# Unified Fundamental Formulas for Static Analysis of Pin-Jointed Bar Assemblies

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Basic Theory

_{k}, so that the axial force vector of the assembly can be defined as

**n**= [n

_{1}, n

_{2}, …, n

_{b}]

^{T}∊ ℜ

^{b}. The equilibrium equations at nodes of the structure are written in the form [14]:

**A**∊ ℜ

^{d}

^{×b}is the equilibrium matrix of the structure containing the orientations of members;

**q**∊ ℜ

^{d}is nodal loads vector made up of the x-, y-, and z-components of the external loads applied at each node.

**l**= [l

_{1}, l

_{2}, …, l

_{b}]

^{T}∊ ℜ

^{b},

**L**= diag(l

_{1}, l

_{2}, …, l

_{b}) ∊ ℜ

^{b}

^{×b}and

**t**= [n

_{1}/l

_{1}, n

_{2}/l

_{2}, …, n

_{b}/l

_{b}]

^{T}∊ ℜ

^{b}respectively. The vector

**n**containing the axial forces of members can be expressed by

**AL**and gave an equivalent expression using Kronecher product notation “$\otimes $” as

**I**is the three-order identity matrix for a three-dimensional structure;

**x**is a vector contains x-, y-, and z-components of the coordinates for each node; the symmetric matrix Ω can be written directly by means of the force densities [19]. The (i, j)-component Ω (i, j) of Ω is given by

**AL**only contains the components of nodal coordinates or be zero alternatively, δ(

**AL**) can be obtained by replacing coordinate components with the corresponding incremental ones in

**AL**. Then, it is clear that the following equations hold by considering Equation (4)

**x**is a vector of incremental coordinates, describing infinitesimal movements of nodes. In Equation (7), the coefficient matrix $\mathbf{I}\otimes \mathsf{\Omega}$ is equivalent to the so-called geometric stiffness matrix

**K**

_{G}in conventional finite element formulas since it corresponds to the stiffness resulting from the reorientation of stressed members. For simplicity, Equation (7) is rewritten as

**t**=

**L**

^{−1}

**n**is considered. Noting that the differential of

**L**

^{−1}yields

**T**= diag(n

_{1}/l

_{1}, n

_{2}/l

_{2}, …, n

_{b}/l

_{b}) = diag(

**t**) ∊ ℜ

^{b}

^{×b}, we also have the relationship

**l**is an elongation vector, consisting of infinitesimal extension of each member. Substitution of Equation (12) into Equation (11) yields

**x**to δ

**l**hold as [14]

**J**is a symmetric matrix since

**K**

_{G}and ATA

^{T}are symmetric matrices.

**q**for assemblies made from pin-jointed bars. Particularly, the resistance resulting from the slight change of nodal coordinates can only play a role for prestressed structures.

**l**and δ

**n**are related by

**e**contains the imposed elongation e of each member; the axial flexibility matrix

**F**is defined as

**F**= diag[l

_{1}/(E

_{1}A

_{1}), l

_{2}/(E

_{2}A

_{2}),…,l

_{b}/(E

_{b}A

_{b})], where E

_{k}, A

_{k}, l

_{k}denote Young’s modulus, cross-sectional area and length of k-th member, respectively.

## 3. A Unified Form of Fundamental Formulas

**n**and δ

**x**are coupled with each other. Without loss of generality, type IV assemblies are employed as illustrative examples in this section. Then, the techniques of linear algebra and generalized inverse matrix are applied to decouple δ

**n**and δ

**x**, thereby obtaining a new set of fundamental equations, which have the similar form with the ones of the traditional force method.

