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Understanding 2D Structural Analysis by A.W.C. Oreta : Pinned &Fixed Support Conditions 6 - 1
CHAPTER 6
PINNED AND FIXED SUPPORT CONDITIONS
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BACKGROUND
Supports are used to attach structures to the ground to restrict their movement
due to external loads. The loads tend to move the structure; but the supports
prevent the movements by exerting reactions to neutralize the effects of the
forces; thereby keeping the structure under equilibrium. The type of reaction a
support exerts on a structure depends on the type of supporting device used and
the type of movement it
prevents.
represents
Figure
models
6.1
of
supports for plane structures.
Consider first the idealized
models at the left portion of
the figure. A roller support
prevents translation normal
Figure 6.1 Models for Supports
to the plane of the roller and produces a corresponding normal reactive force,
while a pinned or hinged support prevents translation in any direction but allows
rotation and thus produces reaction forces. A fixed support prevents rotation and
translation and thus produces reaction forces and a moment.
The pinned (or roller) and fixed support conditions are idealized models of
support conditions. What type of model for the support should you use when
you want to represent the actual support conditions? The answer to this question
depends on degree of constraints provided by the foundation. One factor which
affects the constraints at the support is the type and detail of the connection
between the column and the footing. Figure 6.2 shows two
examples of
connections at the footing and the corresponding idealized models. The steel
column is welded to a base plate and the base plate is connected to a concrete
Understanding 2D Structural Analysis by A.W.C. Oreta : Pinned &Fixed Support Conditions 6 - 2
footing by bolts. The degree of fixity
between the column and the footing will
depend on the bolted connections. Figure
6.3 is a detail of a rolling expansion
bearing which consists of a hinge on top
of a pedestal whose base rests on a
Pinned
Fixed
Figure 6.2 Types of Connections
series of rollers. In the building in Figure
6.4,
the
frame
consists
of
tapered
columns pinned at the base. As the top of
the columns is rigidly built into a stiff
beam, the columns are effectively fixed at the top and pinned at the base.
Schodek (1998) provides more examples of different types of connections and
idealized models. Another factor which affects the constraints or fixity at the
supports is the soil condition. A column supported on a relatively small footing
and resting on compressible soil may be assumed to be hinged at the end, since
such soils offer but little resistance to rotation of the footing. On the other hand, a
footing resting on solid rock, or a column supported by a pile foundation may be
assumed to have sufficient fixity to prevent rotation and a fixed support may be
assumed. Columns supported by a continuous foundation mat should likewise be
assumed fixed (Nilson et al 2004).
Figure 6.3 End Bearing Detail
(http://nisee.berkeley.edu.ph/godden)
Figure 6.4 Fixed-Pinned Columns
(http://nisee.berkeley.edu.ph/godden)
Understanding 2D Structural Analysis by A.W.C. Oreta : Pinned &Fixed Support Conditions 6 - 3
How important is the assumed model of the support in the behavior and
response of the structure? This chapter aims to explore the effects of the support
conditions on the response of a structure.
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CASE STUDY : What are the implications of pinned and fixed support
conditions to structural design?
Two identical steel gabled frames with different support conditions similar to
Figure 6.2 will be analyzed subjected to two basic load cases – dead load and
wind load. Compare the behavior and response of the two structures.
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Things to Do
1. Draw two identical frames with different support conditions - one frame
with pin supports and the other frame with fixed supports.
2. Apply dead load (WL = 0.5 k/ft) and display the diagrams for the bending
moment, shear and axial forces.
3. Apply the wind loads as shown acting on the windward and leeward walls
and the roofs. Display the diagrams for the bending moment, shear and
axial forces.
4. Apply combination load case : 0.9 DL + 1.3 WL
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Observation
Displacements: Figures 6.5 and 6.6 show the deformed shapes of the two
frames due to dead and wind loads. Which frame has relatively larger
displacements? If you view the nodal displacements at the nodes, you will find
that the nodal displacements for the pinned supported frame are almost twice
that of the fixed supported frame. As an example, for the top node, the vertical
displacements due to dead load is about 2.4 in for the pinned case, while 1.28 for
the fixed case. On the other hand, for the same node the vertical displacements
due to wind load is about 12.0 in for the pinned case, while 6.40 for the fixed
case.
