs and underground installation practices. 

DETERMINATION OF ECCENTRICITY IN PROP SUPPORT

LOADING 

A program to assist a manufacturer to develop a new prop

support was undertaken by NIOSH that provided an ideal opportunity 

to evaluate the proposed method of analysis. Tests were conducted

on props at 7-, 9- and 11-foot heights, using the vertical only, biaxial, 

and eccentric loading profiles. The props under development were a

non-yielding support with a design capacity of 50 tons. Each unit

included a threaded spindle arrangement (see figure 4), to 

accommodate entry height variations of up to one-foot and to generate 

a preload to secure the prop against the mine roof during installation.

The procedure utilized in the previous example was used to calculate 

the performance parameters and initial eccentricity that coincides with 

the measured peak loads for each of the test specimens.

Height Effects 

The Parametric Column Analysis method was used to evaluate

the performance of the new prop design at three different heights when 

subjected to a 3:1 vertical-to-horizontal displacement load profile. 

Since the props are non-yielding and very stiff, the yield load was 

achieved in about 0.3 inches of vertical convergence. This means that 

the horizontal motion applied to the prop was about 0.1 inches and

was insignificant in this particular case. The results of the analysis are

shown in table 1 (see Appendix). 

The calculated stability parameters predicted that the spindle 

would fail first for the 7- and 9-ft props (Ψspindle>Ψprop) and the pipe

section would fail first for the 11-ft prop (Ψprop>Ψspindle). The test results 

confirmed that the less stable component failed for these

configurations. The values calculated for the initial eccentricity 

corresponding to the 7-, 9- and 11-ft props were 0.072, 0.085 and

0.089 inches, respectively. Using these eccentricity values, it is 

estimated that the failure load of the pipe section was 168 kips for the

7-ft prop and 152 kips for the 9-ft prop. Likewise, the failure load for 

the pipe in the 11-ft design was 111.8 kips, while the spindle failure

load was estimated to be 115 kips with the same eccentricity. All of

these specimens exceeded the 50-ton capacity design goal. 

Boundary Condition Effects 

Non-yielding props are typically installed using a header board or

a yield dish at the roof interface to increase the amount of entry closure

required to cause the prop to fail. In this case, an oak header board

measuring 20 x 9.5 x 2 inches was used and the yield dish was a

prototype made from steel that was approximately 18 inches in

diameter. Figure 5 shows the laboratory test results to compare the

performance of props with these different boundary conditions. An

additional 0.7 inches of displacement was required to fail the prop with 

the header board and an additional 2.3 inches was required to buckle

the prop with the dish header. Another benefit of the yield devices is to

distribute the support load over a wider area of the roof. 

A parametric column analysis for these initial boundary conditions 

was completed to determine the eccentricity. The results are shown in

table 2. The props were identical and therefore the stability parameter 

was the same for all the specimens. Notice that the initial eccentricity

increased from 0.073 inches for the full contact specimen to 0.099 

inches for the oak header board and increased to 0.124 inches for the

dish header. The result of this increase in eccentricity was that the

capacity of the support with an oak header was reduced by 13.6

percent and by 23.6 percent for the dish header compared to the full 

contact condition provided by the 8x8-in flat steel head plate. The full 

contact prop and the prop using a header board exceeded the 50-ton 

capacity design requirement specified by the manufacturer. The prop

using the prototype dish header failed at 96 kips (48 tons). 

0

20

40

60

80

100

120

140

160

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.

VERTICAL DISPLACEMENT, inches

VERTICAL LOAD, kips

0

Full Contact

Oak Header

 Dish Header

Figure 5. The effects of boundary conditions on prop performance.

Table 2. Impact of boundary condition on eccentricity. 

Boundary Condition Flat Oak Dish 

Component Analyzed Spindle Spindle Spindle 

Prop Height, ft 7 7 7 

Spindle Height, ft 1 1 1 

PARAMETRIC COLUMN ANALYSIS

Ψ - Stability Parameter 0.588 0.588 0.588

θ - Strength Parameter 0.550 0.475 0.420

η - Eccentricity Ratio 0.294 0.396 0.495

SUPPORT/COMPONENT FAILURE ASSESSMENT

Component Failure Spindle Spindle Spindle 

Measured Prop Capacity, kips 126.1 108.9 96.3 

REQUIRED ECCENTRICITY TO PRODUCE 

MEASURED SUPPORT CAPACITY

Initial Eccentricity, in 0.073 0.099 0.124

Eccentric Loading Analysis 

An analysis comparing the performance of two 11-ft props was 

completed to evaluate the affect of a severe eccentric load application 

due to a single wood wedge between the prop and the roof (figure 6). 