**x**is looked upon as the basic unknown variable of the system, the following consistent condition should be satisfied to ensure the existence of the solutions in Equation (21):

**S**satisfying

**AS**=

**0**. If Equation (23) holds, δ

**x**can be solved as

**H**satisfying

**A**

^{T}

**H**=

**0**. The matrices

**S**and

**H**can be determined by using Singular Value Decomposition on

**A**[20]. The superscript “+” in Equation (24) represents Moore–Penrose generalized inverse. The vector

**β**gives the coefficient of each of the basis mechanisms. According to the properties of Moore-Penrose generalized inverse, the following equation holds [21]:

**J**=

**K**

_{G}−

**ATA**

^{T}, the following equation holds

**n**is regarded as the basic unknown variable of the system, starting from Equation (19), we can get:

**J**is a symmetric matrix, then

**n**can be solved as

**α**contains the coefficient of each of the basis state of self-stress. Substitute Equation (36) into Equation (21) and tidy up

**B**’ for simplicity, i.e.,

**A**as r

**,**it is easy to verify that

**B**is a square matrix of (d + b − r) order. The submatrix

**K**

_{G}

**H**of

**B**consists of the so-called product-force vector [14] resulting from each independent mechanism, arranged by columns. On the other hand, the submatrix -

**S**

^{T}

**F**of

**B**is comprised of the bar-elongation vector arising from each independent state of self-stress, arranged by rows and taking negative sign. When all the mechanisms of the structure can be stiffened by initial internal forces, the matrix

**B**has full rank. Then $\overline{\mathsf{\delta}\mathbf{n}}$ and $\overline{\mathsf{\delta}\mathbf{x}}$ can be solved readily through Equations (41) and (42), that is, the solutions of δ

**n**,

**β**, δ

**x**and

**α**are obtained simultaneously.

**A**δ

**n =**δ

**q**’ and compatibility equations ‘

**A**

^{T}δ

**x =**δ

**l = FA**

^{-1}δ

**q + e**’ (the equilibrium matrix

**A**also has full rank), which are consistent with Equations (41) and (42) in form. Particularly, when Equations (41) and (42) are used for type I assemblies, the matrix

**B**is reduced to

**A**, while $\overline{\mathsf{\delta}\mathbf{q}}$ and $\overline{\mathsf{\delta}\mathbf{L}}$ degenerate into δ

**q**and δ

**l,**respectively. Therefore, they can be regarded as generalized equilibrium equations and compatibility equations implying constitutive relations. Since the derivation of Equations (41) and (42) does not involve the unique properties of type IV assemblies, they are applicable to types I, II, and III assemblies as well. In order to facilitate the readers to follow the above derivation process, Figure 1 gives a road map for development of the unified formulas presented in this section, while Figure 2 shows the calculation steps to guide the designers to apply the proposed formulas.

## 4. Comparison with Other Formulations

#### 4.1. Matrix Force Method

**x**can be regarded as the basic unknown variable of the system. When the consistent condition given by Equation (23) is satisfied, δ

**x**can be solved form Equation (24). Furthermore, δ

**x**may be divided as follows:

**β**

^{I}and

**β**

^{II}are the vectors that need to be determined further, satisfying

**β**

^{I}+

**β**

^{II}=

**β**. In Equation (43), δ

**x**

^{I}related to the elastic elongations of the bars is called ‘extensional deformation’, while δ

**x**

^{II}only associated with the infinitesimal mechanism corresponds to ‘inextensional deformation’ [14].

**A**δ

**n**+

**J**δ

**x =**δ

**q**—is investigated. As mentioned before, the second item on the left side of Equation (19) represents the resistance arising from small nodal displacements when the internal forces of the bars are kept constant. Thus, they are called “product forces” in Pellegrino and Calladine [14]. If the product forces caused by δ

**x**

^{I}are ignored and the relations given by Equation (29) (

**JH**=

**K**

_{G}

**H**) are considered, Equation (19) can be rewritten as:

**n**and

**β**

^{II}can be obtained.