Understanding 2D Structural Analysis by A.W.C. Oreta : Pinned &Fixed Support Conditions 6 - 4
CASE STUDY 6
Rafter
W 21 x 68
15 ft
Column
W 27 x 84
20 ft
90 ft
Section Properties (Ref. AISC manual)
W 21 x 68 A = 20.0 in2 I = 1480 in4 d = 21.13 in
W 27 x 84 A = 24.8 in2 I = 2830 in4 d = 26.69 in
Material Properties (A36 steel)
E = 29 x 103 ksi
Specific weight = 0.284 lb/in3
Coefficient of Thermal Expansion = 6.5 x 10-6 /F
DL = 0.5 k/ft
Uplift = 3.0 k/ft
Uplift = 3.0 k/ft
windward
wall
0.35 k/ft
Dead Load
Leeward
wall
0.25 k/ft
Wind Load
Understanding 2D Structural Analysis by A.W.C. Oreta : Pinned &Fixed Support Conditions 6 - 5
(a) Pinned
(b) Fixed
Figure 6.5 Dead Load and Deformation Diagram
(a) Pinned
(b) Fixed
Figure 6.6 Wind Load and Deformation Diagram
Understanding 2D Structural Analysis by A.W.C. Oreta : Pinned &Fixed Support Conditions 6 - 6
(a) Pinned
(b) Fixed
Figure 6.7 Bending Moments due to Dead Load
(a) Pinned
(b) Fixed
Figure 6.8 Bending Moments due to Wind Load
Understanding 2D Structural Analysis by A.W.C. Oreta : Pinned &Fixed Support Conditions 6 - 7
Bending Moments : Figures 6.7 and 6.8 shows the bending moment diagrams
for each frame for the two basic load cases. The bending moments for the
pinned-base frame are relatively larger than the fixed-base frame for both loading
conditions. The maximum end moment of the column for the pinned case due to
dead load is 259.6 kip-ft at the top end compared to 240.5 kip-ft at the bottom
end for the fixed case. The maximum end moment at the rafter due to dead load
is 259.6 kip-ft for the pinned case and only 196.7 kip-ft for the fixed case. Similar
observations can be found for the bending moments due to wind load. If the
loads are now combined using appropriate load factors as shown in Figure 6.9,
the end moments in the pinned case are about 12% more in the columns and
about 25% more in the rafters compared with the fixed case.
(a) Pinned
(b) Fixed
Figure 6.9 Bending Moments for Combination Load Case : 0.9 DL + 1.3 WL
Understanding 2D Structural Analysis by A.W.C. Oreta : Pinned &Fixed Support Conditions 6 - 8
(a) Pinned
(b) Fixed
Figure
Shear Forces
Forces due
due to
to Dead
Dead Load
Load
Figure6.10
6.5 Shear
(a) Pinned
(b) Fixed
Figure 6.11 Shear Forces due to Wind Load
Understanding 2D Structural Analysis by A.W.C. Oreta : Pinned &Fixed Support Conditions 6 - 9
(a) Pin Supports
(b) Fixed Supports
Figure 6.12 Axial Forces due to Dead Load
(a) Pinned
(b) Fixed
Figure 6.13 Axial Forces due to Wind Load
Understanding 2D Structural Analysis by A.W.C. Oreta : Pinned &Fixed Support Conditions 6 - 10
Shear Forces: Compare now the shear forces in Figures 6.10 and 6.11. The
magnitudes of the shear forces in the columns in the fixed-base condition are
greater than the pinned-base condition, but the shear forces in the rafters in the
pinned case are greater than the fixed case. This is true for both dead and wind
loading conditions.
Axial Forces: In Figures 6.12 and 6.13 are shown the comparison of the axial
forces. There is not much of a difference between the magnitudes of the axial
forces in the columns between the two frames, although the axial forces in the
fixed supported frame are slightly larger for the rafters.
What are the implications of the observations about the two gabled frames with
different support conditions? In the design of these structures, the size of the
members is determined based on the internal moments and forces. The size of
the rafters is usually determined based on the critical moments, while the size of
the columns is obtained for the combined effects of the moments and axial forces.
Based on the member size obtained, the shear requirements are checked. As
observed earlier, the maximum moments developed in the frame which has
fixed-base connections are relatively less than those developed in the pinned
supported frame. This means that the members of the fixed supported frame
may be designed with smaller sections. Moreover, there is a reduction in
deflections in the fixed case.
