Mechanics of Building Structural Materials Based on Lagrangian Mathematical Model Analysis

This paper approximately analyzes the dynamic response mode of building materials under impact load according to the Lagrange equation. We use the redundant constraint force of the statically indeterminate structure to list its strain energy expression. At the same time, the paper introduces the Lagrange multiplier and combines the static equilibrium equation to obtain the internal force of the statically indeterminate structure. The calculation results show that the load duration and peak value are important factors determining structural members' failure mode. This study lays the foundation for the research on the structural building material's dynamic control and mechanism performance. At the same time, our research results also provide methods and ideas for the rigid body dynamics modeling of other structural redundant parallel mechanisms


Introduction
Building foundation piles are mainly affected by horizontal loads.Vertical bearing capacity and settlement are generally not controlled [1].Both building and pile foundation codes require the following formula to check the horizontal bearing capacity: ik H is the classic combination of horizontal force at the top of the foundation pile.h R is the characteristic value of horizontal bearing capacity of foundation pile determined by static load test or formula.Using the formula (1) to check the horizontal bearing capacity is reasonable.The methods for calculating horizontally loaded piles mainly include the p-y curve method and the three-dimensional finite element method.The m method is a linear elastic method.This method is only used when the horizontal displacement is less than 10mm.Theoretically, the three-dimensional finite element method can better simulate the problem of pile-soil contact, but its modeling is more complicated [2].This method is inconvenient to reflect the continuous change of soil parameters.The nonlinear p-y curve method is a common method for analyzing horizontally loaded piles.This method is theoretically rigorous and can reflect the gradual change of soil parameters with depth.The main forms of the p-y curve are the American API standard method, Resee method, Hohai University method, Tongji University method, ideal elastic-plastic method, sand strain wedge model, etc.The method for calculating the deflection of the pile body by the p-y curve is mainly the finite difference method, and the nonlinear finite element method and the mathematical programming method are also used.This paper introduces the Lagrangian multiplier of horizontal force and moment balance to derive the Lagrangian equation of the deflection curve array.At the same time, we use Newton's method to solve it.The method herein can be used for any form of the p-y curve.The typical stratum in my country's coastal waters is soft clay on the surface and sandy soil underneath [3].The API specification gives detailed recommended methods for the p-y curves of these two soils.The soft soil p-y curve is shown in Figure 1.As the lateral displacement increases, the horizontal resistance first increases nonlinearly.This part will not change after reaching the ultimate resistance.The p-y curve of shallow soft clay also has a descending segment.In Figure 1 Where A is a coefficient describing static or cyclic loads.K is the initial modulus 3 ( / ) kN m of the sand foundation reaction force.It is obtained from a look-up table for the effective internal friction angle.

Variational Form of the Deflection Curve Problem
The horizontal load pile and the coordinate axis setting are shown in Figure 2. X axis downward is positive [4].The direction of deflection w is positive if it is the same as the direction of the external load.When subjected to external loads, the configuration of the pile body will minimize the potential energy of the pile-soil-load system.The deflection curve function represents the pile body configuration ( ) w x .The system potential energy includes the following three parts: 1 E is the potential energy of the external load. 2 E is the potential energy stored by the bending of the pile body.3 E is the reaction potential energy of the deformation of the foundation soil.When the pile foundation is not displaced, the potential energy of the external force is 0, then there is F is the external force.h is the load height., w w are the displacement and the tangent of the rotation angle of the pile at the soil surface, respectively.1 w and ' 1 w in the positive direction of the coordinate axis lead to the decrease and increase of the potential energy of the external force, respectively.Therefore, the sign on the right side of formula ( 4) is negative and positive.( ) 2 E is the elastic modulus of the pile body material.I is the moment of inertia along the direction of the external force.L is the length of the pile below the mud surface.The bending of the pile body above the mud surface causes 1 E and 2 E to each increase by a constant.The pile is an elongated structure [5].The reaction potential energy of the foundation soil is equal to the work done by the foundation soil to reset from the deflection w of the pile foundation: ( ) p y is the reaction load concentration when the displacement of the pile body is y .This value is obtained from the p-y curve of the foundation soil.The kinetic energy of the pile foundation can be ignored in the working state.The pile foundation needs to meet the horizontal force balance conditions: The moment equilibrium condition at the mud surface is expressed as follows: M is the bending moment of the load on the soil surface.There is M Fh  under equivalent concentrated load conditions.A positive ( ) p w will result in a moment opposite to the load moment, so the second term in the above equation is taken as a negative sign.The deflection curve of the pile body can be obtained by solving the extreme value of the functional ' 1 ( , , ') T E w w w under the constraints ( 7) and (8).