The material and geometric parameters for the props were identical to

the previous examples, so the stability parameter for the whole prop 

(Ψprop = 0.807) and for the spindle section (Ψspindle = 0.588) as shown in 

table 3, remained the same as used in previous analyses. 

Table 3. Impact of wedging or eccentricity.

Support Configuration Flat End Plate Wedged End Plate

Prop/Component Prop Spindle Prop Spindle

Height, ft 11 1 11 1 

Boundary Condition Flat Flat Wedged Wedged

PARAMETRIC COLUMN ANALYSIS

Ψ - Stability Parameter 0.807 0.588 0.807 0.588

θ - Strength Parameter 0.588 0.510 0.419 0.345

η - Eccentricity Ratio 0.111 0.344 0.220 0.680

SUPPORT/COMPONENT FAILURE ASSESSMENT 

Measured Prop Capacity, 

kips 

113.4 79.2

Component Failure Pipe Spindle 

REQUIRED ECCENTRICITY TO PRODUCE 

MEASURED SUPPORT CAPACITY

Initial Eccentricity, in 0.086 0.170 
SME Annual Meeting 

Feb. 25-Feb. 28, 2007, Denver, CO 

6 Copyright © 2007 by SME 

Figure 6. Wedged prop for eccentric loading. 

The prop with uniform steel plate contact buckled in the pipe

section with a calculated eccentricity of 0.086 inches. The wedge at

the roof contact for the second support configuration created an initial 

eccentricity value of 0.170 inches. Figure 7 shows the effect of

increased eccentricity on the computed failure load of the two prop

components. If the eccentricity exceeds 0.113 inches, the prop will fail 

to meet the design load capacity, and if the eccentricity exceeds 0.130

inches, the spindle will fail at a lower load than the pipe. The effect of

the severe eccentric load on the performance of the prop was a 30

percent reduction in support capacity compared to the full contact test

and buckling failure of the spindle rather than the pipe. 

60

70

80

90

100

110

120

130

140

150

0.050 0.075 0.100 0.125 0.150 0.175 0.200 0.225 0.250

ECCENTRICITY, inches

COMPONENT CAPACITY, kips

Pipe Load

Spindle Load

Design capacity

Full contact

Wedge contact

Figure 7. Effect of eccentricity on 11-ft prop. 

Application of Parametric Column Analysis to 

Estimate Prop Capacity 

The range of values for the initial eccentricity calculated from the

laboratory test results in the previous section can now be used for 

estimating prop capacity. The initial eccentricity ranged from 0.072 to

0.089 inches for uniform contact conditions of props ranging in height 7

to 11 feet. The initial eccentricity for props with a header board at the

roof interface was 0.099 inches and 0.124 inches with an 18-indiameter steel dish at the roof contact. The application of biaxial 

displacements had minimal effect on the calculated eccentricity for 

non-yielding props. For the 7-foot height, the eccentricity was 0.073

inches for vertical only loading and 0.072 inches for biaxial load 

application. For the 11-foot height, the eccentricity was 0.086 inches 

for vertical only loading and 0.089 inches for biaxial loading. 

Based on this range of initial eccentricities calculated from the

MRS testing results, a value of 0.075 inches will be used to evaluate

the performance of a yieldable prop design that was tested with

uniform contact, using both the vertical only and biaxial load 

application. First, the stability parameter is calculated for the yieldable 

prop using equation 3 and the eccentricity ratio is calculated using 

equation 5 for the assumed initial eccentricity of 0.075 inches. For 

these test specimens, the stability parameter was computed as 0.165 

and the eccentricity ratio was 0.076. Using these values and equation

6, the strength ratio for the prop can be computed as 0.914.

Multiplying the strength ratio by the material strength and the crosssectional area of the prop gives an ultimate capacity of about 183 kips. 

The performance curves from the laboratory tests for both the

vertical only and the 3:1 biaxial loading and the capacity estimated 

from the parametric column analysis are shown in figure 8.

Comparison of the ultimate capacity with the yield capacity of the prop

indicates a factor of safety of 1.5 for the support load in this case. 