However, to achieve these advantages of
minimizing moments and reducing deflections in the gabled frame using fixed
supports, special attention should be given in the design of the foundation so that
full fixity of the column will be achieved. Does this mean a fixed supported frame
is more superior than a pin supported frame? Not really! There are cases where
the design of the foundation is a problem and full fixity at the base is difficult to
achieve. In this case, a pinned-base connection may be the best overall solution.
Besides, there also advantages in a pinned supported frame. The foundation for
a pinned-base frame need not be designed to provide moment resistance.
Horizontal thrusts associated with vertical loads are usually smaller in a pinned
Understanding 2D Structural Analysis by A.W.C. Oreta : Pinned &Fixed Support Conditions 6 - 11
condition. Each specific design must be evaluated in its own context to see which
approach proves most desirable.
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Things to Ponder
The modeling of the supports of any structure should be based on actual
conditions at the site. The engineer should make sure what is assumed in the
modeling, analysis and design of the supports should be realized when the
structure is constructed. When full fixity is assumed in the modeling and analysis
of the structure, this condition should be assured when the design and
construction of the foundation is done. Similarly, the condition of no moment
resistance at the base should likewise be assured in the foundation design and
construction to simulate a pinned-base assumption. If the assumptions and
actual conditions are entirely different, the outcome may be catastrophic.
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Things to Try
1. Analyze the frame shown. Compare the results between pinned-supported
frame and fixed-supported frame. Consider the effects of vertical loads (WDL =
15 kN/m and WLL = 7 kN/m) applied fully and earthquake loads applied at
each floor (F4 = 80 kN, F3 = 60 kN, F2 = 40 kN and F1 = 20 kN), separately.
And then combine the loads (1.4 DL + 1.7 LL and 1.3 DL + 1.1 LL + 1.1 EQ).
Assume the following material properties:
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Modulus of elasticity = 20,500 N/mm2
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Unit weight = 24 kN/m3
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Coefficient of thermal expansion = 0.00099 / oC
2. Make sketches of possible details of column and footing connections for
reinforced concrete structures. When can the footing be assumed pinned or
fixed?
Understanding 2D Structural Analysis by A.W.C. Oreta : Pinned &Fixed Support Conditions 6 - 12
4 @3.0 m
= 12.0 m
3 @ 5.0 m = 15.0 m
Frame geometry
350 mm
400 mm
350 mm
Column cross-section
250 mm
Beam cross-section
Understanding 2D Structural Analysis by A.W.C. Oreta : Pinned &Fixed Support Conditions 6 - 13
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References
http://nisee.berkeley.edu.ph/godden, Godden Structural Engineering Slide
Library
Kassimali, A. (1999). Structural Analysis, 2nd Edition, Section 3.3, Brooks-Cole
Publishing Co., USA
Nilson, A.H., Darwin, D. and Dolan, C.W. (2004). Design of Concrete Structures,
13th Edition, Section 12.5, McGraw-Hill, Inc. NY, USA
Schodek, D.L. (1998). Structures. Section 3-3-2, Prentice-Hall, Inc. New Jersey,
USA
Understanding 2D Structural Analysis by A.W.C. Oreta : Soil Effects on Foundations 7 - 1
CHAPTER 7
SOIL EFFECTS ON FOUNDATIONS
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BACKGROUND
Foundations of structures are supported by the soil. The effect of the soil on the
behavior of the structure is significant especially for soft soils since the required
fixity between the column and footing may be difficult to realize. A simple isolated
footing may rotate, settle or
shift
sideways
by
some
amount depending on the
load and soil conditions.
Modeling
the
Isolated Footing
foundation
considering the soil stiffness
falls between the pinned or
fixed conditions. When the
effect of the soil in the
structural
model
is
considered, this becomes a
“soil–structure
Pile Foundation
Figure 7.1 Modeling of Foundations (Anwar 1998)
interaction”
problem. One popular and simple approach of modeling the soil is by the used of
“springs”. An isolated footing or a pile foundation may be represented by three
springs – one for vertical settlement, one for rotation and one for lateral
movement
(Figure
7.1).
GRASP provides an option
of
representing
the
constraint at a support by
springs as shown in Figure
7.2. You first choose a basic
support condition from the
six idealized models shown
Figure 7.2 Spring Models in GRASP
Understanding 2D Structural Analysis by A.W.C. Oreta : Soil Effects on Foundations 7 - 2
at the left and then modify the restraint at one or more degrees of freedom by
spring models by inputting the appropriate spring stiffness. The stiffness of the
spring can be derived by the modulus of sub-grade reaction of the soil or by the
method suggested by Gazetas (1991) which is adapted by ATC-40 (1996), where
the footing dimensions, depth of embedment and soil properties (modulus of
elasticity, shear modulus, poisson’s ratio) are parameters. This chapter explores
the option of modeling foundations using springs and compares the results to the
idealized pinned or fixed conditions.