**β**

^{I}(or δ

**x**

^{I}) will be determined. From the compatibility Equation (21), we can see that ${\mathbf{A}}^{\mathrm{T}}(\mathsf{\delta}{\mathbf{x}}^{\mathrm{I}}+\mathsf{\delta}{\mathbf{x}}^{\mathrm{II}})=\mathbf{e}+\mathbf{F}\xb7\mathsf{\delta}\mathbf{n}$ holds. Because of ${\mathbf{A}}^{\mathrm{T}}\mathsf{\delta}{\mathbf{x}}^{\mathrm{II}}={\mathbf{A}}^{\mathrm{T}}\mathbf{H}{\mathsf{\beta}}^{\mathrm{II}}=0$, the following formula is obtained

**β**

^{I}is optional. In order to determine the unique

**β**

^{I}, the new constraint conditions have to be considered further.

**H**

^{T}simultaneously:

**x**satisfies Equation (35), i.e.,

**K**

_{G}

**H**is made up of product-force vectors arranged in columns (it is consistent with the constraint equations derived from the principle of virtual work in Pellegrino and Calladine [14]). Then, Equations (49) and (46) are combined and written in the following matrix form:

**x**

^{I}can be obtained.

**x**

^{I}can be ignored. Specifically, it can be divided into two situations: (1) the absolute values of the product forces caused by δ

**x**

^{I}are small enough; (2) the values of the product forces caused by δ

**x**

^{I}are relatively small compared with the ones resulting from δ

**x**

^{II}. It is worth mentioning that the type IV assembles—statically and kinematically indeterminate ones—usually belong to the latter situation. Therefore, the static analysis results obtained from the two methods are often close to each other. Although the fundamental formulas proposed in this paper are more complex than the ones of the matrix force method, they are applicable to static analysis of a general pin-jointed assembly based on the assumption of small deformation and linear elasticity, which will be further verified through several numerical examples in the next section.

#### 4.2. Displacement-Based Method

**n**was employed to express δ

**x**from Equation (21) in Section 3. When

**A**is a rectangular matrix and does not have full row rank, the expression of the general solution of δ

**x**includes the generalized inverse of

**A**and the matrix

**H**consisting of a full basis for the left nullspace of

**A**(see Equation (24)). Meanwhile, the consistent condition to ensure the existence of the solutions of δ

**x**in Equation (21) arises, given by Equation (23). However, if the variable δ

**x**is employed to express δ

**n**from Equation (21), the expression is much simpler as:

**G**=

**F**

^{−1}is called as axial stiffness matrix of bars. Substitute Equation (51) into Equation (19) and tidy up:

**K**

_{E}. Then Equation (52) can be further written as:

**K**is the so-called tangent stiffness matrix of the structure.

**K**; (2) make use of Equation (53) to calculate the nodal displacements in step i—δ

**x**

^{i}; (3) update the nodal coordinates in terms of δ

**x**

^{i}; (4) assemble the tangent stiffness matrix and the unbalanced nodal loads vector based on the new configuration; (5) substitute them into Equation (53) to calculate the nodal displacements in step i + 1—δ

**x**

^{i}

^{+1}. Repeat the above procedures until the nodal displacements or the unbalanced nodal loads are small enough to satisfy the preset accuracy requirements.

## 5. Numerical Examples

^{4}N. Two equal weights were hung as shown in Figure 3. All dimensions are in mm. Consider the situation that the cable segment 1 is shortened by 10 mm (it is obtained by letting 10 mm of cable slide through the left-hand pin), while the lengths of cable segments 2 and 3 are fixed. How to determine the final configurations when W = 30 N and W = 3000 N are imposed, respectively?

**A**is 3. Hence, the numbers of independent mechanisms and self-stress states are m = 1 and s = 0 respectively, which means that this system belongs to type III—statically determinate and kinematically indeterminate assemblies. The unique mechanism composing the matrix

**H**can be obtained by Singular Value Decomposition of the matrix

**A**, which is normalized as:

**K**

_{G}is obtained from the following formula:

**A**,

**T,**and

**K**

_{G}are substituted into Equation (18) to get the matrix

**J**. The axial flexibility matrix

**F**is also assembled. Then the matrix $\widehat{\mathbf{A}}$ can be obtained by substituting

**F**and

**J**into Equation (28).

**B**can be obtained from Equation (40):

**B**has full rank, and the initial nodal loads vector

**q**is orthogonal to the mechanism

**H**. In other words, the mechanism is stiffened by the external loads. The system becomes a relatively stable structure that can withstand the external disturbance.

