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Description of the problem

In many cases, it is required to determine the settlement under an embankment, a metal plate foundation of a liquid tank, loads on small isolated plates or a raft of thin thickness. In these cases, the foundation will be assumed as flexible foundation.

The Figure shows an ore heap on thin concrete pavement slabs. The pavement slabs are connected with each other by movable joints. Consequently, the pavement slabs are considered as completely flexible foundation. The unit weight of the ore material is γ = 30 [kN/m3].

The foundation base under the ore heap has the dimension of 13×13 [m2], while the top area of the ore heap has the dimensions of 9×9 [m2]. The height of the ore heap is 4.0 [m] (Figure a).

It is required to determine the expected settlement due to the ore heap.

 

 Example 22: Settlement calculation under flexible foundation of an ore heap

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Description of the problem

In many cases, it is required to determine the settlement under an abutment, a bridge pier, a building core or a raft of thick thickness. In these cases, the foundation will be assumed as rigid foundation.

As an example for rigid rafts, consider the rectangular raft of a core from concrete walls shown in Figure 55 as a part of 93.0 [m] structure. The length of the raft is L = 28.0 [m], while the width is B = 25.0 [m]. Due to the lateral applied wind pressure, the raft subjected to an eccentric vertical load of P = 142000 [kN]. Figure 55 shows section elevation through the raft and subsoil, while Figure 56 shows a plan of the raft, load, dimensions and mesh.

It is required to estimate the expected settlement if the raft is considered as perfectly rigid.

 

 Example 23: Settlement calculation for a rigid raft subjected to an eccentric load

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Description of the problem

To verify the mathematical model of ELPLA for computing plane stresses, results of a cantilever beam having a thin rectangular cross section introduced by Timoshenko/ Goodier (1970) (Example 21, page 41) are compared with those obtained by ELPLA. The cantilever carries a point load of P = 150 [kN] applied at the end as shown in Figure 58.

 

 

 Example 24: Deflection of a thin cantilever beam

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Description of the problem

To verify the mathematical model of ELPLA for determining pile forces of pile groups under a pile cap, results of a pile group obtained by Bakhoum (1992) (Example 5.19, page 592) are compared with those obtained by ELPLA.

A pile cap on 24 vertical piles is considered as shown in Figure 59. It is required to determine the force in each pile of the group due to a vertical load of N = 8000 [kN] acting on the pile cap with eccentricities ex = 1.4 [m] and ey = 1.8 [m] in both x- and y-directions.

 

 

 Example 25: Forces in piles of a pile group

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Description of the problem

To verify the mathematical model of ELPLA for analyzing continuous beams, results of a continuous beam introduced by Harry (1993) (Examples 10.2, 10.4 and 10.5, pages 399, 409 and 411) are compared with those obtained by ELPLA.

A continuous beam of length L = 35 [m] is chosen as shown in Figure 60. The beam is subjected to a point load of P = 500 [kN] at the center. The beam cross section yields Moment of Inertia I = 0.003 [m4]. Young's modulus of the beam is E = 2.0×108[kN/m2].

For the comparison, three different cases are considered as follows:

Case a: Continuous beam with a point load P at the center on supports at points a, b, d and e.

Case b: Instead of the point load P at the center of the beam, points a, b, d and e have the following support settlements: Δa = -2.75 [cm], Δb = -4.75 [cm], Δd = -2.2 [cm] and Δe = -1.0 [cm].

Case c: Points c and d are supported by elastic springs that have stiffness of ksb = ksd = 3600 [kN/m].

 

 Example 26: Analysis of a continuous beam

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Description of the problem

To verify the mathematical model of ELPLA for analyzing unsymmetrical closed frames, moments in an unsymmetrical closed frame introduced by Wang (1983) (Example 15.10.1, page 574) are compared with those obtained by ELPLA.

An unsymmetrical closed frame ABCD is considered as shown in Figure 63. The frame is subjected to a point load of P = 24 [kN] at the center of the member BC and a distributed load of q = 2 [kN/m] on the member AD.

 

 

 Example 27: Moment in an unsymmetrical closed frame

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Description of the problem

To verify the mathematical model of ELPLA for analyzing plane trusses, results of plane truss introduced by Werkle (2001) (Example 3.1, page 61) are compared with those obtained by ELPLA.

A plane truss of 4 nodes and 6 members is considered as shown in Figure 65. Members 5 and 6 are unconnected in their intersection point. The truss is subjected to vertical and horizontal point loads at node 2, each of 10 [kN].

 

 Example 28: Analysis of a plane truss

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