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CASE STUDY : How can footings resting on soil be modeled to
incorporate soil-structure interaction?
Two identical steel gabled frames with different support conditions will be
analyzed. The supports will be represented by three springs – one frame
resting on dense soil and the other frame resting on soft soil. Two basic load
cases – dead load and wind load – will be applied. Observe the behavior and
response of the two structures supported by springs resting on two types of
soils and then compare the results with the frames supported by idealized
pinned and fixed supports in Chapter 6.
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Things to Do
1. Draw two identical frames supported by three springs.
2. Input the stiffness of the springs for two types of soils : (a) dense soil and
(b) soft soil
3. Apply dead load on the rafters.
4. Apply the wind loads as shown acting on the windward and leeward walls
and the roofs.
5. Apply combination load case : 0.9 DL + 1.3 WL
6. Perform analysis and display the diagrams for the bending moment, shear
and axial forces. Compare the results with the case study in Chapter 6.
Understanding 2D Structural Analysis by A.W.C. Oreta : Soil Effects on Foundations 7 - 3
CASE STUDY 7
Rafter
W 21 x 68
15 ft
Column
W 27 x 84
20 ft
90 ft
Dense Soil
Section Properties (Ref. AISC manual)
W 21 x 68 A = 20.0 in2 I = 1480 in4 d = 21.13 in
W 27 x 84 A = 24.8 in2 I = 2830 in4 d = 26.69 in
Kx = 4500 kip/in
Ky = 1500 kip/in
Kz = 200,000 kip-ft/rad
Material Properties (A36 steel)
Soft Soil
E = 29 x 103 ksi
Specific weight = 0.284 lb/in3
Coefficient of Thermal Expansion = 6.5 x 10-6 /F
Kx = 240 kip/in
Ky = 100 kip/in
Kz = 12,800 kip-ft/rad
DL = 0.5 k/ft
Uplift = 3.0 k/ft
Uplift = 3.0 k/ft
windward
wall
0.35 k/ft
Dead Load
Leeward
wall
0.25 k/ft
Wind Load
Understanding 2D Structural Analysis by A.W.C. Oreta : Soil Effects on Foundations 7 - 4
(a) Dense Soil
(b) Soft Soil
Figure 7.3 Dead Load and Deformation Diagram
(a) Dense Soil
(b) Soft Soil
Figure 7.4 Wind Load and Deformation Diagram
Understanding 2D Structural Analysis by A.W.C. Oreta : Soil Effects on Foundations 7 - 5
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Observation
Displacements: Figures 7.3 and 7.4 show the deformed shapes of the two
frames due to dead and wind loads. Which frame has relatively larger
displacements? If you view the nodal displacements at the nodes, you will find
that the frame resting on soft soil is more flexible and had displacements about
40% more than that of the frame resting on dense soil. Observe for example the
top node. The vertical displacements due to dead load is about 2.5 in for the soft
soil case, while 1.5 in for the dense soil case. On the other hand, the vertical
displacement due to wind load is about 12.6 in for the soft soil condition, while
7.6 in for the dense soil condition.
Bending Moments : Figures 7.5 and 7.6 shows the bending moment diagrams
for each frame for the two basic load cases. The bending moments for the frame
resting on soft soil are relatively larger than the dense soil condition for both
loading conditions. The magnitude of the maximum end moments of the rafters
(a) Dense Soil
(b) Soft Soil
Figure 7.5 Bending Moments due to Dead Load
Understanding 2D Structural Analysis by A.W.C. Oreta : Soil Effects on Foundations 7 - 6
(a) Dense Soil
(b) Soft Soil
Figure 7.6 Bending Moments due to Wind Load
(a) Dense Soil
(b) Soft Soil
Figure 7.7 Bending Moments for Combination Load Case : 0.9 DL + 1.3 WL
Understanding 2D Structural Analysis by A.W.C. Oreta : Soil Effects on Foundations 7 - 7
(a) Dense Soil
(b) Soft Soil
Figure 7.8 Shear Forces due to Dead Load
(a) Dense Soil
(b) Soft Soil
Figure 7.9 Shear Forces due to Wind Load
Understanding 2D Structural Analysis by A.W.C. Oreta : Soil Effects on Foundations 7 - 8
(a) Dense Soil
(b) Soft Soil
Figure 7.10 Axial Forces due to Dead Load
(a) Dense Soil
(b) Soft Soil
Figure 7.11 Axial Forces due to Wind Load
Understanding 2D Structural Analysis by A.W.C. Oreta : Soil Effects on Foundations 7 - 9
and columns for the soft soil case due to dead load is 246.6 kip-ft compared to
209.5 kip-ft for the dense soil case. On the other hand, The magnitude of the
maximum end moments of the rafters and columns for the soft soil case due to
wind load is 1,400 kip-ft compared to 1,100 kip-ft for the dense soil case. The
maximum end moments for the combined dead and wind loads in Figure 7.7, for
the soft soil case are about 13% more than the moments in the dense soil case.