**e**= [−10 0 0]

^{T}, while the increments of the nodal loads satisfy δ

**q**=

**0**. The generalized nodal loads vector (incremental form) and bar elongation vector are given as:

**q**=

**0**holds in this example, we can get the solutions of δ

**n**=

**0**and

**β**

^{II}=

**0**(i.e., δ

**x**= δ

**x**

^{I}) from the above formula, substituting which into Equation (50) gives

**x**solved from the above formula is irrelevant to the initial nodal load W.

**A**is 11. Hence, the numbers of independent mechanisms and self-stress states are m = 1 and s = 1 respectively, which means that this system belongs to type IV—statically and kinematically indeterminate assemblies. The mechanism of this structure can be stiffened by introducing a certain amount of prestress. In this example, the prestress state can be determined by multiplying the unique normalized self-stress state by a coefficient μ, so that μ represents the level of prestress. Assume that μ = 8000 N holds. Meanwhile, the products of Young’s modulus and the cross-section area for cables and struts satisfy (EA)

_{cable}= 1.836 × 10

^{4}N and (EA)

_{strut}= 5.796 × 10

^{6}N. Let us consider the following two situations respectively:

**A**is 12. Therefore, s = 4 and m = 0 hold, which means that this system belongs to type II—statically indeterminate and kinematically determinate assemblies.

**x**

^{I}are relatively small compared with the ones resulting from δ

**x**

^{II}and, therefore, considering (like Method 1) or ignoring (like Method 2) the contribution of this part has little effect on the final results. However, when Method 2 is employed to deal with type II assemblies in the second situation, the fundamental Equations (45) and (50) are reduced to:

**S**

_{i}(i = 1, 2, 3, 4) can be obtained through the Singular Value Decomposition of the equilibrium matrix, each of which is formed as:

**S**

_{i}, the first 12 ones correspond to the original 12 members, while the last 4 ones correspond to the 4 added cables. It is found that

**S**

_{i}is featured by the fact that the sum of the last 4 entries is zero, i.e., the following equation holds

**S**

^{T}

**e**=

**0**holds. Moreover, due to δ

**q**=

**0**, then:

**n**=

**0**and $\mathsf{\delta}\mathbf{x}={({\mathbf{A}}^{\mathrm{T}})}^{+}(\mathbf{e}+\mathbf{F}\xb7\mathsf{\delta}\mathbf{n})={(\mathbf{A}{\mathbf{A}}^{\mathrm{T}})}^{-1}\mathbf{A}\xb7\mathbf{e}$, which will keep unchanged with the variation of the prestress level μ.

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 5.**A prestressed cable-strut structure with 4 struts and 8 cables. (

**a**) Top view (mm); (

**b**) perspective view; (

**c**) side view (mm).

**Figure 6.**A prestressed cable-strut structure with 4 struts and 12 cables (the lengths of 4 added cables are adjustable). (

**a**) Top view (mm); (

**b**) perspective view.

**Figure 7.**The increments of axial forces δn with respect to different levels of prestress μ. (

**a**) The members numbered from ① to ④; (

**b**) the members numbered from ⑤ to ⑧; (

**c**) the members numbered from ⑨ to ⑫; (

**d**) the members numbered from ⑬ to ⑯.

Assembly Type | Statical and Kinematical Properties |
---|---|

I | Statically determinate and kinematically determinate |

II | Statically indeterminate and kinematically determinate |

III | Statically determinate and kinematically indeterminate |

IV | Statically indeterminate and kinematically indeterminate |

W | Cable Segments | n (Initial Axial Force) | δn (the Increments of n) | ||||
---|---|---|---|---|---|---|---|