0

20

40

60

80

100

120

140

160

180

200

0 2 4 6 8 10 12 14 16 18 20

DISPLACEMENT, inches

VERTICAL FORCE, kips

VERTICAL ONLY LOADING

3 : 1 BIAXIAL LOADING

BUCKLING CAPACITY

Figure 8. Estimated buckling load for a 50-ton yielding prop support. 

The next goal is to calculate the maximum eccentricity that the

yielding prop can withstand at the yield load. First the strength

parameter (θ) is calculated by dividing the yield load by the prop area

to determine the direct stress and then dividing by the material yield 

strength. The stability parameter is computed using equation 3. The

eccentricity ratio can be computed from the strength and stability

parameters using equation 7. Then using equation 8, the initial 

eccentricity can be calculated. From this analysis, the eccentricity 

required to buckle this yieldable prop would be 0.530 inches.

It is important to recognize that if the yield capacity of the prop is 

designed too close to the limit of instability, unexpected changes in 

eccentricity can cause premature buckling failure of yieldable supports.

The prop support analyzed in this case has a capacity margin of about

60 kips between the yielding load and the buckling load, and the 

analysis shows that the design could withstand severe eccentricity. In

addition, the biaxial test demonstrated successful performance under 

an even more severe load condition, where the base of the prop was 

displaced 6 inches horizontally relative to the top of the support during

the test. 

Key Points Derived From Parametric Column 

Analysis of Prop Supports 

The Parametric Column Analysis methodology is a useful tool for 

analyzing the capacity reduction of prop supports due to eccentric 

loads and various boundary conditions. The values for the stability

parameter and initial eccentricity that were obtained by the analysis of 

full-scale tests accurately reflected the observed strength performance

of the supports. The findings listed here were particularly noteworthy 

and are important key points. 

1. The stability parameter is a valid predictor of prop 

performance based completely on the physical parameters 

of the structure. This value can be used for a direct

comparison of the stability of different prop designs. All 

props should be designed to provide a stability parameter 

less than 0.75. In this range, even with an eccentricity ratio

of 0.2, the strength performance will be at least 50 percent of

the material capability (see figure 3). 
SME Annual Meeting 

Feb. 25-Feb. 28, 2007, Denver, CO

7 Copyright © 2007 by SME

2. Laboratory test conditions for full, uniform contact at the 

ends of the prop produced initial eccentricity values that 

range from 0.072 inches at the 7-foot height to 0.086 inches 

at the 11-foot height. A header board increased the 

eccentricity to 0.099 for a 7-foot prop. 

3. The relatively small horizontal displacement applied during 

biaxial loading was insignificant for the non-yielding props 

examined in this study. Additional research is needed to 

evaluate the impact of the biaxial load condition. During 

biaxial loading, the load axis is not offset from the central 

axis of the support as shown in figure 2, but rather rotates 

away from the central axis at the ends. Further analysis is 

needed to evaluate the effect of this condition compared to 

the offset loading assumed in the current analysis. 

4. One laboratory test induced an extreme initial eccentricity of 

0.170 inches, which caused a 30 percent decrease in the 

support capacity. This configuration, which only used a 

single wedge driven from one side, created a condition that 

can be easily avoided underground by driving wedges from 

both sides. Wedges should be used to fill voids as 

completely as possible, and should be installed to minimize 

eccentric contact. 

5. Yieldable props should be designed with excess buckling 

capacity. If the yield capacity is too close to the buckling 

load, the props are susceptible to eccentric loads and 

premature failure. 

Conclusions and Additional Research Recommendations

The options for replacement of bulky, crib-type secondary 

standing supports have been expanded by the development of new 

prop support designs in recent years. Although cribs can meet the 

need for support, there are many disadvantages to their use. The 

volume of material that must be transported underground, the labor 

required for installation, and the injuries associated with material 

handling and construction to name a few. Prop supports, that can 

match the capacity of cribs have been developed, and these supports 

can reduce the material handling requirements and the associated 

installation injuries. It is critical that any new support system 

introduced has the strength, stiffness and stability demanded by the 

application. 