In all loading cases, smaller moments at the bottom end of the columns occur in
the frame resting on soft soil.
Shear Forces: Compare now the shear forces in Figures 7.8 and 7.9. The
magnitudes of the shear forces in the columns in the dense soil condition are
greater than the soft soil condition, but the shear forces in the rafters in the soft
soil case are greater than the dense soil case. This is true for both dead and
wind loading conditions.
Axial Forces: In Figures 7.10 and 7.11 are shown the comparison of the axial
forces. There is not much of a difference between the magnitudes of the axial
forces in the columns between the two frames, although the axial forces in the
rafters for dense soil case are slightly larger than for the soft soil case. The axial
forces for both soil conditions are almost the same.
Comparing with pinned and fixed conditions: How do the results of the
analysis of the frames supported by spring models compare with the idealized
pinned-base and fixed-base conditions in Chapter 6? By simply comparing the
diagrams, we can see that the response of the frame resting under soft soil
conditions is similar to the pinned-base frame. The only difference between the
two models is that moments are developed at the bottom ends of the columns
for the spring model compared to zero moments for the pinned case. As a result,
the maximum end moments under the soft soil condition are slightly smaller than
the pinned-base condition. The response of the frame under the dense soil
condition is very similar to the fixed-base frame. However, the maximum end
Understanding 2D Structural Analysis by A.W.C. Oreta : Soil Effects on Foundations 7 - 10
moments of the columns for the frame under dense soil condition are slightly
smaller than the fixed-base condition. On the other hand, the rafter moments are
slightly larger than the dense soil case than the fixed-base case.
Things to Ponder
A pinned-base support assumes zero moment resistance at the base, while a
fixed-base support assumes a rigid base connection. Foundations rest on soil
whose properties are variable from soft clay to hard rock. Depending on the soil
and load conditions, the actual restraint developed at the base may fall between
the pinned-base and fixed-base conditions. The modeling of the supports of any
structure should be based on actual conditions at the site. Spring models to
represent the soil may be a simple approach when soil-structure interaction is
considered. However, one of the problems that the designer should confront
when considering the soil effects is the soil property, particularly what appropriate
value of soil stiffness to use in the model. An unreasonable assumption of the
soil stiffness may lead to an unconservative design. In the absence of information
about the soil, the pinned-base or fixed-base models may be used appropriately
with the designer introducing additional safety factors in the design (e.g.,
introducing some moment at the base even if a pinned-base support is assumed).
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Things to Try
1. Analyze the same gabled frame of the case study. Instead of using three
springs to model the soil, represent the support by a pin with a rotational
spring. Use the soil stiffness, kz values given for the dense and soft soil.
Compare the results of the “pin-rotational spring” supported frames with the
“three-spring” supported frames for both types of soil.
Understanding 2D Structural Analysis by A.W.C. Oreta : Soil Effects on Foundations 7 - 11
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References and related readings
Anwar, N. (1998). “Modeling of Foundations,” ACECOMS News & Views, April –
June, page 17, AIT, Thailand
Gazetas, G. (1991). Foundation Vibrations, Principles and Practices. PrenticeHall, New Jersey, USA
Applied Technology Council (1996), “Foundation Effects”, Seismic Evaluation
and Retrofit of Concrete Buildings (ATC-40), Vol. 1, Chapter 10
De La Cruz, L., Florendo, C. and Santiago, H. (2003). The Effect of Soil-Spring
Modeling in 2D Frame Analysis. Undergraduate Thesis, De La Salle University,
Manila
Obrien E. and Keogh, D. (1999). Bridge Deck Analysis., Chapter 4, E & FN Spon,
London
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