Lagrangian equations in discrete form
The actual p-y curves are mostly piecewise functions.It is difficult to obtain analytical solutions for the functional extremum problems represented by equations ( 3) to (8).We subdivide the pile length in soil into N segments of equal length [6].We mark the length of each segment as / ds L N  , then the total potential energy can be expressed as i EI is the stiffness of the pile body in section i .( ) i p y is the reaction force function at the depth of soft clay in section i .The second derivative of deflection, i w , can be expressed as a linear combination of the deflections of adjacent pile segments: When we use equation (10) to calculate the binding energy of the first and second pile sections, 1 w ″ and 2 w ″ are out of the range of the array.Since 1 1 / w M EI  ″ is a constant, the bending energy of the first pile section is not considered [7].The bending energy of the second pile section can be taken as the average value of the first and third pile sections.In the actual calculation, only half of the bending energy of the third pile section is taken.This eliminates the need to calculate the terms 1 w ″ and 2 w ″ in equation ( 9).' 1 w in formula ( 9) is substituted into 2 1 ( ) / w w ds  for calculation.The discretization constraints are 1 ( ) 0 Since ds is small, it does not matter whether we take the depth i x of the pile segment as the top or bottom of the segment.We take i x at the top of the pile segment, so 1 x is right at the mud surface.At this point, we introduce the Lagrangian multipliers  and  , then the Lagrangian function of the system potential energy is We take the partial derivative of the function L with respect to i w to obtain the Lagrange equations as ( ) ( ) ( ) ( ) 0, 1, 2, , In formula ( 14),  is abbreviated as function ( ) i f w , and it can be known from the first four items of formula (13) that ( ) i f w is a linear combination of deflection array w .The combination coefficient is related to the pile stiffness and load.It contains only a few terms around i w .From equations ( 11), ( 12) and ( 14), a total of 2 N  equations can be used to obtain the deflection array w and the Lagrange multipliers  and  , a total of 2 N  unknowns.The establishment of equation ( 14) requires that the reaction force function ( ) i i p w is piecewise derivable.In the API specification, the p-y curve of soft clay is a cubic square root.Its derivative is infinite at 0 i w  .This does not indicate that equation ( 14) fails.The infinite derivative of the p-y curve means that the stiffness of the foundation spring is infinite when no displacement occurs.We can solve this problem by replacing the small displacement segment of the curve with a linear function.In the actual calculation, we take the 50 / 0.1 y y  segment as a straight line, and this difference cannot be reflected in Figure 1.After we obtain the array w , we can further obtain the reaction force array p and the bending moment of the pile body.

Equation Solving
The Newton iteration method can solve the Lagrangian equation ( 14) and the constraints.We denote the multiplier ,   as l  in the Laplace equation vector l .The desired system of equations is abbreviated as The 2 N  dimensional zero vector 0 (0, 0, , 0) T w   is taken as the initial deflection vector.We, using the iterative equation: ) Where the superscripts k and 1 k  denote the number of iteration steps.J is the Jacobi matrix of vectors l to w .From equations ( 11), ( 12), ( 14), it can be known that the element of matrix J subscript ( , ) i j is: , It can be seen that the bulk of matrix J depends on element / function of a few components in the vector w , so / i j f w   constitutes a constant matrix.Only elements near the main diagonal are non-zero.This paper uses Matlab matrix operations to solve equation ( 16).We divide the pile body into hundreds of segments, and the iterative calculation is also very fast at this time [8].We take N100 or 500 to calculate the very small difference.When the displacement difference is less than 0.6mm, the angle difference is less than 0.003°.The calculation error mainly depends on the accuracy of the p-y curve.In the actual calculation, the pile body is divided into 200 segments. is the nonlinear segment of the 1/ 3 y form.The derivative function of the p-y curve is also divided into 3 segments accordingly.Two of them are constants, and one is in the form of 2/ 3 y  .In the actual calculation, it is found that the method in this paper is completely unaffected by the nonlinearity and piece wiseness of the p-y curve [9].At this point, the deflection array quickly converges.We used the static load p-y curve recommended by the API specification to obtain the soil surface displacement and the maximum bending moment curve of the pile body under different loads.The calculation results are shown in Figure 3.The red steel city pile foundation in the sand is within the load range of Figure 3.Its loading curve is linear.The loading curves of Matlock piles in soft clay are also significantly nonlinear.