Method 1 | Error (%) | Method 2 | Error (%) | Method 3 | |||

W = 30 | ① | 67.08 | 7.621 | −19.192 | 0 | −100 | 9.431 |

② | 60.00 | 8.199 | −18.902 | 0 | −100 | 10.113 | |

③ | 67.08 | 7.045 | −21.082 | 0 | −100 | 8.927 | |

W = 3000 | ① | 6708 | 256.076 | −1.424 | 0 | −100 | 259.778 |

② | 6000 | 255.767 | −1.578 | 0 | −100 | 259.930 | |

③ | 6708 | 201.454 | −2.657 | 0 | −100 | 207.046 |

W | Nodes | Directions | Method 1 | Error (%) | Method 2 | Error (%) | Method 3 |
---|---|---|---|---|---|---|---|

W = 30 N | 1 | x-direction | −5.193 | 0.562 | −5.160 | −0.077 | −5.164 |

y-direction | −11.809 | −4.217 | −12.040 | −2.352 | −12.332 | ||

2 | x-direction | −5.122 | 0.787 | −5.160 | 1.535 | −5.082 | |

y-direction | −10.090 | −7.176 | −10.320 | −5.060 | −10.870 | ||

W = 3000 N | 1 | x-direction | −6.000 | −0.150 | −5.160 | −14.129 | −6.009 |

y-direction | −4.782 | 1.810 | −12.040 | 156.334 | −4.697 | ||

2 | x-direction | −3.771 | 0.506 | −5.160 | 37.527 | −3.752 | |

y-direction | −3.153 | 1.187 | −10.320 | 231.194 | −3.116 |

Situations | Members | n (Initial Axial Force) | δn (the Increments of n) | ||||
---|---|---|---|---|---|---|---|

Method 1 | Error (%) | Method 2 | Error (%) | Method 3 | |||

1 | ① to ④ | 1311.3 | 46.254 | −1.996 | 45.592 | −3.398 | 47.196 |

⑤ to ⑧ | 2389.5 | −10.390 | 8.173 | −10.243 | 6.641 | −9.605 | |

⑨ to ⑫ | −2853.3 | −119.708 | −1.039 | −118.328 | −2.180 | −120.965 | |

2 | ① to ④ | 1311.3 | 30.325 | −4.869 | 0 | −100 | 31.877 |

⑤ to ⑧ | 2389.5 | −6.755 | 15.946 | 0 | −100 | −5.826 | |

⑨ to ⑫ | −2853.3 | −65.954 | −3.548 | 0 | −100 | −68.380 | |

⑬ to ⑯ | 0 | 61.137 | −1.397 | 0 | −100 | 62.003 |

Situations | Directions | Method 1 | Error (%) | Method 2 | Error (%) | Method 3 |
---|---|---|---|---|---|---|

1 | x-direction | 4.319 | 0.512 | 4.327 | 0.698 | 4.297 |

y-direction | −5.388 | 2.045 | −5.380 | 1.894 | −5.280 | |

z-direction | −3.495 | 1.569 | −3.486 | 1.308 | −3.441 | |

2 | x-direction | 5.077 | −0.743 | 9.111 | 78.123 | 5.115 |

y-direction | −5.778 | 1.120 | −9.111 | 59.450 | −5.714 | |

z-direction | −3.635 | 0.665 | −5.467 | 51.399 | −3.611 |

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## Share and Cite

**MDPI and ACS Style**

Zhang, P.; Xiong, H.; Chen, J.
Unified Fundamental Formulas for Static Analysis of Pin-Jointed Bar Assemblies. *Symmetry* **2020**, *12*, 994.
https://doi.org/10.3390/sym12060994

**AMA Style**

Zhang P, Xiong H, Chen J.
Unified Fundamental Formulas for Static Analysis of Pin-Jointed Bar Assemblies. *Symmetry*. 2020; 12(6):994.
https://doi.org/10.3390/sym12060994

**Chicago/Turabian Style**

Zhang, Pei, Huiting Xiong, and Junsheng Chen.
2020. "Unified Fundamental Formulas for Static Analysis of Pin-Jointed Bar Assemblies" *Symmetry* 12, no. 6: 994.
https://doi.org/10.3390/sym12060994