The Parametric Column Analysis methodology is an effective tool 

for the evaluation of props. It provides direct measurements of 

structural stability and the impact of eccentric loading on strength 

performance. These measurements allow the direct comparison of the 

stability of different prop designs and reinforce the importance of 

installation practices that minimize eccentric loading. The strength 

parameter from the analysis can be used to calculate a factor of safety 

for support design. In order to facilitate this design approach, 

eccentricity values for support loading must be established. The 

values calculated for the initial eccentricity from the laboratory tests 

thus far are limited to one non-yielding support. Additional tests must 

be conducted on other prop designs to determine if the eccentricity 

values are consistent or vary by design. 

More testing is also required to develop complete installation 

guidelines. However, the installation should always minimize 

eccentricity to preserve capacity. Using Parametric Column Analysis 

to evaluate a yielding prop also showed the effectiveness of designing 

the yield load capacity well below the ultimate buckling capacity to 

enable the prop to withstand severe eccentric loading. 

New testing protocols for props should be developed to induce 

bending moments at the ends of the props to simulate roof sag or floor 

heave. This test configuration will also be used to analyze the effect of 

the opposing moments created by horizontal displacement and inclined 

installation of the prop. An outcome from this continued research will 

be to determine if and when a spherical or pinned end condition is 

beneficial because of the reduction of bending moments. 

The advantages of using prop supports to reduce material 

handling injuries will only be realized if the supports meet the 

requirements for ground control. The determination of the engineering 

limitations of props is crucial to ensure that the supports have the 

capabilities to meet or exceed the demands of the application. 

References 

Barczak, T.M. (2000), "NIOSH Safety Performance Testing Protocols 

for Standing Roof Supports and Longwall Shields," In the 

Proceedings: New Technology for Coal Mine Roof Support, 

Pittsburgh, PA: U.S. Department of Health and Human Services, 

Public Health Service, Centers for Disease Control and 

Prevention, National Institute for Occupational Safety and Health, 

DHHS (NIOSH) Publication No. 2000-151, IC 9453, pp. 207-221. 

Timoshenko, S. and G.H. MacCullogh, (1949), Elements of Strength of 

Materials, Third Edition, D. Van Nostrand Co. Inc., New York, pp. 

291-292 

Dishongh, BE (2002). Universal Column Formula. In: ASCE Practice 

Periodical on Structural Design and Construction, Vol 7 No 3, 

August 1, 2002. pp 118-121. 

Wright, Douglas (1999), "Notes on Design and Analysis of Machine 

Elements," 

http://www.mech.uwa.edu.au/DANotes/buckling/columns/columns

.html DANotes: Buckling – Practical Columns; Updated 2005. 

Department of Mechanical and Materials Engineering, the 

University of Western Australia. 
SME Annual Meeting 

Feb. 25-Feb. 28, 2007, Denver, CO 

8 Copyright © 2007 by SME 

Appendix 

Table 1. Impact of height on eccentricity. 

7 ft Prop 9 ft Prop 11 ft Prop Design Parameter Full Prop Spindle Full Prop Spindle Full Prop Spindle 

Height, ft 7 1 9 1 11 1 

Outer diameter, in 3.563 2.00 3.563 2.00 3.563 2.00 

Wall thickness, in 0.254 1.00 0.254 1.00 0.254 1.00 

Material strength, ksi 73 73 73 73 73 73 

Component length, in 84 12 108 12 132 12 

K – end factor 0.5 2.0 0.5 2.0 0.5 2.0 

Effective length, in 42 24 54 24 66 24 

Area, sq in 2.64 3.142 2.64 3.142 2.64 3.142 

Moment of inertia, in43.634 0.785 3.634 0.785 3.634 0.785 

Radius of gyration, in 1.173 0.500 1.173 0.500 1.173 0.500 

Slenderness ratio 35.80 48.00 46.03 48.00 56.29 48.00 

Critical buckling stress, ksi 223.3 124.2 135.1 124.2 90.4 124.2 

SUPPORT/COMPONENT FAILURE ASSESSMENT 

Component failure Spindle Spindle Pipe 

Measured support capacity, kips 127.4 117.4 111.8 

PARAMETRIC COLUMN ANALYSIS 

ψ - Stability parameter 0.327 0.588 0.540 0.588 0.807 0.588 

θ - Strength parameter 0.870 0.555 0.786 0.512 0.580 0.502 

η - Eccentricity ratio 0.093 0.287 0.110 0.342 0.115 0.356 

REQUIRED ECCENTRICITY TO PRODUCE MEASURED SUPPORT CAPACITY 

Initial eccentricity, in 0.072 0.085 0.089