Load height-deformation curve of the horizontally loaded pile
This paper studies the variation of soil surface displacement and rotation angle under different load heights.We can realize a series of load height change processes by adding a loop about the parameter h in the Matlab program.We compare the calculation results of different pile foundations.We perform a dimensionless treatment of the load height-deformation curve [10].The article takes the ratio of the load height to the pile length / h L as the load height ratio.The ratio of the soil surface rotation angle when the load height is h to the soil surface rotation angle when applied to the soil surface is the rotation angle ratio.We examine the relationship between the load-to-height ratio and the displacement or angle ratio.In this way, the influence of different bending moment values on the pile foundation properties under the same horizontal load can be known.This paper calculated the load-height-deformation curves of the two pile foundations in the previous section (Fig. 4).

Figure 4 Load height-deformation curves of two pile foundations
The displacement ratio-height ratio curve varies with pile-soil conditions and loads.There is no fixed relationship between the two.The relationship between the angle ratio and / h L for small load heights seems to be independent of the pile-soil conditions [11].In case / 0.5 h L  , the corners of each pile foundation are about 7 times that of the soil surface load.There is no clear quantitative relationship between the displacement and load-to-height ratios.More than 90% of pile foundation deformation is caused by pile top bending.Deformation caused by horizontal loads is only a very small part.

Parameter design example of pile foundation
Matlab as mentioned above is used to study design parameters for horizontally loaded piles with large bending moments.We changed the cycle parameter to the outer diameter D of the pipe pile.The horizontal load is 2.3MN, and the bending moment at the top of the pile is 100MN•m.The parameter design needs to meet the deformation control requirements under the normal service limit state.After the parameter design is completed, we carry out the check calculation of the limit state of the bearing capacity.20 / kN m .The sandy soil was judged to be non-liquefiable.The design pile foundation adopts a steel pipe pile.The elastic modulus of steel is 210GPa.Note the diameter of the hollow part of the pipe pile as d.The common void ratio of steel pipe piles is F / d D  0.977.This section calculates the pile top deformation of steel pipe piles with different diameters under normal service and bearing capacity conditions.The pile length ranges from 30 to 60 m.The wind and wave loads acting on the offshore piles are all long-term cyclic loads.Therefore, we use the p-y curve of the periodic load in the API specification.Figures 5 and 6 show pile top deformation calculation results under normal and bearing capacity conditions, respectively.In this example, the pile length L can be taken as 40m.The pile diameter is determined by the intersection of the deformation curve of L  40m and the control line in Figures 5 and 6.The horizontal dotted line in the figure is the control line.It can be seen from Figure 5 that the minimum pile diameter is 5m under the requirement of this turning angle.Although the displacement of the pile top increases significantly under the bearing capacity condition, the minimum pile diameter required by the experiment is also 5m.When the soil surface displacement is limited to 30mm, the minimum pile diameter is 5.8m.The minimum pile diameter is 6.25m when the displacement is limited to 25mm.The rotation angle and settlement requirements are relatively easy to meet, but the limit of horizontal displacement is not easy to meet.This indicates that more explicit horizontal displacement criteria are required for building foundation design.A horizontal displacement limit of 25mm will result in a mud surface turning angle that is half the value allowed by the specification.In this paper, the horizontal displacement limit is 30mm, and the design pile diameter is 5.8m.The soil surface rotation angles under the normal service limit and bearing capacities are 0.14°and 0.30°, respectively.


, x is the depth, y is the lateral displacement, and R X is the critical depth.50 y is the lateral displacement when the soil around the pile reaches half of the ultimate resistanceis the strain at the half of the maximum principal stress difference on the UU test curve of the undisturbed soil.D is the outer diameter of the pile.The dimensionless number / u p p is multiplied by the ultimate resistance u p to obtain the reaction load concentration.The following formula expresses the p-y curve of sand:

Figure 1
Figure 1 API specification soft clay p-y curve tanh u u Kx p Ap y Ap

Figure 2
Figure 2 Deflection of a horizontally loaded pile The bending potential energy of the pile body in the mechanics of materials is 2 2 0  in the augmented deflection vector w .At the same time, we denote the constraints(11) and (12) as 1 N l  and 2 N

Figure 3
Figure 3 Calculated and measured results of pile foundation static load test

Figure 5
Figure 5 Deformation calculation at the limit state of normal service

Figure 6
Figure 6 Deformation calculation under the load-carrying capacity conditionThe rotation angle and settlement requirements are relatively easy to meet, but the limit of horizontal displacement is not easy to meet.This indicates that more explicit horizontal displacement criteria are required for building foundation design.A horizontal displacement limit of 25mm will result in a mud surface turning angle that is half the value allowed by the specification.In this paper, the horizontal displacement limit is 30mm, and the design pile diameter is 5.8m.The soil surface rotation angles under the normal service limit and bearing capacities are 0.14°and 0.30°, respectively.
Taking the static load test of Matlock pile foundation in Table 1 as an example, it has converged The soil layer is layered soil.The upper layer is soft clay with a layer thickness of 10m.Undrained shear strength 30