Understanding 2D Structural Analysis by A.W.C. Oreta : Pinned &Fixed Support Conditions 6 - 1 CHAPTER 6 PINNED AND FIXED SUPPORT CONDITIONS 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. 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. 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 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. 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. 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: Modulus of elasticity = 20,500 N/mm2 Unit weight = 24 kN/m3 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 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 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. 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. 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 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). 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 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
