Simplification Errors in the Classic Truss Model: Are They Still Tolerable in This Age of Microcomputers?
Primus V. Mtenga, P.E., Member ASCE and Edward Lloyd
Recent developments in computational abilities plus the availability of powerful low-priced microcomputers have driven the need for more efficient analysis of trusses. The classical truss model provides no recognition of semirigid behavior of truss joints. A model to predict axial forces and moments in trusses which deviate from the classic truss model assumptions is presented. Results show significant reduction in moments in the continuous chords of a truss as the rigidity of the truss member joints increases.
Key Words: semirigid joints, joint slippage, continuous chords, spring coefficients, classical truss model
Introduction and Background
The classical truss model is a structure composed of individual members joined together to form a series of triangles. To analyze the truss structure the following simplifying assumptions are made:
a) The truss members are connected together by frictionless pins. This assumption implies that as the members change their lengths due to axial forces, they will be free to rotate without causing any bending in the truss members;
b) Truss members are straight, and thus there is no bending moments due to P-D effect;
c) The structural system is arranged in such a manner that all loads are transmitted to the joints only.
In real life, however, there are deviations from these assumptions. For example, multiple bolt joints and welded joints are far away from the frictionless pin assumption. Therefore, bending moments and associated shear forces will arise, whenever these rigid or semi-rigid joints resist the rotation of the members as they change in length due to axial forces. The fact that some of the truss members (such as the top and bottom chords) are continuous is a deviation from the ideal truss assumption. This deviation in addition to the fact that not all loads are nodal, such as the top chord trusses with sheathing attached (both floor and roof trusses), will cause bending moments in the truss members. The bending of the compression members is particularly critical since it will lead to the P-D effect.
The aforementioned simplifications were very important in the days when our computational abilities were limited. However, recent developments in computer applications in the field of structural analysis have made it very easy to operate with models which are much closer to real life situations. It will take some extra effort, on the part of software developer, to include semi-rigid joints as well as the modeling of continuous chords in his analysis. However, it will take almost no extra effort for the software user to incorporate the extra data required.
A number of methods have been used in an attempt to account for the deviations from the classical model. For example, the Truss Plate Institute(TPI) (Truss.. 85) Specification for the design of metal plate connected wood trusses requires the consideration of moments, in addition to the axial forces obtained from the classical truss model. Models dealing with these types of trusses have been proposed. Such models include the Purdue Plane Structures Analyzer (PPSA) (Suddarth and Wolfe, 1984), in which fictitious analog members are introduced to model the rigidity of the joints. On the other hand the Wisconsin Model (Cramer et. al. 1993), uses a more refined connector model based on connector behavior proposed by Foschi (1977). All these models, which are closer to real life situation than the classical model, rely heavily on recent developments in computation capabilities.
In recent times a number of methods of dealing with semirigid joints have been proposed. These include studies by Seif et. al. (1981), Wang (1983), and Maraghechi and Itani (1984). In these studies the semirigid effect was accounted for by introducing springs with coefficients varying from zero to one. Study on the effect of bolt slippage is reported by Kitipornchai et. al. (1994). Results of this study illustrate that joint slippage has significant effects on deflections.
Model
A member with semi-rigid connections can be decomposed into two parts as shown in Fig.1; a beam element and a truss element.
The local element stiffness matrix will then be:
[ke] = [kbl] + [ktl] (1)
where
[ke] = local element stiffness matrix
[kbl] = local stiffness matrix for the beam part of the element
[ktl] = local stiffness matrix for the truss part of the element
The modified element local stiffness matrix for the beam part (Fig.1 b) can be derived from what has been proposed by Wang (1993) as follows:
[kbl] = [A] [S] [AT] (2)
where
1 0
[A] = . - -
. .
(3)
0 1
[S] =. sii sij
.
(4)
sji sjj
sii = , sij = sji = and sjj = (5)
pi and pj are the fixity factors at member ends i and j respectively. These factors are zero for hinged connections and equal 1 for rigid connections.
For the truss part of the element (Fig.1a), there is a possibility of connection slippage along the axis of the member. Slippage can occur in two possible ways: 1) gradual slippage increasing in proportion to the axial load, or 2) sudden slippage taking place after a load sufficient to overcome the friction between the connected members is reached. Presented in Fig.2 are possible truss member load-elongation curves for the two slippage mechanisms described above. Stresses occurring at the connectors are always higher than those at other locations of a truss member. Because of these higher stresses local deformations around the connectors will be higher than what is considered to be the deformation of a bar in axial loading. In such a scenario, the gradual slippage model will be a better approximation to reality. The gradual slippage model was used in this study.
i j
Springs to represent gradual joint slippage
a)Truss Element
Rotational springs
b)Beam Element
Fig.1. Model of a Member in a Truss With Semi-rigid Joints
axial
force
F
Fs
gradual slippage
d sudden slippage
member deformation D
Fig.2. Truss Member Joint Slippage Characteristics
Following is the derivation of the truss element stiffness characteristics under the assumption of gradual connection slippage.
Let axial stiffness of the member be
am = (6)
and the stiffness of the gradual slippage springs at member ends i and j be ai and aj respectively.
The equivalent stiffness, ae, for these three springs in series can then be written as:
ae = (7) The stiffness of the joints can be expressed as multiples of the member axial stiffness, as follows:
ai = xi am and aj = xj am (8)
where the factors xi and xi will have a value of infinity for joints with no axial flexibility. With these factors Eq.7 will be of the form
ae = (9)
Defining the axial fixity of the member as
pa = (10)
then the axial stiffness of the member will be
ae = pa (11)
The local element stiffness matrix for the truss part will therefore be:
[ktl] = . pa - pa
(12)
- pa pa
Presented in Fig. 3 is a sketch of a truss analyzed using the model described above. This truss has a span of 8.4 m and a slope of 1 to 2. It was analyzed at a load of 1.44 KN/m (2.4 KN/m2 applied to trusses placed at 600 mm on center), which is the design load for a wood truss of this configuration.
Fig. 3 A Sketch of one of the Trusses Used in this Study.
The axial force and moments at the top-chord at a location indicated in Fig. 3 are presented and discussed below.
Results and Discussions
Presented in Fig.4 are the variations of axial forces and moments with respect to the rotational rigidity coefficients in the truss members. It can be seen that as the rigidity coefficients increase, both the axial force and the moment at the chosen location in the top-chord decrease. This may be explained by the fact that for non-zero rigidity coefficients, the truss members take part in restraining the rotation of the joints. Consequently, the overall rotations of the joints are less, and thus there is less bending in the top-chord.
a)Axial force variation with respect to rotational spring coefficients
b)Relationship between moments and rotational spring coefficients
Fig. 4 Variation of Axial Force and Moments in a Top-Chord of a Fink Truss (shown in Fig. 3)
In Fig.4 b) the percent moment reduction at selected rigidity coefficients are computed. From these values it can be seen that even at low rigidity coefficients there are significant reduction in bending moments. For example at a coefficient of 0.2, a value that may be quite conservative for heavy bolted connections or connections with gusset plates, the moment reduction is more than 15 percent.
Conclusion
The results presented in this paper suggest that there is need for including the semi-rigid nature of "truss" joints in the analysis of trusses. The classical truss model gives results that are quite different from reality. There is need for software developers and researchers to take advantage of recent developments in computer applications in the field of structural analysis by developing codes and databases that incorporate the semi-rigid nature of joints in the analysis of trusses. It will take little extra effort on the part of a software developer to include these effects in the code, however, it will take almost no extra effort for the software user to incorporate the extra data required.
References
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